# Patent

## Title
Extended active disturbance rejection controller

## Abstract

Multiple designs, systems, methods and processes for controlling a system or plant using an extended active disturbance rejection control (ADRC) based controller are presented. The extended ADRC controller accepts sensor information from the plant. The sensor information is used in conjunction with an extended state observer in combination with a predictor that estimates and predicts the current state of the plant and a co-joined estimate of the system disturbances and system dynamics. The extended state observer estimates and predictions are used in conjunction with a control law that generates an input to the system based in part on the extended state observer estimates and predictions as well as a desired trajectory for the plant to follow.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation of U.S. application Ser. No. 13/464,558 filed May 4, 2012, which is a continuation of U.S. application Ser. No. 12/195,353 filed Aug. 20, 2008, both of which are hereby incorporated by reference. This application claims the benefit of U.S. Application No. 60/965,359 filed 20 Aug. 2007, which is hereby incorporated by reference. This application is a continuation in part and claims the benefit of the following applications: U.S. application Ser. No. 12/067,141 filed 17 Mar. 2008; that claims priority to the following: Patent Cooperation Treaty Appl. No. PCT/US2006/036156, filed 18 Sep. 2006; U.S. Appl. No. 60/718,393, filed 19 Sep. 2005; U.S. Appl. No. 60/718,581, filed 19 Sep. 2005; U.S. Appl. No. 60/718,899, filed 20 Sep. 2005; U.S. Appl. No. 60/728,928, filed 20 Oct. 2005; and U.S. Appl. No. 60/728,929 filed, 20 Oct. 2005; and U.S. application Ser. No. 10/351,664 filed, 27 Jan. 2003 that claims priority to U.S. Appl. No. 60/373,404, filed 18 Apr. 2002, all of which are hereby incorporated by reference.


This work was supported at least in part by NASA under NASA contract number GT3-52387. Accordingly, the United States government may have certain rights herein.


TECHNICAL FIELD
The subject matter described herein relates to controllers, systems, and methods for feedback control of various systems.
BACKGROUND
A feedback (closed-loop) control system 10, as shown in Prior Art FIG. 1, is widely used to modify the behavior of a physical process, denoted as the plant 110, so it behaves in a specific desirable way over time. For example, it may be desirable to maintain the speed of a car on a highway as close as possible to 60 miles per hour in spite of possible hills or adverse wind; or it may be desirable to have an aircraft follow a desired altitude, heading and velocity profile independently of wind gusts; or it may be desirable to have the temperature and pressure in a reactor vessel in a chemical process plant maintained at desired levels. All of these industrial tasks are accomplished today by using traditional feedback control, and the above are examples of what automatic control systems are designed to do, without human intervention.
The key component in a feedback control system is the controller 120, which determines the difference between the output “y” of the plant 110, (e.g., the temperature or position) and its desired value, and produces a corresponding control signal “u” (e.g., modulating the power input to a heater or motor).
The goal of controller design is to make the difference between the actual output of the plant and the desired output as small as possible as rapidly as possible. Today, controllers are employed in a large number of industrial control applications and in areas like robotics, aeronautics, astronautics, motors, motion control and thermal control, just to name a few.
INDUSTRIAL APPLICABILITY
The present system and method has potential applicability to a wide range of different industrial and commercial applications. The following brief synopsis is intended only to provide background on some exemplary applications to assist a person of ordinary skill in the art in understanding this disclosure more fully. Additional background information on a select number of exemplary industrial applications are also provided.
BACKGROUND OF THE INVENTION
Classic Control Theory provides a number of techniques for an engineer to use in controller design. Existing controllers for linear, time invariant, and single-input single-output plants are categorized in one taxonomy into three forms: the proportional/integral/derivative (PID) controllers, transfer function based (TFB) controllers, and state feedback (SF) controllers. The PID controller is defined by the equation:

u=KPe+KI∫e+KDė  Equation (1)

where (u) is the control signal and (e) is the error between the set point and the process output (y) being controlled. As commonly used, the notation of a dot ({dot over ( )}) above a variable indicates the use of a derivative of the variable where the order corresponds to the number of dots.

The PID-type of controller has been employed in engineering and other applications since the early 1920s. It is an error-based controller that does not require an explicit mathematical model of the plant 110.
Transfer Function Based Controller
The transfer function based (TFB) controller is given in the form of




  
    
      
        
          
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where U(s) and E(s) are Laplace Transforms of u and e defined above, and n(s) and d(s) are polynomials in s. The TFB controller can be designed using methods in control theory based on the transfer function model of the plant, Gp(s). A PID controller can be considered a special case of a TFB controller because it has an equivalent transfer function of s.





  
    
      
        
          
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State Feedback (SF) Controller

The State Feedback (SF) controller can be defined by

u=r+Kx  Equation (4)

where u is the control input, r is the setpoint for the output to follow and x is the state vector associated with the system, based on the state space model of the plant 110:

{dot over (x)}(t)=Ax(t)+Bu(t)  Equation (5)

y(t)=Cx(t)+Du(t)  Equation (6)

where x(t) is the state vector, y(t) is the output vector, u(t) is the input or control vector, A is the state matrix, B is the input matrix, C is the output matrix and D is the feed-through matrix. When the state x is not accessible, a state observer (SO):

{circumflex over (x)}=A{circumflex over (x)}+Bu+L(y−ŷ))  Equation (7)

is usable to estimate the state x, where the (^) symbol is used to denote an estimate or observed value of a given variable and L denotes the observer coefficient matrix.

State Observers

Observers extract real-time information of a plant's 110 internal state from its input-output data. The observer usually presumes precise model information of the plant 110, since performance is largely based on its mathematical accuracy. Most prior art closed loop controllers require both types of information. Such presumptions, however, often make the method impractical in engineering applications, since the challenge for industry remains in constructing these models as part of the design process. Another level of complexity is added when gain scheduling and adaptive techniques are used to deal with nonlinearity and time variance, respectively.
Disturbance Observers and Disturbance Rejection
Recently, disturbance rejection techniques have been used to account for uncertainties in the real world and successfully control complex nonlinear systems. The premise is to solve the problem of model accuracy in reverse by modeling a system with an equivalent input disturbance d that represents any difference between the actual plant P and a derived/selected model Pn of the plant, including external disturbances w. An observer is then designed to estimate the disturbance in real time and provide feedback to cancel it. As a result, the augmented system acts like the model Pn at low frequencies, making the system behave like Pn and allowing a controller to be designed for Pn.
The most common of these techniques is the disturbance observer (DOB) structure (Endo, S., H. Kobayashi, C J. Kempf, S. Kobayashi, M. Tomizuka and Y. Hori (1996). “Robust Digital Tracking Controller Design for High-Speed Positioning Systems.” Control Eng Practice, 4:4, 527-536; Kim, B. K., H.-T. Choi, W. K. Chung and I. H. Suh (2002). “Analysis and Design of Robust Motion Controllers in the Unified Framework.” J of Dynamic Systems, Measurement, and Control, 124, 313-321; Lee, H. S. and M. Tomizuka (1996). “Robust Motion Controller Design for High-Accuracy Positioning Systems.” IEEE Trans. Ind. Electron., 43:1, 48-55; Tesfaye, A., H. S. Lee and M. Tomizuka (2000). “A Sensitivity Optimization Approach to Design of a Disturbance Observer in Digital Motion Control.” IEEE/ASME Trans, on Mechatronics, 5:1, 32-38; Umeno, T. and Y. Hori (1991). “Robust Speed Control of DC Servomotors Using Modern Two Degrees of-Freedom Controller Design”. IEEE Trans. Ind. Electron., 38:5, 363-368). It uses a disturbance observer consisting of a binomial Q-filters and the inverse model of the plant. And the controller is called parameterized in the sense that the Q-filter is treated as a parameter. A model deliberately different from P is suggested in E. Schrijver and J. Van Dijk, “Disturbance Observers for Rigid Mechanical Systems: Equivalence, Stability, and Design,” Journal of Dynamic Systems, Measurement, and Control, vol. 124, no. 4, pp. 539-548, 2002 to facilitate design, but no guidelines are given other than it should be as simple as possible, while cautioning stability and performance may be in danger if the model is significantly different from the plant. Another obstacle is that a separate state observer must be designed to provide state feedback to the controller. In existing research, derivative approximates are used in this way but their effect on performance and stability is questionable. Furthermore, the controller design is dependent on the DOB design, meaning that derivative approximates can not be arbitrarily selected.
Multiple DOBs were used to control a multivariable robot by treating it as a set of decoupled single-input single-output (SISO) systems, each with disturbances that included the coupled dynamics (Bickel, R. and M. Tomizuka (1999). “Passivity-Based Versus Disturbance Observer Based Robot Control: Equivalence and Stability.” J. of Dynamic Systems, Measurement, and Control, 121, 41-47; Hori, Y., K. Shimura and M. Tomizuka (1992). “Position/Force Control of Multi-Axis Robot Manipulator Based on the TDOF Robust Servo Controller For Each Joint.” Proc. of ACC, 753-757; Kwon, S. J. and W. K. Chung (2002). “Robust Performance of the Multiloop Perturbation Compensator.” IEEE/ASME Trans. Mechatronics, 7:2, 190-200; Schrijver, E. and J. Van Dijk (2002) Disturbance Observers for Rigid Mechanical Systems: Equivalence, Stability, and Design.” J. of Dynamic Systems, Measurement, and Control, 124, 539-548.)
Another technique, referred to as the unknown input observer (UIO), estimates the states of both the plant and the disturbance by augmenting a linear plant model with a linear disturbance model (Burl, J. B. (1999). Linear Optimal Control, pp. 308-314. Addison Wesley Longman, Inc., California; Franklin, G. F., J. D. Powell and M. Workman (1998). Digital Control of Dynamic Systems, Third Edition, Addison Wesley Longman, Calif.; Johnson, C D. (1971). “Accommodation of External Disturbances in Linear Regulator and Servomechanism Problems.” IEEE Trans. Automatic Control, AC-16:6, 635-644; Liu, C-S., and H. Peng (2002). “Inverse-Dynamics Based State and Disturbance Observer for Linear Time-Invariant Systems.” J. of Dynamic Systems, Measurement, and Control, 124, 375-381; Profeta, J. A. III, W. G. Vogt and M. H. Mickle (1990). “Disturbance Estimation and Compensation in Linear Systems.” IEEE Trans. Aerospace and Electronic Systems, 26:2, 225-231; Schrijver, E. and J. Van Dijk (2002) “Disturbance Observers for Rigid Mechanical Systems: Equivalence, Stability, and Design.” J. of Dynamic Systems, Measurement, and Control, 124, 539-548). Unlike the DOB structure, the controller and observer can be designed independently, like a Luenberger observer. However, it still relies on a good mathematical model and a design procedure to determine observer gains. An external disturbance w is generally modeled using cascaded integrators (l/sh). When they are assumed to be piece-wise constant, the observer is simply extended by one state and still demonstrates a high degree of performance.
Extended State Observer (ESO)
In this regard, the extended state observer (ESO) is quite different. Originally proposed by Han, J. (1999). “Nonlinear Design Methods for Control Systems.” Proc. 14th IFAC World Congress, in the form of a nonlinear UIO and later simplified to a linear version with one tuning parameter by Gao, Z. (2003). “Scaling and Parameterization Based Controller Tuning.” Proc. of ACC, 4989-4996, the ESO combines the state and disturbance estimation power of a UIO with the simplicity of a DOB. One finds a decisive shift in the underlying design concept as well. The traditional observer is based on a linear time-invariant model that often describes a nonlinear time-varying process. Although the DOB and UIO reject input disturbances for such nominal plants, they leave the question of dynamic uncertainty mostly unanswered. The ESO, on the other hand, addresses both issues in one simple framework by formulating the simplest possible design model Pd=l/sn for a large class of uncertain systems. The design model Pd is selected to simplify controller and observer design, forcing P to behave like the design model Pd at low frequencies rather than Pn. As a result, the effects of most plant dynamics and external disturbances are concentrated into a single unknown quantity. The ESO estimates this quantity along with derivatives of the output, giving way to the straightforward design of a high performance, robust, easy to use and affordable industrial controller.
Active Disturbance Rejection Control (ADRC)
Originally proposed by Han, J. (1999). “Nonlinear Design Methods for Control Systems.” Proc. 14th IFAC World Congress, a nonlinear, non-parameterized active disturbance rejection control (ADRC) is a method that uses an ESO. A linear version of the ADRC controller and ESO were parameterized for transparent tuning by Gao, Z. (2003). “Scaling and Parameterization Based Controller Tuning.” Proc. of ACC, 4989-4996. Its practical usefulness is seen in a number of benchmark applications already implemented throughout industry with promising results (Gao, Z., S. Hu and F. Jiang (2001). “A Novel Motion Control Design Approach Based on Active Disturbance Rejection.” Proc. of 40th IEEE Conference on Decision and Control; Goforth, F. (2004). “On Motion Control Design and Tuning Techniques.” Proc. of ACC; Hu, S. (2001). “On High Performance Servo Control Solutions for Hard Disk Drive.” Doctoral Dissertation, Department of Electrical and Computer Engineering, Cleveland State University; Hou, Y., Z. Gao, F. Jiang and B. T. Boulter (2001). “Active Disturbance Rejection Control for Web Tension Regulation.” Proc. of 40th IEEE Conf. on Decision and Control; Huang, Y., K. Xu and J. Han (2001). “Flight Control Design Using Extended State Observer and Nonsmooth Feedback.” Proc. of 40th IEEE Conf on Decision and Control; Sun, B and Z. Gao (2004). “A DSP-Based Active Disturbance Rejection Control Design for a 1 KW H-Bridge DC-DC Power Converter.” IEEE Trans. on Ind. Electronics; Xia, Y., L. Wu, K. Xu, and J. Han (2004). “Active Disturbance Rejection Control for Uncertain Multivariable Systems With Time-Delay., 2004 Chinese Control Conference. It also was applied to a fairly complex multivariable aircraft control problem (Huang, Y., K. Xu and J. Han (2001). “Flight Control Design Using Extended State Observer and Nonsmooth Feedback. Proc. of 40th IEEE Conf on Decision and Control).
What is needed is a control framework for application to systems throughout industry that are complex and largely unknown to the personnel often responsible for controlling them. In the absence of required expertise, less tuning parameters are needed than current approaches, such as multi-loop PID, while maintaining or even improving performance and robustness.
INDUSTRIAL APPLICATION
A selection of industrial applications is presented herein to provide a background on some selected applications for the present extended active disturbance rejection controller.
Tracking Control
Tracking Control refers to the output of a controlled system meeting design requirements when a specified reference trajectory is applied. Oftentimes, it refers to how closely the output y compares to the reference input r at any given point in time. This measurement is known as the error e=r−y.
Control problems are categorized in two major groups; point-to-point control and tracking control. Point-to-point applications usually call for a smooth step response with small overshoot and zero steady state error, such as when controlling linear motion from one position to the next and then stopping. Since the importance is placed on destination accuracy and not on the trajectory between points, conventional design methods produce a bandwidth limited controller with inherent phase lag in order to produce a smooth output. Tracking applications require a good tracking of a reference input by keeping the error as small as possible at all times, not just at the destination. This is particularly common in controlling a process that never stops. Since the importance is placed on accurately following a changing reference trajectory between points, phase lags are avoided as much as possible because they may lead to significant errors in the transient response, which lasts for the duration of the process in many applications. A common method of reducing phase lag is to increase the bandwidth, at the cost of increased control effort (energy) and decreased stability margin.
Various methods have been used to remove phase lag from conventional control systems. All of them essentially modify the control law to create a desired closed loop transfer function equal to one. As a result, the output tracks the reference input without much phase lag and the effective bandwidth of the overall system is improved. The most common method is model inversion where the inverse of the desired closed loop transfer function is added as a prefilter. Another method proposed a Zero Phase Error Tracking Controller (ZPETC) that cancels poles and stable zeros of the closed loop system and compensates for phase error introduced by un-cancelable zeros. Although it is referred to as a tracking controller, it is really a prefilter that reduces to the inverse of the desired closed loop transfer function when unstable zeros are not present. Other methods consist of a single tracking control law with feed forward terms in place of the conventional feedback controller and prefilter, but they are application specific. However, all of these and other previous methods apply to systems where the model is known.
Model inaccuracy can also create tracking problems. The performance of model-based controllers is largely dependent on the accuracy of the model. When linear time-invariant (LTI) models are used to characterize nonlinear time-varying (NTV) systems, the information becomes inaccurate over time. As a result, gain scheduling and adaptive techniques are developed to deal with nonlinearity and time variance, respectively. However, the complexity added to the design process leads to an impractical solution for industry because of the time and level of expertise involved in constructing accurate mathematical models and designing, tuning, and maintaining each control system.
There have been a number of high performance tracking algorithms that consist of three primary components: disturbance rejection, feedback control, and phase lag compensation implemented as a prefilter. First, disturbance rejection techniques are applied to eliminate model inaccuracy with an inner feedback loop. Next, a stabilizing controller is constructed based on a nominal model and implemented in an outer feedback loop. Finally, the inverse of the desired closed loop transfer function is added as a prefilter to eliminate phase lag. Many studies have concentrated on unifying the disturbance rejection and control part, but not on combining the control and phase lag compensation part, such as the RIC framework. Internal model control (IMC) cancels an equivalent output disturbance. B. Francis and W. Wonham, “The Internal Model Principal of Control Theory,” Automatica, vol 12, 1976, pp. 457-465. E. Schrijver and J. Van Dijk, “Disturbance Observers for Rigid Mechanical Systems: Equivalence, Stability, and Design,” Journal of Dynamic Systems, Measurement, and Control, vol. 124, December 2002, pp. 539-548 uses a basic tracking controller with a DOB to control a multivariable robot. The ZPETC has been widely used in combination with the DOB framework and model based controllers.
Thus, having reviewed prior art in tracking control, the application now describes example systems and methods of tracking control employing predictive ADRC.
Web Processing Applications
Web tension regulation is a challenging industrial control problem. Many types of material, such as paper, plastic film, cloth fabrics, and even strip steel are manufactured or processed in a web form. The quality of the end product is often greatly affected by the web tension, making it a crucial variable for feedback control design, together with the velocities at the various stages in the manufacturing process. The ever-increasing demands on the quality and efficiency in industry motivate researchers and engineers alike to explore better methods for tension and velocity control. However, the highly nonlinear nature of the web handling process and changes in operating conditions (temperature, humidity, machine wear, and variations in raw materials) make the control problem challenging.
Accumulators in web processing lines are important elements in web handling machines as they are primarily responsible for continuous operation of web processing lines. For this reason, the study and control of accumulator dynamics is an important concern that involves a particular class of problems. The characteristics of an accumulator and its operation as well as the dynamic behavior and control of the accumulator carriage, web spans, and tension are known in the art.
Both open-loop and closed-loop methods are commonly used in web processing industries for tension control purposes. In the open-loop control case, the tension in a web span is controlled indirectly by regulating the velocities of the rollers at either end of the web span. An inherent drawback of this method is its dependency on an accurate mathematical model between the velocities and tension, which is highly nonlinear and highly sensitive to velocity variations. Nevertheless, simplicity of the controller outweighs this drawback in many applications. Closing the tension loop with tension feedback is an obvious solution to improve accuracy and to reduce sensitivity to modeling errors. It requires tension measurement, for example, through a load cell, but is typically justified by the resulting improvements in tension regulation.
Most control systems will unavoidably encounter disturbances, both internal and external, and such disturbances have been the obstacles to the development of high performance controller. This is particularly true for tension control applications and, therefore, a good tension regulation scheme must be able to deal with unknown disturbances. In particular, tension dynamics are highly nonlinear and sensitive to velocity variations. Further, process control variables are highly dependent on the operating conditions and web material characteristics. Thus, what are needed are systems and methods for control that are not only overly dependent on the accuracy of the plant model, but also suitable for the rejection of significant internal and external disturbances.
Jet Engine Control Applications
A great deal of research has been conducted towards the application of modern multivariable control techniques on aircraft engines. The majority of this research has been to control the engine at a single operating point. Among these methods are a multivariable integrator windup protection scheme (Watts, S. R. and S. Garg (1996). “An Optimized Integrator Windup Protection Technique Applied to a Turbofan Engine Control,” AIAA Guidance Navigation and Control Conf.), a tracking filter and a control mode selection for model based control (Adibhatla S. and Z. Gastineau (1994). “Tracking Filter Selection And Control Mode Selection For Model Based Control.” AIAA 30th Joint Propulsion Conference and Exhibit), an H, method and linear quadratic Gaussian with loop transfer recovery method (Watts, S. R. and S. Garg (1995).“ A Comparison Of Multivariable Control Design Techniques For A Turbofan Engine Control.” International Gas Turbine and Aeroengine Congress and Expo.), and a performance seeking control method (Adibhatla, S. and K. L. Johnson (1993). “Evaluation of a Nonlinear Psc Algorithm on a Variable Cycle Engine.” AIAA/SAE/ASME/ASEE 29th Joint Propulsion Conference and Exhibit). Various schemes have been developed to reduce gain scheduling (Garg, S. (1997). “A Simplified Scheme for Scheduling Multivariable Controllers.” IEEE Control Systems) and have even been combined with integrator windup protection and Hm (Frederick, D. K., S. Garg and S. Adibhatla (2000). “Turbofan Engine Control Design Using Robust Multivariable Control Technologies. IEEE Trans. on Control Systems Technology).
Conventionally, there have been a limited number of control techniques for full flight operation (Garg, S. (1997). “A Simplified Scheme for Scheduling Multivariable Controllers.” IEEE Control Systems; and Polley, J. A., S. Adibhatla and P J. Hoffman (1988). “Multivariable Turbofan Engine Control for Full Conference on Decision and Control Flight Operation.” Gas Turbine and Expo). However, there has been no development of tuning a controller for satisfactory performance when applied to an engine. Generally, at any given operating point, models can become inaccurate from one engine to another. This inaccuracy increases with model complexity, and subsequently design and tuning complexity. As a result, very few of these or similar aircraft design studies have led to implementation on an operational vehicle.
The current method for controlling high performance jet engines remains multivariable proportional-integral (PI) control (Edmunds, J. M. (1979). “Control System Design Using Closed-Loop Nyquist and Bode Arrays.” Int. J on Control, 30:5, 773-802, and Polley, J. A., S. Adibhatla and P J. Hoffman (1988). “Multivariable Turbofan Engine Control for Full Conference on Decision and Control. Flight Operation.”Gas Turbine and Expo). Although the controller is designed by implementing Bode and Nyquist techniques and is tunable, a problem remains due to the sheer number of tuning parameters compounded by scheduling.
SUMMARY OF INVENTION
The present system, method, process, and device for controlling systems is applicable for a variety of different application and has utility in a number of different industrial fields. The following brief summary section is provided to facilitate a basic understanding of the nature and capabilities of the present system and method. This summary section is not an extensive nor comprehensive overview and is not intended to identify key or critical elements of the present systems or methods or to delineate the scope of these items. Rather this brief summary is intended to provide a conceptual introduction in simplified form as an introduction for the more detailed description presented later in this document.
The present application describes selected embodiments of the ADRC controller that comprise one or more computer components, that extend, build upon and enhance the use of ADRC controllers and provide enhanced performance and utility. Specifically, in one aspect the ADRC controller utilizes a predictive computer component or module that in one embodiment predicts future values of the plant output and in a second embodiment predicts future estimates of the system state and generalized disturbance. The generalized disturbance includes dynamics of the plant itself such that the plant is effectively reduced to a cascaded integral plant. In the various embodiments a variety of estimates are available and introduced.
In still another aspect, the ADRC controller further includes a model of the plant dynamics. The model is used in one embodiment to improve the state estimator or predictor. In a second embodiment the model is used to provide an enhanced control law that provides improved performance. In both aspects of the embodiments discussed in this paragraph, the generalized disturbance captures model errors and discrepancies and allows the system to eliminate errors caused by model errors, discrepancies, and assumptions. In the multiple embodiments, a variety of models are available for use by one of ordinary skill in the art to achieve the desired performance relative to the type of plant being controlled.
In yet another aspect, the ADRC controller further includes a non-linear, or discrete time optimal control law. The non-linear control law of the present embodiment utilizes non-linear control laws to improve the overall performance of an ADRC controller as compared to an ADRC controller with a linear proportional-derivative based control law.
In still another aspect, additional knowledge of derivative plant output gathered from high quality sensor information or direct measurement sensors (e.g. velocity and acceleration sensors) is used to reduce the order of the system state and disturbance estimator.



BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying figures depict multiple embodiments of a device and method for the separation of constituents from a flow. A brief description of each figure is provided below. Elements with the same reference numbers in each figure indicate identical or functionally similar elements. Additionally, the left-most digit(s) of a reference number identifies the drawings in which the reference number first appears.
FIG. 1 is a block diagram of a Prior Art feedback controller.
FIG. 2 is a block diagram of an ADRC configuration for a 2nd order plant.
FIG. 3 is a block diagram of a predictive ADRC configuration.
FIG. 4 is a block diagram of a 2 degree of freedom transfer function based (TFB) control structure.
FIG. 5 is a block diagram of an implementation of a PADRC configuration based on output prediction.



The present invention is described with reference to block diagrams and operational flow charts. It is to be understood that the functions/acts noted in the blocks may occur out of the order noted in the operational illustrations. For example, two blocks shown in succession may in fact be executed substantially concurrently or the blocks may sometimes be executed in the reverse order, depending upon the functionality/acts involved, including for example executing as asynchronous threads on a processor. Although some of the diagrams include arrows on communication paths to show a primary direction of communication, it is to be understood that communication may occur in the opposite direction to the depicted arrows.
DETAILED DESCRIPTION
Multiple embodiments of a system, device, and method for the control of systems are presented herein. Those of ordinary skill in the art can readily use this disclosure to create alternative embodiments using the teaching contained herein.
Lexicon
The following terms used herein have the meanings as follows.
As used herein the term “computer component” refers to a computer and elements of a computer, such as hardware, firmware, software, a combination thereof, or software in execution. For example, a computer component can include by way of example, a process running on a processor, a processor, an object, an executable, an execution thread, a program, and a computer itself. One or more computer components can in various embodiments reside on a server and the server can be comprised of multiple computer components. One or more computer components are in some cases referred to as computer systems whereby one or more computer components operate together to achieve some functionality. One or more computer components can reside within a process and/or thread of execution and a computer component can be localized on one computer and/or distributed between two or more computers.
“Software”, as used herein, includes but is not limited to, one or more computer readable and/or executable instructions that cause a computer or other electronic device to perform functions, actions and/or behave in a desired manner. The instructions may be embodied in various forms like routines, algorithms, modules, methods, threads, and/or programs. Software may also be implemented in a variety of executable and/or loadable forms including, but not limited to, a stand-alone program, a function call (local and/or remote), a servelet, an applet, instructions stored in a memory, part of an operating system or browser, and the like. It is to be appreciated that the computer readable and/or executable instructions can be located in one computer component and/or distributed between two or more communicating, co-operating, and/or parallel processing computer components and thus can be loaded and/or executed in serial, parallel, massively parallel and other manners. It will be appreciated by one of ordinary skill in the art that the form of software may be dependent on, for example, requirements of a desired application, the environment in which it runs, and/or the desires of a designer/programmer or the like.
“Computer communications”, as used herein, refers to a communication between two or more computers and can be, for example, a network transfer, a file transfer, an applet transfer, an email, a hypertext transfer protocol (HTTP) message, a datagram, an object transfer, a binary large object (BLOB) transfer, and so on. A computer communication can occur across, for example, a wireless system (e.g., IEEE 802.11), an Ethernet system (e.g., IEEE 802.3), a token ring system (e.g., IEEE 802.5), a local area network (LAN), a wide area network (WAN), a point-to-point system, a circuit switching system, a packet switching system, and so on.
An “operable connection” is one in which signals and/or actual communication flow and/or logical communication flow may be sent and/or received. Usually, an operable connection includes a physical interface, an electrical interface, and/or a data interface, but it is to be noted that an operable connection may consist of differing combinations of these or other types of connections sufficient to allow operable control.
As used herein, the term “signal” may take the form of a continuous waveform and/or discrete value(s), such as digital value(s) in a memory or register, present in electrical, optical or other form.
The term “controller” as used herein indicates a method, process, or computer component adapted to control a plant (i.e. the system to be controlled) to achieve certain desired goals and objectives.
As used herein the following symbols are used to describe specific characteristics of the variable. A variable with a “^” overtop the variable label, unless other indicated to the contrary, indicates that the variable is an estimate of the actual variable of the same label. A variable with a “{dot over ( )}” overtop the variable label, unless otherwise indicated, indicates that the variable is an nth derivative of that variable, where n equals the number of dots.
To the extent that the term “includes” is employed in the detailed description or the claims, it is intended to be inclusive in a manner similar to the term “comprising” as that term is interpreted when employed as. a transitional word in a claim.
To the extent that the term “or” is employed in the claims (e.g., A or B) it is intended to mean “A or B or both”. When the author intends to indicate “only A or B but not both”, then the author will employ the term “A or B but not both”. Thus, use of the term “or” in the claims is the inclusive, and not the exclusive, use.
Active Disturbance Rejection Control (ADRC)
Active Disturbance Rejection Control (“ADRC”) is a method or process and, when embedded within a computer system, a device (i.e., a controller that processes information and signals using the ADRC algorithms), that effectively rejects both unknown internal dynamics of a system of an arbitrary order to be controlled, or plant 110, and external disturbances in real time without requiring detailed knowledge of the plant 110 dynamics, which is required by most existing control design methods.
The only information needed to configure an ADRC controller is knowledge of the relative order of the plant 110 and the high frequency gain of the plant 110. In comparison, traditional model-based control methods require accurate mathematical models to conform with prevailing model-based design methods. In some embodiments, the ADRC controller leverages knowledge gained from model-based controller designs to improve control performance. The ADRC approach is unique in that it first forces an otherwise nonlinear, time-varying and uncertain plant 110 to behave as a simple linear cascade integral plant that can be easily controlled. To this end, the unknown, nonlinear, and time-varying internal dynamics is combined with the external disturbances to form what is denoted as the generalized disturbance, which is then estimated and rejected, thus greatly simplifying the control design.
The embodiments of the ADRC controller design are founded on the basic theory that uncertainties in dynamics and unknown external disturbances in the operating environment are a key challenge to be dealt with in control engineering practice. Traditional control systems are incapable of providing a solution to controlling and mitigating the impact of these uncertainties. The ADRC controller in contrast is specifically adapted to mitigate and eliminate the impact of these uncertainties, thus providing a dynamically simplified system for control.
One previous major characteristic of the ADRC controller is the parameterization of the controller as a function of control loop bandwidth. This parameterization makes all of the ADRC controller parameters functions of the control loop bandwidth; see, U.S. patent application Ser. No. 10/351,664, titled, “Scaling and Parameterizing A Controller” to Zhiqiang Gao, which is hereby incorporated by reference in its entirety.
An embodiment of a parameterized ADRC controller applied to a second-order plant 202 is illustrated in FIG. 2. The system to be controlled is a second-order plant 202 and represents an arbitrary plant 110 with predominantly second-order dynamics. The second-order plant 202 has an output (y) in response to an input that comprises a control signal (u) with superimposed external disturbances (d).
Consider a general uncertain, nonlinear and time-varying second-order plant 202 of the form:

ÿ=f(y,{dot over (y)},d,t)+bu  Equation (8)

where y is system output, u is the control signal, b is a constant, d is external input disturbance, f(y,{dot over (y)},d,t) is treated as the generalized disturbance. When this generalized disturbance is cancelled, the system is reduced to a simple double-integral plant with a scaling factor b, which can be easily controlled. Here we assume that the approximate value of b, denoted as {circumflex over (b)}, is given or estimated. As shown in FIG. 2, the scaling factor ({circumflex over (b)}) is used to scale 208, the control output (uft). In other embodiments, the scaling factor is implicit within the linear PD control law 206.

To capture the information of the generalized disturbance f(y,{dot over (y)},d,t) and cancel it from the system dynamics leads us to the Extended State Observer (ESO) 204, which is now presented. Let x1=y, x2={dot over (y)}, and x3=f, the description of the above plant 202 is rewritten in state space form with f (y,{dot over (y)},d,t), or simply f, treated as an additional state:




  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        x
                        .
                      
                      =
                      
                        
                          
                            A
                            x
                          
                          ⁢
                          x
                        
                        +
                        
                          
                            B
                            x
                          
                          ⁢
                          u
                        
                        +
                        
                          E
                          ⁢
                          
                            f
                            .
                          
                        
                      
                    
                  
                
                
                  
                    
                      y
                      =
                      
                        
                          C
                          x
                        
                        ⁢
                        x
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              
                A
                x
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              B
              x
            
            =
            
              [
              
                
                  
                    0
                  
                
                
                  
                    b
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            E
            =
            
              [
              
                
                  
                    0
                  
                
                
                  
                    0
                  
                
                
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              C
              x
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          9
          )
        
      
    
  




The corresponding state observer is




  
    
      
        {
        
          
            
              
                
                  z
                  .
                
                =
                
                  
                    
                      A
                      x
                    
                    ⁢
                    z
                  
                  +
                  
                    
                      B
                      z
                    
                    ⁢
                    u
                  
                  +
                  
                    L
                    ⁡
                    
                      (
                      
                        
                          y
                          m
                        
                        -
                        
                          
                            C
                            x
                          
                          ⁢
                          z
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  f
                  ^
                
                =
                Vz
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          10
          )
        
      
    
  




where ym is the measured system output, Bz=[0 {circumflex over (b)} 0]T, V=[0 0 1], and L=[l1 l2 l3]T is the observer gain vector. With the parameterization technique as described more fully in U.S. patent application Ser. No. 10/351,664, L is obtained by solving the equation λ(s)=|sI−Ax+LCx|=(s+ωo)3; that is, all the eigenvalues of the observer are placed at −ωo with

l1=3ωo,l2=3ωo2,l3=ωo3  Equation (11)

With a well-tuned extended state observer 204, z1, z2, and z3 will closely track y, {dot over (y)}, and f respectively, i.e. z1≈y, z2≈{dot over (y)}, and z3≈f
Using the information provided by the extended state observer 204, the generalized disturbance f is rejected by the control law

u=(u0−z3)/{circumflex over (b)}  Equation (12)

which reduces the plant 110 to a unity gain, double integral plant

ÿ=uo  Equation (13)

that can be easily controlled with linear proportional-derivative (PD) with feedforward

uo=gpd(e,ė,ωc)+{umlaut over (r)}  Equation (14)

where gpd(e,ė,ωc) is the parameterized linear PD controller with the loop bandwidth of ωc,

gpd(e,ė,ωc)=ωc2e+2ωcė  Equation (15)

with the error term defined as

e=r−z1ė={dot over (r)}−z2  Equation (16)

where the desired trajectory (r) is provided by the user.

Note that the linear PD control law 206 in Equation (14) corresponds to the common PD gains of

kp=ωc2 and kd=2ωc  Equation (17)

Using these details, the ADRC controller is generalized in its various embodiments to control an arbitrary nth-order system y(n)=f(y, {dot over (y)}, . . . , y(n-1), d, t)+bu, in which the external disturbances and unknown internal dynamics are rejected. In a similar manner, the plant 110 is reduced to a cascaded integral plant y(n)=u, which is then controlled by a generalized proportional-derivative (PD) control law, also described as a linear PD control law 206 similar to Equation (14).
Use of Enhanced Sensor Information to Improve Control Performance
In other embodiments of the ADRC controller, the linear PD control law 206 is updated to enhance performance by using direct measurement of the system output states (ym) instead of the estimated states (z1, z2). Specifically, when the measured output ym(t) is sufficiently noise free and {dot over (y)}m(t) is obtainable, either via direct measurement or from ym(t), the control law in Equation (14) can be implemented with

e=r−ymė={dot over (r)}−{dot over (y)}m  Equation (18)

That is, the output measurement and its derivative are used in the control law, instead of their estimates, z1 and z2. Similarly for the nth-order plant 110, the output measurement ym and its derivatives from the 1st to the (n−1)th-order available, can also be used directly in the PD control law, also referred to as the linear PD control law 206, instead of their estimates from the Extended State Observer (ESO) 204. Using the direct measurement of the output measurement state ym and its derivatives reduces the phase lag and improve control system performance, if the output measurement and its derivatives are not corrupted by noise.
Multiple Extended States
The Extended State Observer 204 in alternative embodiments includes more than one extended states, such as embodiments where the 2nd order plant 202 is replaced with a higher order plant 110. In one embodiment, when augmented with two extended states, x3=f and x4={dot over (f)}, the extended plant state space model can be written as follows.




  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        x
                        .
                      
                      =
                      
                        
                          
                            A
                            x
                          
                          ⁢
                          x
                        
                        +
                        
                          
                            B
                            x
                          
                          ⁢
                          u
                        
                        +
                        
                          E
                          ⁢
                          
                            f
                             ̈
                          
                        
                      
                    
                  
                
                
                  
                    
                      y
                      =
                      
                        
                          C
                          x
                        
                        ⁢
                        x
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              
                A
                x
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              B
              x
            
            =
            
              [
              
                
                  
                    0
                  
                
                
                  
                    b
                  
                
                
                  
                    0
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            E
            =
            
              [
              
                
                  
                    0
                  
                
                
                  
                    0
                  
                
                
                  
                    0
                  
                
                
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              C
              x
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          19
          )
        
      
    
  




With z1, z2, z3, and z4 tracking x1, x2, x3, and x4 respectively, the 4th-order ESO 204 is employed to estimate f.




  
    
      
        {
        
          
            
              
                
                  z
                  .
                
                =
                
                  
                    
                      A
                      x
                    
                    ⁢
                    z
                  
                  +
                  
                    
                      B
                      z
                    
                    ⁢
                    u
                  
                  +
                  
                    L
                    ⁡
                    
                      (
                      
                        
                          y
                          m
                        
                        -
                        
                          
                            C
                            x
                          
                          ⁢
                          z
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  f
                  ^
                
                =
                Vz
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          20
          )
        
      
    
  




where Bz=[0 {circumflex over (b)} 0 0]T and V=[0 0 1 0].

In this manner, any embodiment of an ADRC controller is adaptable for use with an arbitrary number (h) extended state observer states with is generalized to control an nth-order system.
Discrete Implementation
ADRC algorithms are implemented in discrete time, with ESO 204 discretized in both Euler and ZOH, and in both the current estimator and predictive estimator form.
In discrete implementation, the ESO 204 pole location is determined in the Z domain by solving the equation λ(z)=|zI−φ+LpCx=(Z−β)3, where β=e−ωoT.
Predictive Discrete Extended State Observer (PDESO) is in predictive estimator form, as shown in the following equation.




  
    
      
        {
        
          
            
              
                
                  z
                  ⁡
                  
                    [
                    
                      k
                      +
                      1
                    
                    ]
                  
                
                =
                
                  
                    Φ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      z
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      u
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                  
                  +
                  
                    
                      L
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            m
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        -
                        
                          
                            z
                            1
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    z
                    _
                  
                  ⁡
                  
                    [
                    k
                    ]
                  
                
                =
                
                  z
                  ⁡
                  
                    [
                    k
                    ]
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          21
          )
        
      
    
  




Current Discrete Extended State Observer (CDESO) is in current estimator form, as shown in the following equation.




  
    
      
        {
        
          
            
              
                
                  z
                  ⁡
                  
                    [
                    
                      k
                      +
                      1
                    
                    ]
                  
                
                =
                
                  
                    Φ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      z
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      u
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                  
                  +
                  
                    
                      L
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            m
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        -
                        
                          
                            z
                            1
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    z
                    _
                  
                  ⁡
                  
                    [
                    k
                    ]
                  
                
                =
                
                  
                    z
                    ⁡
                    
                      [
                      k
                      ]
                    
                  
                  +
                  
                    
                      L
                      c
                    
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            m
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        -
                        
                          
                            z
                            1
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          22
          )
        
      
    
  




The gain matrices of different ADRC discrete implementations are shown in the following Table I.







TABLE I







PDESO AND CDESO COEFFICIENT MATRICES











Type
Φ
Γ
Lp
Lc





Euler




   
  
    [
    
      
        
          1
        
        
          T
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          T
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  








   
  
    [
    
      
        
          0
        
      
      
        
          
            
              
                b
                ⁢
                
                  
                
              
              ^
            
            ⁢
            T
          
        
      
      
        
          0
        
      
    
    ]
  








   
  
    [
    
      
        
          
            3
            ⁢
            
              (
              
                1
                -
                β
              
              )
            
          
        
      
      
        
          
            3
            ⁢
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              2
            
            ⁢
            
              /
            
            ⁢
            T
          
        
      
      
        
          
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              3
            
            ⁢
            
              /
            
            ⁢
            
              T
              2
            
          
        
      
    
    ]
  








   
  
    [
    
      
        
          
            1
            -
            
              β
              3
            
          
        
      
      
        
          
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              2
            
            ⁢
            
              (
              
                2
                +
                β
              
              )
            
            ⁢
            
              /
            
            ⁢
            
              
            
            ⁢
            T
          
        
      
      
        
          
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              3
            
            ⁢
            
              /
            
            ⁢
            
              T
              2
            
          
        
      
    
    ]
  






 











ZOH




   
  
    [
    
      
        
          1
        
        
          T
        
        
          
            
              T
              2
            
            2
          
        
      
      
        
          0
        
        
          1
        
        
          T
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  








   
  
    [
    
      
        
          
            
              b
              ^
            
            ⁢
            
              
                T
                2
              
              2
            
          
        
      
      
        
          
            
              
                b
                ⁢
                
                  
                
              
              ^
            
            ⁢
            T
          
        
      
      
        
          0
        
      
    
    ]
  








   
  
    [
    
      
        
          
            3
            ⁢
            
              (
              
                1
                -
                β
              
              )
            
          
        
      
      
        
          
            3
            ⁢
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              2
            
            ⁢
            
              
            
            ⁢
            
              (
              
                5
                +
                β
              
              )
            
            ⁢
            
              /
            
            ⁢
            2
            ⁢
            
              
            
            ⁢
            T
          
        
      
      
        
          
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              3
            
            ⁢
            
              /
            
            ⁢
            
              T
              2
            
          
        
      
    
    ]
  








   
  
    [
    
      
        
          
            1
            -
            
              β
              3
            
          
        
      
      
        
          
            3
            ⁢
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              2
            
            ⁢
            
              (
              
                1
                +
                β
              
              )
            
            ⁢
            
              /
            
            ⁢
            2
            ⁢
            
              
            
            ⁢
            T
          
        
      
      
        
          
            
              
                (
                
                  1
                  -
                  β
                
                )
              
              3
            
            ⁢
            
              /
            
            ⁢
            
              T
              2
            
          
        
      
    
    ]
  













This discrete implementation of ADRC is also generalizable to nth-order plants 110.
Predictive Active Disturbance Rejection Controller
The thrust of the many of the embodiments presented here is the addition of the predictive functions to extend the capabilities of the original components of the ADRC. This extends the ADRC controller and generates embodiments of predictive versions of the ADRC, such as the predictive ADRC 300, depicted in block diagram form in FIG. 3 In this embodiment, the linear PD control law 206 in the predictive ADRC 300 is now a predictive control law 306. Similarly, the observer, or more specifically the extended state observer 204 from the traditional ADRC 200 is now embodied as a predictive state and disturbance observer 304. In one embodiment the predictive state and disturbance observer 304 is a predictive extended state observer (PESO) where the prediction occurs within the bounds of the ESO 204 to where the estimated states zn(t) are predicted to form the predictive state and disturbance observer 304, and outputs the prediction of the estimated states zn(t+τ). In a second embodiment, a state predictor method is used to extend the design of the ESO 204 to estimate the states zn(t) and form the predictive state and disturbance observer 304.
In other embodiments, other estimators and predictors, either in an integrated form or in combination are used to generate similar information as that generated by the predictive state and disturbance observer 304 and used by the embodiments predictive ADRC 300. One of these other embodiments adopts a disturbance observer (DOB) estimator structure to estimate the disturbance and a state observer and a predictor element that together form the predictive state and disturbance observer 304. In a second other embodiment, the predictive state and disturbance observer 304 utilizes an unknown input observer (UIO) estimator. Other estimators, including combinations of estimators such as the combination of a state observer and a DOB, are suitable for estimation of both the plant 110 output and the known part of the general disturbance (fn) are adaptable using the disclosure provided herein to those of ordinary skill in the art to serve as a predictive state and disturbance observer 304.
The challenge in constructing the various embodiments of a predictive controller within the general purpose ADRC controller architecture is how the various signals used by the ADRC algorithm are predicted and used. The signals used by the ADRC algorithm originate from a multitude of sources in the various embodiments, including in one embodiment the extended state observer 204 (ESO). In another embodiment, a simplified, reduced-order ESO (RESO) is used when either: (i) quality measurements of the output signals allows numerical differentiation to estimate derivatives of the output signal; or (ii) direct measurement of derivatives are possible (e.g. velocity and acceleration sensors), are available.
In another embodiment, a nonlinear proportional-derivative (NPD) control law is used to further optimize the performance of the present ADRC system. The RESO aspect of the ADRC controller provides alternative embodiments of ESO 204 and RESO; a general form of RESO with multiple extended states for an nth-order plant 110, and incorporation of the known plant 110 dynamics into ESO 204 and control law.
Furthermore, to facilitate practical implementations of the new techniques, we have provided at selected locations herein, the transfer function equivalent predictive ADRC 400, with H(s) 402 and Gc(s) 404, in a two-degree-of-freedom (2dof) structure shown in FIG. 4. Similarly, for a transfer function equivalent predictive ADRC 400, the plant dynamics are represented as Gp(s) 402. For various forms of the Predictive ADRC 400 described herein, these transfer function expressions, together with the NPD and RESO, are novel and significant even when the amount of prediction is set to zero.
All of these new developments help to make the predictive ADRC 300 possible and to provide the means of implementation the present embodiments in a diverse range of industrial machinery, and they are illustrated in details in the following sections.
Basic Design Principles for Predictive ADRC
In the real world, the sensor feedback, the generation of the control signal and its arrival at the actuator, and the estimation of the states and disturbances all contain time delays and phase lags, which can endanger system stability. To accommodate such delays and lags, we propose the following Predictive ADRC control law:

u(t)=(g(e(t+le),ė(t+le),ωc)+{umlaut over (r)}(t+lu)−{circumflex over (f)}(t+lf))/{circumflex over (b)}  Equation (23)

where g(e,ė,ωc) is the parameterized linear PD controller 206, or nonlinear proportional derivative (NPD) controller, Equation (68), le, lė, lu, and lf are positive real numbers representing the prediction horizons, a special case of which is le=lė=lu=lf=τ and the corresponding control law is

u(t)=(g(e(t+τ),ė(t+τ),ωc)+{umlaut over (r)}(t+τ)−{circumflex over (f)}(t+τ))/{circumflex over (b)}  Equation (24)

Also, if a part of the plant 110 dynamics, fn, is given, it can be incorporated into ADRC, as shown later in this disclosure. In the context of Predictive ADRC 300, the knowledge of fn can be used to help predict its future value, fn(t+lfn). The corresponding control law for Equation (23) and Equation (24) are

u(t)=(g(e(t+le),ė(t+lė),ωc)+{umlaut over (r)}(t+lu)−fn(t+lfn)−{circumflex over (f)}u(t+lfu))/{circumflex over (b)}  Equation (25)

and

u(t)=(g(e(t+τ),ė(t+τ),ωc)+{umlaut over (r)}(t+τ)−fn(t+τ)−{circumflex over (f)}(t+τ))/{circumflex over (b)}  Equation (26)

respectively.

Described below are various methods to obtain the prediction of e(t), ė(t), and {circumflex over (f)}(t), respectively, so that these predictive ADRC 300 control laws can be implemented.
Prediction of e and ė
Two embodiments of methods for predicting the values of error e(t) and its derivative are presented herein.
Method 1:
Using Taylor Series, the prediction of e(t), ė(t) can be obtained as



  
    
      e
      ⁡
      
        (
        
          t
          +
          
            l
            e
          
        
        )
      
    
    =
    
      
        e
        ⁡
        
          (
          t
          )
        
      
      +
      
        
          
            e
            .
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          l
          e
        
      
    
  




  or




  
    
      
        e
        ⁡
        
          (
          
            t
            +
            
              l
              e
            
          
          )
        
      
      =
      
        
          e
          ⁡
          
            (
            t
            )
          
        
        +
        
          
            
              e
              .
            
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            l
            e
          
        
        +
        
          
            
              
                e
                 ̈
              
              ⁡
              
                (
                t
                )
              
            
            2
          
          ⁢
          
            l
            e
            2
          
        
      
    
    ,
    
      

    
    ⁢
    and
  




  
    
      
        e
        .
      
      ⁡
      
        (
        
          t
          +
          
            l
            
              e
              .
            
          
        
        )
      
    
    =
    
      
        
          e
          .
        
        ⁡
        
          (
          t
          )
        
      
      +
      
        
          
            e
             ̈
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          
            l
            
              e
              .
            
          
          .
        
      
    
  



Here ė(t) can be obtained directly from {dot over (r)}−{dot over (y)}, or from {dot over (r)}−z2 using regular ESO, or {dot over (r)}−z1 using RESO. Here ë(t) can be obtained as ë(t)={umlaut over (r)}(t)−ÿ(t)≈{umlaut over (r)}(t)−(z3(t)+{circumflex over (b)}u(t)) with ESO or ë(t)={umlaut over (r)}(t)−ÿ(t)≈{umlaut over (r)}(t)−(z2(t)+{circumflex over (b)}u(t)) with RESO.

Method 2:
Since e(t+τ)=r(t+τ)−z1(t+τ) and ė(t+τ)={dot over (r)}(t+τ)−z2(t+τ), e(t+τ) and ė(t+τ) can be obtained directly using the future reference signals r(t+τ) and {dot over (r)}(t+τ), which are generally known, and the predicted output of the ESO z1(t+τ) and z2(t+τ), which need to be obtained approximately.
Prediction of {circumflex over (f)} in ESO and RESO
The primary method of predicting {circumflex over (f)} is the 1st or 2nd-order Taylor Series approximation, i.e.




  
    
      
        
          
            
              f
              ^
            
            ⁡
            
              (
              
                t
                +
                
                  l
                  f
                
              
              )
            
          
          ≈
          
            
              
                f
                ^
              
              ⁡
              
                (
                t
                )
              
            
            +
            
              
                
                  f
                  
                    .
                    ^
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                l
                f
              
            
          
        
        ,
        
          

        
        ⁢
        or
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          27
          )
        
      
    
  
  
    
      
        
          
            f
            ^
          
          ⁡
          
            (
            
              t
              +
              
                l
                f
              
            
            )
          
        
        ≈
        
          
            
              f
              ^
            
            ⁡
            
              (
              t
              )
            
          
          +
          
            
              
                f
                
                  .
                  ^
                
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              l
              f
            
          
          +
          
            
              
                
                  f
                  
                     ̈
                    ^
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              2
            
            ⁢
            
              l
              f
              2
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          28
          )
        
      
    
  




which requires the 1st and/or 2nd order derivatives of {circumflex over (f)}. To this end, a new form and a new way of implementation of extended state observer (ESO) 204 are introduced first, followed by the discussion on how to use them to obtain the predicted estimate of {circumflex over (f)} in Equation (27) and Equation (28)

Reduced-Order ESO

In the observer-based control methods, the phase lag introduced by the observer decreases the phase margin of the control loop and is therefore undesired. To reduce the phase lag in the ESO 204, an embodiment of a reduced-order ESO (RESO) described here.
Consider the plant 202 in Equation (9), if the derivative of the measured output ym ({dot over (y)}m), is given, a RESO can be constructed with z1 and z2 estimating x2={dot over (y)} and x3=f respectively. The correspondence between the augmented plant state space model and the RESO is shown as follows.




The shadowed blocks show how the gain matrices of the RESO are obtained from the augmented plant state space model. Equivalently, the RESO can be represented as




  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        z
                        .
                      
                      =
                      
                        
                          
                            A
                            z
                          
                          ⁢
                          z
                        
                        +
                        
                          
                            B
                            z
                          
                          ⁢
                          u
                        
                        +
                        
                          L
                          ⁡
                          
                            (
                            
                              
                                
                                  y
                                  .
                                
                                m
                              
                              -
                              
                                
                                  C
                                  z
                                
                                ⁢
                                z
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        f
                        ^
                      
                      =
                      Vz
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                A
                z
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              B
              z
            
            =
            
              [
              
                
                  
                    
                      b
                      ^
                    
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            L
            =
            
              [
              
                
                  
                    
                      l
                      1
                    
                  
                
                
                  
                    
                      l
                      2
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              C
              z
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            V
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              and
              ⁢
              
                

              
              ⁢
              
                l
                1
              
            
            =
            
              2
              ⁢
              
                ω
                o
              
            
          
          ,
          
            

          
          ⁢
          
            
              l
              2
            
            =
            
              
                ω
                o
                2
              
              .
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          30
          )
        
      
    
  




With RESO replacing the ESO 204, the corresponding linear PD control law 206 in ADRC 400 is

u=(gpd(e,ė,ωc)+{umlaut over (r)}−z2)/{circumflex over (b)}  Equation (31)

where

e=r−ym,ė={dot over (r)}−z1  Equation (32)

If {dot over (y)}m is clean it can be also fed back into the control law instead of z1, the Equation (33) changes to

e=r−ym,ė={dot over (r)}−{dot over (y)}m  Equation (34)

Note here that the parameterized linear PD control law 206 can be replaced by the parameterized NPD control law 506, which will be introduced later.

The order of ESO 204 is reduced further if ÿ is also given, as the double derivative of ym. According to U.S. patent application Ser. No. 10/351,664, the total disturbance is obtained as x3=f=ÿ−bu and it is susceptible to measurement noise. In an alternative embodiment, a first order RESO can be constructed with the only state z tracking x3=f. The correspondence between the augmented plant state space model and the new RESO is shown as follows.




The shadowed blocks show how the gain matrices of the RESO are obtained from the augmented plant state space model. Equivalently, the RESO can be represented as




  
    
      
        {
        
          
            
              
                
                  z
                  .
                
                =
                
                  
                    
                      A
                      z
                    
                    ⁢
                    z
                  
                  +
                  
                    
                      B
                      z
                    
                    ⁢
                    u
                  
                  +
                  
                    L
                    ⁡
                    
                      (
                      
                        
                          
                            y
                             ̈
                          
                          m
                        
                        -
                        
                          
                            b
                            ^
                          
                          ⁢
                          u
                        
                        -
                        
                          
                            C
                            z
                          
                          ⁢
                          z
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  f
                  ^
                
                =
                Vz
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          35
          )
        
      
    
  




where Az=0, Bz=0, L=l=ωo, Cz=1, V=1. The corresponding linear PD control law 206 with this RESO is

u=(gpd(e,ė,ωc)+{umlaut over (r)}−z)/{circumflex over (b)}  Equation (36)

where e=r−ym, ė={dot over (r)}−{dot over (y)}m. Note here again that the parameterized linear PD control law 206 can be replaced by the parameterized NPD control law 506 to be introduced.

A New ESO Implementation

Consider the same second-order plant 202 as described in Equation (9). The ESO 204 described in U.S. patent application Ser. No. 10/351,664 can be divided into two parts.




  
    
      
        {
        
          
            
              
                
                  [
                  
                    
                      
                        
                          
                            z
                            .
                          
                          1
                        
                      
                    
                    
                      
                        
                          
                            z
                            .
                          
                          2
                        
                      
                    
                  
                  ]
                
                =
                
                  
                    
                      [
                      
                        
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            0
                          
                          
                            0
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                z
                                .
                              
                              1
                            
                          
                        
                        
                          
                            
                              
                                z
                                .
                              
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  +
                  
                    
                      [
                      
                        
                          
                            0
                          
                        
                        
                          
                            
                              b
                              ^
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    u
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              l
                              1
                            
                          
                        
                        
                          
                            
                              l
                              2
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      (
                      
                        
                          y
                          m
                        
                        -
                        
                          z
                          1
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  z
                  3
                
                =
                
                  ∫
                  
                    
                      
                        l
                        3
                      
                      ⁡
                      
                        (
                        
                          
                            y
                            m
                          
                          -
                          
                            z
                            1
                          
                        
                        )
                      
                    
                    ⁢
                    
                      d
                      t
                    
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          37
          )
        
      
    
  




The first part is a Luenberger observer for the double integral plant in Equation (13), and the second part shows that the estimation of the general disturbance can be obtained by integrating the observer error with a gain of l3. This implementation is equivalent to the ESO 204.
When implemented in ZOH current estimator form, the Luenberger observer is




  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        z
                        ⁡
                        
                          [
                          
                            k
                            +
                            1
                          
                          ]
                        
                      
                      =
                      
                        
                          Φ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            z
                            ⁡
                            
                              [
                              k
                              ]
                            
                          
                        
                        +
                        
                          Γ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            u
                            ⁡
                            
                              [
                              k
                              ]
                            
                          
                        
                        +
                        
                          
                            L
                            p
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  y
                                  m
                                
                                ⁡
                                
                                  [
                                  k
                                  ]
                                
                              
                              -
                              
                                
                                  z
                                  1
                                
                                ⁡
                                
                                  [
                                  k
                                  ]
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          z
                          _
                        
                        ⁡
                        
                          [
                          k
                          ]
                        
                      
                      =
                      
                        
                          z
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        +
                        
                          
                            L
                            c
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  y
                                  m
                                
                                ⁡
                                
                                  [
                                  k
                                  ]
                                
                              
                              -
                              
                                
                                  z
                                  1
                                
                                ⁡
                                
                                  [
                                  k
                                  ]
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              Φ
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    T
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            Γ
            =
            
              [
              
                
                  
                    
                      
                        b
                        ^
                      
                      ⁢
                      
                        
                          T
                          2
                        
                        2
                      
                    
                  
                
                
                  
                    
                      
                        b
                        ^
                      
                      ⁢
                      T
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              L
              c
            
            =
            
              [
              
                
                  
                    
                      1
                      -
                      
                        β
                        2
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          1
                          +
                          
                            β
                            2
                          
                          -
                          
                            2
                            ⁢
                            
                              β
                              1
                            
                          
                        
                        )
                      
                      /
                      T
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              L
              p
            
            =
            
              [
              
                
                  
                    
                      2
                      -
                      
                        2
                        ⁢
                        
                          β
                          1
                        
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          1
                          +
                          
                            β
                            2
                          
                          -
                          
                            2
                            ⁢
                            
                              β
                              1
                            
                          
                        
                        )
                      
                      /
                      T
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              and
              ⁢
              
                

              
              ⁢
              
                β
                1
              
            
            =
            
              
                e
                
                  
                    
                      -
                      3
                    
                    2
                  
                  ⁢
                  
                    ω
                    o
                  
                  ⁢
                  T
                
              
              ⁢
              
                cos
                (
                
                  
                    
                      3
                    
                    2
                  
                  ⁢
                  
                    ω
                    o
                  
                  ⁢
                  T
                
                )
              
            
          
          ,
          
            

          
          ⁢
          
            
              β
              2
            
            =
            
              
                e
                
                  
                    -
                    2
                  
                  ⁢
                  
                    ω
                    o
                  
                  ⁢
                  T
                
              
              .
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          38
          )
        
      
    
  




Similarly, the RESO in Equation (30) can be divided into two parts:




  
    
      
        {
        
          
            
              
                
                  
                    z
                    .
                  
                  1
                
                =
                
                  
                    
                      b
                      ^
                    
                    ⁢
                    u
                  
                  +
                  
                    
                      l
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            .
                          
                          m
                        
                        -
                        
                          z
                          1
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  z
                  2
                
                =
                
                  ∫
                  
                    
                      
                        l
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            
                              y
                              .
                            
                            m
                          
                          -
                          
                            z
                            1
                          
                        
                        )
                      
                    
                    ⁢
                    
                      d
                      t
                    
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          39
          )
        
      
    
  




where the first part is a first order Luenberger observer, and the second part is an integrator. When implemented in ZOH current estimator form, this Luenberger observer is





  
    
      
        {
        
          
            
              
                
                  
                    z
                    1
                  
                  ⁡
                  
                    [
                    
                      k
                      +
                      1
                    
                    ]
                  
                
                =
                
                  
                    Φ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        z
                        1
                      
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                  
                  +
                  
                    Γ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      u
                      ⁡
                      
                        [
                        k
                        ]
                      
                    
                  
                  +
                  
                    
                      L
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              y
                              .
                            
                            m
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        -
                        
                          
                            z
                            1
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      z
                      _
                    
                    1
                  
                  ⁡
                  
                    [
                    k
                    ]
                  
                
                =
                
                  
                    
                      z
                      1
                    
                    ⁡
                    
                      [
                      k
                      ]
                    
                  
                  +
                  
                    
                      L
                      c
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              y
                              .
                            
                            m
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                        -
                        
                          
                            z
                            1
                          
                          ⁡
                          
                            [
                            k
                            ]
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          40
          )
        
      
    
  




where Φ=1, Γ={circumflex over (b)}T, Lc=Lp=1−β, and β=e−2ωoT.

Prediction of {circumflex over (f)}

With the new implementation of ESO 204, {circumflex over (f)}=∫l3(ym−z1)dt, {dot over ({circumflex over (f)}=l3(ym−z1) and {umlaut over ({circumflex over (f)}=l3({dot over (y)}m−z2), and this enables us to determine {circumflex over (f)}(t+lf) based on the 1st or 2nd-order Taylor series approximation shown in




  
    
      
        
          
        
        ⁢
        
          
            
              
                f
                ^
              
              ⁡
              
                (
                
                  t
                  +
                  
                    l
                    f
                  
                
                )
              
            
            ≈
            
              
                ∫
                
                  
                    
                      l
                      3
                    
                    ⁡
                    
                      (
                      
                        
                          y
                          m
                        
                        -
                        
                          z
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    t
                  
                
              
              +
              
                
                  l
                  f
                
                ⁢
                
                  
                    l
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        y
                        m
                      
                      -
                      
                        z
                        1
                      
                    
                    )
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          or
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          41
          )
        
      
    
  
  
    
      
        
          
            f
            ^
          
          ⁡
          
            (
            
              t
              +
              
                l
                f
              
            
            )
          
        
        ≈
        
          
            ∫
            
              
                
                  l
                  3
                
                ⁡
                
                  (
                  
                    
                      y
                      m
                    
                    -
                    
                      z
                      1
                    
                  
                  )
                
              
              ⁢
              
                d
                t
              
            
          
          +
          
            
              l
              f
            
            ⁢
            
              
                l
                3
              
              ⁡
              
                (
                
                  
                    y
                    m
                  
                  -
                  
                    z
                    1
                  
                
                )
              
            
          
          +
          
            
              
                
                  l
                  f
                  2
                
                ⁢
                
                  l
                  3
                
              
              2
            
            ⁢
            
              (
              
                
                  
                    y
                    .
                  
                  m
                
                -
                
                  z
                  2
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          42
          )
        
      
    
  




With RESO, {circumflex over (f)}(t+lf) is obtained as

{circumflex over (f)}(t+lf)≈∫l2({dot over (y)}m−z1)dt+lfl2({dot over (y)}m−z1)  Equation (43)

Equivalent 2dof Transfer Functions with Prediction in ESO

Using prediction in ESO 204, with the 1st-order Taylor series approximation of future value of {circumflex over (f)} used in control law 404, the equivalent 2dof transfer functions are




  
    
      
        
          C
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            1
            
              b
              ^
            
          
          ⁢
          
            
              
                
                  l
                  f
                
                ⁢
                
                  l
                  3
                
                ⁢
                
                  s
                  3
                
              
              +
              
                
                  (
                  
                    
                      
                        k
                        p
                      
                      ⁢
                      
                        l
                        1
                      
                    
                    +
                    
                      
                        k
                        d
                      
                      ⁢
                      
                        l
                        2
                      
                    
                    +
                    
                      l
                      3
                    
                  
                  )
                
                ⁢
                
                  s
                  2
                
              
              +
              
                
                  (
                  
                    
                      
                        k
                        p
                      
                      ⁢
                      
                        l
                        2
                      
                    
                    +
                    
                      
                        k
                        d
                      
                      ⁢
                      
                        l
                        3
                      
                    
                  
                  )
                
                ⁢
                s
              
              +
              
                
                  k
                  1
                
                ⁢
                
                  l
                  3
                
              
            
            
              
                s
                3
              
              +
              
                
                  (
                  
                    
                      l
                      1
                    
                    +
                    
                      k
                      d
                    
                  
                  )
                
                ⁢
                
                  s
                  2
                
              
              +
              
                
                  (
                  
                    
                      l
                      2
                    
                    +
                    
                      k
                      p
                    
                    +
                    
                      
                        k
                        d
                      
                      ⁢
                      
                        l
                        1
                      
                    
                    -
                    
                      
                        l
                        f
                      
                      ⁢
                      
                        l
                        3
                      
                    
                  
                  )
                
                ⁢
                s
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          44
          )
        
      
    
  
  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              (
              
                
                  s
                  3
                
                +
                
                  
                    l
                    1
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    l
                    2
                  
                  ⁢
                  s
                
                +
                
                  l
                  3
                
              
              )
            
            ⁢
            
              (
              
                
                  s
                  2
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  s
                
                +
                
                  k
                  p
                
              
              )
            
          
          
            
              
                l
                f
              
              ⁢
              
                l
                3
              
              ⁢
              
                s
                3
              
            
            +
            
              
                (
                
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      1
                    
                  
                  +
                  
                    
                      k
                      d
                    
                    ⁢
                    
                      l
                      2
                    
                  
                  +
                  
                    l
                    3
                  
                
                )
              
              ⁢
              
                s
                2
              
            
            +
            
              
                (
                
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      2
                    
                  
                  +
                  
                    
                      k
                      d
                    
                    ⁢
                    
                      l
                      3
                    
                  
                
                )
              
              ⁢
              s
            
            +
            
              
                k
                1
              
              ⁢
              
                l
                3
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          45
          )
        
      
    
  




where l1, l2, l3 are observer gains defined in Equation (11) and kp and kd are the proportional-derivative (PD) gains defined in Equation (17).

When the 2nd-order Taylor series approximation of {circumflex over (f)} is used in control law 404, the equivalent 2dof transfer functions are



  
    
      
    
    ⁢
    
      Equation
      ⁢
      
        
      
      ⁢
      
        (
        46
        )
      
    
  




  
    
      C
      ⁡
      
        (
        s
        )
      
    
    =
    
      
        1
        
          b
          ^
        
      
      ⁢
      
        
          
            
              
                
                  
                    
                      
                        l
                        f
                        2
                      
                      ⁢
                      
                        l
                        3
                      
                    
                    2
                  
                  ⁢
                  
                    s
                    4
                  
                
                +
                
                  
                    (
                    
                      
                        
                          l
                          f
                        
                        ⁢
                        
                          l
                          3
                        
                      
                      +
                      
                        
                          
                            l
                            f
                            2
                          
                          ⁢
                          
                            l
                            1
                          
                          ⁢
                          
                            l
                            3
                          
                        
                        2
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    3
                  
                
                +
              
            
          
          
            
              
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          1
                        
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        l
                        3
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          3
                        
                      
                    
                    )
                  
                  ⁢
                  s
                
                +
                
                  
                    k
                    1
                  
                  ⁢
                  
                    l
                    3
                  
                
              
            
          
        
        
          
            s
            3
          
          +
          
            
              (
              
                
                  l
                  1
                
                +
                
                  k
                  d
                
                -
                
                  
                    
                      l
                      f
                      2
                    
                    ⁢
                    
                      l
                      3
                    
                  
                  2
                
              
              )
            
            ⁢
            
              s
              2
            
          
          +
          
            
              (
              
                
                  l
                  2
                
                +
                
                  k
                  p
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  
                    l
                    1
                  
                
                -
                
                  
                    l
                    f
                  
                  ⁢
                  
                    l
                    3
                  
                
                -
                
                  
                    
                      l
                      f
                      2
                    
                    ⁢
                    
                      l
                      1
                    
                    ⁢
                    
                      l
                      3
                    
                  
                  2
                
              
              )
            
            ⁢
            s
          
        
      
    
  




  
    
      
    
    ⁢
    
      Equation
      ⁢
      
        
      
      ⁢
      
        (
        47
        )
      
    
  




  
    
      
    
    ⁢
    
      
        H
        ⁡
        
          (
          s
          )
        
      
      =
      
        
          
            (
            
              
                s
                3
              
              +
              
                
                  l
                  1
                
                ⁢
                
                  s
                  2
                
              
              +
              
                
                  l
                  2
                
                ⁢
                s
              
              +
              
                l
                3
              
            
            )
          
          ⁢
          
            (
            
              
                s
                2
              
              +
              
                
                  k
                  d
                
                ⁢
                s
              
              +
              
                k
                p
              
            
            )
          
        
        
          
            
              
                
                  
                    
                      
                        l
                        f
                        2
                      
                      ⁢
                      
                        l
                        3
                      
                    
                    2
                  
                  ⁢
                  
                    s
                    4
                  
                
                +
                
                  
                    (
                    
                      
                        
                          l
                          f
                        
                        ⁢
                        
                          l
                          3
                        
                      
                      +
                      
                        
                          
                            l
                            f
                            2
                          
                          ⁢
                          
                            l
                            1
                          
                          ⁢
                          
                            l
                            3
                          
                        
                        2
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    3
                  
                
                +
              
            
          
          
            
              
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          1
                        
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        l
                        3
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          3
                        
                      
                    
                    )
                  
                  ⁢
                  s
                
                +
                
                  
                    k
                    1
                  
                  ⁢
                  
                    l
                    3
                  
                
              
            
          
        
      
    
  



Equivalent 2dof Transfer Function with Prediction in RESO

The equivalent 2dof transfer functions of ADRC-RESO with prediction in {circumflex over (f)} are as follows.




  
    
      
        
          
            G
            c
          
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            1
            
              b
              ^
            
          
          ⁢
          
            
              
                
                  l
                  f
                
                ⁢
                
                  l
                  2
                
                ⁢
                
                  s
                  3
                
              
              +
              
                
                  (
                  
                    
                      k
                      p
                    
                    +
                    
                      
                        k
                        d
                      
                      ⁢
                      
                        l
                        1
                      
                    
                    +
                    
                      l
                      2
                    
                  
                  )
                
                ⁢
                
                  s
                  2
                
              
              +
              
                
                  (
                  
                    
                      
                        k
                        p
                      
                      ⁢
                      
                        l
                        1
                      
                    
                    +
                    
                      
                        k
                        d
                      
                      ⁢
                      
                        l
                        2
                      
                    
                  
                  )
                
                ⁢
                s
              
              +
              
                
                  k
                  p
                
                ⁢
                
                  l
                  2
                
              
            
            
              
                s
                2
              
              +
              
                
                  (
                  
                    
                      k
                      d
                    
                    +
                    
                      l
                      1
                    
                    -
                    
                      
                        l
                        f
                      
                      ⁢
                      
                        l
                        2
                      
                    
                  
                  )
                
                ⁢
                s
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          48
          )
        
      
    
  
  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              (
              
                
                  s
                  2
                
                +
                
                  
                    l
                    1
                  
                  ⁢
                  s
                
                +
                
                  l
                  2
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  p
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  s
                
                +
                
                  s
                  2
                
              
              )
            
          
          
            
              
                l
                f
              
              ⁢
              
                l
                2
              
              ⁢
              
                s
                3
              
            
            +
            
              
                (
                
                  
                    k
                    p
                  
                  +
                  
                    
                      k
                      d
                    
                    ⁢
                    
                      l
                      1
                    
                  
                  +
                  
                    l
                    2
                  
                
                )
              
              ⁢
              
                s
                2
              
            
            +
            
              
                (
                
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      1
                    
                  
                  +
                  
                    
                      k
                      d
                    
                    ⁢
                    
                      l
                      2
                    
                  
                
                )
              
              ⁢
              s
            
            +
            
              
                k
                p
              
              ⁢
              
                l
                2
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          49
          )
        
      
    
  




where l1, l2 are observer gains defined in Equation (35).

State Predictor Method for Predictive Extended State Observer

Another embodiment for forming a predictive state and disturbance observer 304 module is to use a state predictor method to extend the performance of a baseline ESO 204 to predict state estimates τseconds in the future. The basic state predictor method approach was published by T. Oguchi, H. Nijmeijer, Prediction of Chaotic Behavior, IEEE Trans. on Circuits and Systems-I, Vol. 52, No. 11, pp. 2464-2472, 2005, which is hereby incorporated by reference. The state predictor method is combined in the embodiments presented herein with the regular ESO 204 of the form




  
    
      
        
          
            z
            .
          
          =
          
            Az
            +
            Bu
            +
            
              L
              ⁡
              
                (
                
                  y
                  -
                  
                    y
                    ^
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            y
            ^
          
          =
          Cz
        
        ⁢
        
          

        
        ⁢
        where
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          50
          )
        
      
    
  
  
    
      
        
          A
          =
          
            [
            
              
                
                  0
                
                
                  1
                
                
                  θ
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        ,
        
          

        
        ⁢
        
          B
          =
          
            [
            
              
                
                  0
                
              
              
                
                  
                    b
                    ^
                  
                
              
              
                
                  0
                
              
            
            ]
          
        
        ,
        
          

        
        ⁢
        
          C
          =
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        ,
        
          

        
        ⁢
        
          L
          =
          
            [
            
              
                
                  
                    3
                    ⁢
                    
                      ω
                      o
                    
                  
                
              
              
                
                  
                    3
                    ⁢
                    
                      ω
                      o
                      2
                    
                  
                
              
              
                
                  
                    ω
                    o
                    3
                  
                
              
            
            ]
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          51
          )
        
      
    
  




to form the predictive ESO is in the form of

ż(t)=Az(t)+Bu(t)+L(y(t)−ŷ(t))

ŷ(t)=Cz(t−τ)  Equation (52)

Then ŷ(t)=z1(t−τ) will track y(t) while z1 (t)=ŷ(t+τ) is its prediction τ seconds ahead. The ideas is that since the observer design ensures that the observer error goes to zero, or becomes very small, if that error is defined as the difference between y(t) and z1(t−τ) and z1(t−τ) will approach y(t) and z1(t) will approach y(t+τ). Therefore z1(t) can be used as a prediction of y(t)τ second ahead.
Building a Predictive State and Disturbance Observer Using an Output Prediction Mechanism
Perhaps a simpler method to obtain z(t+τ) is to leave the observer unchanged but replace its input ym(t) with the predicted output ŷ(t+τ) as shown in FIG. 5. This configuration is used both to obtain the predicted error and its derivatives as well as the predicted disturbance estimation. Also depicted in FIG. 5 is a prediction module computer component or more generally a prediction module 508. The prediction module 508 is adapted to accept the measured system output (ym(t)) and calculated estimated values of the system output (ŷ(t+τ)) from the present time to a time (τ) in the future.
A state and disturbance observer 504, shown in FIG. 5, accepts the predicted and current values of the plant output (ym) and the control commands (u(t)) to estimate the future estimated states (zn(t+τ)) where (n) is the number of estimated states plus an estimate of the generalized disturbance (fn). The state and disturbance observer 504 in one embodiment is an ESO 204. In a second embodiment, the state and disturbance observer 504 is a disturbance observer (DOB) structure. In yet another embodiment, the state and disturbance observer 504 is an unknown input observer (UID). It is apparent that the combination of the state and disturbance observer 504 in combination with the prediction module 508 provides a similar output and performance a similar function within the framework of an output predictive ADRC controller 500 as the predictive state and disturbance observer 304 of the predictive ADRC controller 300. In this manner, the various embodiments of the predictive ADRC controller 300 and output predictive ADRC controller 500 are correlated.
Also shown in FIG. 5, a control law 506 is provided that uses the estimated future states of the plant 110 coupled with the desired trajectory (r(n)(t+τ)) to generate a current control input (u(t)). The embodiment of the control law 506 depicted in FIG. 5 incorporates the additive inverse of the estimated future values of the estimated disturbance (fn).
Combine the Output Prediction with Regular ESO
To compensate for the delay, y(t) is replaced with the approximately predicted output ŷ(t+τ) as the input to the ESO 204. Again, the approximated prediction, ŷ(t+τ), is obtained from the Taylor Series approximation. The 1st-order Taylor series approximation and the 2nd-order Taylor series approximation of ŷ(t+τ) are as shown in the following equations respectively.




  
    
      
        
          
            y
            ^
          
          ⁡
          
            (
            
              t
              +
              τ
            
            )
          
        
        =
        
          
            
              y
              m
            
            ⁡
            
              (
              t
              )
            
          
          +
          
            
              
                
                  y
                  .
                
                m
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            τ
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          53
          )
        
      
    
  
  
    
      
        
          
            y
            ^
          
          ⁡
          
            (
            
              t
              +
              τ
            
            )
          
        
        =
        
          
            
              y
              m
            
            ⁡
            
              (
              t
              )
            
          
          +
          
            
              
                
                  y
                  .
                
                m
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            τ
          
          +
          
            
              
                
                  
                    y
                     ̈
                  
                  m
                
                ⁡
                
                  (
                  t
                  )
                
              
              2
            
            ⁢
            
              τ
              2
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          54
          )
        
      
    
  




Then the predicted system output and the current control signal are used as the inputs to the ESO 204 as follows.




  
    
      
        {
        
          
            
              
                
                  
                    z
                    .
                  
                  ⁡
                  
                    (
                    
                      t
                      +
                      τ
                    
                    )
                  
                
                =
                
                  
                    
                      A
                      z
                    
                    ⁢
                    
                      z
                      ⁡
                      
                        (
                        
                          t
                          +
                          τ
                        
                        )
                      
                    
                  
                  +
                  
                    
                      B
                      z
                    
                    ⁢
                    
                      u
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  +
                  
                    L
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            ^
                          
                          ⁡
                          
                            (
                            
                              t
                              +
                              τ
                            
                            )
                          
                        
                        -
                        
                          
                            C
                            z
                          
                          ⁢
                          
                            z
                            ⁡
                            
                              (
                              
                                t
                                +
                                τ
                              
                              )
                            
                          
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    f
                    ^
                  
                  ⁡
                  
                    (
                    
                      t
                      +
                      τ
                    
                    )
                  
                
                =
                
                  Vz
                  ⁡
                  
                    (
                    
                      t
                      +
                      τ
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          55
          )
        
      
    
  




which will provide both the predicted states and the disturbance estimation in z(t+τ).

Equivalent 2dof Transfer Functions

When the 1st-order Taylor series approximation is used to obtain ŷ(t+τ), the equivalent 2dof transfer functions are




  
    
      
        
          C
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            1
            
              b
              ^
            
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          l
                          3
                        
                        +
                        
                          
                            k
                            d
                          
                          ⁢
                          
                            l
                            2
                          
                        
                        +
                        
                          
                            l
                            1
                          
                          ⁢
                          
                            k
                            p
                          
                        
                      
                      )
                    
                    ⁢
                    
                      s
                      2
                    
                  
                  +
                  
                    
                      (
                      
                        
                          
                            k
                            p
                          
                          ⁢
                          
                            l
                            2
                          
                        
                        +
                        
                          
                            l
                            3
                          
                          ⁢
                          
                            k
                            d
                          
                        
                      
                      )
                    
                    ⁢
                    s
                  
                  +
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      3
                    
                  
                
                )
              
              ⁢
              
                (
                
                  1
                  +
                  
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    s
                  
                
                )
              
            
            
              s
              ⁡
              
                (
                
                  
                    s
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          l
                          1
                        
                        +
                        
                          k
                          d
                        
                      
                      )
                    
                    ⁢
                    s
                  
                  +
                  
                    (
                    
                      
                        l
                        2
                      
                      +
                      
                        k
                        p
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          1
                        
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          56
          )
        
      
    
  
  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              (
              
                
                  s
                  3
                
                +
                
                  
                    l
                    1
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    l
                    2
                  
                  ⁢
                  s
                
                +
                
                  l
                  3
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  p
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  s
                
                +
                
                  s
                  2
                
              
              )
            
          
          
            
              (
              
                
                  
                    (
                    
                      
                        l
                        3
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        
                          l
                          1
                        
                        ⁢
                        
                          k
                          p
                        
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        
                          l
                          3
                        
                        ⁢
                        
                          k
                          d
                        
                      
                    
                    )
                  
                  ⁢
                  s
                
                +
                
                  
                    k
                    p
                  
                  ⁢
                  
                    l
                    3
                  
                
              
              )
            
            ⁢
            
              (
              
                1
                +
                
                  τ
                  ⁢
                  
                    
                  
                  ⁢
                  s
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          57
          )
        
      
    
  




When the 2nd-order Taylor Series approximation is used to obtain ŷ(t+τ), the equivalent 2dof transfer functions are




  
    
      
        
          C
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            1
            
              b
              ^
            
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          l
                          3
                        
                        +
                        
                          
                            k
                            d
                          
                          ⁢
                          
                            l
                            2
                          
                        
                        +
                        
                          
                            l
                            1
                          
                          ⁢
                          
                            k
                            p
                          
                        
                      
                      )
                    
                    ⁢
                    
                      s
                      2
                    
                  
                  +
                  
                    
                      (
                      
                        
                          
                            k
                            p
                          
                          ⁢
                          
                            l
                            2
                          
                        
                        +
                        
                          
                            l
                            3
                          
                          ⁢
                          
                            k
                            d
                          
                        
                      
                      )
                    
                    ⁢
                    s
                  
                  +
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      3
                    
                  
                
                )
              
              ⁢
              
                (
                
                  1
                  +
                  
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    s
                  
                  +
                  
                    
                      
                        τ
                        2
                      
                      2
                    
                    ⁢
                    
                      s
                      2
                    
                  
                
                )
              
            
            
              s
              ⁡
              
                (
                
                  
                    s
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          l
                          1
                        
                        +
                        
                          k
                          d
                        
                      
                      )
                    
                    ⁢
                    s
                  
                  +
                  
                    (
                    
                      
                        l
                        2
                      
                      +
                      
                        k
                        p
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          1
                        
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          58
          )
        
      
    
  
  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              (
              
                
                  s
                  2
                
                +
                
                  
                    l
                    1
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    l
                    2
                  
                  ⁢
                  s
                
                +
                
                  l
                  3
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  p
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  
                    s_s
                    2
                  
                
              
              )
            
          
          
            
              (
              
                
                  
                    (
                    
                      
                        l
                        3
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        
                          l
                          1
                        
                        ⁢
                        
                          k
                          p
                        
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          2
                        
                      
                      +
                      
                        
                          l
                          3
                        
                        ⁢
                        
                          k
                          d
                        
                      
                    
                    )
                  
                  ⁢
                  s
                
                +
                
                  
                    k
                    p
                  
                  ⁢
                  
                    l
                    3
                  
                
              
              )
            
            ⁢
            
              (
              
                1
                +
                
                  τ
                  ⁢
                  
                    
                  
                  ⁢
                  s
                
                +
                
                  
                    
                      τ
                      2
                    
                    2
                  
                  ⁢
                  
                    s
                    2
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          59
          )
        
      
    
  




Combine the Output Prediction with RESO

With the RESO configuration, the prediction is obtained as

{dot over (ŷ)}(t+τ)={dot over (y)}m(t)+ÿm(t)τ  Equation (60)

and the new RESO is constructed as





  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        
                          z
                          .
                        
                        ⁡
                        
                          (
                          
                            t
                            +
                            τ
                          
                          )
                        
                      
                      =
                      
                        
                          
                            A
                            z
                          
                          ⁢
                          
                            z
                            ⁡
                            
                              (
                              
                                t
                                +
                                τ
                              
                              )
                            
                          
                        
                        +
                        
                          
                            B
                            z
                          
                          ⁢
                          
                            u
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                        +
                        
                          L
                          ⁡
                          
                            (
                            
                              
                                
                                  y
                                  
                                    .
                                    ^
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    +
                                    τ
                                  
                                  )
                                
                              
                              -
                              
                                
                                  C
                                  z
                                
                                ⁢
                                
                                  z
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      +
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          f
                          ^
                        
                        ⁡
                        
                          (
                          
                            t
                            +
                            τ
                          
                          )
                        
                      
                      =
                      
                        Vz
                        ⁡
                        
                          (
                          
                            t
                            +
                            τ
                          
                          )
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              
                A
                z
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              B
              z
            
            =
            
              [
              
                
                  
                    
                      b
                      ^
                    
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            L
            =
            
              [
              
                
                  
                    
                      l
                      1
                    
                  
                
                
                  
                    
                      l
                      2
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              C
              z
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            V
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              and
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              
                l
                1
              
            
            =
            
              2
              ⁢
              
                ω
                o
              
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              l
              2
            
            =
            
              
                ω
                o
                2
              
              .
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          61
          )
        
      
    
  




which again provides the predicted state and disturbance estimation in z(t+τ).

Equivalent 2 dof Transfer Functions

With the above the 1st-order Taylor Series approximation {dot over (ŷ)}(t+τ) is used as the input to the RESO, the equivalent 2dof transfer functions are:




  
    
      
        
          C
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            1
            
              b
              ^
            
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          k
                          p
                        
                        +
                        
                          
                            k
                            d
                          
                          ⁢
                          
                            l
                            1
                          
                        
                        +
                        
                          l
                          2
                        
                      
                      )
                    
                    ⁢
                    
                      s
                      2
                    
                  
                  +
                  
                    
                      (
                      
                        
                          
                            k
                            p
                          
                          ⁢
                          
                            l
                            1
                          
                        
                        +
                        
                          
                            l
                            2
                          
                          ⁢
                          
                            k
                            d
                          
                        
                      
                      )
                    
                    ⁢
                    s
                  
                  +
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      2
                    
                  
                
                )
              
              ⁢
              
                (
                
                  1
                  +
                  
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    s
                  
                
                )
              
            
            
              s
              ⁡
              
                (
                
                  s
                  +
                  
                    (
                    
                      
                        l
                        1
                      
                      +
                      
                        k
                        d
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          62
          )
        
      
    
  
  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              (
              
                
                  s
                  2
                
                +
                
                  
                    l
                    1
                  
                  ⁢
                  s
                
                +
                
                  l
                  2
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  p
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  s
                
                +
                
                  s
                  2
                
              
              )
            
          
          
            
              (
              
                
                  
                    (
                    
                      
                        k
                        p
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          1
                        
                      
                      +
                      
                        l
                        2
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          1
                        
                      
                      +
                      
                        
                          l
                          2
                        
                        ⁢
                        
                          k
                          d
                        
                      
                    
                    )
                  
                  ⁢
                  s
                
                +
                
                  
                    k
                    p
                  
                  ⁢
                  
                    l
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                1
                +
                
                  τ
                  ⁢
                  
                    
                  
                  ⁢
                  s
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          63
          )
        
      
    
  




Combine the Output Prediction with RESO

With the RESO configuration, the prediction is obtained as

{dot over (ŷ)}(t+τ)={dot over (y)}m(t)+ÿm(t)τ  Equation (64)

and the new RESO is constructed as





  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        
                          z
                          .
                        
                        ⁡
                        
                          (
                          
                            t
                            +
                            τ
                          
                          )
                        
                      
                      =
                      
                        
                          
                            A
                            z
                          
                          ⁢
                          
                            z
                            ⁡
                            
                              (
                              
                                t
                                +
                                τ
                              
                              )
                            
                          
                        
                        +
                        
                          
                            B
                            z
                          
                          ⁢
                          
                            u
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                        
                        +
                        
                          L
                          ⁡
                          
                            (
                            
                              
                                
                                  y
                                  
                                    .
                                    ^
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    t
                                    +
                                    τ
                                  
                                  )
                                
                              
                              -
                              
                                
                                  C
                                  z
                                
                                ⁢
                                
                                  z
                                  ⁡
                                  
                                    (
                                    
                                      t
                                      +
                                      τ
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          f
                          ^
                        
                        ⁡
                        
                          (
                          
                            t
                            +
                            τ
                          
                          )
                        
                      
                      =
                      
                        Vz
                        ⁡
                        
                          (
                          
                            t
                            +
                            τ
                          
                          )
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              
                A
                z
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              B
              z
            
            =
            
              [
              
                
                  
                    
                      b
                      ^
                    
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            L
            =
            
              [
              
                
                  
                    
                      l
                      1
                    
                  
                
                
                  
                    
                      l
                      2
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              C
              z
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            V
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              and
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              
                l
                1
              
            
            =
            
              2
              ⁢
              
                ω
                a
              
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              l
              2
            
            =
            
              
                ω
                o
                2
              
              .
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          65
          )
        
      
    
  




which again provides the predicted state and disturbance estimation in z(t+τ).

Equivalent 2 Dof Transfer Functions

With the above the 1st-order Taylor Series approximation {dot over (ŷ)}(t+τ) is used as the input to the RESO, the equivalent 2dof transfer functions are




  
    
      
        
          C
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            1
            
              b
              ^
            
          
          ⁢
          
            
              
                (
                
                  
                    
                      (
                      
                        
                          k
                          p
                        
                        +
                        
                          
                            k
                            d
                          
                          ⁢
                          
                            l
                            1
                          
                        
                        +
                        
                          l
                          2
                        
                      
                      )
                    
                    ⁢
                    
                      s
                      2
                    
                  
                  +
                  
                    
                      (
                      
                        
                          
                            k
                            p
                          
                          ⁢
                          
                            l
                            1
                          
                        
                        +
                        
                          
                            l
                            2
                          
                          ⁢
                          
                            k
                            d
                          
                        
                      
                      )
                    
                    ⁢
                    s
                  
                  +
                  
                    
                      k
                      p
                    
                    ⁢
                    
                      l
                      2
                    
                  
                
                )
              
              ⁢
              
                (
                
                  1
                  +
                  
                    τ
                    ⁢
                    
                      
                    
                    ⁢
                    s
                  
                
                )
              
            
            
              s
              ⁡
              
                (
                
                  s
                  +
                  
                    (
                    
                      
                        l
                        1
                      
                      +
                      
                        k
                        d
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          66
          )
        
      
    
  
  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              (
              
                
                  s
                  2
                
                +
                
                  
                    l
                    1
                  
                  ⁢
                  s
                
                +
                
                  l
                  2
                
              
              )
            
            ⁢
            
              (
              
                
                  k
                  p
                
                +
                
                  
                    k
                    d
                  
                  ⁢
                  s
                
                +
                
                  s
                  2
                
              
              )
            
          
          
            
              (
              
                
                  
                    (
                    
                      
                        k
                        p
                      
                      +
                      
                        
                          k
                          d
                        
                        ⁢
                        
                          l
                          1
                        
                      
                      +
                      
                        l
                        2
                      
                    
                    )
                  
                  ⁢
                  
                    s
                    2
                  
                
                +
                
                  
                    (
                    
                      
                        
                          k
                          p
                        
                        ⁢
                        
                          l
                          1
                        
                      
                      +
                      
                        
                          l
                          2
                        
                        ⁢
                        
                          k
                          d
                        
                      
                    
                    )
                  
                  ⁢
                  s
                
                +
                
                  
                    k
                    p
                  
                  ⁢
                  
                    l
                    2
                  
                
              
              )
            
            ⁢
            
              (
              
                1
                +
                
                  τ
                  ⁢
                  
                    
                  
                  ⁢
                  s
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          67
          )
        
      
    
  




A Parameterized Nonlinear PD Control Law, Replacing the Linear PD Controller

The parameterized linear PD control law 206 gpd (e,ė,ωc) used above can be replaced by the following parameterized Nonlinear PD (NPD) control law 506 gnpd(e,ė,ωc):




  
    
      
        
          y
          =
          
            
              
                ω
                c
                2
              
              ⁢
              e
            
            +
            
              
                ω
                c
              
              ⁢
              
                e
                .
              
            
          
        
        ,
        
          

        
        ⁢
        
          x
          =
          
            {
            
              
                
                  
                    
                      
                        
                          
                            
                              ω
                              c
                            
                            ⁢
                            
                              e
                              .
                            
                          
                          +
                          
                            
                              
                                
                                  
                                    R
                                    ⁡
                                    
                                      (
                                      
                                        R
                                        +
                                        
                                          8
                                          ⁢
                                          
                                            
                                            y
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                R
                              
                              2
                            
                            ⁢
                            
                              sign
                              ⁡
                              
                                (
                                y
                                )
                              
                            
                          
                        
                        ,
                      
                    
                    
                      
                        
                          
                          y
                          
                        
                        >
                        R
                      
                    
                  
                  
                    
                      
                        
                          
                            ω
                            c
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  ω
                                  c
                                
                                ⁢
                                e
                              
                              +
                              
                                2
                                ⁢
                                
                                  e
                                  .
                                
                              
                            
                            )
                          
                        
                        ,
                      
                    
                    
                      
                        
                          
                          y
                          
                        
                        ≤
                        R
                      
                    
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                    g
                    npd
                  
                  ⁡
                  
                    (
                    
                      e
                      ,
                      
                        e
                        .
                      
                      ,
                      
                        ω
                        c
                      
                    
                    )
                  
                
              
              =
              
                {
                
                  
                    
                      
                        
                          
                            
                              
                                R
                                ⁢
                                sign
                              
                              ⁡
                              
                                (
                                x
                                )
                              
                            
                            ,
                          
                        
                        
                          
                            
                              
                              x
                              
                            
                            >
                            R
                          
                        
                      
                      
                        
                          
                            x
                            ,
                          
                        
                        
                          
                            
                              
                              x
                              
                            
                            ≤
                            R
                          
                        
                      
                    
                    ⁢
                    
                      

                    
                    ⁢
                    where
                    ⁢
                    
                      

                    
                    ⁢
                    
                      sign
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                  =
                  
                    {
                    
                      
                        
                          
                            1
                            ,
                          
                        
                        
                          
                            x
                            ≥
                            0
                          
                        
                      
                      
                        
                          
                            
                              -
                              1
                            
                            ,
                          
                        
                        
                          
                            x
                            <
                            0.
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          68
          )
        
      
    
  




Equation (68) shows that, within the region |x|≦R and |y|≦R, this parameterized NPD control law 506 is ωc2e+2ωcė, which is the same as the parameterized linear PD control law 206 in Equation (15); outside this region, however, the NPD control law 506 is nonlinear. This new NPD control law 506 can be used to replace the linear PD control law 206 in the original ADRC 200 algorithm to achieve better performance. In one embodiment the NPD control law 506 is used in conjunction with a predicted state estimate zn(t+τ). In a second embodiment, the NPD control law 506 is uses only the current state estimate (zn(t).
A General Form of Reduced-Order Multiple Extended-State ESO for an nth-Order Plant
Although the Predictive ADRC 300 has been presented with an ESO 204 or RESO with only one extended state, the same technique can be readily extended to a general nth-order plant with an ESO 240 or RESO that has multiple extended states. To facilitate such development, we present here a general form of the reduced-order multiple extended states ESO 204 for an nth-order plant of the form

y(n)=f(y,y, . . . ,y(n-1),d,t)+bu  Equation (69)

Let the n states be x1=y, x2={dot over (x)}1={dot over (y)}, . . . , xn={dot over (x)}n−1=y(n-1), and the h extended states be xn+1=f, xn+2={dot over (x)}n+1=f, . . . , xn+h={dot over (x)}n+h-1=f(h-1). The plant 110 in Equation (69) can now be expressed in the augmented form:




or equivalently





  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        x
                        .
                      
                      =
                      
                        
                          
                            A
                            x
                          
                          ⁢
                          x
                        
                        +
                        
                          
                            B
                            x
                          
                          ⁢
                          u
                        
                        +
                        
                          Ef
                          
                            (
                            h
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      y
                      =
                      
                        
                          C
                          x
                        
                        ⁢
                        x
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              
                A
                x
              
            
            =
            
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                    
                      ⋱
                    
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                    
                      ⋱
                    
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                    
                      ⋱
                    
                    
                      ⋱
                    
                    
                      ⋱
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                
                ]
              
              
                
                  (
                  
                    n
                    +
                    h
                  
                  )
                
                ×
                
                  (
                  
                    n
                    +
                    h
                  
                  )
                
              
            
          
          ,
          
            

          
          ⁢
          
            
              B
              x
            
            =
            
              
                [
                
                  
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      b
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                  
                
                ]
              
              
                
                  (
                  
                    n
                    +
                    h
                  
                  )
                
                ×
                1
              
            
          
          ,
          
            

          
          ⁢
          
            E
            =
            
              
                [
                
                  
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      1
                    
                  
                
                ]
              
              
                
                  (
                  
                    n
                    +
                    h
                  
                  )
                
                ×
                1
              
            
          
          ,
          
            

          
          ⁢
          
            
              C
              x
            
            =
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                
                ]
              
              
                1
                ×
                
                  (
                  
                    n
                    +
                    h
                  
                  )
                
              
            
          
          ,
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          70
          )
        
      
    
  




and the nonzero element in Bx is in the nth row.

Assume that y, {dot over (y)}, . . . , y(n+h-m) are measured or otherwise available as ym, {dot over (y)}m, . . . , ym(n+h-m), then the approximate states x1, x2, . . . , xn+h-m+1 are available (If m=h, the last state can be obtained from its definition: xn+1=f=ym(n)−{circumflex over (b)}u). Also assume f(n) does not change rapidly and can be handled by the observer. The ESO 204 order can be reduced to m, where h≦m≦n+h. With xn+h-m+1 and u as inputs, and with z1, . . . , zm estimating xn+h-m+1, . . . , xn+h respectively, the general ESO 204 form is




or in state space form





  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        z
                        .
                      
                      =
                      
                        
                          
                            A
                            z
                          
                          ⁢
                          z
                        
                        +
                        
                          
                            B
                            z
                          
                          ⁢
                          u
                        
                        +
                        
                          L
                          ⁡
                          
                            (
                            
                              
                                x
                                
                                  n
                                  +
                                  h
                                  -
                                  m
                                  +
                                  1
                                
                              
                              -
                              
                                
                                  C
                                  z
                                
                                ⁢
                                z
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        f
                        ^
                      
                      =
                      Vz
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              where
              ⁢
              
                

              
              ⁢
              
                A
                z
              
            
            =
            
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                    
                      ⋱
                    
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                    
                      ⋱
                    
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                    
                      ⋱
                    
                    
                      ⋱
                    
                    
                      ⋱
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                
                ]
              
              
                m
                ×
                m
              
            
          
          ,
          
            

          
          ⁢
          
            L
            =
            
              
                [
                
                  
                    
                      
                        l
                        1
                      
                    
                  
                  
                    
                      
                        l
                        2
                      
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      
                        l
                        m
                      
                    
                  
                
                ]
              
              
                m
                ×
                1
              
            
          
          ,
          
            

          
          ⁢
          
            
              C
              z
            
            =
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                
                ]
              
              
                1
                ×
                m
              
            
          
          ,
          
            

          
          ⁢
          
            
              if
              ⁢
              
                

              
              ⁢
              m
            
            =
            h
          
          ,
          
            

          
          ⁢
          
            
              
                B
                z
              
              =
              
                
                  [
                  0
                  ]
                
                
                  m
                  ×
                  1
                
              
            
            ;
            
              

            
            ⁢
            
              
                
                  if
                  ⁢
                  
                    

                  
                  ⁢
                  m
                
                >
                
                  h
                  ⁢
                  
                    

                  
                  ⁢
                  
                    B
                    z
                  
                
              
              =
              
                
                  [
                  
                    
                      
                        0
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        0
                      
                    
                    
                      
                        
                          b
                          ^
                        
                      
                    
                    
                      
                        0
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        0
                      
                    
                  
                  ]
                
                
                  m
                  ×
                  1
                
              
            
          
          ,
          
            

          
          ⁢
          
            V
            =
            
              
                [
                
                  
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      1
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      ⋮
                    
                  
                  
                    
                      0
                    
                  
                
                ]
              
              
                m
                ×
                1
              
              T
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          71
          )
        
      
    
  




Thus, when m>h the nonzero elements of Bz and V are in the (m−h)th and (m−h+1)th. column respectively.

For timing simplicity, the observer gains l1, . . . , lm are chosen as




  
    
      l
      i
    
    =
    
      
        (
        
          
            
              m
            
          
          
            
              i
            
          
        
        )
      
      ⁢
      
        ω
        o
        i
      
    
  
  ,
  
    

  
  ⁢
  
    0
    <
    i
    ≤
    
      m
      .
      
        
      
      ⁢
      
        (
        
          
            
              m
            
          
          
            
              i
            
          
        
        )
      
    
  




denotes the Binomial Coefficient





  
    
      m
      !
    
    
      
        i
        !
      
      ⁢
      
        
          (
          
            m
            -
            i
          
          )
        
        !
      
    
  
  .




Also, to make the transfer function derivation easier, define





  
    l
    0
  
  =
  
    
      
        (
        
          
            
              m
            
          
          
            
              0
            
          
        
        )
      
      ⁢
      
        ω
        o
        0
      
    
    =
    1.
  




With the state zm-h+1 in the ESO 204 to estimate f, a control law, u=(u0−zm-h+1)/{circumflex over (b)}, is applied to reduce the plant to a pure nth-order cascade integral plant y(n)=u0, which can be easily controlled using the parameterized linear PD control law 206 of the form




  
    
      
        
          
            u
            0
          
          =
          
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  n
                  +
                  1
                
              
              ⁢
              
                
                  k
                  i
                
                ⁢
                
                  r
                  
                    (
                    
                      i
                      -
                      1
                    
                    )
                  
                
              
            
            -
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  n
                  +
                  h
                  -
                  m
                
              
              ⁢
              
                
                  k
                  i
                
                ⁢
                
                  y
                  m
                  
                    (
                    
                      i
                      -
                      1
                    
                    )
                  
                
              
            
            -
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  m
                  -
                  h
                  +
                  1
                
              
              ⁢
              
                
                  k
                  
                    n
                    +
                    h
                    -
                    m
                    +
                    i
                  
                
                ⁢
                
                  z
                  i
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          
            
              k
              i
            
            =
            
              
                (
                
                  
                    
                      m
                    
                  
                  
                    
                      
                        i
                        -
                        1
                      
                    
                  
                
                )
              
              ⁢
              
                ω
                c
                
                  n
                  +
                  1
                  -
                  i
                
              
            
          
          ,
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          72
          )
        
      
    
  




0≦i≦n+1, and the middle term,





  
    
      ∑
      
        i
        =
        1
      
      
        n
        +
        h
        -
        m
      
    
    ⁢
    
      
        k
        i
      
      ⁢
      
        y
        
          (
          
            i
            -
            1
          
          )
        
      
    
  
  ,




is corresponding to the order reduction (it disappears when the inputs of ESO 204 are ym and u). In the case where the measured output ym is clean and its derivatives can be obtained from ym, ADRC 200 can also be applied as





  
    
      
        
          u
          0
        
        =
        
          
            
              ∑
              
                i
                =
                1
              
              
                n
                +
                1
              
            
            ⁢
            
              
                k
                i
              
              ⁢
              
                r
                
                  (
                  
                    i
                    -
                    1
                  
                  )
                
              
            
          
          -
          
            
              ∑
              
                i
                =
                1
              
              n
            
            ⁢
            
              
                k
                i
              
              ⁢
              
                y
                m
                
                  (
                  
                    i
                    -
                    1
                  
                  )
                
              
            
          
          -
          
            z
            
              m
              -
              h
              +
              1
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          73
          )
        
      
    
  




The corresponding transfer function representation of ADRC 400, in a 2dof structure (as shown in FIG. 4) is




  
    
      
        
          H
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              Ψ
              
                0
                ,
                m
              
              m
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  1
                
                
                  n
                  +
                  1
                
              
              ⁢
              
                
                  k
                  j
                
                ⁢
                
                  s
                  
                    j
                    -
                    1
                  
                
              
            
          
          
            
              
                Ψ
                
                  0
                  ,
                  m
                
                m
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    n
                    +
                    h
                    -
                    m
                  
                
                ⁢
                
                  
                    k
                    j
                  
                  ⁢
                  
                    s
                    
                      (
                      
                        j
                        -
                        1
                      
                      )
                    
                  
                
              
            
            +
            
              
                ∑
                
                  j
                  =
                  1
                
                
                  m
                  -
                  h
                  +
                  1
                
              
              ⁢
              
                
                  k
                  
                    n
                    +
                    h
                    -
                    m
                    +
                    j
                  
                
                ⁢
                
                  s
                  
                    n
                    +
                    h
                    -
                    1
                    -
                    m
                    +
                    j
                  
                
                ⁢
                
                  Ψ
                  
                    j
                    ,
                    m
                  
                  m
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          74
          )
        
      
    
  
  
    
      
        
          
            
              G
              c
            
            ⁡
            
              (
              s
              )
            
          
          =
          
            
              1
              
                b
                ^
              
            
            ⁢
            
              
                
                  
                    Ψ
                    
                      0
                      ,
                      m
                    
                    m
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        1
                      
                      
                        n
                        +
                        h
                        -
                        m
                      
                    
                    ⁢
                    
                      
                        k
                        j
                      
                      ⁢
                      
                        s
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      j
                      =
                      1
                    
                    
                      m
                      -
                      h
                      +
                      1
                    
                  
                  ⁢
                  
                    
                      k
                      
                        n
                        +
                        h
                        -
                        m
                        +
                        j
                      
                    
                    ⁢
                    
                      s
                      
                        n
                        +
                        h
                        -
                        1
                        -
                        m
                        +
                        j
                      
                    
                    ⁢
                    
                      Ψ
                      
                        j
                        ,
                        m
                      
                      m
                    
                  
                
              
              
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    m
                    -
                    h
                    +
                    1
                  
                
                ⁢
                
                  
                    k
                    
                      n
                      +
                      h
                      -
                      m
                      +
                      j
                    
                  
                  ⁢
                  
                    s
                    
                      h
                      -
                      1
                      +
                      j
                      -
                      m
                    
                  
                  ⁢
                  
                    Ψ
                    
                      0
                      ,
                      
                        j
                        -
                        1
                      
                    
                    m
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              Ψ
              
                x
                ,
                y
              
              m
            
            =
            
              
                ∑
                
                  i
                  =
                  x
                
                y
              
              ⁢
              
                
                  l
                  i
                
                ⁢
                
                  s
                  
                    m
                    -
                    i
                  
                
              
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            0
            ≤
            x
            ≤
            
              y
              .
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          75
          )
        
      
    
  




ADRC with Partially Known Plant Dynamics

ADRC controller does not require the detail plant model information. However, if the whole or part of the plant dynamics is known, various embodiments of the ADRC controller can be modified to utilize the information to estimate and reject disturbance more effectively, either by control law modification or ESO 204 modification. With a part of the plant dynamics known, the known part of the generalized disturbance can be generated and rejected in the control law directly, while the rest of the generalized disturbance is estimated by ESO 204 and then rejected in the linear PD control law 206 as previously shown in the original ADRC 200 framework. The other way to utilize the partially known plant dynamics is adapting the dynamics into ESO 204, where ESO 204 still estimates the generalized disturbance.
Consider the second-order plant 202 in Equation (8), with the knowledge of the plant information, part of the generalized disturbance is known, denoted as fn(y,{dot over (y)}). The rest is unknown part of the generalized disturbance, denoted as fu(y,{dot over (y)},d,t). The plant in Equation (9) can now be described as

ÿ=fn(y,{dot over (y)})+fu(y,{dot over (y)},d,t)+bu  Equation (76)

and there are two ways to take advantage of new information in fn(y,{dot over (y)}) and incorporate it into ADRC 200. One way is to improve ESO 204 by combining fn(x) with Axx+Bxu in the plant space description in Equation (76). The augmented state space form of the plant is





  
    
      
        {
        
          
            
              
                
                  x
                  .
                
                =
                
                  
                    
                      A
                      x
                    
                    ⁢
                    x
                  
                  +
                  
                    E
                    ⁢
                    
                      
                        
                          f
                          .
                        
                        n
                      
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                  
                  +
                  
                    
                      B
                      x
                    
                    ⁢
                    u
                  
                  +
                  
                    E
                    ⁢
                    
                      
                        f
                        .
                      
                      u
                    
                  
                
              
            
          
          
            
              
                y
                =
                
                  
                    C
                    x
                  
                  ⁢
                  x
                
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          77
          )
        
      
    
  




and the corresponding ESO 204 is





  
    
      
        {
        
          
            
              
                
                  z
                  .
                
                =
                
                  
                    
                      A
                      x
                    
                    ⁢
                    z
                  
                  +
                  
                    E
                    ⁢
                    
                      
                        
                          f
                          .
                        
                        n
                      
                      ⁡
                      
                        (
                        z
                        )
                      
                    
                  
                  +
                  
                    
                      B
                      x
                    
                    ⁢
                    u
                  
                  +
                  
                    L
                    ⁡
                    
                      (
                      
                        
                          y
                          m
                        
                        -
                        
                          
                            C
                            x
                          
                          ⁢
                          z
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  f
                  ^
                
                =
                Vz
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          78
          )
        
      
    
  




Then a new ESO 204 can be constructed to estimate the generalized disturbance f based on the new state space description of the plant 110 in Equation (77). For example, consider the plant ÿ=−a1{dot over (y)}+bu where a1 is given, i.e. fn(y,{dot over (y)})=−a1{dot over (y)}. The known part of the generalized disturbance is obtained from the model information: {dot over (f)}n=−a1ÿ=−a1(−a1{dot over (y)}+bu). In this case where fn(y,{dot over (y)}) is a linear function, the augmented state space representation of the plant 110 can be written as




  
    
      
        {
        
          
            
              
                
                  
                    x
                    .
                  
                  =
                  
                    
                      
                        
                          A
                          _
                        
                        x
                      
                      ⁢
                      x
                    
                    +
                    
                      
                        
                          B
                          _
                        
                        x
                      
                      ⁢
                      u
                    
                    +
                    
                      E
                      ⁢
                      
                        
                          f
                          .
                        
                        u
                      
                    
                  
                
              
            
            
              
                
                  y
                  =
                  
                    
                      C
                      x
                    
                    ⁢
                    x
                  
                
              
            
          
          ,
          
            

          
          ⁢
          
            
              where
              ⁢
              
                

              
              ⁢
              
                
                  A
                  _
                
                x
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    
                      a
                      1
                      2
                    
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              
                B
                _
              
              x
            
            =
            
              [
              
                
                  
                    0
                  
                
                
                  
                    b
                  
                
                
                  
                    
                      
                        -
                        
                          a
                          1
                        
                      
                      ⁢
                      b
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          79
          )
        
      
    
  




and the new ESO is ż=Āxz+ Bzu+L(ym−Cxz), where Bz=[0 {circumflex over (b)} −a1{circumflex over (b)}]T.

The other way of taking advantage of the additional information in fn(y,{dot over (y)}) is to use it to cancel known plant dynamics in the control law directly. There are three ways to do this:

u=(g(e,ė,ωc)+{umlaut over (r)}−fn(ym,{dot over (y)}m)−z3/{circumflex over (b)}  Equation (80)

or ym and {dot over (y)}m can be replaced by z1 and z2 in the control law,

u=(g(e,ė,ωc)+{umlaut over (r)}−fn(z1,z2)−z3)/{circumflex over (b)}  Equation (81)

or ym and {dot over (y)}m can be replaced by r and {dot over (r)} respectively and rejected in the control law in a feedforward manner.

u=(g(e,ė,ωc)+{umlaut over (r)}−fn(r,{dot over (r)})−z3)/{circumflex over (b)}  Equation (82)

where g(e,ė,ωc) is the parameterized PD controller 206 or NPD controller defined in Equation (15) or Equation (68) respectively. Of course, any combination of the above three methods are available for use by one of ordinary skill in the art.

INDUSTRIAL APPLICATIONS
The present active disturbance rejection controller, and its associated processes, methods and devices as disclosed herein possesses a number of unique attributes and industrial applicability, including for example, utility as a controller for tracking control, web processing applications, and jet engine control applications as described in greater detail above.
For example, in the case of tracking or motion control applications, the various predictive methods presented herein, particularly the ones that use the knowledge of the plant dynamics, offer reduced phase lag in the observer and compensate for the time delay in the plant itself, all of which especially benefit tracking applications that employ a desired output trajectory, sometimes called motion profile.
CONCLUSION
While various embodiments of the present system and method for feedback control of systems have been described above, it should be understood that the embodiments have been presented by the way of example only, and not limitation. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined. Thus, the breadth and scope of the present invention should not be limited by any of the above described exemplary embodiments.

## Claims

1. A multiple-input, multiple-output controller for controlling a system, comprising:
a discretized system model describing one or more distinct states, where each input has a distinct output and disturbance;
an extended state estimator implemented in a computer component that receives a signal from a sensor that is operably connected to the system; and
at least one correction term in the extended state estimator, where the at least one correction term is determined as a function of a single tuning parameter and the at least one correction term and the signal estimates system states and extended states of one or more orders.

2. The multiple-input, multiple-output controller of claim 1, where the number of inputs is greater than or equal to the number of outputs.

3. The multiple-input, multiple-output controller of claim 1, where the number of inputs and the number of outputs are the same.

4. The multiple-input, multiple-output controller of claim 1, where the number of outputs is three.

5. The multiple-input, multiple-output controller of claim 1, where the controller is implemented as a plurality of single-input, single-output controllers.

6. The multiple-input, multiple-output controller of claim 1, where the system is a turbofan.

7. The multiple-input, multiple-output controller of claim 1, where the system is one of a chemical process system, a mechanical process system, or an electrical process system.

8. The multiple-input, multiple-output controller of claim 1, where the system is one of a chemical process system, a flight control system for an airplane, a flight control system for a missile, a CNC machine control system, a robotics system, a magnetic bearing system, a satellite attitude control system, or a process control system.