# New Iterative Learning Control Algorithm Using Learning Gain Based onσ Inversion for Nonsquare Multi-Input Multi-Output Systems

Authors: Leila Noueili; Wassila Chagra; Moufida Ksouri
Journal: Modelling and Simulation in Engineering
 (2018)

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## Abstract

Model inversion Iterative Learning Control (ILC) for a class of nonsquare linear time variant/invariant multi-input multi-output (MIMO) systems is considered in this paper. A new ILC algorithm is developed based onσ-right inversion of nonsquare learning gain matrices to resolve the matrix inversion problems appeared in the direct model inversion of nonsquare MIMO systems. Furthermore, a sufficient and necessary monotonic convergence condition is established. With rigorous analysis, the proposed ILC scheme guarantees the convergence of the tracking error. To prove the effectiveness and to illustrate the performance of the proposed approach for linear time-invariant (LTI) and time-varying nonsquare systems, two illustrative examples are simulated.


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## Body

## 1. Introduction

Iterative Learning Control is an intelligent control strategy to deal with repetitive processes. The aim of the ILC is to control systems which execute the same task over a finite duration, such as the industrial robot manipulator [1]. The basic principle behind ILC is that the data generated from previous trial are used to adapt the control input for current trial. The control input in each trial is adjusted by using the tracking error obtained from the previous trial. As the iterations continue, the control system eventually learns the task and follows the desired trajectory with minimized tracking errors. The concept has been well developed in terms of both the fundamental theory and experimental applications that were accomplished in [2–6]. Thus, ILC research has considerably coped with mechanical systems such as multijoint hand-arm robots [7], the station stop control of train [8], and wafer stage motion systems [9]. The most of ILC schemes in the literature focus on proportional type learning law [10] and Optimal Iterative Learning Control (OILC) [11–14] where in [13] the learning approach has been applied to a rapid thermal processing. However, the majority of proposed algorithms were based on the notion of direct model inversion where the learning gain is obtained by Markov matrix inverse which represents the input-output map of the system [15]. However, this kind of ILC cannot be developed for general nonsquare (rectangular) MIMO systems that are systems in which the numbers of inputs and outputs are unequal. Indeed, a problem of the matrix inversion is encountered.Many industrial processes require nonsquare MIMO mapping, especially chemical plants such as crude distillation process [16], mixing tank process [17], three-tank systems [18], distillation column [19], and chemical mechanical planarization process [20]. Therefore, the need of control schemes treating this type of systems is of major interest. The model inverse design and its related applications in control design have been widely studied in [21–24].In the literature, there are control works that treated the nonsquare MIMO systems such as a Proportional Integrator Derivator (PID) controller cited in [25] which has been applied to a voltage model of Proton Exchange Membrane Fuel Cell (PEMFC); Model Predictive Control (MPC) was studied in [26] and has been applied to a Shell Heavy Oil Fractionator (SHOF); Minimum Variance Control (MVC) which is an inverse model control was studied in [27] and other examples of control strategies applied to nonsquare systems [28]. In the above cited works [25, 26], the control schemes were achieved without a need to matrix inversion. However, in [27] the synthesis of the inverse model in the control procedure required a matrix inversion based on left and right inverses which are developed in [29–31].However, all the above mentioned methods could not deal with repetitive systems that require a learning strategy based on inversion model to achieve the control of repetitive nonsquare MIMO systems. Therefore, developing an ILC procedure for nonsquare systems remains an open problem.Based on nonsquare polynomial matrix inversions [32], we propose the design and the analysis of an ILC scheme based on the model inversion called In-ILC for linear time-invariant and varying-time nonsquare MIMO systems. The main contribution of this work is to prove the monotonic convergence of the proposed scheme where the tracking error trial-to-trial will converge to zero even though the system has the initial resetting state. Through the simulation results, we prove the effectiveness of the proposed method.The rest of this paper is organized as follows: In Section2, the problem formulation is presented. The proposed inverse learning gain is developed in Section 3. A sufficient and necessary monotonic convergence of the In-ILC is established in Section 4. Further, the proposed In-ILC law is extended to time-varying systems in Section 5. Simulation results are illustrated in Section 6 to prove the effectiveness of the scheme for nonsquare MIMO systems. Finally, conclusions and an outlook on future work are given in Section 7.


## 2. Problem Formulation

The state-space representation of an LTI discrete-time system is given by (8):(1)xjk+1=Axjk+Bujk,yjk=Cxjk,xj+10=dj+1,where j∈ℑ≜1,2,…,nj−1 denotes the iteration numbers and nj is the total number of trials. k∈Λ≜0,1,…,N−1 represents the number of the discrete-time sampling steps and N is the total number of discrete-time steps at each trial. xjk∈ℜp is the jth iteration vector of system states, ujk∈ℜnu is the jth iteration vector of the system inputs which will be recursively generated by an iterative learning algorithm, yjk∈ℜny is the jth iteration of the system outputs and xj+10 is a fixed initial state. Further, A, B, and C are constant matrices with appropriate dimensions, and CB≠0. Therefore, this state-space system can be described as follows:(2)yj=Muj+dj,where M is a Markov matrix of rank N and whose terms are Markov parameters of the plant as cited in [33] and(3)djk=CAkxj0,M=CB0⋯0CABCB⋯0⋮⋮⋱⋮CAN−1BCAN−2B⋯CB∈RnyN×nuN.The control objective is to find a control sequenceujk with the ability to reduce tracking error for the whole trajectory based on the past tracking experiences until j→∞, the system tracking error limit is(4)limj→∞ej=limj→∞yc−yj∞=0,where ∞ is the infinite norm.The main formulation of the ILC design problem is to achieve an update mechanism for the control input trajectory of a new cycle based on the information from previous cycles, so that the output trajectory converges asymptotically to the reference trajectory [34–36]. This idea is depicted in the block diagram form in Figure 1, which shows the next trial’s control input uj+1 to be calculated from the previous trials control input uj and transient output error ej.Figure 1
Basic ILC structure.In this paper, we focus on the nonsquare systems when the number of input variablesnu is different from the number of output variables ny. The goal is to make the system outputs track a given desired reference trajectory.The first-order ILC algorithms update the input trajectoryu with the following equation:(5)uj+1k=ujk+Hejk,where we designate by H the learning gain and by ej the error between the set-point reference yc and the jth output as(6)ej=yc−yj.The most proposed algorithms were based on the direct model inversion, that isH=M−1, assuming that M−1 is invertible. M−1 represents the input-output map of the process [37, 38]. However, in nonsquare system case, the learning gain based on the model inversion cannot be calculated. In fact, the adaptation law in (5) cannot be satisfied for general nonsquare MIMO processes. To overcome this problem, a new method is proposed to design a realizable controller based on the direct model inversion I-ILC for nu×ny linear nonsquare system process called In-ILC.Theorem 1
(stability of ILC).Let an LTI discrete-time system be described by (1) is asymptotically stable if and only if the polynomial matrix M is right invertible [39, 40].The ILC law (5) can be immediately written as(7)uj+1k=ujk+MRejk,where MR is the right inverse of Markov matrix M.


## 3. Inverses of Nonsquare Polynomial Matrices andIn-ILC Design

Several methods of right/left inverses of nonsquare polynomial matrices, such as the classical minimum-norm right/left inverse, calledT-inverse and τ-σ-S-inverses methods, have been studied in [27, 32]. In this work, we focus on σ-inverse as follows:Corollary 1.
Let the polynomial matrixMq−1=m0+m1q−1+⋯+msq−s be a full normal rank ny (or nu) and let zrψz−1=ψz∈ℝny×nuz of full normal rank ny (or nu) be arbitrary, including an arbitrary order r. The product M¯q−1ψT¯Tq−1 must be of full normal rank ny. Then a σ-inverse can be defined as(8)MR¯q−1=ψT¯q−1M¯q−1ψT¯q−1−1.
For right invertible system (1), MR¯ denotes an infinite number of right inverse of M.Definition 1.
LetM∈ℜny×nu and ψ∈ℜny×nu are a full rank ny<nu where M is right invertible and ψ is selected such that ψopt=argminψ∑i=0Nuj+1Tiuj+1i. Then, the right inverse of M is(9)MR=ψTMψT−1.
Using previous Definition 1, we can employ theσ-inverse in In-ILC (7) to obtain(10)uj+1k=ujk+ψTq−1Mq−1ψTq−1−1ejk.
Using (2) and (3), the tracking error (6) can be rewritten as(11)ej+1k=yck−yj+1k=yck−Muj+1k−CAkxj+10=yck−Mujk−MψTMψT−1ejk−CAkxj+10=yck−Mujk−CAkxj+10−MψTMψT−1ejk.
Adding and subtractingCAkxj0 into (11), we get(12)ej+1k=yck−Mujk−CAkxj0+CAkxj0−CAkxj+10−MψTMψT−1ejk=yck−yjk−CAkxj+10−xj0−MHejk=In−MHej−CAkxj+10−xj0=GILCnej−CAkxj+10−xj0.


## 4. Convergence Analysis of the Proposed Learning Gain

The control design problem is to determine a new ILC law based on the model inversion such that the trial-to-trial error convergence occurs inj; that is, limj→∞ej=0 [41].Theorem 2.
Assume that In-ILC is applied to the linear time-invariant systems (1). Suppose xj+10=xj0, for all j∈ℑ≜1,2,…,nj−1. Then, the propositions(13)∑k=0+∞ej+1k2<∑k=0+∞ejk2,(14)limj→∞∑k=0+∞ej+1k2=0,hold if and only if(15)In−MHl=GILCnl<1.Proof (sufficiency). Based on initial condition, the initial states xj+10≡j→∞xj0, then (12) becomes(16)ej+1k=GILCnejk.LetSj=∑k=0+∞ejk2. By taking the assumption (13), it is deduced that 0≤Sj+1<Sj<⋯<S1. Then, the sequence Sj+1 is strictly decreasing and lower-bounded by zero. This means that limj→∞Sj+1 exists. Therefore, the series Sj=∑k=0+∞ejk2 is convergent.Then, we prove thatlimj→∞Sj+1=0 by reduction to absurdity. Suppose that limj→∞Sj+1=S>0. Therefore, for τ1=S/2, there exists a finite positive integer j0 so that, for all j>j0, Sj+1>S−τ1>S/2. However, Sj is convergent, so for all j>j0, the limit of the partial sequence Sj+1r=∑k=0rej+1k2 exists. This means that, for a given constant τ2=S/4, there exists an integer N0 so that, for all r>N0, we have(17)Sj+1r=∑k=0rej+1k2>S−τ1−τ2,in particular,(18)∑k=02N0ej+1k2>S−τ1−τ2,then, for k0∈0,2N0, we obtain(19)∑k=02N0ej+1k02>S−τ1−τ2,thus,(20)2N0ej+1k02>S−τ1−τ2.Hence, we have(21)ej+1k02>S−τ1−τ22N0=S8N0>0.On the other hand, by recursion, (16) reduces to(22)ej+1k0=GILCnje1k0.Although, by considering the assumption (15), we obtain(23)limj→∞ej+1k0=0.This is contradictory to inequality (19). This contradiction means that the propositions (13) and (14) are true. This proves the sufficiency assumption.Proof (necessity). We assume that the inequality (15) is not always true. So, there exists at least a number k0 such that(24)GILCnk0l≥1.The equality (16) gives(25)ej+1k0=GILCnje1k0≥e1k0,thus,(26)∑k=0+∞ej+1k2≥ej+1k02≥e1k02.Finally, we obtain(27)limj→∞∑k=0+∞ej+1k2≥limj→∞e1k02=e1k02.It is evident to chooseu1k and yck such that e1k0>0, which contradicts to the assumption (14). This proves the necessity of the assumption.The sufficient and necessary assumption for the monotonic convergence of the givenIn-ILC algorithm shows that ILC trial-to-trial error convergence requires that all of the initial states and tracking errors are reset.


## 5. Extension to Time-Varying Systems

In this part, the proposedIn-ILC scheme is extended to linear time-varying systems as follows:(28)xjk+1=Akxjk+Bkujk,yjk=Ckxjk,where Ak, Bk, and Ck are time-varying matrices with appropriate dimensions and CkBk≠0. Therefore, this state-space system can be described as follows:(29)yjk=Mujk+djk,where(30)M=C1B0⋯0⋮⋱⋮CN∏r=1N−1AN−rB0⋯CNBN−1,such that M∈RnyN×nuN and djk=Ck∏r=0N−1AN−rxj0. Thus the output can rewritten as(31)yjk=Ck∏r=0N−1AN−rxj0+Ck∑i=1N∏r=1i−1AN−rBN−iujk.Using theσ-inverse (8), the In-ILC to control time-varying MIMO systems is similar to control law (10).Theorem 3.
For the discrete linear varying-time system (28), the In-ILC is chosen such that, for any constant 0≤δ≤1,(32)supk∈1,…,NIn−HkM≤δ.
Therefore, the tracking errorsejk will converge to zero where j→∞ if and only if the previous conditions (13) and (14) are proved.Proof. The proof of Theorem 3 can be completed identically as in the proof of Theorem 2 such that the tracking error is given by(33)ej+1k=GILCnkejk.Following Proof 1, we can conclude thatlimj→∞ej=0 where all of initial states xj0 are reset.


## 6. Illustrative Examples

In order to show the effectiveness of the proposed ILC based on the direct model inversion to deal with nonsquare systems, two examples are considered.Example 1: Time-Invariant System. Consider a three-output two-input discrete-time Linear Time-Invariant (LTI) system:(34)A=1.10000010.80000001.1000000.50.80000001.10000010.9,B=0.50.5111.50.50.20.30.30.21.50.30.10.30.10.20.60.1,C=00.200000000.100,D=0,where xj0=0.1,0.2,0,0.2,0.1,0T,j∈ℑ≜1,2,…,25.The reference trajectories are chosen as(35)yc1=2sin2πk5+sin50πk,yc2=sin2πk5+2sin3πk5+sin50πk,where k∈Λ≜1,2,…,30; thus N=30 and the initial input vector is chosen as u0=0,0,0T.By applying the proposedIn-ILC law (10), the evolutions of transient outputs y1 and y2 profiles with different iteration numbers j=2,j=5, and j=25 are given in Figures 2 and 3, respectively. The evolutions of terminal inputs are shown in Figure 4. It is clear that the proposed scheme is well to be used for linear nonsquare MIMO systems to track time-varying reference. The In-ILC approach converges quickly and the performance keeps well even the reference changes. Furthermore, the performance of the absolute initial and terminal tracking error profiles e1,j and e2,j versus the iteration numbers are illustrated in Figures 5 and 6. It shows that the tracking e1,j and e2,j will converge within 15 and 20 iterations. Moreover, from Figures 7 and 8 which illustrated the first and the second components of tracking error profiles at 5th, 10th, 15th, and 25th iterations. It is clear that the fast convergence of errors in the iteration domain is obvious for the proposed algorithm.Figure 2
Outputsy1 at the 2nd, 5th, and 25th iterations.Figure 3
Outputsy2 at the 2nd, 5th, and 25th iterations.Figure 4
Evolutions of all inputs with trial lengthnj=25.Figure 5
Initial tracking errorse1,j and e2,j versus the iteration number j.Figure 6
Terminal tracking errorse1,j and e2,j versus the iteration number j.Figure 7
Tracking errors profiles of ILCe1,j.Figure 8
Tracking errors profiles of ILCe2,j.Example 2: Time-Variant System. In order to demonstrate the effectiveness of our proposed algorithm, let the discrete-time linear time-varying system as follows.(36)A=0.2sink0000010.60000002000000.80.5000000e−k/1000000000.4,B=0.50.5111.50.50.20.30.30.21.50.30.10.30.10.20.60.1,C=00.200000000.100,D=0,where xj0=0,0,0,0,0,0T,j∈ℑ≜1,2,…,100. k∈Λ≜1,2,…,80; thus N=80 and the initial input vector is chosen as u0=0,0,0T. yc1 (dashed line) and the first component of output y1 (solid line) are depicted in Figure 9. yc2 (dashed line) and the first component of output y2 (solid line) are depicted in Figure 10. The performance of the absolute tracking errors e1,j and e2,j is illustrated in Figure 11, where the tracking errors will converge within 20 and 30 iterations. Moreover, Figures 12 and 13 show the tracking error profiles for 20th, 50th, and 100th iterations. It is clear that the tracking errors converge to zero in more than 70 iterations, where xj0=0,0,0,0,0,0T,j∈ℑ≜1,2,…,100.Figure 9
Outputy1 at the 20th, 50th, and 100th iterations.Figure 10
Outputy2 at the 20th, 50th, and 100th iterations.Figure 11
Absolute values of errorse1,j and e2,j versus the iteration number j.Figure 12
Tracking errors profiles of ILCe1,j.Figure 13
Tracking errors profiles of ILCe2,j.


## 7. Conclusion

In this paper, a new model inversion Iterative Learning Control calledIn-ILC is proposed to deal with nonsquare MIMO systems where the numbers of inputs and outputs are unequal. The proposed scheme is based on σ-right-inverse learning gain in order to resolve a major problem appeared in ILC based on inversion model (I-ILC). The convergence condition of the learning algorithm has been derived. It is shown that under some given conditions, the tracking error of In-ILC law converges to zero through a sufficient and necessary stability condition. Then, the convergence properties are established. Through simulation results, we proved that the performances offered by the proposed method in terms of tracking error convergence after few trials are achieved. Therefore, robustness of the In-ILC scheme to perturbations and parametric uncertainties, especially with less knowledge of plant model, remains a challenging topic which will be addressed in future works.



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*Source: 4195938-2018-06-07.xml*