# Accelerated Iterative Learning Control for Linear Discrete Time Invariant Switched Systems

Authors: Yongping Wang; Saleem Riaz; Ziyun Bao; Wenqian Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

For a class of linear discrete time invariant stochastic switched systems with repetitive operation characteristics, a novel P-type accelerated iterative learning control algorithm with association correction is proposed. Firstly, the concrete form of accelerated learning law is given, and the generation of correction in the algorithm is explained in detail; secondly, on the premise that the switching sequence does not change along the iteration axis but only along the time axis, the convergence of the algorithm is strictly mathematically proved by using the super vector method, and the sufficient conditions for the convergence of the algorithm are given; finally, the theoretical results show that the convergence speed mainly depends on the controlled object, the proportional gain of the control law, the switching sequence, the correction coefficient, the association factor, and the size of the learning interval. Simulation results show that the proposed algorithm has a faster convergence speed than the traditional open-loop P-type algorithm under the same conditions.


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## Body

## 1. Introduction

In recent years, switching system has gradually become one of the research hotspots in the control field [1]. Switching system is a kind of important hybrid system, which is composed of a set of differential equations or difference equations and switching rules. According to the changes of actual environmental factors, the whole system can be switched between different subsystems through switching rules so as to meet the requirements of different working conditions and improve the system performance [2]. Therefore, switching systems are widely used in practical engineering systems, such as a traffic control system [3], power system, circuit system [4], and network control system [5]. At present, a large number of research results on switched systems are focused on the stability of switched systems [6], but the research results on the output tracking control of switched systems are very limited [7]. The reason is that it is much more difficult to realize the tracking control of switched systems than the stabilization and stability problems [8,9]. It is worth noting that the ILC algorithm and its optimal robust control [10]can be used in many industrial applications such as robotics control [11], industrial automation [12], power and energy sectors [13–15], chemical plant processing [16], servo control [17], and many other automation industries [18].Iterative learning control [19], as a model-free control, has a simple structure and does not need specific model parameters. After enough repeated iterations in a limited interval, the behavior of the execution object can meet the expected requirements [18]. Due to the above characteristics, the learning algorithm has been favored by many scholars in recent years, which makes it widely used in the control of a robot rigid manipulator [12,20], batch processing in industries [21], aerodynamic system [22], traffic control system [23], electrical power system, etc. [24–26]. However, at present, most scholars have only performed a lot of research on the control of nonswitched systems, while there are few research results on the iterative learning control of switched systems [27].Xu-Hui et al. [28] proposed an open-loop P-type iterative learning control algorithm for a class of linear discrete stochastic switched systems on the premise that the switching sequence is randomly determined and analyzed the convergence of the algorithm by using the super vector method, realizing the complete tracking of the desired trajectory in the whole running finite time interval. For linear and nonlinear discrete switched systems, the convergence of an open-loop D-type ILC algorithm is analyzed in detail by using the compression mapping principle in references [29, 30]. In reference [31], an open-loop D-type ILC algorithm is proposed for a class of continuous random switched systems. Zhou et al. [32] proposed a discrete P-type iterative learning control algorithm for the trajectory tracking problem of a class of time-varying discrete switched systems with arbitrary switching sequences. The convergence of each subinterval was strictly proved by using compressed mapping, and the sufficient conditions of norm form for the convergence of the algorithm were given. Finally, it was compared with the state feedback robust control. Simulation results show the superiority of the iterative learning algorithm. Cao and Sun [33] proposed an iterative learning control algorithm with an attenuation factor, which reduces the influence of measurement noise during trajectory tracking. Bu et al. [34] analyzed the stability and convergence of a high-order PID iterative learning control algorithm in switched systems with state delay. In references [17, 35], aiming at the problem of model uncertainty and external noise of discrete switched systems, the PD type iterative learning algorithm is used to analyze the uncertainty and robustness.It is important to note that the issue discussed in this article is analogous to the ILC for time-varying linear systems. There are, nevertheless, some critical distinctions. While the time-varying system accepts solutions that are parameterized merely by the initial condition, the switched system admits solutions that are parameterized by both the initial condition and the switching signal. The time-varying linear system in the super vector formulation has a constant lower-triangular Markov parameter matrix throughout the timespan, and the starting state is the same for the entire system. While in a linear switched system, the lifting matrix is defined by both the subsystems and the switching signal. Additionally, the initial condition is same for only the first subsystem, not for all subsystems, resulting in a different convergence approach than for time-varying systems.Novel Points: The autoassociative ILC proposed in this paper is based on the traditional ILC, namely, using the current information to estimate the future input. Compared to traditional ILC, the new algorithm is characterized as follows: in each trial, the unlearned time is precorrected with the current time information. The algorithm can reduce the number of iterations and accelerate the learning convergence speed.The algorithm proposed in this paper differs from the traditional discrete closed-loop algorithm and the higher-order algorithm as follows:(1)
Although the algorithm proposed in this paper is similar in form to the traditional closed-loop iterative learning algorithm, the principle is completely different from that of the traditional discrete closed-loop P-type algorithm (feedback algorithm). The traditional discrete closed-loop ILC algorithm is to correct the control input of the current time directly with the error of the previous time in the same trial. The algorithm proposed in this paper uses the error of the current time to pre-estimate the amount of control after which it does not occur at all times and plays the role of precorrection.(2)
Although the associative iterative learning algorithm proposed in this paper is similar in form to the traditional higher-order discrete learning algorithm, the learning process is completely different from the traditional higher-order iterative learning algorithm. The traditional high-order ILC is the algebraic overlay of the control information of the previous two or more trials at the corresponding time. The new iterative learning algorithm proposed in this paper is to precorrect the subsequent unoccurred time with the error value of the current time in the same trial.The switched system is a piecewise controlled system. When the switching rules of the switched system are arbitrarily selected in the initial iteration and remain unchanged in the learning process, the dynamic structure of the switched system remains unchanged in the learning process, which meets the basic requirements of iterative learning control for the controlled system. On this premise, when the switched system runs repeatedly in a fixed finite time interval and meets certain conditions, iterative learning control can be used to realize the accurate control of the switched system. The proposed algorithms can obtain the consultations exactly in the piecewise means. Strict theoretical proof and corresponding experimental simulation are carried out, which will be published one after another.Based on the above analysis, for a class of discrete time invariant arbitrary switching systems that perform repeated tracking tasks on the desired trajectory in a finite time interval, under the premise that the switching sequence is randomly determined and the iteration is unchanged, using the characteristics of iterative learning control, taking p-type as an example, a discrete iterative learning control algorithm with backward error association and correction of the control quantity of the next iterative learning is proposed. Combined with the theory of super vector and spectral radius, the convergence of the algorithm is discussed, and sufficient conditions for the convergence of the algorithm are given in theory.


## 2. Problem Formulation

A class of single input single output linear discrete time invariant switched systems with repetitive operation in a finite time period is considered as follows:(1)xkt+1=Aσtxkt+Bσtukt,ykt+1=Cσtxkt+1,where discrete time t∈0,1,…,N−1, xkt∈Rm, ukt∈R, ykt∈R are the state vector, input vector, and output vector of the system; AσtandBσtandCσt are the constant matrices with appropriate dimension; σt is a random switching sequence, defined as σt:0,1,2,…⟶M=1,2,…,m, namely, σt takes value in a finite set M=1,2,…,m; m>1 is the number of subsystems; when σt=i, the i subsystem is activated; the subscript k indicates the number of iterations.Hypothesis 1.
In each iteration of the switched system, the initial state is equal to the ideal initial state, namely,xk0=xd0.Hypothesis 2.
Desired trajectoryydt+1,t∈0,1,…,N−1 is predetermined, and it is not related to the number of iterations. Since udj exists uniquely, the uniform convergence of the control profile uj to udj implies that the state and output tracking errors will be vanished. It is a reasonable assumption that the task for control should be feasible.Hypothesis 3.
For any given desired trajectoryydt+1, there are desired states xdt and desired control signal udt so that a desired output yd(t) is given, and there is only desired control input ud(t) which generates xd(t) and yd(t), where xd(t) and yd(t) are desired state vector and desired output vector, respectively.(2)xdt+1=Aσtxdt+Bσtudt,ydt+1=Cσtxdt+1,t∈0,1,…,N−1.Hypothesis 4.
After the random switching sequenceσt is randomly determined in the first iteration, the sequence is used in the subsequent iterations.
By system (1), the output signals at each time in array 1,N during the k-th iteration can be expressed as(3)yk1=Cσ0xk1=Cσ0Aσ0xk0+Cσ0Bσ0uk0,yk2=Cσ1xk2=Cσ1Aσ1Aσ0xk0+Cσ1Aσ1Bσ0uk0+Cσ1Bσ1uk1,⋮,ykN=CσN−1xkN=CσN−1∏j=0N−1Aσjxk0+CσN−1∏j=1N−1AσjBσ0uk0+CσN−1∏j=2N−1AσjBσ1uk1+⋯+CσN−1AσN−1BσN−2ukN−2+CσN−1BσN−1ukN−1.
For the convenience of description, the above formula is written in the form of super vector and introduced into super vector.(4)Uk=uk0,uk1,…,ukN−1T,Yk=yk1,yk2,…,ykNT.
The above formula can be written as(5)Yk=GUk+V,where(6)G=Cσ0Bσ000⋯0Cσ1Aσ1Bσ0Cσ1Bσ10⋯0Cσ2Aσ2Aσ1Bσ0Cσ2Aσ2Bσ1Cσ2Bσ2⋯⋮⋮⋮⋮⋱0CσN−1∏j=1N−1AσjBσ0CσN−1∏j=1N−1AσjBσ1⋯CσN−1AσN−1BσN−2CσN−1BσN−1,V=Cσ0∏j=00Aσjxk0,Cσ1∏j=01Aσjxk0,Cσ2∏j=02Aσjxk0,⋯…,CσN−1∏j=0N−1Aσjxk0T.
A control law with backward error association and correction of subsequent control variables is considered as follows:(7a)u˜kt=L∑i=0teki+1eKi−t+1/N2,t∈0,1,…,N−1,(7b)uk+1t=ukt+u˜kt+Γekt+1,where uk+1t is the control quantity at the time of the K+1th iteration process, ukt is the control quantity at time t of the k-th iteration process, u˜kt is the correction of the control quantity by the error at the time and before the k-th iteration process, Γ is the iterative proportional gain, ekt+1=ydt+1−ykt+1 is the tracking error. The goal of iterative learning control is to find a control signal sequence through a learning algorithm ukt and make the output trajectory ykt+1 of the controlled system (1) under this sequence control, as the number of iterations increases, progressive tracking of desired trajectory ydt+1, namely,(8)limk⟶∞ekt+1=0,t∈0,1,…,N−1.
As shown in Figure1, in the k-th iterative learning process, the error ek1 at point 1 will correct the control amount at N time in the (k+1)th iterative process, and the correction amount is shown in Table 1.
The errorek2 at point 2 will correct the control amount at N-1 times in the (k + 1)th iteration, as shown in Figure 2, and the correction amount is shown in Table 2.
According to this method, up to point N, its error isekN, as shown in Figure 3, and it only corrects the control quantity of point N in the K + 1th iteration, and the correction quantity is LekNeK1/N2.
For the discrete switched system (1) meeting the assumptions 1–4, combined with the characteristics of the switched system, the random switching sequenceσt is added, and the P-type accelerated iterative learning control law for the random switched system is proposed on the basis of the control law (4):(9a)u˜kt=Lσt∑i=0teki+1eKσti−t+1/N2,t∈0,1,…,N−1,(9b)uk+1t=ukt+u˜kt+Γσtekt+1,where σt is the random switching sequence.Figure 1
Errorek1 correction of the control term at N times in the (k+1)th iteration.Table 1
Associative correction of errorek1 to the control quantity at N times of the next iteration.
123⋯N−1NLek1eK1/N2Lek1eK0/N2Lek1eK1/N2⋯Lek1eKN−3/N2Lek1eKN−2/N2Figure 2
Errorek2 correction of the control term at N-1 time in the (k+1)th iteration.Table 2
Associative correction of errorek2 to later time control quantity.
234⋯N−1NLek2eK1/N2Lek2eK0/N2Lek2eK1/N2⋯Lek2eKN−4/N2Lek2eKN−3/N2Figure 3
ErrorekN up to point N in the K and (K + 1)th iteration.


## 3. Convergence Analysis

We have developed an accelerated algorithm of P-type ILC strategy with kernel-based auto-associative memory for a linear discrete system. During each iterative learning process, the control value of the current time is corrected. Meanwhile, the amount of control in the subsequent time is precorrected by association, which speeds up the learning process. The iterative learning algorithm based on the kernel function is revealed theoretically in depth. The convergence speed of the association algorithm is illustrated by simulation results, which is related with the proportional learning gain, the learning interval, and the exponential factor. This research method can be widely used to extend to study the influence of association depth and industrial control.Theorem 1.
A linear discrete time invariant switched system with single input and single output (1) is considered, if assumptions 1–4 are satisfied, and a P-type accelerated iterative learning control algorithm with associative correction (5) is adopted, if the selected learning parameter matrix satisfies(10)ρ=maxt1−LσteKσt1/N2+ΓσtCσtBσt<1.The output trajectory uniformly converges to the desired trajectory, whenk⟶∞, ykt+1⟶ydt+1,t∈0,1,…,N−1。Proof. According to the iterative learning control algorithm (5), in thek+1th iterative learning process, the control variables at each time in the interval 0,N−1 are respectively(11)uk+10=uk0+Lσ0ek1eKσ01/N2+Γσ0ek1,uk+11=uk1+Lσ1ek1eKσ10/N2+ek2eKσ11/N2+Γσ1ek2,uk+12=uk2+Lσ2ek1eKσ2−1/N2+ek2eKσ20/N2+ek3eKσ21/N2+Γσ2ek3,⋮,uk+1N−1=ukN−1+LσN−1ek1eKσN−1N−2/N2+ek2eKσN−1N−3/N2+⋯+ekN−1eKσN−10/N2+ekNeKσN−11/N2+ΓσN−1ekN.The super vector is introduced as follows:(12)Ek=ek1,ek2,…,ekNT.Then,(13)Uk+1=Uk+LH+ΓEk,where(14)L=Lσ0Lσ1Lσ1Lσ2Lσ2Lσ2⋮⋮⋱⋮LσN−2LσN−2⋯LσN−2LσN−1LσN−1LσN−1LσN−1LσN−1,H=eKσ01/N2eKσ10/N2eKσ11/N2eKσ21/N2eKσ20/N2eKσ21/N2⋮⋮⋱⋱⋮eKσN−2−N−3N2eKσN−2−N−4N2⋯eKσN−20/N2eKσN−21/N2eKσN−1−N−2/N2eKσN−1−N−3/N2eKσN−1−N−4/N2⋯eKσN−10/N2eKσN−11/N2,Γ=Γσ0Γσ1Γσ2⋱⋮ΓσN−2ΓσN−1.According to the equivalent system (3),(15)Ek+1=Yd−Yk+1=Yd−GUk+1+V=Yd−GUk+LH+ΓEk−V=Yd−GUk−V−GLH+ΓEk=Ek−GLH+ΓEk=IN×N−GLH+ΓEk=IN×N−GLH+Γ2Ek−1=⋯=IN×N−GLH+ΓkE0.According to the properties of spectral radius, the necessary and sufficient condition forlimk⟶∞IN×N−GLH+Γk=0 is ρIN×N−GLH+Γ=maxiλi<1. ρM represents the spectral radius of matrix M, and if λi is the eigenvalue of the matrix, then the necessary and sufficient condition for system convergence is(16)ρ=maxt1−LσteKσt1/N2+ΓσtCσtBσt<1.Hence, the theorem is proved.


## 4. Numerical Simulation

In order to verify the effectiveness of the accelerated learning law in this paper, the following discrete switched systems with three subsystems are considered:(17)xkt+1=Aσtxkt+Bσtukt,ykt+1=Cσtxkt,where t∈0,1,…,N−1, σt∈1,2,3 represents a random switching sequence. Any group is selected, as shown in Figure 4.Figure 4
Random switching sequenceσt.The parameters of the three subsystems are as follows:(18)A1=010.125−0.2,B1=01,C1=0.11,A2=−0.2510−0.3,B2=01,C2=−0.21,A3=100.2−0.1,B3=01,C3=0.51.Assuming that the learning parameters remain unchanged during the learning process and will not change with the switching sequence and the number of iterations. The iteration proportional gain is set toΓσi=0.25, the association factor is set to Kσ1=1, the correction coefficient is set to Lσi=0.2, and the discrete time is set to N = 60, and the calculated ρ1=maxt1−LσteKσt1/N2+ΓσtCσtBσt=0.55<1. The convergence condition is met. If Kσt=1 and Lσt=0, the above algorithm degenerates into a traditional P-type iterative learning control algorithm, and its convergence conditions are ρ2=maxt1−ΓσtCσtBσt=0.75<1 and ρ1<ρ2. According to the spectral radius theory, the smaller the convergence radius, the faster the iterative learning algorithm converges.The desired trajectory isydt+1=sin8t/25,t∈0,1,…,59. The initial xk0=0,∀k∈Z+, the initial control vector is u1=0. When using the accelerated P-type learning law proposed in this paper, from the first learning to the 50th time, the changing trend of Ek2 is shown in Figure 5. It can be seen that the algorithm can ensure that Ek2 converges to zero. Figure 6 shows the output of the system after the first, fifth, tenth, and seventeenth iterations, and the convergence process of the algorithm can be seen in more detail.Figure 5
The variation trend of the error 2 norm of the accelerated ILC algorithm with the number of iterations.Figure 6
Output V/S desired trajectory. (a) After the first iteration. (b) After the 5th iteration. (c) After the 10th iteration. (d) After the 17th iteration.
(a)(b)(c)(d)When the traditional P-type learning law is adopted, the proportional learning gainΓσt=0.25, which remains unchanged in the learning process. From the first learning to the 50th, the change trend of Ek2 is shown in Figure 7. In addition, the figure also includes the change trend of Ek2 using the acceleration algorithm proposed in this paper. In the figure, when the allowable error ε=0.01, the traditional P-type algorithm needs 35 iterations and the accelerated P-type iterative learning algorithm needs 17 iterations; given that the allowable error ε=0.001, the traditional p-type algorithm needs 48 iterations and the accelerated P-type iterative learning algorithm needs 15 iterations. It can be seen intuitively that after using the P-type accelerated ILC algorithm proposed in this paper, the convergence speed of the system is obviously accelerated.Figure 7
Traditional ILC VS. accelerated ILC.Table3 shows that the highest tracking error of D, P, and accelerated proposed P-type ILC laws in the first iteration is 6.817316. After the 15th iteration, the error of the P-type law is 0.062823 and the D-type algorithm is 1.07538. Whereas the error of the proposed accelerated P law is 0.003683, from the column of Table 3 to the data, the tracking error of all ILC laws is reduced consecutively with the increase of the iteration number. However, from the horizontal data in Table 3, the tracking error of the proposed accelerated P-type ILC law is the smallest as compared to other ILC laws (D-type and Traditional P-type) under the same iteration number. Therefore, it can be easily observed from Table 3 that the convergence speed of the proposed accelerated P-type law in this paper is significantly higher than that of other traditional laws.Table 3
Comparison of ILC laws with proposed accelerated PD type ILC law.
Iteration numberD-type ILC lawTraditional P-type ILC lawProposed accelerated P-type ILC law1st6.8173165.6173165.52173162nd5.12173163.12173163.01273165th4.5498192.5219281.23208410th2.6035810.2810790.02923815th1.075380.0628230.003683


## 5. Conclusions

In this paper, a class of discrete time invariant switched systems with arbitrary switching sequences is studied. An open-loop P-type accelerated iterative learning algorithm with precorrection function is used, and the convergence conditions are given. The convergence radius of the algorithm is smaller than that of the traditional P-type ILC algorithm. The convergence and rapidity are strictly proved in theory by using the super vector. Numerical simulation examples verify the effectiveness of the proposed algorithm. The results show that for discrete switched systems with arbitrary switching in the time domain, the ILC method can ensure complete tracking control in a finite time period. At the same time, the accelerated ILC algorithm proposed in this paper can reduce the number of iterations when the error reaches the allowable error so as to accelerate the convergence. Since any switching sequence is randomly determined and does not change with the number of iterations, the system is still a determined system from the iteration axis, and ILC can be used to realize complete tracking. In future research, the author will consider the stability and convergence of iterative changing initial conditions, system model uncertainty, and measurement noise.



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