# Robust Iterative Learning Control for 2-D Singular Fornasini–Marchesini Systems with Iteration-Varying Boundary States

Authors: Deming Xu; Kai Wan
Journal: Complexity
 (2021)

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## Abstract

This study first investigates robust iterative learning control (ILC) issue for a class of two-dimensional linear discrete singular Fornasini–Marchesini systems (2-D LDSFM) under iteration-varying boundary states. Initially, using the singular value decomposition theory, an equivalent dynamical decomposition form of 2-D LDSFM is derived. A simple P-type ILC law is proposed such that the ILC tracking error can be driven into a residual range, the bound of which is relevant to the bound parameters of boundary states. Specially, while the boundary states of 2-D LDSFM satisfy iteration-invariant boundary states, accurate tracking on 2-D desired surface trajectory can be accomplished by using 2-D linear inequality theory. In addition, extension to 2-D LDSFM without direct transmission from inputs to outputs is presented. A numerical example is used to illustrate the effectiveness and feasibility of the designed ILC law.


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## Body

## 1. Introduction

Two-dimensional (2-D) singular dynamical systems derived from the discretization of spatiotemporal dynamical systems with singular matrices or singular distributed parameter systems have received much attention due to their extensive applications in physical phenomena and industrial processes, such as electrical circuits [1], nanoelectronics [2], transmission lines in signal propagation [3], and power systems [4]. In recent years, fruitful results on 2-D singular systems in the infinite coordinate domain have been reported, mainly including the detectability and observer design [5], H∞ control [6], and stability analysis [7]. However, in practical industrial applications, 2-D singular systems are often required to execute some specific tracking control tasks repetitively over the finite coordinate domain. For example, in form-closure grasps, the immobilized manipulation of serial chains described by 2-D singular systems could be regarded as a repetitive control problem [8]. Also, in the field of mold processing and material surface manufacturing, it is usually required to obtain high-precision 2-D reference surface by repeated operations of the controlled processing units [9]. Obviously, for the repetitively tracking cases mentioned above, the traditional tracking control approaches for 2-D singular systems in the infinite coordinate domain is difficult to be applicable.Iterative learning control (ILC), as an intelligent control method, is used to address the repetitive trajectory tracking problem for one-dimensional (1-D) dynamical systems with less precise model knowledge, which makes ILC be widely prevalent in practical applications. A large number of ILC research results for 1-D dynamical systems have been reported in the past decades [10–16]. However, only very few ILC results for 2-D dynamical systems is involved in [17–24]. For 2-D linear discrete nonsingular Fornasini–Marchesini systems (2-D LDNFM), a two-gain ILC algorithm is proposed in [20] to address the robust ILC issue under the iteration-varying boundary states, and the detailed proof process of convergence analysis is given. Also, to track a class of nonrepetitive reference surface trajectory represented by a high-order internal model (HOIM) operator, two HOIM-based ILC laws were, respectively, investigated in [21] for 2-D LDNFM by using HOIM-based linear inequality theory, but the ultimate ILC tracking error can only converge to a bounded range. To accomplish the objective of zero tracking error, an adaptive ILC technique is proposed in [24] to deal with the tracking problem of iteration-varying reference surface trajectory. Unfortunately, it requires that the gain matrix of 2-D LDNFM must be positive-definite (or negative-definite), such that the proposed adaptive ILC algorithm, in practical applications, are severely confined. It is worth emphasizing that the aforementioned ILC results have concerned on a nonsingular case. However, in practical life and industry, there exist some 2-D singular dynamical systems, such as electrical circuits, transmission lines in signal propagation, and power systems, which are often required to execute some specific tracking control tasks repetitively. Based on these practical applications, it is essential to exploit ILC techniques for 2-D singular dynamical systems.The main aim of this paper is to investigate the robustness and convergence property of P-type ILC law for two classes of two-dimensional linear discrete singular Fornasini–Marchesini systems (2-D LDSFM) under iteration-varying boundary states. By using singular value decomposition theory, an equivalent dynamical decomposition form of 2-D LDSFM is derived. By using 2-D linear inequality theory, it can guarantee that the ultimate tracking error tends to a bounded range, the bound of which is relevant to the bound parameters of boundary states. The main contributions of this paper relative to the related works are summarized as follows:(1)
In the existing ILC results for 2-D linear discrete systems [17–24], they are concerned on a nonsingular case. To the best of our knowledge, this is the first time to investigate robust ILC algorithms for 2-D discrete singular systems in this paper.(2)
Compared with the adaptive ILC algorithm for 2-D LDNFM in [24], the ILC algorithm proposed in this paper has no restriction on the numbers of system inputs and outputs.(3)
In the study of ILC algorithms for 1-D singular systems [25–27], the λ-norm and identical boundary condition have been widely used to analysis the convergence of the proposed ILC laws. However, the λ-norm might not provide a satisfactory measurement of ILC tracking errors [22], and thus, it is not involved in analysing the proposed ILC law in the paper.The remaining section of this paper is arranged as follows: The problem description is provided in Section2. Sections 3 and 4, respectively, present robustness and convergence analysis of P-type ILC law for 2-D LDSFM (1) and extension to 2-D LDSFM with D=0. In Section 5, a simulation example is introduced. Finally, Section 6 gives a conclusion.


## 2. Problem Description

Consider a ILC issue for the following two-dimensional linear discrete singular Fornasini–Marchesini systems (2-D LDSFM) [28] over finite region n1∈0,1,…,N1−1 and n2∈0,1,…,N2−1:(1)Egxkn1+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∈Rs represent, respectively, control input, system state, and system output and Eg, A1, A2, A3, B, C, and D are real matrices with appropriate dimensions. k∈0,1,2,… denotes the k-th iteration of controlled system (1); n1 and n2 are, respectively, horizontal dynamical index and vertical dynamical index. It is worth noting that as Eg is nonsingular (without loss of generality, let Eg=Ip, where Ip represents identity matrix with (p×p)), (1) is called a regular 2-D LDFFM [28], which has been investigated in [21]. However, in this paper, a singular matrix Eg with rankEg=r<p is considered.Forn1∈0,1,…,N1 and n2∈0,1,…,N2, let an achievable desired surface trajectory and the corresponding tracking error at k-th iteration be denoted as ydn1,n2 and ekn1,n2, respectively. The control objective of ILC for 2-D LDSFM (1) is to generate a control input sequence ukn1,n2, where n1∈0,1,…,N1−1 and n2∈0,1,…,N2−1 with an iterative way, such that the actual tracking output ykn1,n2 can accurately track ydn1,n2, i.e.,(2)limk⟶+∞ykn1,n2=ydn1,n2,where n1∈0,1,…,N1 and n2∈0,1,…,N2.According to the singular value matrix theory [29], there exist two nonsingular matrices P∈Rp×p and Q∈Rp×p such that(3)PEgQ=Ir000,PA1Q=A¯11A¯12A¯21A¯22,PA2Q=A^11A^12A^21A^22,PA3Q=A˜11A˜12A˜21A˜22,PB=B1B2,CQ=C1C2.2-D LDSFM (1) can be transformed into a decomposition form(4)zk1n1+1,n2+1=A¯11zk1n1+1,n2+A˜12zk2n1+1,n2+A^11zk1n1,n2+A^12zk2n1,n2+A˜11zk1n1,n2+1+A˜12zk2n1,n2+1+B1ukn1,n2,(5)0=A¯21zk1n1+1,n2+A¯22zk2n1+1,n2+A^21zk1n1,n2+A^22zk2n1,n2+A˜21zk1n1,n2+1+A˜22zk2n1,n2+1+B2ukn1,n2,(6)ykn1,n2=C1zk1n1,n2+C2zk2n1,n2+Dukn1,n2,where zkn1,n2=zk1n1,n2zk2n1,n2=Q−1xkn1,n2, zk1n1,n2∈Rr, zk2n1,n2∈Rp−r, and Q−1=Q11Q12Q21Q22. For some thermal processes in chemical reactors, heat exchangers and pipe furnaces, zkn1,n2, usually represents temperature at space n1 and time n2 in [6, 30].For the convenience of discussing the ILC problem, Definitions1 and 2 on nonnegative matrix (vector), Assumptions 1–3, and Lemma 1 for 2-D LDSFM (1) are provided.Definition 1.
If every element of a matrix (or vector) is nonnegative, then the matrix (or vector) is said to be nonnegative, i.e., forA=aij∈RL1×L2, if aij≥0, where i∈1,2,…,L1 and j∈1,2,…,L2, then it is denoted that A≥0.Definition 2.
For two matricesP=pij∈RL1×L2 and S=sij∈RL1×L2, P≤S is denoted if pij≤sij for i∈1,2,…,L1 and j∈1,2,…,L2.Assumption 1.
For 2-D LDSFM (1), let boundary states xk0,n2 and xkn1,0 be satisfied as xk0,n2−x00,n2≤Λ1, with n2∈1,2,⋯,N2 and xkn1,0−x0n1,0≤Λ2, with n1∈0,1,⋯,N1, where x00,n2 and x0n1,0 are time-varying functions with respective to n2 and n1, respectively; Λ1 and Λ2 are unknown constants. ⋅ represents the infinite norm of vector/matrix in this paper.Assumption 2.
A˜22 is an invertible matrix.Assumption 3.
2-D LDSFM (1) is regular, if there exists two complex numbers z1 and z2 to make detEgz1z2−A1z1−A2−A3z2≠0.Lemma 1.
Consider the following 3-D linear discrete inequality system forn1∈0,1,…,N1, n2∈0,1,…,N2, and k∈0,1,2,…:(7)θk+1n1,n2≤αθkn1,n2+δkn1,n2,where θkn1,n2∈R and δkn1,n2∈R, respectively, are state and control input and α denotes real constant. Suppose the boundary state θ0n1,n2=fn1,n2, where fn1,n2 is a bounded vector function. When limsupk⟶+∞δkn1,n2≤bδ, if 0<α<1 is satisfied, then(8)lim supk⟶+∞θkn1,n2≤bδ1−α.
The proof process of Lemma1 is similar with [22].Remark 1.
In the existing ILC results for 2-D LDNFM [20, 21, 23], it is usually required that boundary states are satisfied as identical boundary states. However, in practical ILC applications for 2-D systems, it is difficult to obtain in each repetitive operation. To this end, ILC for 2-D LDSFM (1) under iteration-varying boundary states is investigated in this paper. Assumptions 2 and 3 are basic and reasonable assumptions in control theory of the 2-D singular system [28, 31].


## 3. Robustness and Convergence Analysis of P-Type ILC Law for 2-D LDSFM (1)

According to the system characteristic of 2-D LDSFM (1), the following P-type ILC law is proposed for n1∈0,1,…,N1 and n2∈0,1,…,N2:(9)uk+1n1,n2=ukn1,n2+Kekn1,n2,where the learning gain K∈Rm×s is to be designed.Theorem 1.
For 2-D LDSFM (1) under Assumptions 1-3, use the P-type ILC law (9). If there exists the learning gain K satisfying Is−DK<1, then the tracking error ekn1,n2, n1∈0,1,…,N1, n2∈0,1,…,N2 converges to a bounded range, i.e.,(10)lim supk⟶+∞ekn1,n2≤βe,where βe is a certain bound related to the bound parameters Λ1 and Λ2 in Assumption 1.Proof.
Forn1∈0,1,…,N1 and n2∈0,1,…,N2, let(11)δzk+11n1,n2=zk+11n1,n2−zk1n1,n2,(12)δzk+12n1,n2=zk+12n1,n2−zk2n1,n2.
Using (11) and (5), and considering (4) and (12), there is(13)δzk+11n1+1,n2+1=zk+11n1+1,n2+1−zk1n1+1,n2+1=A¯11δzk+11n1+1,n2+A¯12δzk+12n1+1,n2+A^11δzk+11n1,n2+A^12δzk+12n1,n2+A˜11δzk+11n1,n2+1+A˜12δzk+12n1,n2+1+B1uk+1n1,n2−ukn1,n2,0=A¯21δzk+11n1+1,n2+A¯22δzk+12n1+1,n2+A^21δzk+11n1,n2+A^22δzk+12n1,n2+1+A˜21δzk+11n1,n2+1+A^22δzk+12n1,n2+1+B2uk+1n1,n2−ukn1,n2where n1∈0,1,…,N1−1 and n2∈0,1,…,N2−1.
Then, using ILC law (8), we have(14)δzk+11n1+1,n2+1=A¯11δzk+11n1+1,n2+A¯12δzk+12n1+1,n2+A^11δzk+11n1,n2+A^12δzk+12n1,n2+A˜11δzk+11n1,n2+1+A˜12δzk+12n1,n2+1+B1Kekn1,n2,(15)0=A¯21δzk+11n1+1,n2+A¯22δzk+12n1+1,n2+A^21δzk+11n1,n2+A^22δzk+12n1,n2+A˜21δzk+11n1,n2+1+A˜22δzk+12n1,n2+1+B2Kekn1,n2.
According to Assumption2, premultiplying by A˜22−1 on both sides of (15), we obtain(16)δzk+12n1,n2+1=−A˜22−1A¯21δzk+11n1+1,n2−A˜22−1A¯22δzk+12n1+1,n2−A˜22−1A^21δzk+11n1,n2−A˜22−1A^22δzk+12n1,n2−A˜22−1A˜21δzk+11n1,n2+1−A˜22−1B2Kekn1,n2.
Substituting (16) into (12), it yields(17)δzk+11n1+1,n2+1=A¯11−A˜12A˜22−1A¯21δzk+11n1+1,n2+A¯12−A˜12A˜22−1A¯22δzk+12n1+1,n2+A^11−A˜12A˜22−1A^21δzk+11n1,n2+A^12−A˜12A˜22−1A^21δzk+12n1,n2+A˜11−A˜12A˜22−1A˜21δzk+11n1,n2+1+B1K−A˜12A˜22−1B2Kekn1,n2.
On the other hand, according toekn1,n2=ydn1,n2−ykn1,n2 and (6), it generates(18)ek+1n1,n2−ekn1,n2=ydn1,n2−yk+1n1,n2−ydn1,n2+ykn1,n2=−C1zk+11n1,n2−zk1n1,n2−C2zk+12n1,n2−zk2n1,n2−Duk+1n1,n2−ukn1,n2,where n1∈0,1,…,N1 and n2∈0,1,…,N2. From (11)-(12) and the ILC law (9), it yields(19)ek+1n1,n2=−C1δzk+11n1,n2−C2δzk+12n1,n2+Is−DKekn1,n2.
Let(20)δZk1n2=δzk1,T1,n2δzk1,T2,n2…δzk1,TN1,n2T,(21)δZk2n2=δzk2,T1,n2δzk2,T2,n2…δzk2,TN1,n2T,(22)Ekn2=ekT0,n2ekT1,n2…ekTN1,n2T.
From (20)–(22), (17), (16), and (19) can be rewritten as(23)Φ1δZk+11n2+1=Φ2δZk+11n2+Φ3δZk+12n2+Φ4Ekn2+Φ5δzk+110,n2+1+Φ6δzk+110,n2+Φ7δzk+120,n2,(24)Ψ0δZk+12n2+1=Ψ1δZk+11n2+1+Ψ2δZk+11n2+Ψ3δZk+12n2+Ψ4Ekn2+Ψ5δzk+110,n2+1+Ψ6δzk+110,n2+Ψ7δzk+120,n2+Ψ8δzk+120,n2+1,(25)Ek+1n2=Θ1δZk+11n2+Θ2δZk+12n2+Θ3Ekn2+Θ4δzk+110,n2+Θ5δzk+120,n2,where Φi, i∈1,2,…,7, Ψj, j∈0,1,2,…,8, and Θh, h∈1,2,…,5, are given in the next page. Since Φ1 is a nonsingular block matrix, and premultiplying Φ1−1 on both sides of (23), we get(26)δZk+11n2+1=Φ1−1Φ2δZk+11n2+Φ1−1Φ3δZk+12n2+Φ1−1Φ4Ekn2+Φ1−1Φ5δzk+110,n2+1+Φ1−1Φ6δzk+110,n2+Φ1−1Φ7δzk+120,n2.
Substituting (26) into (24), we have(27)Ψ0δZk+12n2+1=Ψ1Φ1−1Φ2+Ψ2δZk+11n2+Ψ1Φ1−1Φ3+Ψ3δZk+12n2+Ψ1Φ1−1Φ4+Ψ4Ekn2+Ψ1Φ1−1Φ5+Ψ5δzk+110,n2+1+Ψ1Φ1−1Φ6+Ψ6δzk+110,n2+Ψ1Φ1−1Φ7+Ψ7δzk+120,n2+Ψ8δzk+120,n2+1.
Taking norms on both sides of (26), (27), and (25), respectively, and considering Ψ0=1, there is(28)δZk+11n2+1≤Φ1−1Φ2δZk+11n2+Φ1−1Φ3δZk+12n2+Φ1−1Φ4Ekn2+Φ1−1Φ5δzk+110,n2+1+Φ1−1Φ6δzk+110,n2+Φ1−1Φ7δzk+120,n2,(29)δZk+12n2+1≤Ψ1Φ1−1Φ2+Ψ2δZk+11n2+Ψ1Φ1−1Φ3+Ψ3δZk+12n2+Ψ1Φ1−1Φ4+Ψ4Ekn2+Ψ1Φ1−1Φ5+Ψ5δzk+110,n2+1+Ψ1Φ1−1Φ6+Ψ6δzk+110,n2+Ψ1Φ1−1Φ7+Ψ7δzk+120,n2+Ψ8δzk+120,n2+1,(30)Ek+1n2≤Θ1δZk+11n2+Θ2δZk+12n2+Θ3Ekn2+Θ4δzk+110,n2+Θ5δzk+120,n2,where(31)Φ1=Ir00…0−A˜11−A˜12A˜22−1A˜21Ir0…00−A˜11−A˜12A˜22−1A˜21Ir⋱⋮⋮⋱⋱⋱00…0−A˜11−A˜12A˜22−1A˜21Ir,Φ2=A¯11−A˜12A˜22−1A¯2100…0A^12−A˜12A˜22−1A^21A¯11−A˜12A˜22−1A¯210⋱00A^12−A˜12A˜22−1A^21A¯11−A˜12A˜22−1A¯21⋱⋮⋮⋱⋱⋱00…0A^12−A˜12A˜22−1A^21A¯11−A˜12A˜22−1A¯21,Φ3=A¯12−A˜12A˜22−1A¯2200…0A^12−A˜12A˜22−1A^22A¯12−A˜12A˜22−1A¯220⋱00A^12−A˜12A˜22−1A^22A¯12−A˜12A˜22−1A¯22⋱⋮⋮⋱⋱⋱00…0A^12−A˜12A˜22−1A^22A¯12−A˜12A˜22−1A¯22,Φ4=B1K−A˜12A˜22B2K0…00⋱⋱⋮⋮⋱B1K−A˜12A˜22B2K00…00,Φ5=A˜11−A˜12A˜22−1A˜210⋮0,Φ6=A^11−A˜12A˜22−1A^210⋮0,Φ7=A^12−A˜12A˜22−1A^220⋮0,Ψ0=000…0Ip−r00…00Ip−r⋱⋱⋮⋮⋱⋱000…0Ip−r0,Ψ1=000…0−A˜22−1A˜2100…00−A˜22−1A˜210⋱⋮⋮⋱⋱⋱00…0−A˜22−1A˜210,Ψ2=−A˜22−1A¯2100…0−A˜22−1A^21−A˜22−1A¯210…00−A˜22−1A^21−A˜22−1A¯21⋱⋮⋮⋱⋱⋱00…0−A˜22−1A^21−A˜22−1A¯21,Ψ3=−A˜22−1A¯2200…0−A˜22−1A^22−A˜22−1A¯220…00−A˜22−1A^22−A˜22−1A¯22⋱⋮⋮⋱⋱⋱00…0−A˜22−1A^22−A˜22−1A¯22,Ψ4=−A˜22−1B2K0…00⋱⋱⋮⋮⋱−A˜22−1B2K00…00,Ψ5=−A˜22−1A˜210⋮0,Ψ6=−A˜22−1A^210⋮0,Ψ7=−A˜22−1A^220⋮0Ψ8=Ip−r0⋮0,Θ1=000…0−C100…00−C10⋱⋮⋮⋱⋱⋱00…0−C10,Θ2=000…0−C200⋱⋮0−C20⋱0⋮⋱⋱⋱00…0−C20,Θ3=Is−DK0…00Is−DK⋱⋮⋮⋱⋱00…0Is−DK,Θ4=−C10⋮0,Θ5=−C20⋮0.
LettingδZkn2=δZk1n2δZk2n2T and considering Definitions 1-2, (28)–(30) are reformulated as(32)δZk+1n2+1≤Φ¯1δZk+1n2+Φ¯2Ekn2+Φ¯3δzk+110,n2+1+Φ¯4δzk+110,n2+Φ¯5δzk+120,n2+Φ¯6δzk+120,n2+1,(33)Ek+1n2≤Φ¯7δZk+1n2+Φ¯8Ekn2+Φ¯9δzk+110,n2+Φ¯10δzk+120,n2,where(34)Φ¯1=Φ1−1Φ2Φ1−1Φ3Ψ1Φ1−1Φ2+Ψ2Ψ1Φ1−1Φ3+Ψ3,Φ¯2=Φ1−1Φ4Ψ1Φ1−1Φ4+Ψ4,Φ¯3=Φ1−1Φ5Ψ1Φ1−1Φ5+Ψ5,Φ¯4=Φ1−1Φ6Ψ1Φ1−1Φ6+Ψ6,Φ¯5=Φ1−1Φ7Ψ1Φ1−1Φ7+Ψ7,Φ¯6=0Ψ8,Φ¯7=Θ1Θ2,Φ¯8=Θ3,Φ¯9=Θ4,Φ¯10=Θ5.
From Assumption1, we derive(35)δzk+110,n2=Q11xk+110,n2−xk10,n2+Q12xk+120,n2−xk20,n2≤Λ¯1,δzk+120,n2=Q21xk+110,n2−xk10,n2+Q22xk+120,n2−xk20,n2≤Λ¯2,δzk+11n1,0=Q11xk+11n1,0−xk1n1,0+Q12xk+120,n2−xk20,n2≤Λ¯3,δzk+12n1,0=Q21xk+11n1,0−xk1n1,0+Q22xk+12n1,0−xk2n1,0≤Λ¯4,δZk+110=δzk+111,0δzk+112,0⋮δzk+11N1,0≤Λ¯5,(36)δZk+120=δzk+121,0δzk+122,0⋮δzk+12N1,0≤Λ¯6,where Λ¯1, Λ¯2, Λ¯3, Λ¯4, Λ¯5, and Λ¯6 are dependent on bound parameters Λ1 and Λ2 in Assumption 1. From (35) and (36), we know that δZk+10 is bounded. In addition, E0n2 is bounded for n2∈0,1,…,N2−1 due to the boundedness property of ydn1,n2 and y0n1,n2. Using Lemma 2 in [22], if Φ¯8<1 (equivalently, Is−DK<1), there is(37)lim supk⟶+∞δZk+1n2≤βδZ,n2∈0,1,…,N2,(38)lim supk⟶+∞Ekn2≤βE,n2∈0,1,…,N2−1.
In addition, takingn2=N2 in (19), we have(39)ek+1n1,N2=−C1δzk+11n1,N2−C2δzk+12n1,N2+Is−DKekn1,N2,where n1∈0,1,…,N1. Taking norm on both sides of (39), we have(40)ek+1n1,N2≤C1δzk+11n1,N2+C2δzk+12n1,N2+Is−DKekn1,N2.
From (37), we know δzk+11n1,N2 and δzk+12n1,N2 are bounded. Using Lemma 1, if Is−DK<1 is satisfied, there is(41)lim supk⟶+∞ekn1,N2≤be,n1∈0,1,…,N1.
From (38), (41), and the definition on Ekn2 in (22), it yields(42)lim supk⟶+∞ekn1,n2≤βe,where n1∈0,1,…,N1 and n2∈0,1,…,N2. Theorem 1 is completed.
As iteration-invariant boundary states are imposed on 2-D LDSFM (1), there is Corollary 1.Corollary 1.
For 2-D LDSFM (1) under Assumption 1 with Λ1=0 and Λ2=0 and Assumptions 2-3, use P-type ILC law (9). If there exists the learning gain K satisfying Is−DK<1, then, the tracking error ekn1,n2 converges to zero, i.e.,(43)limk⟶+∞ekn1,n2=0,where n1∈0,1,…,N1 and n2∈0,1,…,N2.


## 4. Extension to 2-D LDSFM withD=0

In this section, under Assumptions1–3, we investigate the ILC issue for 2-D LDSFM with D=0 [28], which is described in the following:(44)Egxkn1+1,n2+1=A1xkn1+1,n2+A2xkn1,n2+A3n1,n2+1+Bukn1,n2,ykn1,n2=Cxkn1,n2,where Eg, A1, A2, A3, B, and C have been described in (1).Based on Assumptions1–3, we will discuss the robustness and convergence property of ILC law (45) for 2-D LDSFM (44). Theorem 2 is given as follows.Theorem 2.
For 2-D LDSFM (44) under Assumptions 1–3, the following P-type ILC law for n1∈0,1,…,N1−1 and n2∈0,1,…,N2−1,(45)uk+1n1,n2=ukn1,n2+Kekn1+1,n2+1,is applied. If there exists the learning gain K satisfying(46)Φ¯8<1,where Φ¯8=IsN1+Θ^1Φ1−1Φ4+Θ^2Ψ1Φ1−1Φ4+Ψ4, where Θ^1, Θ^2, Φ1, Φ4, Ψ1, and Ψ4 are given in (52)–(55); the tracking error ekn1,n2, with n1∈1,2,…,N1 and n2∈1,2,…,N2 converges to a bounded range, i.e.,(47)lim supk⟶+∞ekn1,n2≤βe′,where βe′ is a certain bound relevant with the bound parameters Λ1 and Λ2 in Assumption 1.Proof.
Forn1∈0,1,…,N1 and n2∈0,1,…,N2, let δzk+11n1,n2 and δzk+12n1,n2 be defined as (11) and (12). Using singular value decomposition theory on (43) and following the same procedure as (17) and (16), there is(48)δzk+11n1+1,n2+1=A¯11−A˜12A˜22−1A¯21δzk+11n1+1,n2+A¯12−A˜12A˜22−1A¯22δzk+12n1+1,n2+A^11−A˜12A˜22−1A^21δzk+11n1,n2+A^12−A˜12A˜22−1A^21δzk+12n1,n2+A˜11−A˜12A˜22−1A˜21δzk+11n1,n2+1+B1K−A˜12A˜22−1B2Kekn1+1,n2+1,δzk+12n1,n2+1=−A˜22−1A¯21δzk+11n1+1,n2−A˜22−1A¯22δzk+12n1+1,n2−A˜22−1A^21δzk+11n1,n2−A˜22−1A^22δzk+12n1,n2−A˜22−1A˜21δzk+11n1,n2+1−A˜22−1B2Kekn1+1,n2+1.
On the other hand, according toekn1,n2=ydn1,n2−ykn1,n2 and (44), it generates(49)ek+1n1+1,n2+1−ekn1+1,n2+1=ydn1+1,n2+1−yk+1n1+1,n2+1−ydn1+1,n2+1+ykn1+1,n2+1=−C1zk+11n1+1,n2+1−zk1n1+1,n2+1−C2zk+12n1+1,n2+1−zk2n1+1,n2+1,where n1∈0,1,…,N1−1 and n2∈0,1,…,N2−1.
From (11)-(12), it becomes(50)ek+1n1+1,n2+1−ekn1+1,n2+1=−C1δzk+11n1+1,n2+1−C2δzk+12n1+1,n2+1.
Let(51)E^kn2=ekT1,n2ekT2,n2…ekTN1,n2T.
From (51), (50) can be reformulated as(52)E^k+1n2+1=Θ^1δZk+11n2+1+Θ^2δZk+12n2+1+E^kn2+1,where δZk+11n2 and δZk+12n2 are described in (20) and (21). Θ^1 and Θ^2 are given as(53)Θ^1=−C10…00−C1⋱⋮⋮⋱⋱00…0−C1,Θ^2=−C20…00−C2⋱⋮⋮⋱⋱00…0−C2.
Following the similar deduction process as (26) and (27), it yields(54)δZk+11n2+1=Φ1−1Φ2δZk+11n2+Φ1−1Φ3δZk+12n2+Φ1−1Φ4E^kn2+1+Φ1−1Φ5δzk+110,n2+1+Φ1−1Φ6δzk+110,n2+Φ1−1Φ7δzk+120,n2,(55)Ψ0δZk+12n2+1=Ψ1Φ1−1Φ2+Ψ2δZk+11n2+Ψ1Φ1−1Φ3+Ψ3δZk+12n2+Ψ1Φ1−1Φ4+Ψ4E^kn2+1+Ψ1Φ1−1Φ5+Ψ5δzk+110,n2+1+Ψ1Φ1−1Φ6+Ψ6δzk+110,n2+Ψ1Φ1−1Φ7+Ψ7δzk+120,n2+Ψ8δzk+120,n2+1,where Φi, i∈1,2,3,5,6,7, and Ψj, j∈0,1,2,3,5,6,7,8, have been described in (23) and (24); Φ4 and Ψ4 are given as(56)Φ4=B1K−A˜12A˜22−1B2K0…00B1K−A˜12A˜22−1B2K⋱⋮⋮⋱⋱00…0B1K−A˜12A˜22−1B2K,Ψ4=−A˜22−1B2K0…00−A˜22−1B2K⋱⋮⋮⋱⋱00⋯0−A˜22−1B2K.
Substituting (54) into (52), we have(57)E^k+1n2+1=Θ^1Φ1−1Φ2δZk+11n2+Θ^1Φ1−1Φ3δZk+12n2+IsN1+Θ^1Φ1−1Φ4E^kn2+1+Θ^2δZk+12n2+1+Θ^1Φ1−1Φ5δzk+110,n2+1+Θ^1Φ1−1Φ6δzk+110,n2+Θ^1Φ1−1Φ7δzk+120,n2.
Taking norms on both sides of (54)-(55), respectively, and considering Ψ0=1, there is(58)δZk+11n2+1≤Φ1−1Φ2δZk+11n2+Φ1−1Φ3δZk+12n2+Φ1−1Φ4E^kn2+1+Φ1−1Φ5δzk+110,n2+1+Φ1−1Φ6δzk+110,n2+Φ1−1Φ7δzk+120,n2,(59)δZk+12n2+1≤Ψ1Φ1−1Φ2+Ψ2δZk+11n2+Ψ1Φ1−1Φ3+Ψ3δZk+12n2+Ψ1Φ1−1Φ4+Ψ4E^kn2+1+Ψ1Φ1−1Φ5+Ψ5δzk+110,n2+1+Ψ1Φ1−1Φ6+Ψ6δzk+110,n2+Ψ1Φ1−1Φ7+Ψ7δzk+120,n2+Ψ8δzk+120,n2+1.
Taking norms on both sides of (57), and considering (59), it yields(60)E^k+1n2+1≤Θ^1Φ1−1Φ2+Θ^2Ψ1Φ1−1Φ2+Ψ2δZk+11n2+Θ^1Φ1−1Φ3+Θ^2Ψ1Φ1−1Φ2+Ψ3δZk+12n2+IsN1+Θ^1Φ1−1Φ4+Θ^2Ψ1Φ1−1Φ4+Ψ4E^kn2+1+Θ^1Φ1−1Φ5+Θ^2Ψ1Φ1−1Φ5+Ψ5δzk+110,n2+1+Θ^1Φ1−1Φ6+Θ^2Ψ1Φ1−1Φ6+Ψ6δzk+110,n2+Θ^1Φ1−1Φ7+Θ^2Ψ1Φ1−1Φ7+Ψ7δzk+120,n2+Θ^2Ψ8δzk+120,n2+1.
DefineδZkn2=δZk1n2δZk2n2T, and (58)–(60) can be rewritten as(61)δZk+1n2+1≤Φ¯1δZk+1n2+Φ¯2Ekn2+1+Φ¯3δzk+110,n2+1+Φ¯4δzk+110,n2+Φ¯5δzk+120,n2+Φ¯6δzk+120,n2+1,E^k+1n2+1≤Φ¯7δZk+1n2+Φ¯8E^kn2+1+Φ¯9δzk+110,n2+1+Φ¯10δzk+110,n2+Φ¯11δzk+120,n2+Φ¯12δzk+120,n2+1,where Φ¯i, i∈1,2,…,6, are given in (33)-(34); Φ¯j, j∈7,8,…,12, are shown as(62)Φ¯7=Θ^1Φ1−1Φ2+Θ^2Ψ1Φ1−1Φ2+Ψ2Θ^1Φ1−1Φ3+Θ^2Ψ1Φ1−1Φ3+Ψ3,Φ¯8=IsN1+Θ^1Φ1−1Φ4+Θ^2Ψ1Φ1−1Φ4+Ψ4,Φ¯9=Θ^1Φ1−1Φ5+Θ^2Ψ1Φ1−1Φ5+Ψ5,Φ¯10=Θ^1Φ1−1Φ6+Θ^2Ψ1Φ1−1Φ6+Ψ6,Φ¯11=Θ^1Φ1−1Φ7+Θ^2Ψ1Φ1−1Φ7+Ψ7,Φ¯12=Θ^2Ψ8.
From Assumption1, we obtain δzk+110,n2, δzk+120,n2, and δZk+10 are bounded. Using Lemma 2 in [22], if Φ¯8<1, there is(63)lim supk⟶+∞E^kn2+1≤βE^,n2∈0,1,…,N2−1.
From (51), we obtain(64)lim supk⟶+∞ekn1,n2≤βe′,where n1∈1,2,…,N1 and n2∈1,2,…,N2. Theorem 2 is completed.Corollary 2.
For 2-D LDSFM (44) under Assumption 1 with Λ1=0 and Λ2=0 and Assumptions 2-3, use the P-type ILC law (45). If there exists the learning gain K satisfying (46), then the tracking error ekn1,n2 converges to zero, i.e.,(65)limk⟶+∞ekn1,n2=0,where n1∈1,2,…,N1 and n2∈1,2,…,N2.Remark 2.
It is noted that 2-D LDSFM contains a 2-D linear discrete singular Attasi model as a special case. Also, a 2-D singular Roesser model under some specified coefficient matrix requirements can be converted into 2-D LDSFM [32]. Consequently, depending on the ILC results obtained from 2-D LDSFM, it is easy to extend the 2-D singular Roesser model and singular Attasi model.


## 5. Illustrative Examples

In this section, to demonstrate the feasibility and effectiveness of the designed ILC algorithm for 2-D LDSFM, computer simulation is conducted for mathematical systems.Example 1.
Consider the ILC issue for 2-D LDSFM (44) with the following parameters:(66)Eg=1111,A1=0.10.010.010.03,A2=0.10.020.010.04,A3=0.010.020.10.3,B=11,C=−1−0.5.
TakingN1=N2=20, the desired surface trajectory ydn1,n2 is described by ydn1,n2=sin0.2πn1+n2 for n1∈0,1,…,20 and n2∈0,1,…,20, which is shown in Figure 1. We select two nonsingular transformation matrices P=0.50.50.2−0.2 and Q=0.50.10.5−0.1. In ILC law (45), let the initial control input be given as u0n1,n2=0, where n1∈0,1,…,19 and n2∈0,1,…,19, and the learning gain K be selected as K=−0.4, which satisfies convergence condition (46) in Theorem 2 and Corollary 2. The maximum absolute tracking error index MATEk is adopted to evaluate the ILC tracking performance, which is given as follows:(67)MATEk=maxn1∈1,2,…,20maxn2∈1,2,…,20ydn1,n2−ykn1,n2.Figure 1
The desired surface trajectoryydn1,n2 for n1∈0,1,…,20 and n2∈0,1,…,20.Case 1.
Corresponding to Assumption1 with iteration-varying boundary states, let the boundary states xk0,n2, n2∈1,2,…,20, and xkn1,0, n1∈0,1,…,20, be given as(68)xk0,n2=0sin0.2πn2+m1,km2,k,xkn1,0=−0.5sin0.2πn10+n1,kn2,k,where m1,k, m2,k, n1,k, and n2,k vary randomly at −0.5,0.5, −0.1,0.1, −0.5,0.5, and −0.1,0.1 along the iteration k. Figure 2 displays the profile of MATEk with iteration number k by using ILC law (45). Obviously, robust boundedness of ultimate ILC tracking error is validated from Figure 2.Figure 2
Under the ILC law (38), the profile of tracking error index MATEk with iteration number k.Case 2.
Corresponding to Assumption1 with Λ1=0 and Λ2=0, let the boundary states satisfy(69)xk0,n2=0sin0.2πn2T,n2∈1,2,…,20,xkn1,0=−0.5sin0.2πn10T,n1∈0,1,…,20.
Figure3 displays the profile of MATEk with iteration number k by using ILC law (45). The tracking error surface ekn1,n2 at k=3,5,7,20 is shown in Figure 4. Apparently, a perfect tracking to repetitive reference trajectory ydn1,n2 except the boundaries n1=0 and n2=0 can be observed from Figures 3-4, and the effectiveness of the presented ILC algorithm is illustrated.Figure 3
Under ILC law (45), the tracking error surface ekn1,n2 at k=3,5,7,20.Figure 4
Under ILC law (45), the profile of tracking error index MATEk with iteration number k.


## 6. Conclusions

At present, the ILC tracking issue for 2-D LDSFM under iteration-varying boundary states has not been studied. This paper first introduces singular value decomposition theory into ILC investigations on 2-D LDSFM. In addition, 3-D linear inequality stability theory is first proposed to analysis the convergence property on ILC algorithms, which is a novel analysis method. In the future work, more robust ILC uncertainties for 2-D nonlinear singular systems will be considered.



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