Adaptive control techniques for pH control or control of other industrial processes

This disclosure provides adaptive control techniques for pH control or control of other industrial processes. For example, in one method, a robust stability condition (RSC) value is determined during operation of a process controller, and a characteristic of the process controller is adaptively modified based on the RSC value. The RSC value provides an estimate of performance of the process controller in controlling the industrial process. In another method, one of multiple process controllers is selected based on RSC values associated with the process controllers, and one or more control signals are output from the selected process controller to an industrial process in order to control the industrial process. The RSC values provide estimates of performances of the multiple process controllers in controlling the industrial process.

TECHNICAL FIELD

This disclosure relates generally to industrial process control and automation systems. More specifically, this disclosure relates to adaptive control techniques for pH control or control of other industrial processes.

BACKGROUND

Processing facilities are often managed using industrial process control and automation systems. These types of systems routinely include sensors, actuators, and process controllers. Some of the process controllers typically receive measurements from the sensors and generate control signals for the actuators. Other controllers often perform higher-level functions, such as planning, scheduling, and optimization operations.

Some controllers in an industrial process control and automation system may require tuning or adjustment from time to time. For example, some controllers can implement model predictive control (MPC) or other model-based control techniques, which use models mathematically representing how industrial processes behave in response to changes to their inputs in order to control the industrial processes. As another example, some controllers can implement proportional-integral-derivative (PID) control techniques, which use feedback to identify errors between process variables in industrial processes and desired values in order to minimize the errors over time. If an industrial process changes or a model or PID control loop cannot control the industrial process with enough accuracy, the model or control loop may need to be updated or replaced.

SUMMARY

This disclosure provides adaptive control techniques for pH control or control of other industrial processes.

In a first embodiment, a method includes controlling an industrial process using a process controller. The method also includes determining a robust stability condition (RSC) value during operation of the process controller. The RSC value provides an estimate of performance of the process controller in controlling the industrial process. The method further includes adaptively modifying a characteristic of the process controller based on the RSC value. In a similar manner, a non-transitory computer readable medium could include instructions that, when executed by at least one processing device, cause the at least one processing device to perform the operations of this method.

In a second embodiment, an apparatus includes at least one processing device configured to determine an RSC value during operation of a process controller and adaptively modify a characteristic of the process controller based on the RSC value. The process controller is configured to control an industrial process, and the RSC value provides an estimate of performance of the process controller in controlling the industrial process.

In a third embodiment, a method includes receiving input data associated with an industrial process at multiple process controllers. The method also includes selecting one of the process controllers based on RSC values associated with the process controllers. The RSC values provide estimates of performances of the process controllers in controlling the industrial process. The method further includes outputting one or more control signals from the selected process controller to the industrial process in order to control the industrial process. In a similar manner, a non-transitory computer readable medium could include instructions that, when executed by at least one processing device, cause the at least one processing device to perform the operations of this method.

In a fourth embodiment, an apparatus includes multiple process controllers configured to receive input data associated with an industrial process. Each process controller is also configured to generate one or more control signals for controlling the industrial process. The apparatus also includes a controller configured to select one of the process controllers based on RSC values associated with the process controllers. The RSC values provide estimates of performances of the process controllers in controlling the industrial process. The controller is configured to output the one or more control signals from the selected process controller to the industrial process.

DETAILED DESCRIPTION

As noted above, process controllers in industrial process control and automation systems often need to be tuned or adjusted over time. Ideally, this could be done automatically in order to help ensure more accurate control of industrial processes. “Adaptive control” refers to the concept of adaptively changing how an industrial process is controlled over time, such as to compensate for changes in the industrial process itself or to compensate for inaccuracies or other issues with a process controller. Conventional attempts at providing adaptive control suffer from a number of inherent technical challenges, such as bursting, drifting, structural complexities, monitoring issues, and excitation requirements.

This disclosure provides various solutions for providing adaptive control of industrial processes. In particular, this disclosure describes the use of robust stability condition (RSC) metric-based adaptive control solutions for controlling industrial processes. RSC metrics can provide a good estimate of controller performance and, when used for controller adaptation, can result in substantially uniform controller performance at different operating points of an industrial process.

The RSC metric-based control solutions can be used to support different types of adaptive control, such as direct adaptation of a controller (referred to as “direct” adaptive control) and multi-model switching control (MMSC). Direct adaptive control involves making changes directly to a control loop or controller model in order to alter the performance of a process controller. Multi-model switching control involves executing different controllers (either separate hardware devices or separate instances of controller logic in the same device) and selecting one of the controllers, such as the controller that provides the most accurate results for the specific state of an industrial process. Specific implementations of these approaches are described below, such as in relation to control of a pH neutralization process or other pH control process. Of course, these approaches could be used in any other suitable manner.

FIGS. 1 and 2illustrate example industrial process control and automation systems100and200according to this disclosure. As shown inFIG. 1, the system100includes various components that facilitate production or processing of at least one product or other material. For instance, the system100can be used to facilitate control over components in one or multiple industrial plants. Each plant represents one or more processing facilities (or one or more portions thereof), such as one or more manufacturing facilities for producing at least one product or other material. In general, each plant may implement one or more industrial processes and can individually or collectively be referred to as a process system. A process system generally represents any system or portion thereof configured to process one or more products or other materials in some manner.

InFIG. 1, the system100includes one or more sensors102aand one or more actuators102b. The sensors102aand actuators102brepresent components in a process system that may perform any of a wide variety of functions. For example, the sensors102acould measure a wide variety of characteristics in the process system, such as temperature, pressure, or flow rate. Also, the actuators102bcould alter a wide variety of characteristics in the process system. Each of the sensors102aincludes any suitable structure for measuring one or more characteristics in a process system. Each of the actuators102bincludes any suitable structure for operating on or affecting one or more conditions in a process system.

At least one network104is coupled to the sensors102aand actuators102b. The network104facilitates interaction with the sensors102aand actuators102b. For example, the network104could transport measurement data from the sensors102aand provide control signals to the actuators102b. The network104could represent any suitable network or combination of networks. As particular examples, the network104could represent at least one Ethernet network, electrical signal network (such as a HART or FOUNDATION FIELDBUS network), pneumatic control signal network, or any other or additional type(s) of network(s).

The system100also includes various controllers106. The controllers106can be used in the system100to perform various functions in order to control one or more industrial processes. For example, a first set of controllers106may use measurements from one or more sensors102ato control the operation of one or more actuators102b. A second set of controllers106could be used to optimize the control logic or other operations performed by the first set of controllers. A third set of controllers106could be used to perform additional functions.

Controllers106are often arranged hierarchically in a system. For example, different controllers106could be used to control individual actuators, collections of actuators forming machines, collections of machines forming units, collections of units forming plants, and collections of plants forming an enterprise. A particular example of a hierarchical arrangement of controllers106is defined as the “Purdue” model of process control. The controllers106in different hierarchical levels can communicate via one or more networks108and associated switches, firewalls, and other components.

Each controller106includes any suitable structure for controlling one or more aspects of an industrial process. At least some of the controllers106could, for example, represent proportional-integral-derivative (PID) controllers or multivariable controllers, such as Robust Multivariable Predictive Control Technology (RMPCT) controllers or other types of controllers implementing model predictive control (MPC) or other advanced predictive control. As a particular example, each controller106could represent a computing device running a real-time operating system, a WINDOWS operating system, or other operating system.

Operator access to and interaction with the controllers106and other components of the system100can occur via various operator consoles110. Each operator console110could be used to provide information to an operator and receive information from an operator. For example, each operator console110could provide information identifying a current state of an industrial process to the operator, such as values of various process variables and warnings, alarms, or other states associated with the industrial process. Each operator console110could also receive information affecting how the industrial process is controlled, such as by receiving setpoints for process variables controlled by the controllers106or other information that alters or affects how the controllers106control the industrial process. Each operator console110includes any suitable structure for displaying information to and interacting with an operator.

As noted above, one or more of the process controllers106could implement an RSC metric-based adaptive control approach for controlling one or more industrial processes. As described in more detail below, this could take the form of direct adaptation of a controller106or multi-model switching control within a single controller106or with multiple controllers106.

As shown inFIG. 2, the system200includes various components that support a pH neutralization process or other pH control process. InFIG. 2, a reactor202contains material204that is ideally controlled so that the pH of the material204has a desired value or is within a desired range of values. In this example, the reactor202includes or is associated with at least one stirrer206that mixes different materials in the reactor202so that the resulting material204is substantially uniform throughout. The reactor202includes any suitable structure configured to hold materials, such as a continuous stirred-tank reactor (CSTR).

Various inlet streams of material to the reactor202are provided by pumps208-214. The pump208provides a base material into the reactor202, and the pump210provides an acid material into the reactor202. The pumps212and214provide one or more buffer materials into the reactor202. Note, however, that there could be multiple pumps supplying different acid materials, multiple pumps supplying different base materials, and/or a single pump providing buffer material. An overflow outlet216provides an outlet stream from the reactor202. Each pump208-214includes any suitable structure for providing a flow of material. The overflow outlet216includes any suitable structure for providing material from a reactor.

At least one pH probe218can be inserted into the reactor202and used to measure the pH of the material204in the reactor202. Measurements of pH are provided to a process controller220, which uses the pH measurements to control the operations of the pumps208-214. Ideally, the process controller220uses the pH measurements to control the operation of the pumps208-214so that processed material output from the reactor202has a pH at a desired level or within a desired range (such as a pH of between 6 and 8). Each pH probe218includes any suitable structure for measuring the pH of material. The process controller220includes any suitable structure for controlling components in order to adjust the pH of material. In some embodiments, the process controller220denotes a PID controller. Although not shown, the process controller220could interact with other components, such as by interacting with an operator console110via a network108.

The control of pH can play an important role in various industrial processes, such as wastewater treatment, biotechnical processes like microbial fuel cells, and electro-chemical cells. Because of its nonlinear and time-varying nature, pH control can be very challenging. The nonlinear nature of a pH control process can make linear controllers ineffective at controlling pH over a wide range of values, and the time-varying nature of the pH control process can make linear controllers ineffective even at a single operating point. While some nonlinear controllers have been proposed, these nonlinear controllers generally fail to compensate for the time-varying behavior of the process, such as when there is a change in the buffering capacity of the reactor202. Thus, the process controller220could implement an RSC metric-based adaptive control approach for controlling the pH of material204in a reactor202. As described in more detail below, this could take the form of direct adaptation of the controller220or multi-model switching control within the controller220or with multiple instances of the controller220.

AlthoughFIGS. 1 and 2illustrate examples of industrial process control and automation systems100and200, various changes may be made toFIGS. 1 and 2. For example, industrial control and automation systems come in a wide variety of configurations. The systems100and200shown inFIGS. 1 and 2are meant to illustrate example operational environments in which adaptive control techniques can be used.FIGS. 1 and 2do not limit this disclosure to any particular configuration or operational environment.

FIG. 3illustrates an example device300supporting adaptive control techniques for pH control or control of other industrial processes according to this disclosure. The device300could, for example, denote the controllers106or operator consoles110inFIG. 1or the controller220inFIG. 2. However, the device300could be used in any other suitable system, and the controllers106and220and operator consoles110could be implemented in any other suitable manner.

As shown inFIG. 3, the device300includes at least one processor302, at least one storage device304, at least one communications unit306, and at least one input/output (I/O) unit308. Each processor302can execute instructions, such as those that may be loaded into a memory310. Each processor302denotes any suitable processing device, such as one or more microprocessors, microcontrollers, digital signal processors, application specific integrated circuits (ASICs), field programmable gate arrays (FPGAs), or discrete circuitry.

The memory310and a persistent storage312are examples of storage devices304, which represent any structure(s) capable of storing and facilitating retrieval of information (such as data, program code, and/or other suitable information on a temporary or permanent basis). The memory310may represent a random access memory or any other suitable volatile or non-volatile storage device(s). The persistent storage312may contain one or more components or devices supporting longer-term storage of data, such as a read only memory, hard drive, Flash memory, or optical disc.

The communications unit306supports communications with other systems or devices. For example, the communications unit306could include at least one network interface card or wireless transceiver facilitating communications over at least one wired or wireless network. The communications unit306may support communications through any suitable physical or wireless communication link(s).

The I/O unit308allows for input and output of data. For example, the I/O unit308may provide a connection for user input through a keyboard, mouse, keypad, touchscreen, or other suitable input device. The I/O unit308may also send output to a display, printer, or other suitable output device.

AlthoughFIG. 3illustrates one example of a device300supporting adaptive control techniques for pH control or control of other industrial processes, various changes may be made toFIG. 3. For example, various components inFIG. 3could be combined, further subdivided, or omitted and additional components could be added according to particular needs. Also, computing devices come in a wide variety of configurations, andFIG. 3does not limit this disclosure to any particular configuration of computing device.

The remainder of this patent document provides details regarding example implementations of two adaptive control techniques, namely direct adaptive control and multi-model switching control. The details provided below are for illustration only and do not limit this disclosure to the specific implementations of these two adaptive control techniques.

Direct Adaptive Control

As noted above, direct adaptive control involves making changes directly to a control loop or controller model in order to alter the performance of a process controller. For simplicity, the following discussion describes direct adaptive control as involving a single characteristic of a process controller that is adaptively modified to account for changes to an industrial process, namely a pH control process. However, this is for illustration and explanation only.

FIG. 4illustrates an example control loop400subject to direct adaptive control according to this disclosure. InFIG. 4, the control loop400is used in conjunction with an industrial process402, which could denote any suitable industrial process (such as a pH neutralization process or other pH control process).

As shown inFIG. 4, the control loop400implements a PID control loop in which an output y(t) of the process402is controlled to be at or near a desired setpoint r(t). The control loop400includes a combiner404that identifies a difference or error e(t) between the setpoint r(t) and measurements of the output y(t) (such as from a pH probe218). Three function blocks406-410implement the proportional, integral, and derivative calculations for the control loop400. For example, the function block406accounts for the current value of the error by scaling the error with Kp, which denotes the proportional gain of the control loop400. The function block408accounts for prior values of the error by integrating the error over time and scaling the result with Kiwhich denotes the integral gain of the control loop400. The function block410accounts for potential future values of the error by calculating a derivative of the error and scaling the result with Kd, which denotes the derivative gain of the control loop400. Outputs of the function blocks406-410are combined by a combiner412, which generates a control signal u(t) for adjusting the process402.

The control loop400could be implemented in any suitable manner. For example, in some embodiments, each of the components404-412of the control loop400could be implemented using software or firmware instructions, such as instructions executed by at least one processor302of a controller106or220. In other embodiments, at least some of the components404-412could be implemented using hardware components of a controller106or220.

As described below, the process402could be adaptively controlled by directly altering one or more characteristics of the control loop400. For example, one or more of the gains Kp, Ki, and Kdcould be varied using frequency loop-shaping (FLS), and an RSC metric can be used during the loop-shaping or other tuning. The tuning could be performed by the controller106or220that is executing the control loop400or by another device external to the controller106or220(such as an operator console110). One result of the tuning is that the operation of the control loop400is altered to account for things like setpoint changes or changes to the underlying industrial process402.

AlthoughFIG. 4illustrates one example of a control loop400subject to direct adaptive control, various changes may be made toFIG. 4. For example, in some instances, one of the function blocks406-410could be omitted if its functionality is not required, such as when the function block410is omitted to create a PI control loop instead of a PID control loop.

FIG. 5illustrates a first example method500supporting adaptive control of an industrial process according to this disclosure. In particular,FIG. 5illustrates an example method500for direct adaptive control of an industrial process (such as a pH neutralization process or other pH control process). For ease of explanation, the method500is described as being used with the control loop400ofFIG. 4in the system100ofFIG. 1or the system200ofFIG. 2. However, the method500could be used with any suitable control loop, controller, and system.

As shown inFIG. 5, system identification is performed for an industrial process at step502. This could include, for example, collecting input and output data associated with the industrial process402and identifying a process model representing the industrial process402using the data. One straightforward approach for system identification is to carry out identification experiments and obtain a process model (such as a Laplace transfer function). This can involve perturbing the industrial process402during step testing and fitting a model to recorded input-output time-series data. Many approaches are known for system identification, including tools for fitting a model to process data. Example system identification procedures are described in the following documents (all of which are hereby incorporated by reference in their entirety):Tsakalis et al., “Integrated Identification and Control for Diffusion/CVD Furnaces,” IEEE 6th Int. Conf. ETFA, pp. 514-519, 1997;Tsakalis et al., “Control Oriented Uncertainty Estimation in System Identification,” 17th IASTED Int. Conf. MIC, 1998;Tsakalis et al., “Identification for PID Control,” PID Control in the Third Millennium, pp. 283-317, 2012;Tsakalis et al., “Loop-Shaping Controller Design from Input-Output Data: Application to a Paper Machine Simulator,” IEEE Transactions on Control Systems Technology, pp. 127-136, 2002;Zhan et al., “System Identification for Robust Control,” American Control Conference, 2007, ACC '07, pp. 846-851, 9-13 Jul. 2007;Alvarez et al., “pH Neutralization Process as a Benchmark for Testing Nonlinear Controllers,” Industrial & Engineering Chemistry Research 2001 40 (11), pp. 2467-2473; and{dot over (A)}ström et al., “PID controllers: theory, design, and tuning,” Research Triangle Park, N.C.: Instrument Society of America, 1995.

A characteristic of a process controller to be subjected to direct parameter adaptation is identified at step504. This could include, for example, a user or automated tool identifying a characteristic of the process controller106or220to be controlled. The characteristic could denote any suitable characteristic of the controller106or220. For example, as described below, the characteristic could denote the gain of a controller that controls a pH neutralization process or other pH control process. The characteristic can be identified in any suitable manner, such as by analyzing the process model from the system identification and identifying the primary characteristic of the controller106or220that affects controller accuracy.

A controller is designed to control the industrial process at step506. This could include, for example, executing a design tool to identify the Kp, Ki, and Kdgains for a PID control loop400. Various tools and approaches are known for designing a PID controller or other controller, including those in the various references incorporated above. As part of the controller design, an optimization problem can be solved to identify one or more parameters for the controller106or220. For instance, an optimization problem could be solved to identify the controller gain of the controller106or220. The optimization problem incorporates at least one RSC metric, allowing the parameter(s) of the controller106or220to be directly adapted during operation of the controller.

The controller is placed into operation and used to control the industrial process at step508. This could include, for example, using the identified controller gains or other parameters in the process controller106or220to control the pH of material204in the reactor202. The process controller106or220could implement the control loop400using the identified controller gains or other parameters.

A robust stability control metric is calculated at step510. This could include, for example, the controller106or220calculating any suitable RSC metric. According to the small-gain theorem, a closed-loop system is stable for all stable uncertainties that satisfy the condition:
|Δ(jω)∥M(jω)|<1 for all ω  (1)
where Δ denotes a frequency response of the uncertainty (whose bounds are available) and M denotes the nominal closed-loop transfer function. This inequality is referred to as a robust stability condition for a feedback system. To help guarantee robust stability, a controller C for a nominal process model P0with a multiplicative uncertainty Δmcould be designed such that the robust stability condition in Equation (2) is satisfied.
|Δm(jω)∥T(jω)|<1 for all ω  (2)
This condition implies that the closed-loop bandwidth cannot be higher than that of the inverse multiplicative uncertainty and that excessive peaks (resonances) should be avoided. While exact specifications depend on detailed estimates of the uncertainty, traditional feedback design simplifies the problem by assuming smooth and well-behaved frequency responses, effectively converting the bandwidth constraint into a crossover frequency and the peaking constraint into a phase-margin specification. The choice of design procedure is not important as long as the designed controller satisfies a robust stability condition imposed by uncertainty. In some embodiments, the design of a PID controller can be performed using frequency loop-shaping. In this method, PID parameters are tuned to achieve a loop transfer function close to a chosen target loop. Details of example frequency loop-shaping can be found, for example, in Grassi et al., “Integrated system identification and PID controller tuning by frequency loop-shaping,” IEEE Transactions on Control Systems Technology, vol. 9, no. 2, pp. 285-294, 2001 (which is hereby incorporated by reference in its entirety). Additional details regarding a suitable RSC metric are provided below.

The RSC metric is used to modify the characteristic of the controller at step512. This could include, for example, using the calculated RSC metric as part of the optimization problem in order to identify one or more new controller gains or other controller parameters to be used by the controller. The method500can then return to step508in order to continue controlling the industrial process using the adaptively-updated controller.

AlthoughFIG. 5illustrates a first example of a method500supporting adaptive control of an industrial process, various changes may be made toFIG. 5. For example, while shown as a series of steps, various steps inFIG. 5could overlap, occur in parallel, occur in a different order, or occur any number of times.

Example of Direct Adaptive Control for pH Control Process

The following presents a specific example of how the above-described direct adaptive control technique could be used. It has been shown that the gain for a pH system can change with the system's operating point. It has also been shown that a linear controller destabilizes the system if operated at a different pH than the pH for which the linear controller was designed. One conventional approach involves gain-scheduling based on measured signals, meaning the controller gain is adjusted based on sensor measurements. However, in the case of pH control, the only available sensor measurement may be the pH of material, and gain-scheduling based solely on pH measurements may be insufficient. This is because significant variations in the gain of the pH control process can occur, such as in response to operating point variations or changes in the buffering capacity of the pH process. The disclosure below focuses on adaptive control algorithms that handle gain variations from both operating point variations and buffer variations.

For a pH neutralization process, only the gain of the process may change significantly, so a single parameter adaption (for the controller gain) may be sufficient to compensate for gain variations in the process. The need to adapt a single parameter can be confirmed by modeling the pH neutralization process, such as by using reaction-invariant modeling, and analyzing the model at different pH operating points and with variations in buffer flows. From this, it can be shown that only the gain of the process changes significantly, so only the controller gain may need to be directly adapted. The low excitation requirement of single parameter adaptation and the low computational requirement for direct adaptation of that single parameter make this attractive for implementation in industrial processes.

Consider a 500 mL continuous stirred tank reactor202with inlet streams of a strong acid (1 M HCl), weak acid #1(100 mM H2CO3), weak acid #2(100 mM H3PO4), and a strong base (1 M NaOH). Let q1, q2, q3, and q4denote their respective flow rates. The combination of the strong and weak acids could simulate wastewater, and the flow rate of the strong base can be manipulated to control the pH of the material204in the reactor202. A variation of the buffering capacity of the wastewater can be simulated be varying the flows of the weak acids. The volume of the material204in the reactor202and its temperature can be kept substantially constant. The pH of the material204in the reactor202is measured by the pH probe218, and it is assumed that perfect mixing occurs in the reactor202and that ions are completely soluble. All irreversible reactions and reaction invariants are listed below.

A dynamic model for the pH process can be derived from the component material balance for reaction invariants (Wa5, Wb5, and Wc5) and the algebraic equation relating the pH and reaction invariants. Example operating conditions of the system200are listed in Table 1.

A nonlinear model for this process can be described by mass-balance and charge-balance equations (Equations (3)-(5)) and the relationship between reaction invariants and pH (shown in Equations (A.17) and (A.13) below).

The pump208can be used to pump the base material into the reactor202, and its flow rate can be controlled, such as by varying a voltage across a control terminal of the pump208. The pump208can interface with the controller220using a digital-to-analog converter (DAC) if necessary, such as to convert between digital values (like 12-bit values) used by the controller220and a voltage signal (like 0-5V) used by the pump208. The relation between the base stream flow rate and the DAC values could be described as:

Base⁢⁢flow⁢⁢(mL⁢/⁢min)=7.45×DAC⁢⁢Value4095(6)
In the following discussion, “plant input” refers to the DAC value, and “plant output” refers to the pH of the material204in the reactor202.

The nonlinear model of the pH process can be linearized at different pH operating points and for the variations of buffering capacity. The following example cases are considered for the buffer flow rates and are used to simulate variations of the buffering capacity.Case 1: q2=2.5 mL/min and q3=2.5 mL/minCase 2: q2=0 mL/min and q3=2.5 mL/minCase 3: q2=2.5 mL/min and q3=0 mL/minCase 4: q2=0 mL/min and q3=0 mL/min
Transfer functions of the linearized plants for these four cases can be of the following structure.

P⁡(s)=Kpτ⁢⁢s+1⁢e(-s/6)(7)
Here, Kpis the gain and τ is the time constant of a linearized plant. The values of the gains of the linearized plants for all cases are listed in Table 2, and the values of the time constants for all cases are listed in the Table 3.

The gain values of the linearized plants are directly related to the slope of a titration curve. Thus, it can be shown that the gain of the linearized plants is a function of the operating pH and buffering capacity of the material204in the reactor202. Moreover, it can be shown that the variation in the gain with respect to the operating pH and buffering capacity is significantly high. On the other hand, the time constants of the plants are directly related to retention time. The variation in the time constants is minimal compared to the gain variation, and the time constant of all of the linearized plants could be significantly shorter than the settling time of a pH controller. As a result, the variation in the time constant could have little or no impact on the performance of a control loop. It could therefore be concluded that only the gain of the system changes significantly, and direct adaptation of only the controller gain could be sufficient to compensate for gain variations of the plant.

Based on this example, the following provides a description of the formulation of controller gain adaptation as part of a modified approximate H∞loop shaping in a PID parameter adaptation process. A summary of the FLS tuning approach for PID control loops using an RSC metric and estimation of the RSC metric online using plant input-output data is described below.

Let C be a proportional-integral (PI) controller designed for a nominal plant with a transfer function of C(s)=Kp+Ki=s, where Kpand Kiare the proportional and integral gains, respectively. For gain adaptation purposes, the transfer function of the PI controller can be rewritten as:

C⁡(s)=K⁢(s+a)s=K⁢C~(8)
where K=Kp, a=Ki/Kp, and {tilde over (C)}=(s+a)/s. An FLS tuning objective is to determine the gain K of the PI controller so that the open-loop transfer function CG is close to a target L in a weighted L∞sense. The FLS tuning from Equation (B.4) below is adjusted for gain adaptation, and the solution for the gain K of the controller is a solution of the following optimization problem:

K=arg⁢⁢minK∈M⁢S⁡(GK⁢C~-L)∞(9)
where S=(1+L)−1is the target sensitivity of the controller and M is the set of controller gain values. M can be defined as the closed interval M=[Kmin, Kmax], where Kminis the minimum controller gain and Kmaxis the maximum controller gain.

The online direct adaptive algorithm in Equation (9) can be modified into a minimization of an RSC estimate as described in Equation (B.6) below. The modified equation can be expressed as:

K=arg⁢⁢minK∈M⁢maxi⁢KS⁢C~⁢Fi⁢y-TFi⁢u2,δ-dFi⁢u2,δ(10)
The optimization problem in Equation (10) can be solved recursively for each time step k as follows.

Various optimization solvers can be used to solve the above optimization problem. One example solver is described in Tsakalis et al., “Approximate H∞Loop Shaping in PID Parameter Adaptation,” International Journal of Adaptive Control and Signal Processing 27 (1-2) (2013), pg. 136-152 (which is hereby incorporated by reference). However, changes are made to the approach described in that document as follows. For faster convergence (apart from the forgetting factor), a co-variance term P is reset periodically if one of the following conditions is met: the setpoint is changed more than a threshold, the time from the last reset is more than a threshold, or there is a significant change in the pH readings (for disturbances). Also, the value of the co-variance term P represents confidence (sufficiency of excitation) in the parameter estimate, and the estimated parameter is used in the controller only if the value of the sum of squares of P for all filters is more than a threshold. In addition, to avoid bias in the parameter estimate because of disturbances and noise, the plant input-output pair used in the adaptation algorithm is filtered using a band-pass filter that attenuates frequencies of these signals outside the frequency of interest.

Initially, system identification and PI controller design for a nominal plant can occur. The nominal plant model can be obtained at a specific operating pH (such as pH=6) by fitting a model to input-output data from a system identification experiment, and a nominal PI controller can be designed for the nominal plant model.

Various system identification algorithms are known and can use input-output data from a system identification experiment to fit a model, and an estimation error can be used to obtain an estimate of uncertainty. For system identification to work properly, an input signal used in the identification experiment could have persistence of excitation. There are different methods of designing “plant-friendly” inputs for an identification experiment with persistence of excitation. One common method involves the use of pseudo-random binary sequence (PRBS) signals. Input-output data can be used to identify the parameters of a transfer function for the plant, and a PI controller can be designed for the identified plant, such as by using an FLS algorithm. The parameters for the nominal PI controller can then be obtained, such as by minimizing an RSC metric for a chosen loop shape.

As a specific example of this functionality, assume that input-output data is used to generate a transfer function Gesp6as follows:

Gexp⁢⁢6⁡(s)=7.6305⨯10-5⁢(-s+1.53)(s+4.261)⁢(s+0.01822)(12)
A PI controller (Cexp6) can be designed for this plant using an online FLS algorithm. A target loop (Lexp6) chosen for the PI controller design with a closed-loop bandwidth specification of 0.6 rad/min can be expressed as:

Cexp⁢⁢6⁡(s)=21243⁢(s+0.273)s(14)
To validate this controller design, a co-prime factor uncertainty estimate can be used, and a plot of the small gain condition (the RSC) can be created. If the small gain condition value is less than one, the controller design for the nominal plant is robust enough to handle the uncertainty that corresponds to the estimation error. In addition, the optimization problem in Equation (10) can be solved recursively for each time step k in order to adapt the controller design.

Experimental and simulation results confirm that this approach can function adequately to provide effective control over a pH process. To provide excitation for adaptation, a setpoint signal identifying a desired pH target can be superimposed with a small magnitude square wave or other wave. Through these results, it can be seen that the direct adaptive control approach is able to adapt to process gain changes at different operating points (different pH levels) and buffering capacities. After controller gain convergence, the closed-loop system response can be very close to the target response, so it can be concluded that the direct adaptive control approach can achieve substantially uniform performance as process conditions vary. It can also be concluded from these results that, using a single data-driven linear model (obtained using a system identification experiment) along with the direct adaptive control approach, it is possible to control pH over a wide operating region. Moreover, the implementation of the direct adaptive control algorithm can be computationally inexpensive, allowing the algorithm to be used in a wide range of systems and devices.

From the experimental and simulation results, it can also be shown that, following a setpoint transition to the gain of the process controller220, the process controller220is adapted to a steady-state value that gives performance close to the desired target. It may take several cycles of the superimposed square wave or other wave to obtain convergence for the controller gain parameter. However, this still establishes that the direct adaptive control algorithm can perform well in achieving substantially uniform performance and that small excitations (such as a small square wave) are sufficient to achieve this substantially uniform performance.

In conclusion, the direct adaptive control approach avoids problems associated with conventional approaches, such as bursting and drifting. Because there is only one parameter (such as controller gain) being updated, the excitation requirement for updating the parameter can be modest. Furthermore, substantially the same level of performance can be obtained by a process controller at various operating points. While many controllers use objective functions that do not have meaningful performance criteria, one or more performance targets are explicitly considered in this approach via use of the RSC metric, which helps to improve controller performance and stability at various operating points.

Multi-Model Switching Control

As noted above, multi-model switching control involves executing different controllers (either separate hardware devices or separate instances of controller logic in the same device) and selecting one of the controllers. The selected controller could, for example, denote the controller that provides the most accurate results for the specific state of an industrial process. As the state of the industrial process changes, a different controller could provide the most accurate results and be selected. For simplicity, the following discussion describes multi-model switching control as involving one or more characteristics that are adaptively modified to account for changes to an industrial process, namely a pH control process. However, this is for illustration and explanation only.

FIG. 6illustrates an example multi-model switching control loop600supporting adaptive control of an industrial process according to this disclosure. InFIG. 6, the control loop600is used in conjunction with an industrial process602, which could denote any suitable industrial process (such as a pH neutralization process or other pH control process).

The control loop600here includes a controller bank containing multiple controllers604a-604n. Each of the controllers604a-604nreceives input data (such as one or more setpoints and one or more feedback signals) and operates to generate one or more output signals for the industrial process602. However, the controllers604a-604nare configured differently so that their control calculations and their output signals are different. For example, different controllers604a-604ncould denote PID controllers with different controller gains. As another example, different controllers604a-604ncould denote MPC or other model-based controllers that operate using different process models or different optimization functions.

The output signals from the controllers604a-604nare provided to a controller or control selector606. The control selector606operates to provide the output signal(s) from a selected one of the controllers604a-604nto the industrial process602. Multi-model switching control relies upon some metric to perform switching between the controllers604a-604nin the controller bank. A high correlation between RSC metrics and the performances of closed-loop systems can make the RSC metrics useful in monitoring controller performance. Thus, the control selector606can use an RSC metric to identify the controller output to be provided to the industrial process602.

AlthoughFIG. 6illustrates one example of a multi-model switching control loop600supporting adaptive control of an industrial process, various changes may be made toFIG. 6. For example, the control loop600could include any suitable number of controllers in the controller bank, and those controllers could have any suitable differences.

FIG. 7illustrates a second example method700supporting adaptive control of an industrial process according to this disclosure. In particular,FIG. 7illustrates an example method700for multi-model switching control of an industrial process (such as a pH neutralization process or other pH control process). For ease of explanation, the method700is described as being used with the control loop600ofFIG. 6in the system100ofFIG. 1or the system200ofFIG. 2. However, the method700could be used with any suitable control loop, controller, and system.

As shown inFIG. 7, system identification is performed for an industrial process at step702. This could include, for example, collecting input and output data associated with the industrial process602and identifying a process model representing the industrial process602using the data. The techniques described above with respect toFIG. 5could be used here to support the system identification.

An initial controller is designed to control the industrial process at step704. This could include, for example, executing a design tool to identify the Kp, Ki, and Kdgains for a PID control loop or to identify different parameters for an MPC control loop. Again, the techniques described above with respect toFIG. 5could be used here to support the controller design. Various tools and approaches are also known for designing an MPC controller or other model-based controller.

One or more parameters of the initial controller are varied to create a bank of controllers at step706. This could include, for example, creating a set of controllers604a-604nby varying the controller gain(s) or other parameter(s) of the initial controller. The controllers604a-604nin the set have different gains or other parameter values that could be selected in any suitable manner (such as variations in fixed or variable steps).

A controller is selected from the bank of controllers at step708. This could include, for example, selecting the initial controller that was originally designed in step704. Of course, another controller could also be selected. Control signals are generated using the controllers in the bank at step710, and the control signal or signals from the selected controller are used to control the industrial process at step712. This could include, for example, generating one or more control signals using each of the controllers604a-604nin the bank. This could also include the control selector606outputting the control signal(s) from the selected controller to the industrial process602.

A robust stability control metric is calculated for each controller in the bank at step714, and the RSC metrics are used to select a different controller if justified at step716. This could include, for example, the controller106or220calculating any suitable RSC metric. This could also include the controller106or220determining whether an unselected controller604a-604nin the controller bank has a better RSC metric than the selected controller. This could further include the controller106or220selecting the controller604a-604nhaving the best RSC metric for future use in controlling the industrial process602. The method700can then return to step712in order to continue controlling the industrial process using the newly-selected controller or the previously-selected controller (depending on the determination in step716).

Note that a hysteresis value could be used in step716as part of the determination whether to select a new controller. The hysteresis value could, for example, require that the RSC metric for an unselected controller needs to exceed a specified amount before a controller switch occurs. Among other things, this could help to ensure that an excessive amount of controller switching does not occur.

AlthoughFIG. 7illustrates a second example of a method700supporting adaptive control of an industrial process, various changes may be made toFIG. 7. For example, while shown as a series of steps, various steps inFIG. 7could overlap, occur in parallel, occur in a different order, or occur any number of times.

Example of Multi-Model Switching Control for pH Control Process

The following presents a specific example of how the above-described multi-model switching control technique could be used. The following example describes the formulation of multi-model switching control using an RSC metric for the frequency loop-shaping technique of Equation (B.6) described below.

Assume [Gj] is a bank of plants and [Cj] is a bank of controllers corresponding to the plants [Gj], where j=1 . . . n. The plants need not represent different physical plants and can instead represent different states of the same plant. The controllers can be designed, for example, using an FLS algorithm with the same target L. An RSC estimate using input-output data for each controller j could be calculated as:

RSCj≤maxi⁢SCj⁢Fi⁢y-TFi⁢u2,δ-dFi⁢u2,δ(15)
An objective of multi-model adaptive switching control can be to switch to the controller that results in a minimum estimate of the RSC metric. The selection of the controller can thus be formulated as:

For some pH control processes, the controller structure can be restricted to a PI controller, and the adaptation can be restricted to the controller gain as described above. The bank of controllers604a-604ncould therefore be formed by varying the gain of a nominal PI controller designed for a nominal plant model. The transfer function of a controller in the bank could be expressed as:
Cj(s)=Kl{tilde over (C)}(17)
where {tilde over (C)}=(s+a)/s, a=Ki/Kp, and Kpand Kiare the proportional and integral gains of the PI controller designed for the nominal plant. To minimize the number of controller switches, a hysteresis parameter h can be introduced. In some embodiments, the hysteresis parameter could have a value between 0.2 and 1.0.

The multi-model switching control algorithm could occur as follows. API controller C0can be designed for a nominal plant G0. A bank of controllers can be formed by varying the gain Kjor other parameter(s) of the nominal controller C0by different amounts. An initial controller Cselectedcan be chosen, such as by selecting the original controller C0. For each time step k, an RSC metric can be computed for each controller in the bank. The RSC metric RSCjfor controller j could be calculated as:

RSCj=maxi⁢SCj⁢Fi⁢y-TFi⁢u2,δ-dFi⁢u2,δ(18)
A decision can be made to switch controllers if the RSC metric RSCselectedfor the selected controller satisfies the following condition:
RSCselected>(1+h)minjRSCj(k)  (19)
If that condition is satisfied, the controller Cj* can be selected, where j*=arg minjRSCj(k). Otherwise, the currently-selected controller can remain in use.

The multi-model switching control algorithm described above can use an RSC value estimated from plant input-output data, which in turn could be identified through calculation of the data's exponentially-weighted two-norm. Computation of an exponentially-weighted two-norm is described below. The choice of the exponential weight (δ) is important in the computation of the exponentially-weighted two-norm. Smaller δ values can result in more accurate RSC values but higher transient times, while higher δ values can result in lower transient times but less accurate RSC values.

The RSC computation can also handle a lack of excitation in given circumstances. For example, the disturbance threshold term described in Tsakalis et al., “Multivariable Controller Performance Monitoring Using Robust Stability Conditions,” Journal of Process Control 17 (9) (2007), pp. 702-714 (which is hereby incorporated by reference) could be used in the RSC computation to avoid large bias in the RSC values because of the lack of excitation. To avoid bias in the parameter estimate because of disturbances and noise, the plant input-output data used in the adaptation algorithm can be filtered, such as by using a band pass filter that only allows signals at frequencies of interest. The target loop used in the design of the nominal controller can be used in computing the RSC for this algorithm.

Simulation results confirm that this approach can function adequately to provide effective control over a pH process or other industrial process. For example, the simulation results can show that, after a setpoint transition, the gain of a controller adapts to a steady-state value that gives performance close to a target. Note that a mismatch between a target and an actual control loop could exist as a result of quantization of the controller parameter values in the bank. The quantization error can be reduced or minimized by increasing the number of controller parameter values used in the bank. However, an increase in the number of controller parameter values can also result in an increase in the computational power required to implement an adaptive loop. Setpoint changes could also result in controller saturation in one or more instances, which again could be avoided by increasing the number and range of controller parameter values in the bank.

In conclusion, the MMSC adaptive control approach avoids problems associated with conventional approaches, such as bursting and drifting. Only one parameter (such as controller gain) may be varied in the controllers, or multiple parameters can be varied in the controllers. These multiple controllers can provide a range of operation, which could be important in many practical applications. The switching design approach can eliminate various structural limitations placed on conventional systems, and the use of an RSC metric allows this approach to take the dynamics and uncertainties of the various controllers into consideration.

Industrial Implementation Notes

As noted above, when a controller's setpoint changes in a large step, it is possible that a large overshoot could occur in the controller's output. Ramp signals may therefore be better suited for the transition between setpoints as they can result in reduced or minimal overshoot. Controller parameter adaptation also uses some excitation so that a parameter converges to a desired value. A ramp signal coupled with some pulses (such as at the end of the ramp signal) may allow a closed-loop system to transition between two operating points smoothly while providing excitation for controller parameter adaptation. The magnitude of the pulses can be chosen so that the pulses are higher than noise levels and small enough to operate the system at the provided setpoint.

Example Irreversible Reactions and Reaction Invariants of a pH Process

This section provides a summary of irreversible reactions and reaction invariants for an example pH neutralization process. The details of this pH process are examples only, and other pH control processes could be used.

The reversible reactions in the example pH neutralization process can include the following.
H2CO3H++HCO3−(A.1)
HCO3−H++CO32−(A.2)
H3PO4H++H2PO4−(A.3)
H2PO4−H++HPO42−(A.4)
HPO42−H++PO43−(A.5)
H2OH++OH−(A.6)
Reaction constants of these reactions can include the following.

Ka⁢⁢1=[HCO3-]⁡[H+][H2⁢CO3](A⁢.7)Ka⁢⁢2=[CO32-]⁡[H+][HCO3-](A⁢.8)Ka⁢⁢3=[H2⁢PO4-]⁡[H+][H3⁢PO4](A⁢.9)Ka⁢⁢4=[HPO42-]⁡[H+][H2⁢PO4-](A⁢.10)Ka⁢⁢5=[PO43-]⁡[H+][HPO42-](A⁢.11)Ka⁢⁢6=[H+]⁡[OH-](A⁢.12)
The pH of the solution is the negative logarithm of the hydrogen ion concentration, which can be expressed as follows.
pH=−log10([H+])  (A.13)

Chemical equilibrium can be modeled based on the concept of reaction invariants. Three reaction invariants (Warefers to a charge-related quantity, Wbrefers to a concentration of carbonate ions, and Wcrefers to a centration of phosphate ions) can be involved for each stream in this system. The reaction invariants can be expressed as follows.
Wai=[H+]i−[OH−]i−[HCO3−]−2[CO32−]−2[HPO42−]−3[PO43−]  (A.14)
Wbi=[H2CO3]+[HCO3−]+[CO32−]  (A.15)
Wci=[H3PO4]+[H2PO4−]+[HPO42−]+[PO43−]  (A.16)
The relationship between hydrogen ion concentration and reaction invariants can therefore be given as follows.

Example Online FLS PID Tuning Using RSC Metric

PID tuning is a well-researched topic, and one known method of PID tuning is frequency-loop shaping. In FLS, closed-loop design specifications are defined in terms of a target open-loop transfer function, and PID parameters can be selected by minimizing the distance between a target loop and an actual open-loop transfer function. A weighted H∞norm that corresponds to the RSC metric provides a good measure of distance between the actual and target loops. Offline FLS algorithms for PID tuning can be formulated as an RSC minimization problem, and the quality of a controller design can be assessed by looking at the RSC value. If the RSC value is less than one, stability of the closed-loop system can be guaranteed. Also, lower RSC values correlate with closer actual performance to a target. The offline tuning approach can be extended to online tuning and used in adaptive control.

Let G represent a plant or process to be controlled. One objective of the FLS algorithm can be to find a controller C that minimizes the distance between an open-loop transfer function GC and a target loop L. One way of representing distance between GC and L involves using the RSC metric and the derivation of the RSC metric with the use of the small gain theorem as shown below.

FIGS. 8A through 8Cillustrate example representations of a closed-loop system used to demonstrate application of a robust stability condition metric for adaptive control of an industrial process according to this disclosure. Consider a diagram800of a closed-loop system as shown inFIG. 8A. The system can be rearranged into a diagram820as shown inFIG. 8B. If the transfer function (GC−L) is stable, it can be viewed as having stable uncertainty, and the small gain theorem can be used to derive an RSC metric. Reduction of the diagram820inFIG. 8Binto a diagram840convenient for application of the small gain theorem is shown inFIG. 8C. The small gain theorem condition for the diagram840inFIG. 8Ccan be expressed as follows.

z⁡(GC-L)⁢11+L∞<1(B⁢.1)
The left side of Equation (B.1) is the RSC metric. By rearranging this equation, the RSC metric can be rewritten as follows.
RSC=∥S(CG−L)∥∞(B.2)
where S=1/(1+L) is the target sensitivity. An RSC metric that is less than one can guarantee closed-loop stability and can have a high correlation to closed-loop performance. Formulation of an FLS algorithm for a controller with a PID structure as an RSC minimization is shown below.

Consider the PID parameterization of a controller, where the transfer function of the controller is shown below.

C⁡(s)=Kp+Kis+Kd(τ⁢⁢s+1)(B⁢.3)
Here, Kp, Ki, and Kdare the proportional, integral, and derivative gains, respectively, and τ is a time constant of a filter of the derivative function block. For this PID parameterization, the RSC minimization problem is convex and can be solved, such as by using a standard convex optimization solver. An expression of FLS as an RSC minimization for PID tuning can be expressed as follows:

θ*=arg⁢⁢minθ∈M⁢S(GC⁡(θ)-L∞(B⁢.4)
where θ represents the PID parameters (Kp, Ki, and Kd).

In cases where PID parameters are adaptively tuned online with the help of an RSC estimate, the RSC values shown in Equation (B.4) can be estimated using input-output data. Let (u,y) denote an input-output pair of a plant, and consider an error signal e=S(CG−L)u. Because y=Gu, the error signal e can be rewritten as e=SCy−Tu. The RSC metric could then be expressed in terms of input-output data as follows:

RSC=S⁡(GC-L)∞=supu≠0⁢e2-du2(B⁢.5)
where ∥d∥ is a norm bound on the disturbance.

Estimation of RSC in Equation (B.5) could involve decomposition of signals e and u into frequency components. A Fast Fourier Transform (FFT) is one of the methods to decompose a signal into its frequency components. However, it results in circular convolution problems for signals with finite intervals. Another way of approximate decomposition of signals into frequency components involves using a filter bank of band-pass filters Fi. When a signal is passed through the filter bank, the energy of the filtered signal is computed using an exponentially-weighted average. The relation between RSC in Equation (B.5) and its estimate can be expressed as:

RSC≈⁢maxi⁢SCFi⁢y-TFi⁢u2-dFi⁢u2≤⁢maxi⁢SCFi⁢y-TFi⁢u2,δ-dFi⁢u2,δ(B⁢.6)
where ∥.∥2and ∥.∥2,δare the two-norm and exponentially-weighted two-norm, respectively. There are different ways of computing an exponentially-weighted two-norm, such as by using the method described below. This RSC estimate can provide one example basis for the proposed adaptive control algorithms discussed above.

Consider a signal v(t). The exponentially-weighted two-norm of the signal can be defined as follows.

v2,δ=(∫-∞∞⁢e-δ⁢⁢t⁢v2⁡(t)⁢d⁢⁢t)12(C⁢.1)
To compute the exponentially-weighted two-norm for data, it is assumed that the signal is from a casual system v(t)=0, ∀t<0. Thus, the lower limit of the integral in Equation (C.1) can be changed to 0, and the upper limit of the integral in Equation (C.1) can be changed to the current time T of the signal v(t). Equation (C.1) can therefore be modified as follows.

v2,δ=(∫0T⁢e-δ⁢⁢t⁢v2⁡(t)⁢d⁢⁢t)12(C⁢.2)
To compute the norm in Equation (C.2), a signal v2(t) can be filtered through a filter with a transfer function of 1/(s+δ), with zero as an initial condition of the state. The square root of the output of the filter can give the value of the exponentially-weighted two-norm value as described in Equation (C.2).

CONCLUSION

In conclusion, this patent document has provided different formulations of adaptive control, including direct adaptive control and multi-model switching control. Both approaches could use FLS or other tuning, along with an RSC cost function obtained from online input-output data. These algorithms can be used to support controller gain adaptation or other parameter adaptation in order to obtain substantially uniform performance over a wide operating range.