# Nonlinear Tracking Control Algorithm for Dynamical Output Systems Manipulated by the Hardly Constrained Oscillatory Actuator

Authors: M. R. Homaeinezhad; M. H. Saeidi Mostaghim
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

In this article, a nonlinear control algorithm has been presented, in order to reduce vibrational movements of a flexible beam cantilevered in a cart supported by nonlinear stiffness, actuated by a hardly constrained second-order dynamic system. In the presented control algorithm, the interactive model of the solid body and a flexible beam has been extracted and utilized to design an algorithm, based on which the nonlinear vibrations of the tip point of the beam whose function is smooth and nonoscillatory tracking of the predefined desired trajectory are diminished. To design a nonlinear vibration control system of the beam, it has been assumed that the feedback system includes three modules: a vibrational beam cantilevered on a base, a base affected by nonlinear stiffness and actuator’s control force, and a force imposing system with constraints of bandwidth frequency limitation and actuation saturation boundary. In order to design this control system, the generalized tracking error function has been defined to nonlinearly amplify the vibrationally induced error and its rate, which by properly adjusting them, a new quantity can be extracted that satisfies the control system design constraints. To design a tracking control system for a nonlinear vibrating beam for the aforementioned three-modulus system, first, the generalized tracking error for the three modules is separately defined and parametrized, and then, by simultaneously adjusting all the defined parameters systematically, the predominant system presence in Lyapunov stability conditions is guaranteed. To illustrate the multimodule imitative (MMI) controller design algorithm, a moving support connected to a nonlinear spring and damper is considered, which carries a flexible cantilevered beam. The applied actuation system has second-order linear dynamics with the presence of command control saturation boundaries. For each of the abovementioned modules, a generalized tracking error is defined, and then, it is explained to how simultaneously adjust the parameters based on the stability and actuation constraints. The MMI controller is applied to the mentioned mechanical system modeled in the ANSYS® Mechanical APDL environment, and then, the necessary conclusions are discussed about the performance of the control system in eliminating the vibrations of the flexible arm, considering the actuation constraints while possessing the dominant Lyapunov stability.


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## Body

## 1. Introduction

In many mechanical engineering applications in the field of robotics, haptics, industrial machining, instrumentation, and the optical industry, vibration-sensitive equipment might be mounted on one or more flexible co-operative arms carried by the moving support [1–5]. Generally, ignoring mechanical phenomena, such as deformation and vibration, causes poor performance or even the controller instability, when using flexible components in mechanical systems such as robotic arms [6–8]. In order to control the vibrations of flexible systems, from the point of view of installation (mounting) techniques of sensors and actuators, the command control input can be applied into the system as a distributed input or as a concentrated (point) input. Logically, the approach of applying the distributed input to a flexible object leads to better responses, while the numerous sensors and actuators are required to acquire data and execute control input demanding complex and sophisticated equipment. On the other hand, in the approach of the concentrated command control input, sensors and actuators are usually installed at a limited number of points within the boundaries of the flexible system, and it is possible that stabilization and tracking control can be performed successfully by controlling the boundaries. Among the most important methods of mathematical modeling and control of flexible systems, modal methods, the method of assume modes, the linear or nonlinear finite elements method, the method of lines, and variable separation methods, can be mentioned [7, 8]. Another technique of mechanical systems’ control with flexible arms is the boundary control method, in which the dynamic equations of the system that are in the form of PDE are used directly by methods such as functional analysis, operators’ techniques, and the semigroup theory, and the differential geometry calculus is included in the design of the control system [9, 10]. The subject of controlling systems with flexible components, in the presence of nonlinear effects of stiffness and damping in the system configuration, is acutely more difficult than in the case where the mathematical model is considered as linear operators. Also, if the force and torque generating components (actuators) have a linear or nonlinear dynamic model, the control problem will be much more problematic. In this situation, an important concern for the plausible performance of the specified objective is to minimize the vibrations transmitted to the equipment mounted on the flexible arm. In such cases, because the control forces are applied to the moving support and are not applied directly to the equipment coordinate, the defined output of the control system has linear or nonlinear dynamics based on the measured state variables [5, 11]. However, in many feedback control systems, the relationship between output and system state variables is often algebraic and linear. In addition, in actual feedback systems, the actuators mounted on the support, which are responsible for generating the necessary forces and torques, have linear first- or second-order dynamics, and if the actuator dynamics are not taken into the account in the system design, the possibility of eliminating vibrations in practical applications based on logics obtained from computer simulation will not result in the satisfactory results [5]. Once a suitable dynamic is defined for the actuator connected to the base, the open-loop system, including the moving support dynamics and actuator with two important control constraints, namely, the bandwidth frequency limitation and the saturation boundaries of the control commands, will be imposed to the control system design. The presence of the bandwidth frequency limitation of the actuator, as well as the high-order dynamics between the control command and the output of the actuator, makes the problem of vibration control an important and significant issue [12–14]. Vibrations generated in a continuous system contain multiple frequencies where the experiments show that the smallest frequency has the greatest impact on the overall system vibration. Therefore, with the problem of the presence of actuator bandwidth limitation, the maximum vibrational frequency that can be eliminated according to the mentioned actuation restrictions should be examined [12, 15]. Finally, it can be concluded that the recent research studies in the field of designing nonlinear feedback control systems mainly focus on solving the problem of designing Lyapunov stable control systems in the presence of hard constraints such as system switching, the output constraint, the small gain theorem, unmodeled dynamics, the dead zone of the actuator, the restrictions imposed due to the communication between the sensors, and the presence of time delay in measuring the quantities or in commanding the actuators [11, 13, 14, 16]. This article is also among the research studies carried out with this point of view in order to give more possibility of practical implementation. In general, the innovation of the presented study is to provide an algorithm which is robust to unmodeled dynamics to solve the complicated problem of the feedback control of a system with an actuator with constrained oscillatory dynamics. In this research, the following limitations are considered for designing the feedback control system [17–19]:(i)
The system under control is a nonlinear system with both modeling and unmodeled dynamic uncertainties.(ii)
The desired trajectory that should be followed by the system has a countable number of discontinuities in terms of time.(iii)
A linear second-order differential equation governs the relationship between the output of the system and the measured state variables of the system.(iv)
Between the command input to the actuator and its output, there is a linear second-order oscillatory dynamics.(v)
The actuator output has a saturation limit and any value determined by the Lyapunov stability law cannot be produced by the actuator.(vi)
The actuator command input also has a saturation level limit.The aforementioned assumptions are embedded in the design of the presented control algorithm so as to propose a new and unique design procedure for feedback control of such a system. To solve the control system design problem due to such limitations, optimal control methods, model predictive control, and methods based on the artificial intelligence might be used. Although by using such methods, it is possible to provide solutions for the control system design, but the most important issues that arise as side effects are as follows:(i)
In the design of controllers based on optimization methods and model prediction algorithms, the computational load increases so much that the real-time implementation of such algorithms becomes practically impossible and these methods remain at the theoretical level [11].(ii)
In the design of control systems based on artificial intelligence, if the adaptive mechanism adjusts the parameters of the artificial intelligent system in real-time, it may train the network in such a way that the intelligence of the system forgets the previous training due to the occurrence of nonlinear behaviors of the system [11].The innovation proposed in this article is a boundary control problem in which a moving base supports a flexible arm and also the dynamics of the system actuator is a second-order linear differential equation. The purpose of the control system design is to calculate the control input to the base so that, first, the tip of the flexible arm has minimum vibrations while tracking for the smooth or nonsmooth desired trajectory. Second, the dynamic model of the actuator with existing constraints, namely, actuation bandwidth frequency limitation, the oscillatory behaviors in the actuator dynamics, and the presence of the saturation boundaries at the command control input, should be considered in the design of the control systems. Third, under abovementioned limitations and circumstances, the validity of the Lyapunov stability theorem should remain possible. In other words, the three modular components, namely, the actuator, the base, and the flexible beam form a cascade dynamic system in which a bounded control input is applied to a second-order system where outputs are measured from the supporting base and the system output which is the position of the tip of the flexible beam should track a predetermined desired trajectory with the least possible vibrations. In order to control the vibrations and track the desired trajectory by a flexible component of the system, in the problem of the Lyapunov stability of the control system, it was necessary to use a new quantity called the generalized tracking error instead of the tracking error. The quantity of the generalized tracking error is a function of the predictive tracking error and its rate and it is parameterized by four amplification parameters, two of them are gains, and the other two are exponents of the predictive error. The generalized tracking error function incorporates an abstraction of the behavior of Newtonian mechanical systems in such a way that all defined control objectives are achieved indirectly. For this purpose, by considering an illustrative example of the design method, in which a moving base supports a flexible beam and interacts with the nonlinear spring and damper, first, the desired trajectory of the tip of the flexible beam is filtered by the generalized tracking error function. By applying the generalized tracking error function to the desired trajectory at the tip of the beam, the desired trajectory of the supporting base of the beam is determined. In the next step, by reapplying the generalized tracking error function to the base dynamic variables, the desired control force is obtained to be generated by an actuator with second-order dynamics. Finally, by setting another generalized tracking error function for the actuator system dynamics, the command control input is calculated in such a way that the actuator output precisely tracks the desired force determined in the previous step, with the presence of the actuation constraints, under the Lyapunov stability conditions. This article describes how to simultaneously adjust the parameters of the generalized tracking error functions used in the sliding mode control structure, in order to hit all the defined control targets while statistically and numerically examines the reliability of the adjustment techniques of the presented parameters. This article is arranged as follows: In Section2, the generalized tracking error is defined and the method of the design of the controller for a cascade dynamic system is explained. In Section 3, the simulation results are presented, explaining confronted issues in the design procedure, proving the asymptotic stability of the MMI control algorithm and its validation using the finite elements analysis environment of the ANSYS® APDL. In Section 4, the main and principal conclusions obtained from this research and also some suggestions for future works are stated.


## 2. Multimodule Imitative (MMI) Control System Design Procedure

### 2.1. Introductory Definition of the Three-Module Oscillatory System

The MMI control system is designed and developed for systems in which the command control input is applied to the system through a linear second-order dynamics, and on the other hand, the relationship between the output of the system and its supporting system dynamics is a linear or nonlinear differential equation. The MMI control system is a generalization of the output tracking control methods in which a conventional algebraic relation is replaced by a linear or nonlinear dynamic differential equation. The schematic of an open-loop system with a linear actuator and a dynamic output is shown in Figure1. Equation (1) expresses the equations of the motion of the system shown in Figure 1, where a base carrying a flexible beam which is excited by a second-order linear actuator, and on the other hand, the output of the system is considered to be the end point of the flexible beam. The flexible beam is equalized by a linear spring and the effective mass based on Rayleigh’s quotient [5]. Therefore, by designing the control system based on such simplification, during the control process, the unmodeled dynamics of the flexible beam will also affect the performance of the control system. In order to facilitate the process of the MMI control system design, a vibrational system is considered as shown in Figure 1. In this system, the support is connected to an impedance configuration including a nonlinear spring and damper while supporting a flexible beam. A point mass is attached to the tip of the flexible beam and the aim of the MMI tracking control system is to minimize and if possible, to eliminate the nonlinear vibrations of the base and the flexible beam. The exemplary system is a mechanical system in which, first, a control input with a dynamic is applied to the base, while there is a second-order dynamic relationship between the system output and the support state variables.Figure 1
Illustration of a mechanical system including three modules, module 1, the dynamic output; module 2, the main system; and module 3, constrained actuators with a dynamic.


### 2.2. Modeling of a Equivalent Rigid Body of the Moving Support System Carrying a Flexible Beam

According to the theorems of flexible beam vibrations, in the simplest case, instead of a beam, a spring with a stiffness coefficient based on the beam deflection equation can be considered. According to the well-known theories of Euler–Bernoulli or Timoshenko [11], this equivalent stiffness coefficient corresponds to the first vibrational mode of the flexible beam system. On the other hand, since the beam support base is connected to a nonlinear impedance configuration, it is no longer possible to analytically determine and solve the governing equations, which are a set of coupled PDE and ODE equations, by the method of separation of variables or the popular technique of assume modes [5]. Also, since the actuator connected to the base has a second-order linear dynamics, according to theories in the field of the linear dynamics system, such an actuator cannot apply force to the base in a wide bandwidth frequency, and practically like a second-orderlow-pass filter, it is not possible to execute high frequencies of actuator outputs. Therefore, it appears that the control of the first vibration mode, which has the lowest frequency and the highest amplitude value, based on the existing boundary conditions for the beam, is the best and most accessible control solution to solve the problem of tracking control without vibration at the end of the beam. According to the abovementioned characteristics, given that one end of the beam is a cantilever at the supporting base and the other end can be assumed to be connected to a point mass. Based on equation (1), a second-order nonlinear differential equation for the support motion, a second-order nonlinear differential equation for the system output dynamics (w), and a second-order linear differential equation for actuator dynamics are determined.(1)MC+MBx¨=−fNIx,x˙−ksLx−w+f,ML+MBEw¨=+ksLx−w−fdw,w˙;RayleighEffectiveMassMBE=0.2235MB,fNIx,x˙≜k0x+k1x3+c0x˙+c1x˙2signx˙+c2x˙x2−a2,fdw,w˙≜μ0wsignw˙+μ1w˙3,f¨+2ζωf˙+ω2f=ω2fC;fC≤fC0,x¨=−fNIx,x˙MC+MB−ksLMC+MBx−w+1MC+MBf,x¨=gx,x˙,w+bfgx,x˙,w≜−fNIx,x˙MC+MB−ksLMC+MBx−w;b≜1MC+MB.


### 2.3. Defining the Generalized Tracking Error FunctionΨ

According to equation (2), the generalized tracking error function Ψ is a function of the predictive tracking error δ, the predictive tracking error rate δ˙, the amplification gain of the predictive tracking error Cp, the amplification exponent of the predictive tracking error ρp, the amplification gain of the predictive tracking error rate Cv, and the amplification exponent of the predictive tracking error rate ρv, where the main objective is to create a new error function in such a way that a specific prediction with a form of combination of predictive tracking error and its rate is extracted. The structure of the function Ψ is established in such a way that by properly adjusting its set of parameters (Cp,Cv,ρp, and ρv) based on the dynamical characteristics of the system, the maximum tracking error of the control system in a prediction horizon with the allowable length is estimated.(2)Ψδ,δ˙,Cp,ρp,Cv,ρv≜1−e−Cpsgnδδρp+Cvsgnδ˙δ˙ρv1+e−Cpsgnδδρp+Cvsgnδ˙δ˙ρv.In Figures2(a)–2(c), the behavior of the generalized tracking error function Ψ in terms of the amplification exponent ρ is depicted, respectively, for ρ>1, ρ<1 and ρ=1. If ρ is greater than one, greater error intensifies the generalized error. If ρ is less than one, even small errors will result in a large generalized error. Finally, if ρ is equal to one, there is a linear relationship between the error and the generalized tracking error value.Figure 2
Behavior of generalized tracking error functionΨ, in nonlinear amplification of predictive error and its rate, for (a) ρ>1, (b) ρ<1, and (c) ρ=1.
(a)(b)(c)


### 2.4. Determination of Required Filtering for the Desired Trajectory of the Dynamic Outputw

According to equation (1) and the abovementioned actuation constraints, the control objective of defining fC is to eliminate the nonlinear dynamics of the output and the nonlinear vibrations appearing at the output of the system and to track the predetermined desired trajectory with appropriate and acceptable precision over time. According to equation (3), by applying the two-step Adams–Bashforth temporal discretization method, the equations for the system dynamics and output dynamics are calculated in discrete time [11].(3)xk=xk−1+3T2x˙k−1−T2x˙k−2,x˙k=x˙k−1+3T2x¨k−1−T2x¨k−2,wk=wk−1+3T2w˙k−1−T2w˙k−2,w˙k=w˙k−1+3T2w¨k−1−T2w¨k−2,δwk≜wdk−wk−1⇒δ˙wk=w˙dk−w˙k−1,δwk+1=wdk+1−wk−1+3T2w˙k−1−T2w˙k−2,δ˙wk+1=w˙dk+1−w˙k−1+3T2w¨k−1−T2w¨k−2.In order to filter the desired trajectory of the end of the beam, according to the logic which will be stated in Section2.3, it is necessary to define the predictive tracking error and its rate, as well as the same quantities for one step forward, according to the equation (4) of the variable w.(4)w^dk≜wk−1+Ψδwk,δ˙wk,Cpw,ρpw,Cvw,ρvw,w^dk+1≜wk+Ψδwk+1,δ˙wk+1,Cpw,ρpw,Cvw,ρvw,w^dk+1=wk−1+3T2w˙k−1−T2w˙k−2+Ψδwk+1,δ˙wk+1,Cpw,ρpw,Cvw,ρvw.If the desired filtered quantity (w^d) is followed instead of the original prescribed trajectory wd, due to the predictive nature of the generalized tracking error function Ψ, control forces with less amplitude and intensity will be designed for the tracking control problem. If the desired trajectory of the moving base and that of one step forward is considered equal to the filtered desired trajectory w^d and its corresponding one step forward shift, by successfully performing the tracking control of the base, the variable w follows the desired trajectory wd with minimal vibrations.


### 2.5. Trajectory Planning for the Moving Support Based on Dynamic Output Compensation

In order to calculate the predictive tracking error and its rate for the variablex, it is necessary to take the derivative of the quantities xdk and xdk+1 which are defined numerically in the following equation:(5)xdk≜w^dk;xdk+1≜w^dk+1.Numerical time differentiation of a quantity does not usually yield precise results, and if there are high frequency components in the signal corresponding to the quantity, discrete-time differentiation significantly reduces the signal-to-noise ratio because of the nonsmooth shape of the contaminating noise. To avoid this issue, according to equation (6), derivation is performed with a low-pass filter in which the smaller the parameter εw is selected, the higher the frequencies will appear in the result of the derivative. Details of numerical derivation with a low-pass filter are presented in reference [17].(6)x˙d=ddtw^dz˙w=−1εwzw+w^d,zwk=zwk−1+3T2−1εwzwk−1+w^dk−1−T2−1εwzwk−2+w^dk−2,w^˙dk−1=1εww^dk−1−1εwzwk−1,w^˙dk=1εww^dk−1εwzwk=1εww^dk−1εwzwk−1+3T2−1εwzwk−1+w^dk−1−T2−1εzwk−2+w^dk−2,w^˙dk+1=1εww^dk+1−1εw1−3T2εwzwk+3T2w^dk−T2−1εwzwk−1+w^dk−1,w^˙dk+1=1εww^dk+1−1εw1−3T2εwzwk−1+3T2−1εwzwk−1+w^dk−1−T2−1εwzwk−2+w^dk−2+3T2w^dk−T2−1εwzwk−1+w^dk−1.By calculating the quantitiesx˙dk and x˙dk+1 based on equation (7), the filtered quantities x^dk and x^dk+1 are calculated.(7)x˙dk≜w^˙dk;x˙dk+1≜w^˙dk+1.In order to design a discrete-time sliding mode tracking controller for the supporting base of the beam, according to equation (8), a discrete-time sliding mode function is defined with respect to the fact that the dynamics governing the variable x is a second-order differential equation.(8)sxk−1≜x˜k+λxx˜k−1;x˜k−1≜xk−1−xdk−1;λx<1.0,sxk−1=1+λxxk−1+3T2x˙k−1−T2x˙k−2−xdk−λxxdk−1.According to the discrete-time sliding function definition of equation (8), by shifting the quantity sxk−1 one step forward, the actuator output appears in sxk as stated by the following equation:(9)sxk=1+λxxk−1+3T2x˙k−1−T2x˙k−2+3T2x˙k−1+3T2x¨k−1−T2x¨k−2−T2x˙k−1−xdk+1−λxxdk,ξk−1=1+λxxk−1+3T2x˙k−1−T2x˙k−2+3T2x˙k−1+3T2gk−1−T2x¨k−2−T2x˙k−1−xdk+1−λxxdk,sxk=ξk−1+bfdk−1.In equation (9), fdk−1 is the tracking stabilizing desired value of the actuator output which should be defined based on the Lyapunov stability theorem. The command control input needs to be designed in such a way that the actuator output tracks the determined desired force fdk−1. In order to calculate fdk−1, a Lyapunov functional is defined based on equation (10), and by negating its increment in the direction of discrete-time subjected to system dynamics, a convex inequality of equation (11) is obtained.(10)b=9T24b,Vxsxk−1≜sxk−12,sxk=ξk−1+bfdk−1Vxsxk−Vxsxk−1<0⇒−sxk−1<sxk<+sxk−1−sxk−1<ξk−1+bfdk−1<+sxk−1−ξk−1−sxk−1b<fdk−1<−ξk−1+sxk−1b,(11)QxHk−1≜−ξk−1+sxk−1b,QxLk−1≜−ξk−1−sxk−1b.By defining a suitable proximity coefficient according to equation (12), the appropriate value of fdk−1 can be calculated according to equation (13).(12)αxk−1=1−sgnxdk−xk−12,(13)fdk−1=αxk−1QxLk−1+1−αxk−1QxHk−1.Finally, in order to reduce the intensity and amplitude of the command control signal by calculating the predictive tracking error and its rate according to equation (14), the filtered quantity of fdk−1 is calculated according to equation (15).(14)δfk=fdk−1−fk−2,δ˙fk=ddtδfk,z˙f=−1εfzf+δf,zfk=zfk−1+3T2−1εfzfk−1+δfk−1−T2−1εfzfk−2+δfk−2,δ˙fk=1εfδfk−1εfzfk,δ˙fk=1εfδfk−1εfzfk−1+3T2−1εfzfk−1+δfk−1−T2−1εfzfk−2+δfk−2,(15)f^dk−1=fdk−1+Ψδfk,δ˙fk,Cpf,Cvf,ρpf,ρvf.


### 2.6. Constrained Actuator Output Planning Based on Output and System Dynamics

By placing thef^dk−1 in the dynamic equation of the actuator, the value of fCk−1 is calculated. Since the dynamic equation of the actuator is discretized according to equation (16) and is a second-order dynamic similar to that expressed in equations (8)–(10), in order to prevent the repetition of the explanations, the command control signal fCk−1 is calculated based on equation (17). Corresponding to the predictive error of the force generated by the actuator according to equation (17), the appropriate proximity coefficients are determined so as to specify the control signal fCk−1.(16)fk−1=σ2k−1+σ1fCk−1,σ2k−1≜−2T2ω2−8T2ω2+4Tζω+4fk−2−T2ω2−4Tζω+4T2ω2+4Tζω+4fk−3+2T2ω2T2ω2+4Tζω+4fCk−2+T2ω2T2ω2+4Tζω+4fCk−3,σ1=T2ω2T2ω2+4Tζω+4;ζ∈0,∞,(17)f˜k−1=fk−1−f^dk−1,sfk−1≜f˜k−2+λff˜k−3;λf<1.0,sfk−1=fk−2−f^dk−2+λff˜k−3,sfk=f˜k−1+λff˜k−2=fk−1−fdk−1+λff˜k−2=σ2k−1+σ1fCk−1−fdk−1+λff˜k−2,sfk=σ2k−1+σ1fCk−1−f^dk−1+λff˜k−2=ξfk−1+σ1fCk−1,ξfk−1≜σ2k−1−f^dk−1+λff˜k−2−sfk−1<sfk<+sfk−1−sfk−1<ξfk−1+σ1fCk−1<+sfk−1−ξfk−1−sfk−1σ1<fCk−1<−ξfk−1+sfk−1σ1,QfHk−1≜−ξ^fk−1+s^fk−1σ1,QfLk−1≜−ξ^fk−1−s^fk−1σ1,fCk−1=αfk−1QfLk−1+1−αfk−1QfHk−1,αfk−1=1−sgnf^dk−1−fk−22.


### 2.7. Key Features of the MMI Tracking Control Algorithm including Asymptotic Stability, Robust Stability, Actuation Hard Constraints, and Desired Trajectory Tracking

In order to analyze the asymptotic stability of the MMI control algorithm, it should be noted that due to the actuation hard constraints, in unpredictable conditions, the asymptotic stability of the feedback system through the Lyapunov stability theorem is violated.Theorem 1.
If the parameters of the MMI tracking control algorithm, i.e., the set of equation (18) are set in such a way that the histogram and the kernel density estimation (KDE) of the variables of all three modules simultaneously have the following properties, the overall feedback system is internally stable and the desired trajectory will be stably tracked.(18)P≜Cpw,Cvw,ρpw,ρvw,Cpx,Cvx,ρpx,ρvx,Cpf,Cvf,ρpf,ρvf.(i)
Property 1. Histogram and KDE of the spatial variables of all three modules show instability near their desired position and show stability in positions categorized as far from the desired position.(ii)
Property 2. Histogram and KDE of the spatial variables of all three modules show a rapid attraction towards the desired trajectory and at the same time show a weak repulsion to move away from the desired position.In Figures3(a) and 3(b), the histogram and KDE of the variables δw and δ˙w are demonstrated where accordingly, it can be seen that, the variable δw has a broader KDE of stability than instability, in the near-zero distribution. This is also the case with the variable δ˙w, where the KDE of stability has overlapped with the KDE of instability, in two points in the near-zero distribution, which indicates the capability of the feedback system in the stably fluctuating the variable δw around its desired value. This pattern also can be seen in Figures 3(c)–3(f) that illustrate the histogram and the KDE of the second module variables sxands˙x and third module variables sfands˙f, respectively.Definition of equation (18). In equation (18), the set P includes a collection of parameters, which are adjusted so that the properties 1 and 2 remain valid. Based on Figure 3, in the left column, including Figures 3(a), 3(c), and 3(e), if the controlled variable is close to zero and negligible, the control system functions in such a way that it creates an oscillatory behavior around the desired trajectory. In other words, while the system’s trajectory catches its desired trajectory, it starts to oscillate within an acceptable boundary; if the error between the desired trajectory and the current trajectory of the system exceeds this boundary, the system again catches the desired trajectory. The histograms and the KDEs of Figure 3, in Figures 3(a), 3(c), and 3(e), perfectly show this behavior, shown in red. On the other hand, if the system’s trajectory is substantially far from the desired trajectory, the control algorithm would converge the system's reponse to the desired trajectory. The histograms and the KDEs of Figure 3, in Figures 3(b), 3(d), and 3(f), show that if the system’s trajectory is close to the desired trajectory, the repulsion speed is relatively low, shown in red, while if the system’s trajectory is considerably far from the desired trajectory, the attraction speed of these two is relatively high. In fact, Figure 3 shows a statistical description of the system dynamics in distant and nearby points from the desired trajectory. Obviously, if the system’s trajectory starts to oscillate at closer points relative to the desired trajectory and converges in more distant points relative to the desired trajectory, a synthetic Lyapunov stability is obtained. More details regarding the concept of the synthetic Lyapunov stability can be found in the study [11].Figure 3
Histogram and KDE illustrating the behavior of each module of the MMI control system during the feedback algorithm operation, for the (a) variableδw, (b) variable δ˙w, (c) function sx, (d) function s˙x, (e) function sf, and (f) function s˙f.
(a)(b)(c)(d)(e)(f)Theorem 2.
In order to analyze the robust stability of the MMI control algorithm by performing local searches in the parametric uncertainty space indicated by supremum [·] and infimum [·] operators, the convex inequalities shown in equations (11) and (17) are conservatively constricted and robust stability of the MMI tracking control algorithm is proved [11, 17].By considering equations (11) and (17) and based on the definition of the estimated parameter space Φ^ belonging to the uncertainty space Φ, equations (11) and (17) are rearranged and are rewritten according to equations (19) and (20)(19)−ξk−1,Φ−sxk−1b<fdk−1<−ξk−1,Φ+sxk−1b,(20)−ξfk−1,Φ−sfk−1σ1<fCk−1<−ξfk−1,Φ+sfk−1σ1.By taking Sup [⋅] and Inf [⋅] operators from the left and right sides, respectively, of the inequalities in the uncertainty space Φ, the worst case for the estimation of fdk−1 and fCk−1 relative to the estimated parameters space Φ^ is obtained. According to the proofs of the robust stability theorems presented in the studies [11, 17], the obtained worst-case intervals are the subsets of the Lyapunov stability criterion, the overall robust stability of the MMI algorithm is achieved.


## 2.1. Introductory Definition of the Three-Module Oscillatory System

The MMI control system is designed and developed for systems in which the command control input is applied to the system through a linear second-order dynamics, and on the other hand, the relationship between the output of the system and its supporting system dynamics is a linear or nonlinear differential equation. The MMI control system is a generalization of the output tracking control methods in which a conventional algebraic relation is replaced by a linear or nonlinear dynamic differential equation. The schematic of an open-loop system with a linear actuator and a dynamic output is shown in Figure1. Equation (1) expresses the equations of the motion of the system shown in Figure 1, where a base carrying a flexible beam which is excited by a second-order linear actuator, and on the other hand, the output of the system is considered to be the end point of the flexible beam. The flexible beam is equalized by a linear spring and the effective mass based on Rayleigh’s quotient [5]. Therefore, by designing the control system based on such simplification, during the control process, the unmodeled dynamics of the flexible beam will also affect the performance of the control system. In order to facilitate the process of the MMI control system design, a vibrational system is considered as shown in Figure 1. In this system, the support is connected to an impedance configuration including a nonlinear spring and damper while supporting a flexible beam. A point mass is attached to the tip of the flexible beam and the aim of the MMI tracking control system is to minimize and if possible, to eliminate the nonlinear vibrations of the base and the flexible beam. The exemplary system is a mechanical system in which, first, a control input with a dynamic is applied to the base, while there is a second-order dynamic relationship between the system output and the support state variables.Figure 1
Illustration of a mechanical system including three modules, module 1, the dynamic output; module 2, the main system; and module 3, constrained actuators with a dynamic.


## 2.2. Modeling of a Equivalent Rigid Body of the Moving Support System Carrying a Flexible Beam

According to the theorems of flexible beam vibrations, in the simplest case, instead of a beam, a spring with a stiffness coefficient based on the beam deflection equation can be considered. According to the well-known theories of Euler–Bernoulli or Timoshenko [11], this equivalent stiffness coefficient corresponds to the first vibrational mode of the flexible beam system. On the other hand, since the beam support base is connected to a nonlinear impedance configuration, it is no longer possible to analytically determine and solve the governing equations, which are a set of coupled PDE and ODE equations, by the method of separation of variables or the popular technique of assume modes [5]. Also, since the actuator connected to the base has a second-order linear dynamics, according to theories in the field of the linear dynamics system, such an actuator cannot apply force to the base in a wide bandwidth frequency, and practically like a second-orderlow-pass filter, it is not possible to execute high frequencies of actuator outputs. Therefore, it appears that the control of the first vibration mode, which has the lowest frequency and the highest amplitude value, based on the existing boundary conditions for the beam, is the best and most accessible control solution to solve the problem of tracking control without vibration at the end of the beam. According to the abovementioned characteristics, given that one end of the beam is a cantilever at the supporting base and the other end can be assumed to be connected to a point mass. Based on equation (1), a second-order nonlinear differential equation for the support motion, a second-order nonlinear differential equation for the system output dynamics (w), and a second-order linear differential equation for actuator dynamics are determined.(1)MC+MBx¨=−fNIx,x˙−ksLx−w+f,ML+MBEw¨=+ksLx−w−fdw,w˙;RayleighEffectiveMassMBE=0.2235MB,fNIx,x˙≜k0x+k1x3+c0x˙+c1x˙2signx˙+c2x˙x2−a2,fdw,w˙≜μ0wsignw˙+μ1w˙3,f¨+2ζωf˙+ω2f=ω2fC;fC≤fC0,x¨=−fNIx,x˙MC+MB−ksLMC+MBx−w+1MC+MBf,x¨=gx,x˙,w+bfgx,x˙,w≜−fNIx,x˙MC+MB−ksLMC+MBx−w;b≜1MC+MB.


## 2.3. Defining the Generalized Tracking Error FunctionΨ

According to equation (2), the generalized tracking error function Ψ is a function of the predictive tracking error δ, the predictive tracking error rate δ˙, the amplification gain of the predictive tracking error Cp, the amplification exponent of the predictive tracking error ρp, the amplification gain of the predictive tracking error rate Cv, and the amplification exponent of the predictive tracking error rate ρv, where the main objective is to create a new error function in such a way that a specific prediction with a form of combination of predictive tracking error and its rate is extracted. The structure of the function Ψ is established in such a way that by properly adjusting its set of parameters (Cp,Cv,ρp, and ρv) based on the dynamical characteristics of the system, the maximum tracking error of the control system in a prediction horizon with the allowable length is estimated.(2)Ψδ,δ˙,Cp,ρp,Cv,ρv≜1−e−Cpsgnδδρp+Cvsgnδ˙δ˙ρv1+e−Cpsgnδδρp+Cvsgnδ˙δ˙ρv.In Figures2(a)–2(c), the behavior of the generalized tracking error function Ψ in terms of the amplification exponent ρ is depicted, respectively, for ρ>1, ρ<1 and ρ=1. If ρ is greater than one, greater error intensifies the generalized error. If ρ is less than one, even small errors will result in a large generalized error. Finally, if ρ is equal to one, there is a linear relationship between the error and the generalized tracking error value.Figure 2
Behavior of generalized tracking error functionΨ, in nonlinear amplification of predictive error and its rate, for (a) ρ>1, (b) ρ<1, and (c) ρ=1.
(a)(b)(c)


## 2.4. Determination of Required Filtering for the Desired Trajectory of the Dynamic Outputw

According to equation (1) and the abovementioned actuation constraints, the control objective of defining fC is to eliminate the nonlinear dynamics of the output and the nonlinear vibrations appearing at the output of the system and to track the predetermined desired trajectory with appropriate and acceptable precision over time. According to equation (3), by applying the two-step Adams–Bashforth temporal discretization method, the equations for the system dynamics and output dynamics are calculated in discrete time [11].(3)xk=xk−1+3T2x˙k−1−T2x˙k−2,x˙k=x˙k−1+3T2x¨k−1−T2x¨k−2,wk=wk−1+3T2w˙k−1−T2w˙k−2,w˙k=w˙k−1+3T2w¨k−1−T2w¨k−2,δwk≜wdk−wk−1⇒δ˙wk=w˙dk−w˙k−1,δwk+1=wdk+1−wk−1+3T2w˙k−1−T2w˙k−2,δ˙wk+1=w˙dk+1−w˙k−1+3T2w¨k−1−T2w¨k−2.In order to filter the desired trajectory of the end of the beam, according to the logic which will be stated in Section2.3, it is necessary to define the predictive tracking error and its rate, as well as the same quantities for one step forward, according to the equation (4) of the variable w.(4)w^dk≜wk−1+Ψδwk,δ˙wk,Cpw,ρpw,Cvw,ρvw,w^dk+1≜wk+Ψδwk+1,δ˙wk+1,Cpw,ρpw,Cvw,ρvw,w^dk+1=wk−1+3T2w˙k−1−T2w˙k−2+Ψδwk+1,δ˙wk+1,Cpw,ρpw,Cvw,ρvw.If the desired filtered quantity (w^d) is followed instead of the original prescribed trajectory wd, due to the predictive nature of the generalized tracking error function Ψ, control forces with less amplitude and intensity will be designed for the tracking control problem. If the desired trajectory of the moving base and that of one step forward is considered equal to the filtered desired trajectory w^d and its corresponding one step forward shift, by successfully performing the tracking control of the base, the variable w follows the desired trajectory wd with minimal vibrations.


## 2.5. Trajectory Planning for the Moving Support Based on Dynamic Output Compensation

In order to calculate the predictive tracking error and its rate for the variablex, it is necessary to take the derivative of the quantities xdk and xdk+1 which are defined numerically in the following equation:(5)xdk≜w^dk;xdk+1≜w^dk+1.Numerical time differentiation of a quantity does not usually yield precise results, and if there are high frequency components in the signal corresponding to the quantity, discrete-time differentiation significantly reduces the signal-to-noise ratio because of the nonsmooth shape of the contaminating noise. To avoid this issue, according to equation (6), derivation is performed with a low-pass filter in which the smaller the parameter εw is selected, the higher the frequencies will appear in the result of the derivative. Details of numerical derivation with a low-pass filter are presented in reference [17].(6)x˙d=ddtw^dz˙w=−1εwzw+w^d,zwk=zwk−1+3T2−1εwzwk−1+w^dk−1−T2−1εwzwk−2+w^dk−2,w^˙dk−1=1εww^dk−1−1εwzwk−1,w^˙dk=1εww^dk−1εwzwk=1εww^dk−1εwzwk−1+3T2−1εwzwk−1+w^dk−1−T2−1εzwk−2+w^dk−2,w^˙dk+1=1εww^dk+1−1εw1−3T2εwzwk+3T2w^dk−T2−1εwzwk−1+w^dk−1,w^˙dk+1=1εww^dk+1−1εw1−3T2εwzwk−1+3T2−1εwzwk−1+w^dk−1−T2−1εwzwk−2+w^dk−2+3T2w^dk−T2−1εwzwk−1+w^dk−1.By calculating the quantitiesx˙dk and x˙dk+1 based on equation (7), the filtered quantities x^dk and x^dk+1 are calculated.(7)x˙dk≜w^˙dk;x˙dk+1≜w^˙dk+1.In order to design a discrete-time sliding mode tracking controller for the supporting base of the beam, according to equation (8), a discrete-time sliding mode function is defined with respect to the fact that the dynamics governing the variable x is a second-order differential equation.(8)sxk−1≜x˜k+λxx˜k−1;x˜k−1≜xk−1−xdk−1;λx<1.0,sxk−1=1+λxxk−1+3T2x˙k−1−T2x˙k−2−xdk−λxxdk−1.According to the discrete-time sliding function definition of equation (8), by shifting the quantity sxk−1 one step forward, the actuator output appears in sxk as stated by the following equation:(9)sxk=1+λxxk−1+3T2x˙k−1−T2x˙k−2+3T2x˙k−1+3T2x¨k−1−T2x¨k−2−T2x˙k−1−xdk+1−λxxdk,ξk−1=1+λxxk−1+3T2x˙k−1−T2x˙k−2+3T2x˙k−1+3T2gk−1−T2x¨k−2−T2x˙k−1−xdk+1−λxxdk,sxk=ξk−1+bfdk−1.In equation (9), fdk−1 is the tracking stabilizing desired value of the actuator output which should be defined based on the Lyapunov stability theorem. The command control input needs to be designed in such a way that the actuator output tracks the determined desired force fdk−1. In order to calculate fdk−1, a Lyapunov functional is defined based on equation (10), and by negating its increment in the direction of discrete-time subjected to system dynamics, a convex inequality of equation (11) is obtained.(10)b=9T24b,Vxsxk−1≜sxk−12,sxk=ξk−1+bfdk−1Vxsxk−Vxsxk−1<0⇒−sxk−1<sxk<+sxk−1−sxk−1<ξk−1+bfdk−1<+sxk−1−ξk−1−sxk−1b<fdk−1<−ξk−1+sxk−1b,(11)QxHk−1≜−ξk−1+sxk−1b,QxLk−1≜−ξk−1−sxk−1b.By defining a suitable proximity coefficient according to equation (12), the appropriate value of fdk−1 can be calculated according to equation (13).(12)αxk−1=1−sgnxdk−xk−12,(13)fdk−1=αxk−1QxLk−1+1−αxk−1QxHk−1.Finally, in order to reduce the intensity and amplitude of the command control signal by calculating the predictive tracking error and its rate according to equation (14), the filtered quantity of fdk−1 is calculated according to equation (15).(14)δfk=fdk−1−fk−2,δ˙fk=ddtδfk,z˙f=−1εfzf+δf,zfk=zfk−1+3T2−1εfzfk−1+δfk−1−T2−1εfzfk−2+δfk−2,δ˙fk=1εfδfk−1εfzfk,δ˙fk=1εfδfk−1εfzfk−1+3T2−1εfzfk−1+δfk−1−T2−1εfzfk−2+δfk−2,(15)f^dk−1=fdk−1+Ψδfk,δ˙fk,Cpf,Cvf,ρpf,ρvf.


## 2.6. Constrained Actuator Output Planning Based on Output and System Dynamics

By placing thef^dk−1 in the dynamic equation of the actuator, the value of fCk−1 is calculated. Since the dynamic equation of the actuator is discretized according to equation (16) and is a second-order dynamic similar to that expressed in equations (8)–(10), in order to prevent the repetition of the explanations, the command control signal fCk−1 is calculated based on equation (17). Corresponding to the predictive error of the force generated by the actuator according to equation (17), the appropriate proximity coefficients are determined so as to specify the control signal fCk−1.(16)fk−1=σ2k−1+σ1fCk−1,σ2k−1≜−2T2ω2−8T2ω2+4Tζω+4fk−2−T2ω2−4Tζω+4T2ω2+4Tζω+4fk−3+2T2ω2T2ω2+4Tζω+4fCk−2+T2ω2T2ω2+4Tζω+4fCk−3,σ1=T2ω2T2ω2+4Tζω+4;ζ∈0,∞,(17)f˜k−1=fk−1−f^dk−1,sfk−1≜f˜k−2+λff˜k−3;λf<1.0,sfk−1=fk−2−f^dk−2+λff˜k−3,sfk=f˜k−1+λff˜k−2=fk−1−fdk−1+λff˜k−2=σ2k−1+σ1fCk−1−fdk−1+λff˜k−2,sfk=σ2k−1+σ1fCk−1−f^dk−1+λff˜k−2=ξfk−1+σ1fCk−1,ξfk−1≜σ2k−1−f^dk−1+λff˜k−2−sfk−1<sfk<+sfk−1−sfk−1<ξfk−1+σ1fCk−1<+sfk−1−ξfk−1−sfk−1σ1<fCk−1<−ξfk−1+sfk−1σ1,QfHk−1≜−ξ^fk−1+s^fk−1σ1,QfLk−1≜−ξ^fk−1−s^fk−1σ1,fCk−1=αfk−1QfLk−1+1−αfk−1QfHk−1,αfk−1=1−sgnf^dk−1−fk−22.


## 2.7. Key Features of the MMI Tracking Control Algorithm including Asymptotic Stability, Robust Stability, Actuation Hard Constraints, and Desired Trajectory Tracking

In order to analyze the asymptotic stability of the MMI control algorithm, it should be noted that due to the actuation hard constraints, in unpredictable conditions, the asymptotic stability of the feedback system through the Lyapunov stability theorem is violated.Theorem 1.
If the parameters of the MMI tracking control algorithm, i.e., the set of equation (18) are set in such a way that the histogram and the kernel density estimation (KDE) of the variables of all three modules simultaneously have the following properties, the overall feedback system is internally stable and the desired trajectory will be stably tracked.(18)P≜Cpw,Cvw,ρpw,ρvw,Cpx,Cvx,ρpx,ρvx,Cpf,Cvf,ρpf,ρvf.(i)
Property 1. Histogram and KDE of the spatial variables of all three modules show instability near their desired position and show stability in positions categorized as far from the desired position.(ii)
Property 2. Histogram and KDE of the spatial variables of all three modules show a rapid attraction towards the desired trajectory and at the same time show a weak repulsion to move away from the desired position.In Figures3(a) and 3(b), the histogram and KDE of the variables δw and δ˙w are demonstrated where accordingly, it can be seen that, the variable δw has a broader KDE of stability than instability, in the near-zero distribution. This is also the case with the variable δ˙w, where the KDE of stability has overlapped with the KDE of instability, in two points in the near-zero distribution, which indicates the capability of the feedback system in the stably fluctuating the variable δw around its desired value. This pattern also can be seen in Figures 3(c)–3(f) that illustrate the histogram and the KDE of the second module variables sxands˙x and third module variables sfands˙f, respectively.Definition of equation (18). In equation (18), the set P includes a collection of parameters, which are adjusted so that the properties 1 and 2 remain valid. Based on Figure 3, in the left column, including Figures 3(a), 3(c), and 3(e), if the controlled variable is close to zero and negligible, the control system functions in such a way that it creates an oscillatory behavior around the desired trajectory. In other words, while the system’s trajectory catches its desired trajectory, it starts to oscillate within an acceptable boundary; if the error between the desired trajectory and the current trajectory of the system exceeds this boundary, the system again catches the desired trajectory. The histograms and the KDEs of Figure 3, in Figures 3(a), 3(c), and 3(e), perfectly show this behavior, shown in red. On the other hand, if the system’s trajectory is substantially far from the desired trajectory, the control algorithm would converge the system's reponse to the desired trajectory. The histograms and the KDEs of Figure 3, in Figures 3(b), 3(d), and 3(f), show that if the system’s trajectory is close to the desired trajectory, the repulsion speed is relatively low, shown in red, while if the system’s trajectory is considerably far from the desired trajectory, the attraction speed of these two is relatively high. In fact, Figure 3 shows a statistical description of the system dynamics in distant and nearby points from the desired trajectory. Obviously, if the system’s trajectory starts to oscillate at closer points relative to the desired trajectory and converges in more distant points relative to the desired trajectory, a synthetic Lyapunov stability is obtained. More details regarding the concept of the synthetic Lyapunov stability can be found in the study [11].Figure 3
Histogram and KDE illustrating the behavior of each module of the MMI control system during the feedback algorithm operation, for the (a) variableδw, (b) variable δ˙w, (c) function sx, (d) function s˙x, (e) function sf, and (f) function s˙f.
(a)(b)(c)(d)(e)(f)Theorem 2.
In order to analyze the robust stability of the MMI control algorithm by performing local searches in the parametric uncertainty space indicated by supremum [·] and infimum [·] operators, the convex inequalities shown in equations (11) and (17) are conservatively constricted and robust stability of the MMI tracking control algorithm is proved [11, 17].By considering equations (11) and (17) and based on the definition of the estimated parameter space Φ^ belonging to the uncertainty space Φ, equations (11) and (17) are rearranged and are rewritten according to equations (19) and (20)(19)−ξk−1,Φ−sxk−1b<fdk−1<−ξk−1,Φ+sxk−1b,(20)−ξfk−1,Φ−sfk−1σ1<fCk−1<−ξfk−1,Φ+sfk−1σ1.By taking Sup [⋅] and Inf [⋅] operators from the left and right sides, respectively, of the inequalities in the uncertainty space Φ, the worst case for the estimation of fdk−1 and fCk−1 relative to the estimated parameters space Φ^ is obtained. According to the proofs of the robust stability theorems presented in the studies [11, 17], the obtained worst-case intervals are the subsets of the Lyapunov stability criterion, the overall robust stability of the MMI algorithm is achieved.


## 3. Simulation Results and Discussion of the Virtual Experiment of the MMI Tracking Control Algorithm in Eliminating Vibrations of a Moving Support System Carrying a Flexible Beam

A three-modulus system is demonstrated by the Figure4, equivalent to the rigid form of Figure 1, where the technique of designing the MMI controller is described in detail in Section 2. In the current section, the MMI tracking control algorithm is designed for the system illustrated in Figure 4, and its performance is validated in the ANSYS® APDL environment. The parameters used in simulation of the moving support carrying a flexible beam are stated in Table 1. In order to analogize the system demonstrated in Figure 4 by the system shown in Figure 1, the equivalent rigid system stiffness and the effective mass of the beam of its first mode are considered to be ksL=3EI/L3 and MBE=0.2235MB, respectively, as it is assumed that the beam has been cantilevered in the support. The block diagram of the MMI tracking control system is illustrated by the Figure 5, which is designed for a vibrational system with the moving support carrying a flexible beam while the constrained actuator and the system output are both dynamic systems.Figure 4
Demonstration of a vibrational system including a cantilevered flexible beam supported by a moving base interacting with a nonlinear spring and damper whilst the base is actuated by a linear second-order constrained actuator.Table 1
Geometric and inertia parameters of the three-modulus vibrational system and ANSYS® APDL simulation parameters of the MMI control algorithm.
MMI tracking control algorithm operational parametersParametersDescriptionQuantity valuesDimensionsCpw,ρpw,Cvw,andρvwDynamic output trajectory filter parameters[3.5] [1.3] [0.13] [1.85]Not applicableCpx,ρpx,Cvx,andρvxSupport desired trajectory filter parameters[2.5] [1.1] [0.05] [1.25]Not applicableCpf,ρpf,Cvf,andρvfActuator output desired trajectory filter parameters[10.0] [1.5] [0.01] [1.7]Not applicableλxandλfSliding function parameters[0.2] [0.2]Not applicablefC0Saturation level of the command control input[3000.00](N)ζandωDamping ratio and the natural frequency of the actuator[0.7] [157.079][Dimensionless] (rad/sec)TDiscretization time-step[0.001](sec)εwandεfNumerical time differentiation lowpass filter parameters[0.001] [0.001]Not applicablewdt=W0sinωwtBeam tip harmonic desired trajectoryW0 = 0.7, ωw = 4.398(m) (rad/sec)wdt=W0sgnsinωwtBeam tip piecewise rectangular desired trajectoryW0 = 0.7, ωw = 4.398(m) (rad/sec)Moving support and beam geometrical and inertial parametersParametersDescriptionQuantity valuesDimensionsMCCart mass[1.0](kg)MBBeam mass[1.979](kg)MLLoad mass[1.50](kg)LBeam length1.00(m)MBEBeam effective mass0.2235MB(kg)ksLCantilever beam equivalent stiffness3EI/L3(N/m)k0andk1Nonlinear stiffness of foundation[3.0] [4.0](N/m) (N/m3)c0,c1,c2,andaNonlinear damping of foundation[4.0] [3.0] [3.0] [2.0](Ns/m) (Ns2/m2)(Ns/m3) (m)μ0=Cμ0xρμ0,μ1Disturbance force parametersCμ0=0.02,ρμ0=1.100.01(N) (Ns3/m3)ANSYS mechanical APDL environment setting parametersPoint masses typeMASS21Beam typeBEAM188Node number101Beam cross-sectionCircular [d = 6 cm]Beam bulk density700kg/m3Beam young modulus500MPaBeam Poisson’s ratio0.10∗three degrees of the freedom element type for all nodes except node1 = 1 DoF∗cross-section is scaled as a function of the axial stretch∗cubic shape function along the length∗shear stress output is a combination of torsion-related and flexure-relatedtransverse-shear stressFigure 5
Block diagram of the MMI tracking control algorithm designed for the moving support carrying a flexible beam.According to the MMI algorithm design, which is described in Section2, it is necessary to select the parameters of the control system appropriately. To precisely set the parameters of the MMI algorithm (equation (18)), the guideline of Table 2 is suggested. Based on the provided guideline for the MMI control design, to eliminate the vibrations of the beam end point while tracking the trajectory, the control parameters are determined as stated in Table 2.Table 2
The guideline for adjustment of the MMI controller parameters for the vibrational system with the moving support.
Parameters of module # 1 dynamic outputParameters of module # 2 moving supportParameters of module # 3 second-order actuatorCvw∝1/ksL0.75∼1/ksL0.85Cpx∝1/ksL0.65∼1/ksL0.85Cvf∝1/ζω∼3/ζωCpw∝1/ksL0.75∼1/ksL0.85Cvx∝1/ksL0.65∼1/ksL0.85Cpf∝80/ζω∼120/ζωρvw∝1/ksL0.2∼1/ksL0.4ρpx∝1/ksL0.1∼1/ksL0.4ρpf∝100/ζω∼150/ζωρpw∝1/ksL0.2∼1/ksL0.4ρvx∝1/ksL0.1∼1/ksL0.4ρvf∝100/ζω∼150/ζωThe heuristic guideline of Table2, which might not be unique, is derived based on several trial-and-error experiments of the MMI control system in order to obtain suitable stability-instability KDEs plots associated with each module.In order to determine the displacement of the end point of the beam relative to the position of the support with a limited number of strain gauges while keeping the observability of the MMI tracking control designed for this system, it is necessary to study the nodal curve length and nodal transversal displacement to determine the required number of strain gauges mounted on the beam. The nodal curve length and nodal transversal displacement of a finite element beam is depicted by Figure 6, which is processed to determine the required number of strain gauges and their installation position on the beam. Based on Figure6, it seems that with a limited number of strain gauges, it is possible to make an unbiased estimation between the relative displacement of the end of the beam and the measured values obtained from the strain gauges. In order to determine the displacement of the end point of the beam (the beam tip position relative to the support), it is presumed that two one-centimeter strain gauges are correctly installed between the nodes 30–31 and 70–71. The calculation of the relative displacement of the end point of the beam is illustrated by Figure 7, based on the data extracted from the strain gauges 1 and 2 and the estimation presented in equation (21). According to multiple experiments performed in the ANSYS APDL environment, the coefficients of the estimate function presented in equation (21) can be obtained using the least squares method. According to the mechanical properties given in Table 1, the values of the parameters of relative displacement estimation are calculated indicating an unbiased estimation procedure. In equation (21), the variable ϵi is the ith element strain value.(21)ϵw=κ1ϵ30+ϵ70+κ2;κ1=50.0;κ2=0.0.Figure 6
Nodal transversal displacement and nodal curve length of the flexible beam.Figure 7
Estimation of relative displacement of the beam end point, based on the acquired data from strain gauges installed on nodes, 30–31 and 70–71.The performance of the MMI tracking control algorithm in eliminating the vibrations of the end of the beam for harmonic and piecewise step-desired trajectories is shown by Figures8(a) and 8(b), respectively. In the mentioned figures, in addition to the vibrations of the end point of the beam, the vibrations of other nodes of the beam are also shown. Due to the fact that the actuation system connected to the base has linear second-order dynamics and also the command control signal has saturation boundaries, the end point of the beam has a slight offset relative to the desired trajectory, while its translational vibrations are eliminated by accurate and punctual movements of the support. In the mentioned figures, it can also be seen that the support movements are vibrational in such a way that by transferring those vibrations to the beam, the movement of the end point of the flexible beam continues to follow the desired trajectory without fluctuations. In Figures 8(c) and 8(d), the control forces applied to the base are, respectively, shown for the harmonic and piecewise step trajectories with time interval of 0.2 seconds. Given that the actuator dynamics are incorporated into the system dynamics and then based on the dynamics of all three modules, the MMI tracking control algorithm is designed, the control commands are created so that the response lag of the system to the desired trajectories is reduced to an acceptable level, the desired trajectory is followed by the low-magnitude overshoot and undershoot, the saturation boundaries of the control commands are not violated, and finally, the least possible vibrational motion is generated at the end point of the beam, as shown in Figures 8(a) and 8(b).Figure 8
MMI control system performance in application to a three-modulus vibrational system, (a) nodal tracking error of the beam for harmonic trajectory tracking, (b) the nodal tracking error of the beam for piecewise rectangular trajectory tracking, (c) 0.2 sec time interval desired actuator outputfd, command input fC, and actuator output f for harmonic trajectory tracking, and (d) 0.2 sec time interval desired actuator output fd, command input fC, and actuator output f for piecewise rectangular trajectory tracking.
(a)(b)(c)(d)In order to show the internal stability of the MMI control system, in Figures9(a)–9(c), the 2-D KDE of the stability-instability regions of each of the modules of the system is shown. As can be seen in Figures 9(b), the major system behavior is distributed in the second and fourth quarters of the stability-instability plot and occupies a significantly broader area than the system behavior in first and third quarters of the stability-instability plot which indicates the internal stability and overall stability of the MMI control system, based on the perspective of the Monte Carlo analysis [18].Figure 9
Behavior of the MMI control system depicted on the stability-instability plot for modules, (a) system output with variablesδw and δ˙w, (b) major system with functions sx and s˙x, and (c) constrained dynamic actuators with functions sf and s˙f.
(a)(b)(c)A remark on MMI algorithm computational complexity and its performance versus other feedback control strategies.In the present article, since the MMI control algorithm is designed in such a way that by using the functions of equations (10)–(17), the MMI predicts the behavior of the system without using computational optimization or artificial intelligence methods. The required computational complexity is practically so small that the main time of the control loop is devoted to reading the sensors used in the control structure and commanding the actuator [13]. In order to conclude, considering that the proposed MMI algorithm leads to the optimization of the control performance in a noncomputational noniterative way, compensation of the effects of unmodeled dynamics, the dynamic output and the second-order oscillatory bounded actuator, is conducted with the computational load and the complexity of the implementation which is significantly less than that of other algorithms which are designed based on computational optimization, model predictive methods, and methods based on artificial intelligence [11]. This specification along with maintaining the Lyapunov stability in the presence of all design constraints leads to the main innovation of this manuscript.


## 4. Conclusions and Further Future Works

Until now, critical issues such as dynamics of the actuation system, the presence of the saturation boundaries in the control commands, and the existence of dynamics in the system output have not yet been cumulatively studied as an interconnected package by researchers of the design of feedback systems. This has led to a large gap between computer simulations of feedback systems and their implementation in practical situations. In order to solve this problem, the main objective of this research is to provide a universal description of dynamic systems in which three modules of the actuation system, main system, and output with dynamics have formed the open-loop definition. It was shown how by inverting the output dynamics, the desired trajectory of the main system can be filtered so that, first, the actuation hard constraints are satisfied, second, the output of the system tracks the desired trajectory without being affected by the dynamics between output and system states, third, the desired trajectory is followed precisely with an appropriate quality, and fourth, the internal stability and overall stability of the feedback system are guaranteed. To this end, in this article, a moving support base carrying a flexible beam, which is connected to the nonlinear spring and damper, while the applied actuator has a second-order dynamics, is introduced as a three-modulus system. The MMI tracking control algorithm is designed for this system and indicates how to achieve important control objectives by defining a generalized predictive tracking error and accurately adjusting its parameters. Finally, in order to show the feasibility of the MMI control algorithm, a virtual experiment was performed in the ANSYS® APDL environment on the abovementioned system and the advantages and merit of the MMI control algorithm were rendered. In the continuation of this research, future works can be stated as follows:(i)
Development of the MMI control algorithm in which output modules and actuators behave nonlinearly(ii)
Generalization of the MMI control algorithm to multi-input multioutput (MIMO) control systems with linear or nonlinear output and actuation dynamics(iii)
Development of the MMI control algorithm into four-modulus systems and extraction of techniques related to stabilization and tracking control(iv)
Generalization of the MMI algorithm to the cases where the Lyapunov stability theorem is replaced by the power reaching law [19–21].



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*Source: 3799011-2023-02-16.xml*