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

Xu, Deming; Wan, Kai

doi:https://doi.org/10.1155/2021/6686724

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Dynamic Analysis, Learning, and Robust Control of Complex Systems

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Research Article | Open Access

Volume 2021 | Article ID 6686724 | https://doi.org/10.1155/2021/6686724

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

Deming Xu¹and Kai Wan¹

Academic Editor: Heng Liu

Received03 Nov 2020

Revised07 Feb 2021

Accepted22 Feb 2021

Published01 Mar 2021

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.

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], 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 -norm and identical boundary condition have been widely used to analysis the convergence of the proposed ILC laws. However, the -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 Section 2. 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 . 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 and :where , , and represent, respectively, control input, system state, and system output and , , , , , , and are real matrices with appropriate dimensions. denotes the -th iteration of controlled system (1); and are, respectively, horizontal dynamical index and vertical dynamical index. It is worth noting that as is nonsingular (without loss of generality, let , where represents identity matrix with ()), (1) is called a regular 2-D LDFFM [28], which has been investigated in [21]. However, in this paper, a singular matrix with is considered.

For and , let an achievable desired surface trajectory and the corresponding tracking error at -th iteration be denoted as and , respectively. The control objective of ILC for 2-D LDSFM (1) is to generate a control input sequence , where and with an iterative way, such that the actual tracking output can accurately track , i.e.,where and .

According to the singular value matrix theory [29], there exist two nonsingular matrices and such that

2-D LDSFM (1) can be transformed into a decomposition formwhere , , , and . For some thermal processes in chemical reactors, heat exchangers and pipe furnaces, , usually represents temperature at space and time in [6, 30].

For the convenience of discussing the ILC problem, Definitions 1 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., for , if , where and , then it is denoted that .

Definition 2. For two matrices and , is denoted if for and .

Assumption 1. For 2-D LDSFM (1), let boundary states and be satisfied as , with and , with , where and are time-varying functions with respective to and , respectively; and are unknown constants. represents the infinite norm of vector/matrix in this paper.

Assumption 2. is an invertible matrix.

Assumption 3. 2-D LDSFM (1) is regular, if there exists two complex numbers and to make .

Lemma 1. Consider the following 3-D linear discrete inequality system for , , and :where and , respectively, are state and control input and denotes real constant. Suppose the boundary state , where is a bounded vector function. When , if is satisfied, thenThe proof process of Lemma 1 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 and :where the learning gain 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 satisfying , then the tracking error , , converges to a bounded range, i.e.,where is a certain bound related to the bound parameters and in Assumption 1.

Proof. For and , letUsing (11) and (5), and considering (4) and (12), there iswhere and .
Then, using ILC law (8), we haveAccording to Assumption 2, premultiplying by on both sides of (15), we obtainSubstituting (16) into (12), it yieldsOn the other hand, according to and (6), it generateswhere and . From (11)-(12) and the ILC law (9), it yieldsLetFrom (20)–(22), (17), (16), and (19) can be rewritten aswhere , , , , and , , are given in the next page. Since is a nonsingular block matrix, and premultiplying on both sides of (23), we getSubstituting (26) into (24), we haveTaking norms on both sides of (26), (27), and (25), respectively, and considering , there iswhereLetting and considering Definitions 1-2, (28)–(30) are reformulated aswhereFrom Assumption 1, we derivewhere , , , , , and are dependent on bound parameters and in Assumption 1. From (35) and (36), we know that is bounded. In addition, is bounded for due to the boundedness property of and . Using Lemma 2 in [22], if (equivalently, ), there isIn addition, taking in (19), we havewhere . Taking norm on both sides of (39), we haveFrom (37), we know and are bounded. Using Lemma 1, if is satisfied, there isFrom (38), (41), and the definition on in (22), it yieldswhere and . 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 and and Assumptions 2-3, use P-type ILC law (9). If there exists the learning gain satisfying , then, the tracking error converges to zero, i.e.,where and .

4. Extension to 2-D LDSFM with

In this section, under Assumptions 1–3, we investigate the ILC issue for 2-D LDSFM with [28], which is described in the following:where , , , , , and have been described in (1).

Based on Assumptions 1–3, 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 and ,is applied. If there exists the learning gain satisfyingwhere , where , , , , , and are given in (52)–(55); the tracking error , with and converges to a bounded range, i.e.,where is a certain bound relevant with the bound parameters and in Assumption 1.

Proof. For and , let and be defined as (11) and (12). Using singular value decomposition theory on (43) and following the same procedure as (17) and (16), there isOn the other hand, according to and (44), it generateswhere and .
From (11)-(12), it becomesLetFrom (51), (50) can be reformulated aswhere and are described in (20) and (21). and are given asFollowing the similar deduction process as (26) and (27), it yieldswhere , , and , , have been described in (23) and (24); and are given asSubstituting (54) into (52), we haveTaking norms on both sides of (54)-(55), respectively, and considering , there isTaking norms on both sides of (57), and considering (59), it yieldsDefine , and (58)–(60) can be rewritten aswhere , , are given in (33)-(34); , , are shown asFrom Assumption 1, we obtain , , and are bounded. Using Lemma 2 in [22], if , there isFrom (51), we obtainwhere and . Theorem 2 is completed.

Corollary 2. For 2-D LDSFM (44) under Assumption 1 with and and Assumptions 2-3, use the P-type ILC law (45). If there exists the learning gain satisfying (46), then the tracking error converges to zero, i.e.,where and .

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:Taking , the desired surface trajectory is described by for and , which is shown in Figure 1. We select two nonsingular transformation matrices and . In ILC law (45), let the initial control input be given as , where and , and the learning gain be selected as , which satisfies convergence condition (46) in Theorem 2 and Corollary 2. The maximum absolute tracking error index is adopted to evaluate the ILC tracking performance, which is given as follows:

Case 1. Corresponding to Assumption 1 with iteration-varying boundary states, let the boundary states , , and , , be given aswhere , , , and vary randomly at , , , and along the iteration . Figure 2 displays the profile of with iteration number by using ILC law (45). Obviously, robust boundedness of ultimate ILC tracking error is validated from Figure 2.

Case 2. Corresponding to Assumption 1 with and , let the boundary states satisfyFigure 3 displays the profile of with iteration number by using ILC law (45). The tracking error surface at is shown in Figure 4. Apparently, a perfect tracking to repetitive reference trajectory except the boundaries and can be observed from Figures 3-4, and the effectiveness of the presented ILC algorithm is illustrated.

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.

Data Availability

All the data used to support the study have been included within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61573385, the Professorial and Doctoral Scientific Research Foundation of Huizhou University under Grants 2020JB017 and 2020JB011, Natural Science Foundation of Guangdong Province under Grant 2018A030313711, Huizhou Science and Technology Plan Project under Grant 2020SD0406034, the Characteristic Innovation Projects of Universities in Guangdong Province under Grant 2018KTSCX215, and Program for Innovative Research Team of Huizhou University, IRTHZU.

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Copyright

Copyright © 2021 Deming Xu and Kai Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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