# Convergence Analysis of Iterative Learning Control for Two Classes of 2-D Linear Discrete Fornasini–Marchesini Model

Authors: Kai Wan
Journal: Complexity
 (2020)

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## Abstract

This paper first investigates convergent property of two iterative learning control (ILC) laws for two kinds of two-dimensional linear discrete systems described by the first Fornasini–Marchesini model (2-D LDFFM with a direct transmission from inputs to outputs and 2-D LDFFM with input delay). Different from existing ILC results for 2-D LDFFM, this paper provides convergence analysis in a three-dimensional (3-D) framework. By using row scanning approach (RSA) or column scanning approach (CSA), it is theoretically proved no matter which method is adopted, perfect tracking on the desired reference surface is accomplished. In addition, linear matrix inequality (LMI) technique is utilized to computer the learning gain of the ILC controller. The effectiveness and feasibility of the designed ILC law are illustrated through numerical simulation on a practical thermal process.


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## Body

## 1. Introduction

In practical industrial applications for two-dimensional (2-D) dynamical systems, for example, in form-closure grasps, the immobilized manipulation of 2-D serial chains could be regarded as a repetitive control problem [1]. Also, mold processing and material manufacturing usually require repeated operations of the controlled processing units to obtain a high-precision reference surface [2]. With regard to this kind of repetitive tracking problem, the traditional control methods in infinite coordinate domain, such as H∞ control [3], stochastic stability analysis [4], and sliding mode control [5], are difficult to be used because the precise model information is known in advance. Clearly, for the repetitively tracking cases mentioned above, the traditional tracking control approaches for 2-D dynamical systems in infinite coordinate domain are difficult to be applicable. Clearly, for the repetitively tracking cases mentioned above, the traditional tracking control approaches for 2-D dynamical systems in infinite coordinate domain are difficult to be applicable.To the best of our knowledge, iterative learning control (ILC), as a data-driven and unsupervised control approach, does not require accurate knowledge of the controlled system, which makes ILC be widely prevalent in practical applications. A large number of ILC research results reported in the past few decades have fundamentally designed for one-dimensional (1-D) dynamical systems [6–14], only very few results involved in 2-D dynamical systems [15–22], which concentrate on mainly 2-D linear discrete first Fornasini–Marchesini model (2-D LDFFM). An optimal ILC algorithm was proposed in [16], such that the ILC tracking error converges to zero monotonically. Unfortunately, the system parameters of 2-D LDFFM are known. In [15], five ILC algorithms were proposed, the effectiveness and feasibility of which is verified through only numerical example. Afterwards, by using the column scanning approach (CSA), the iterative process of 2-D LDFFM is transformed into a iterative process of 1D systems; rigorously, theoretical proof on convergence and robustness of a two-gain ILC law is provided in [17]. Also, using the CSA as [18], to track a class of nonrepetitive reference surface described by a high-order internal model operator (HOIM), two HOIM-based ILC laws were, respectively, investigated in [23] for 2-D LDFFM by using 2-D HOIM-based linear inequality theory, but the ultimate ILC tracking error can only converge to a bounded range. To this end, adaptive ILC approach was proposed in [21, 22] to identify all unknown system parameters of 2-D LDFFM, and the ILC result of perfect tracking on iteration-varying reference surface can be obtained. Unfortunately, the gain matrix in 2-D LDFFM is required to be positive definite, such that the proposed adaptive ILC algorithm, in practical applications, is greatly restricted. Notably, the aforementioned ILC results have concerned mainly on 2-D LDFFM without input delay.Time delay, recognized as one of the main sources for poor performance or even instability of control systems, is frequently encountered in the signal transmission process of engineering and biological systems [24]. Over the past few years, with respect to 2-D systems with time delay on horizontal direction and vertical direction, some initial results have been reported on the states estimation [25], H∞ control and filtering [26, 27], and nonfragile robust optimal guaranteed cost control [28]. For instance, the authors in [23] have investigated the robust state estimation problem for 2-D linear discrete systems with state delay, and a robust H∞ filter for 2-D discrete systems with time delays has been designed in [23]. A delay-dependent H∞ controller for 2-D discrete state delay systems was designed in [27]. Nonfragile robust optimal guaranteed cost control subject to both state delay and input delay is discussed in [28]. However, no works have considered the ILC tracking problem for 2-D linear discrete systems with input delay, which motivates our current study.This paper first investigates the convergence property of two ILC laws for 2-D LDFFM with a direct transmission from inputs to outputs and 2-D LDFFM with input delay to track the desired reference surface, respectively. The 2-D LDFFM is very typical, and many practical 2-D engineering systems can be represented as the form of 2-D LDFFM [29, 30]. Under a 3-D framework, a 3-D dynamical process is converted into a 2-D dynamical process by using CSA or row scanning approach (RSA), and it is theoretically proved no matter which method is used; perfect tracking on the desired reference surface is fulfilled. Additionally, this paper theoretically provides a linear matrix inequality (LMI) technique to computer the learning gain matrix of the ILC law. The main contributions of this paper are summarized as follows:(1)
All of the existing ILC results for 2-D dynamical systems have not considered input delay. To the best of our knowledge, it is the first time to investigate ILC algorithms for 2-D LDFFM with a direct transmission from inputs to outputs and for 2-D LDFFM with input delay.(2)
Compared with the adaptive ILC algorithm for 2-D LDFFM in [21, 22], two ILC algorithms proposed in this paper have no restriction on the numbers of system inputs and outputs.(3)
Different from the existing ILC work for 2-D LDFFM [17, 18], a 3-D framework learning mechanism is presented in this paper and can reveal the dynamical behavior of the 2-D LDFFM in the horizontal dynamical direction, vertical dynamical direction, and iteration direction.(4)
It is theoretically proved that RSA or CSA can guarantee perfect tracking on the desired reference surface.(5)
Compared with the literature [20], this paper has addressed the ILC issue on nonidentical boundary states. In addition, this paper successfully avoids strong Assumption 2.2 on desired reference surface and Assumption 2.3 on the system matrix, and LMI technique is utilized to computer the learning gain of the ILC controller.The rest of this paper is arranged as follows: Section2 presents problem formulation and some preliminaries. Sections 3 and 4 provide convergence analysis of a symmetrical P-type ILC law and extend to 2-D LDFFM with input delay, respectively. Two simulation examples are introduced in Section 5. Finally, a conclusion on this article is drawn in Section 6. Notations: A≻0 (A≺0) denotes positive-definite (negative-definite) matrix. im1m2 represents i∈m1,m1+1,m1+2,…,m2 with positive integer m2, and i0∞ represents i∈0,1,2,…. Im denotes identity matrix with dimension m×m. ρ⋅ represents spectral radius of matrix.


## 2. Problem Formulation and Some Preliminaries

Consider the following 2-D LDFFM with a direct transmission from inputs to outputs [23], executing designated tracking tasks repetitively over dynamical region n10N1 and n20N2:(1)xkn1+1,n2+1=A1xkn1+1,n2+A2xkn1,n2+A3n1,n2+1+Bukn1,n2,ykn1,n2=Cxkn1,n2+Dukn1,n2,where ukn1,n2∈Rm, xkn1,n2∈Rp, and ykn1,n2∈Rl represent, respectively, control input, system state, and control output; A1, A2, A3, B, C, and D are real matrices with appropriate dimensions. k0∞ denotes the k-th repetitive number (or iteration number) of 2-D LDFFM (1); n1 and n2 are, respectively, discrete indexes along the horizontal dynamical direction and vertical dynamical direction. The independent indexes n1 and n2 in practical 2-D LDFFM, i.e., chemical reactors, heater exchangers, and pipe furnaces, usually represent space locations and time instants, respectively [31]. As D=0, 2-D LDFFM (1) has been investigated in [17, 18].Forn10N1 and n20N2, let the desired trajectory surface and the corresponding tracking error at kth iteration, respectively, be denoted as yrn1,n2 and ekn1,n2, and the control objective of ILC for 2-D LDFFM is to update the control input ukn1,n2 iteratively, such that the system output ykn1,n2 converges to desired reference surface yrn1,n2 asymptotically, i.e.,(2)limk⟶+∞ykn1,n2=yrn1,n2,n11N1,n21N2.For the convenience of discussing the ILC problem for 2-D LDFFM, the property on a block matrix, assumptions, and Lemma is provided as follows.Property 1.
(see [32]). Give the following block matrix H with a nonsingular matrix H1∈Rm×m:(3)H=H100⋯0−H2H10⋯⋮0−H2H1⋱0⋮⋱⋱⋱00⋯0−H2H1∈RmN×mN,and the inverse matrix of which is given as(4)H−1=H¯100⋯0H¯2H¯10⋯⋮H¯3H¯2H¯1⋱0⋮⋱⋱⋱0H¯N⋯H¯3H¯2H¯1∈RmN×mN,with(5)H¯1=H1−1,H¯2=H1−1H2H1−1,H¯3=H1−1H22H1−1,⋮H¯N=H1−1H2N−1H1−1.
In particular, asH1=Im, H−1 becomes(6)H−1=Im00⋯0H2Im0⋯⋮H22H2Im⋱0⋮⋱⋱⋱0H2N−1⋯H22H2Im.Property 2.
(see [32]). For H−1 in Property 1, according to matrix theory, there is ρH−1=ρH¯1, which is irrelevant to H¯2, H¯3, …, and H¯N.Remark 1.
Properties1 and 2 are often used in the ILC convergence analysis for 2-D linear discrete dynamical systems to track reference surface described by a high-order internal model or varying uniformly between k+1-th iteration and k-th iteration [18]. Certainly, Properties 1 and 2 are also suitable to track repetitive reference surface considered in this paper.Assumption 1.
The matrixD is full-row rank.Assumption 2.
The boundary statesxk0,n2 and xkn1,0 of 2-D LDFFM (1) are assumed to satisfy xk0,n2=x00,n2, n21N2 and xkn1,0=x0n1,0, n10N1, where x00,n2 and x0n1,0 are iteration-invariant functions with respective to n2 and n1.Lemma 1.
Consider the following 3-D linear discrete system forn10N1, n20N2, and k0∞:(7)θk+1n1,n2=α1θkn1,n2+α2δkn1,n2,where θkn1,n2∈RL1 and δkn1,n2∈RL2, respectively, are state and control input; α1 and α2 denote real matrices. Suppose the boundary state θ0n1,n2=fn1,n2, where fn1,n2 is a bounded vector function. When limk⟶+∞δkn1,n2=0, if ρα1<1 is satisfied, then the 3-D system is asymptotically stable along the dynamical direction k, i.e.,(8)limk⟶+∞θkn1,n2=0,n10N1,n20N2.
In particular, whenδkn1,n2=0, for n10N1 and n20N2, limk⟶+∞θkn1,n2=0 can still be obtained for n10N1 and n20N2. The proof process of Lemma 1 is shown in Appendix.


## 3. Convergence Analysis of a Symmetrical P-Type ILC Law

The following symmetrical P-type ILC law is applied to 2-D LDFFM (1) for n10N1 and n20N2:(9)uk+1n1,n2=ukn1,n2+Γekn1,n2,where ekn1,n2=yrn1,n2−ykn1,n2. The learning gain Γ∈Rm×l is to be designed.Remark 2.
It is worth noting that 2-D LDFFM (1) is a zero relative degree system; the proposed ILC law (9) is extended on the existing ILC work for 1D linear discrete dynamical systems with zero relative degree [33]. Additionally, 2-D LDFFM (1) and the ILC law (9) are symmetrical on the discrete indexes n10N1 and n20N2; therefore, row scanning approach (RSA) or column scanning approach (CSA) can guarantee the same ILC tracking result, which is demonstrated in Theorem 1 and Appendix.Theorem 1.
Consider the 2-D LDFFM with a direct transmission from inputs to outputs (1) under Assumptions 1 and 2 and use the symmetrical P-type ILC law (9). If the learning gain Γ satisfies(10)ρIl−DΓ<1,then the tracking error ekn1,n2 converges to zero, i.e.,(11)limk⟶+∞ekn1,n2=0,n10N1,n20N2.Proof 1.
Forn10N1−1 and n20N2−1, let(12)δxkn1,n2=xk+1n1,n2−xkn1,n2,δukn1,n2=uk+1n1,n2−ukn1,n2.
Using (1), for n10N1−1 and n20N2−1, there is(13)δxkn1+1,n2+1=xk+1n1+1,n2+1−xkn1+1,n2+1=A1δxkn1+1,n2+A2δxkn1,n2+A3δxkn1,n2+1+Bδukn1,n2.
Applying the ILC law (9) into (13), it generates(14)δxkn1+1,n2+1=A1δxkn1+1,n2+A2δxkn1,n2+A3δxkn1,n2+1+BΓekn1,n2.
On the contrary, according toekn1,n2=yrn1,n2−ykn1,n2 and (1), we have(15)ek+1n1,n2−ekn1,n2=yrn1,n2−yk+1n1,n2−yrn1,n2+ykn1,n2=−Cxk+1n1,n2−xkn1,n2−Duk+1n1,n2−ukn1,n2,where n10N1 and n20N2. Substituting the ILC law (9) into (15), it yields(16)ek+1n1,n2−ekn1,n2=−Cxk+1n1,n2−xkn1,n2−DΓekn1,n2.
Rearranging (16), there is(17)ek+1n1,n2=−Cδxkn1,n2+Il−DΓekn1,n2.
Let(18)δXkn2=δxkT1,n2δxkT2,n2⋯δxkTN1,n2T,(19)Ekn2=ekT0,n2ekT1,n2⋯ekTN1,n2T,where n20N2. According to (18) and (19), using the CSA on horizontal dynamical direction n1, (14) can be rewritten as(20)Φ1δXkn2+1=Φ2δXkn2+Φ3Ekn2+A^2δxk0,n2+A^3δxk0,n2+1,where(21)Φ1=Ip00⋯0−A3Ip0⋯00−A3Ip⋱⋮⋮⋱⋱⋱00⋯0−A3Ip∈RpN1×pN1,Φ2=A100⋯0A2A10⋯00A2A1⋱⋮⋮⋱⋱⋱00⋯0A2A1∈RpN1×pN1,Φ3=BΓ00⋯00BΓ⋱⋱⋮00BΓ00⋮⋱⋱⋱00⋯00BΓ∈RpN1×lN1+1,A^2=A200⋮0∈RpN1×p,A^3=A300⋮0∈RpN1×p.
Then, sinceΦ1 is a nonsingular matrix, premultiplying Φ1−1 on both sides of (20), we get(22)δXkn2+1=Φ¯1δXkn2+Φ¯2Ekn2+Φ1−1A^2δxk0,n2+Φ1−1A^3δxk0,n2+1,where Φ¯1=Φ1−1Φ2 and Φ¯2=Φ1−1Φ3. Similarly, (17) can also rewritten as(23)Ek+1n2=φ1δXkn2+φ2Ekn2−Cδxk0,n2,where(24)φ1=000⋯0−C00⋯00−C0⋱⋮⋮⋱⋱⋱00⋯0−C0∈RlN1+1×pN1,φ2=Il−DΓ0⋯00Il−DΓ⋱⋮⋮⋱⋱00⋯0Il−DΓ∈RlN1+1×lN1+1,C^=C0⋮0∈RlN1+1×p.
From Assumption2, we obtain δxk0,n2=xk+10,n2−xk0,n2=0 for n20N2. Thus, (22) and (23) can be reformulated as(25)δXkn2+1=Φ¯1δXkn2+Φ¯2Ekn2,Ek+1n2=φ1δXkn2+φ2Ekn2,where n20N2−1. In (25), based on Assumption 2, we deduce(26)δXk0=δxkT1,0δxkT2,0⋯δxkTN1,0T=0,and E0n2 is bounded for n20N2−1. Using Lemma 4 in [18], if ρφ2<1 (equivalently, ρIl−DΓ<1, see Property 2), there is(27)limk⟶+∞δXkn2=0,n21N2,limk⟶+∞Ekn2=0,n20N2−1.
Additionally, takingn2=N2 in (17), we have(28)ek+1n1,N2=−Cδxkn1,N2+Il−DΓekn1,N2,where n10N1. From (27) and δxk0,N2=0, we obtain limk⟶+∞δxkn1,N2=0, n10N1. Using Lemma 1, if ρIl−DΓ<1 is satisfied, there is(29)limk⟶+∞ekn1,N2=0,n10N1.
From (27) and (29) and the definition on Ekn2 in (19), it yields(30)limk⟶+∞ekn1,n2=0,n10N1,n20N2.
Theorem1 is completed.Remark 3.
The convergence conditionρIl−DΓ<1 in Theorem 1 only depends on the matrix D and is irrelevant to A1, A2, A3, C, and B, which is in accordance with the ILC convergence condition of the following 1D linear discrete iterative system to track repetitive reference trajectory [33]:(31)xkt+1=Axkt+Bukt,ykt=Cxkt+Dukt,under a P-type ILC law uk+1t=ukt+Γekt and initial iterative state xkt=x0t.Remark 4.
Taking advantage of the estimated information onD, the learning gain matrix Γ of ρIl−DΓ<1 can be determined by using LMI technique. For the following 1D linear discrete dynamical system described by Il−DΓ,(32)xk+1=Il−DΓxk,k0∞.
Asymptotically stable conditionρIl−DΓ<1 is satisfied if and only if there exists a Q≻0 such that(33)Il−DΓTQIl−DΓ−Q≺0.
Using Schur complement theorem [34], it yields(34)−Q−1Il−DΓIl−ΓTDT−Q≺0.
Pre- and postmultiplying bydiagIl,Q−1 on both sides of (34), we have(35)−Q−1Q−1−DΓQ−1Q−1−Q−1ΓTDT−Q−1≺0.
Therefore,ρIl−DΓ<1 is equivalent to solving the following LMI: there exists positive-definite matrix Q∈Rl×l and matrix X∈Rm×l such that(36)−Q−1Q−1−DXQ−1−XTDT−Q−1≺0,with X=ΓQ−1. Therefore, the selection problem of the learning gain matrix Γ is transformed into the feasibility problem on solving LMI (36).Remark 5.
Theorem1 employs the CSA on dynamical direction n1 to obtain ILC result under Assumption 2. As the RSA on dynamical direction n2 is applied to 2-D LDFFM (1) and the ILC law (9), we can still acquire the same ILC result with Theorem 1 due to the symmetrical characteristics on n1 and n2 in 2-D LDFFM (1) and the ILC law (9), and see Appendix.


## 4. Extension to 2-D LDFFM with Input Delay

In this section, under Assumption2, we investigate the ILC issues for 2-D LDFFM with input delay [23], which is described in the following for n10N1−1 and n20N2−1:(37)xkn1+1,n2+1=A1xkn1+1,n2+A2xkn1,n2+A3xkn1,n2+1+Bukn1−τ1,n2−τ2,ykn1,n2=Cxkn1,n2,where τ1 and τ2 are two positive integers. xkn1,n2, ukn1,n2, and ykn1,n2 have been described in (1).Remark 6.
It is worthwhile noting that ILC tracking issues for 2-D LDFFM withτ1=0 and τ2=0 in (37) have been investigated in [17, 18]. It is the first time that ILC issues for 2-D LDFFM with input delay are investigated in this paper.Assumption 3.
The matrixCB is full-row rank.
Next, we discuss the convergence property of the ILC law for 2-D LDFFM with input delay (37) under Assumptions 2 and 3, and there is the following Theorem 2.Theorem 2.
Consider the 2-D LDFFM with input delay (37) under Assumptions 2 and 3 and use the following ILC law for n1−τ1N1−τ1−1 and n2−τ2N2−τ2−1:(38)uk+1n1,n2=ukn1,n2+Γekn1+τ1+1,n2+τ2+1.
If the learning gainΓ satisfies(39)ρIl−CBΓ<1,then the tracking error ekn1,n2 converges to zero, i.e.,(40)limk⟶+∞ekn1,n2=0,n11N1,n21N2.Proof 2.
Forn10N1 and n20N2, let(41)δx^kn1,n2+1=xk+1n1,n2−xkn1,n2.
From (37), there is(42)δx^kn1+1,n2+1=xk+1n1+1,n2−xkn1+1,n2=A1δx^kn1+1,n2+A2δx^kn1,n2+A3δx^kn1,n2+1+Buk+1n1−τ1,n2−τ2−1−ukn1−τ1,n2−τ2−1,where n10N1−1 and n21N2. Inserting the ILC law (38) into (42), it generates(43)δx^kn1+1,n2+1=A1δx^kn1+1,n2+A2δx^kn1,n2+A3δx^kn1,n2+1+BΓekn1+1,n2.
On the contrary, according toekn1,n2=yrn1,n2−ykn1,n2 and using (37), we have(44)ek+1n1+1,n2−ekn1+1,n2=yrn1+1,n2−yk+1n1+1,n2−yrn1+1,n2+ykn1+1,n2=−Cxk+1n1+1,n2−xkn1+1,n2=−Cδx^kn1+1,n2+1,where n10N1−1 and n20N2. Let(45)δX^kn2=δx^kT1,n2δx^kT2,n2⋯δx^kTN1,n2T,(46)Ekn2=ekT1,n2ekT2,n2⋯ekTN1,n2T,where n21N2. From (45) and (46), (43) can be rewritten as(47)ϕ1δX^kn2+1=ϕ2δX^kn2+ϕ3Ekn2+A^2δx^k0,n2+A^3δx^k0,n2+1,where(48)ϕ1=Ip00⋯0−A3Ip0⋯00−A3Ip⋱⋮⋮⋱⋱⋱00⋯0−A3Ip∈RpN1×pN1,ϕ2=A100⋯00A10⋯000A1⋱⋮⋮⋱⋱⋱00⋯00A1∈RpN1×pN1,ϕ3=BΓ00⋯00BΓ0⋯000BΓ⋱⋮⋮⋱⋱⋱00⋯00BΓ∈RpN1×lN1,and A^2 and A^3 have been described in (20). Similarly, (44) can also be reformulated as(49)Ek+1n2=ψ1δX^kn2+1+ψ2Ekn2,where(50)ψ1=−C00⋯00−C0⋱⋮00−C⋱0⋮⋱⋱⋱00⋯00−C∈RlN1×pN1,ψ2=Il00⋯00Il0⋯000Il⋱⋮⋮⋱⋱⋱00⋯00Il∈RlN1×lN1.
Then, sinceϕ1 is a nonsingular matrix, premultiplying ϕ1−1 on both sides of (47), we get(51)δX^kn2+1=ϕ¯1δX^kn2+ϕ¯2Ekn2+ϕ1−1A^2δx^k0,n2+ϕ1−1A^3δx^k0,n2+1,where ϕ¯1=ϕ1−1ϕ2 and ϕ¯2=ϕ1−1ϕ3. Substituting (51) into (49), it yields(52)Ek+1n2=ψ1ϕ¯1δX^kn2+ψ1ϕ¯2+ψ2Ekn2+ψ1ϕ1−1A^2δx^k0,n2+ψ1ϕ1−1A^3δx^k0,n2+1,where n21N2. From Assumption 2, we obtain δx^k0,n2+1=xk+10,n2−xk0,n2=0 for n20N2. Thus, for n21N2, (51) and (52) can be represented as(53)δX^kn2+1=ϕ¯1δX^kn2+ϕ¯2Ekn2,Ek+1n2=ψ1ϕ¯1δX^kn2+ψ1ϕ¯2+ψ2Ekn2.
In (53), based on Assumption 2, we obtain δx^kn1,1=xk+1n1,0−xkn1,0=0. Furthermore,(54)δX^k1=δx^kT1,1δx^kT2,1⋯δx^kTN1,1T=0,and E0n2 is bounded for n21N2 due to the boundedness property of e0n1,n2. Using Lemma 4 in [18], if ρψ1ϕ¯2+ψ2<1 holds (equivalently, ρIl−CBΓ<1, see Remark 7), there is(55)limk⟶+∞Ekn2=0,n21N2.
From (46), we obtain(56)limk⟶+∞ekn1,n2=0,n11N2,n21N2,
Theorem2 is completed.Remark 7.
The convergence conditionρψ1ϕ¯2+ψ2<1 in (53) depends on a complicated matrix ψ1ϕ¯2+ψ2; thus, there is a need to simplify it. From the definition of ψ1ϕ¯2+ψ2 and using Property 1, the simplifying process is shown as follows:(57)ψ1ϕ¯2+ψ2=−C0⋯00−C⋱⋮⋮⋱⋱00⋯0−CIp00⋯0−A3Ip0⋯00−A3Ip⋱⋮⋮⋱⋱⋱00⋯0−A3Ip−1BΓ0⋯00BΓ⋱⋮⋮⋱⋱00⋯0BΓ+Il0⋯00Il⋱⋮⋮⋱⋱00⋯0Il=Il−CBΓ00⋯0−CA3BΓIl−CBΓ0⋯0−CA32BΓ−CA3BΓIl−CBΓ⋱⋮⋮⋱⋱⋱0−CA3N1−1BΓ⋯−CA32BΓ−CA3BΓIl−CBΓ.
From (57) and Property 2, we can know ρψ1ϕ¯2+ψ2=ρIl−CBΓ. Consequently, ρψ1ϕ¯2+ψ2<1 can be replaced by a more convenient condition ρIl−CBΓ<1 without using any knowledge on the system matrix A3 and the interval length N1 in the 2-D LDFFM with input delay (37).Remark 8.
In Remark7, using the estimated information on C and B, the learning gain matrix Γ of ρIl−CBΓ<1 can be decided by using the LMI technique. Therefore, a similar Remark 4 can be obtained.Remark 9.
When the RSA on dynamical directionn2 is applied to 2-D LDFFM with input delay (37) and the ILC law (38), the same ILC result with Theorem 2 can be obtained, the proof process of which is shown in Appendix 7.3.Remark 10.
It is noted that 2-D LDFFM (1) and (37) contain 2-D Attasi model as a special case, i.e., A2=−A1A3=−A3A1. And the 2-D Roesser model under some specified coefficient matrix requirements can also be converted into 2-D LDFFM [29]. Consequently, depending on the ILC results obtained from 2-D LDFFM (1) and (37), it is easy to extend 2-D Roesser model and 2-D Attasi model with input delay.


## 5. Illustrative Examples

To show the effectiveness and feasibility of the proposed ILC algorithms (9) and (38) for the 2-D LDFFM with a direct transmission from inputs to outputs (1) and 2-D LDFFM with input delay (37), two simulation examples are introduced in this section.Example 1.
Consider the following 2-D LDFFM with a direct transmission from inputs to outputs over dynamical regionn1020 and n2020:(58a)xk1n1+1,n2+1xk2n1+1,n2+1=0.010.010.010.03xk1n1+1,n2xk2n1+1,n2+0.010.020.010.04xk1n1,n2xk2n1,n2+0.010.020.010.3xk1n1,n2+1xk2n1,n2+1+10.20.30.5uk1n1,n2uk2n1,n2,(58b)yk1n1,n2yk2n1,n2=10.2−0.21xk1n1,n2xk1n1,n2+2002uk1n1,n2uk2n1,n2,which is required to repetitively track a desired reference surface yrn1,n2=cos2πn1+n2/10sinπn1+n2/10T for n1020 and n2020 shown in Figure 1. Under Assumption 2, the boundary states of 2-D LDFFM (58) are given as xk0,n2=0.1sinn2T, n2120, and xkn1,0=−0.1sinn10T, n1020. Using the proposed ILC law (9) with the initial control input u0n1,n2=00T for n1020 and n2020, the gain matrix Γ is selected as Γ=diag0.30.3, which satisfies the convergence condition ρIl−DΓ=0.6<1 in Theorem 1. In order to evaluate the accuracy of ILC tracking, the following maximum absolute tracking error index(59)MATEkh=maxn1020maxn2020yrhn1,n2−ykhn1,n2,h12,is used. As a result, the ILC tracking error surfaces ek1n1,n2 at k=3,5,7,25, and ek2n1,n2 at k=4,6,8,24 are depicted in Figures 2 and 3, respectively. Also, Figure 4 presents the profiles of the ILC tracking error indexes MATEk1 and MATEk2 in iteration domain k by utilizing the ILC law (9). Obviously, it is observed from Figures 2–4 that the convergent effect of the ILC law (9) against the iteration-invariant boundary states is illustrated.Figure 1
Reference surfacesyr1n1,n2 and yr2n1,n2 for n1020 and n2020.
(a)(b)Figure 2
Tracking error surfaceek1n1,n2 at k=3,5,7,20 by using the ILC law (9).
(a)(b)(c)(d)Figure 3
Tracking error surfaceek2n1,n2 at k=2,4,6,20 by using the ILC law (9).
(a)(b)(c)(d)Figure 4
Tracking error indexesMATEk1 and MATEk2 with iteration number k by using the ILC law (9).
(a)(b)Example 2.
Some practical thermal processes can be represented by the following partial differential equation (PDE) [31]:(60)∂2skw,t∂w∂t=a1∂skw,t∂t+a2∂skw,t∂w+a0skw,t+bfkw−dw,t−dt,where dw and dt, respectively, denote space delay and time delay; a0, a1, a2, and b are real coefficients. The repetitive index k implies that the ILC issue is investigated in the example. Through the discretization approximation on (60), system parameters and delay factors of 2-D LDFFM with input delay (37) are given as follows:(61)A1=1+a2Δt,A2=a0ΔwΔt−a1Δw−a2Δt−1,A3=1+a1Δw,B=bΔwΔt,τ1=intdwΔw,τ2=intdtΔt,where is shown in Figure 5. And the boundary states satisfy xk0,n2=0.5sin2πn2/10, n2120, and xkn1,0=0.5sin2πn1/10, n1020. Under the ILC law (38) with initial control input u0n1,n2=0 for n1−118 and n2−217, let the learning gain Γ be selected as Γ=0.9 and compute ρIl−CBΓ=0.7885<1, which satisfies the convergent condition in Theorem 2. The maximum absolute tracking error index MATEk is adopted to evaluate the ILC tracking performance, which is given as follows:(62)MATEk=maxn1120maxn2120yrn1,n2−ykn1,n2.
And Figure6 displays the profile of MATEk with iteration number k by using the ILC law (38). The tracking error surface ekn1,n2 at k=5,10,15,40 is shown in Figure 7. Apparently, perfect tracking on repetitive reference surface can be observed from Figures 6 and 7.Figure 5
Reference surfaceyrn1,n2 for n1020 and n2020.Figure 6
Tracking error indexMATEk with iteration number k by using the ILC law (38).Figure 7
Tracking error surfaceekn1,n2 at k=5,10,15,40 by using the ILC law (38).
(a)(b)(c)(d)


## 6. Conclusions

To date, ILC tracking issues for 2-D LDFFM with a direct transmission from inputs to outputs and with input delay are first concerned in this paper. Different from existing ILC results for 2-D LDFFM, a 3-D analysis approach is first applied to the convergent property of the proposed ILC law, which supplies a new tool for follow-up ILC development for 2-D systems. Additionally, the LMI technique is used to compute the learning gain. In our future work, reinforcement learning (RL) [35–38] will be incorporated into the ILC designs for 2-D dynamical systems such that more ILC results are made.



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*Source: 6843730-2020-09-14.xml*