Iterative learning control for a class of discrete-time singular systems
- Panpan Gu^{1},
- Senping Tian^{1}Email author and
- Qian Liu^{1}
https://doi.org/10.1186/s13662-018-1471-8
© The Author(s) 2018
Received: 3 October 2017
Accepted: 29 December 2017
Published: 12 January 2018
Abstract
This paper is concerned with the iterative learning control problem for a class of discrete-time singular systems. According to the characteristics of the systems, a closed-loop PD-type learning algorithm is proposed and the convergence condition of the algorithm is established. It is shown that the algorithm can guarantee the system output converges to the desired trajectory on the whole time interval. Moreover, the presented algorithm is also suitable for discrete-time singular systems with state delay. Finally, the validity of the presented algorithm is verified by two numerical examples.
Keywords
1 Introduction
Iterative learning control (ILC) is an effective control strategy to achieve perfect trajectory tracking for repetitive systems in a finite time interval (see [1, 2]). The basic idea of ILC is to improve the current tracking performance by fully utilizing the past control experience. Since the complete iterative learning algorithm was initially proposed by Arimoto et al. [3], it has attracted extensive attention in the field of control theory and many efforts have been made devoted to the progress of ILC in recent years (see [4–8] and the references therein).
Singular systems have essential differences than the normal systems in many aspects, due to the fact that singular systems can preserve the structure of physical systems and impulsive elements. In many practical engineering problems, the systems have singular system models, such as circuit systems, large-scale systems, constrained mechanical systems and robotic systems (see [9, 10]). Hitherto, many significant results based on the theory of normal systems have been successfully extended to singular systems and the related research has been published (see [9–13] and the references therein). Meanwhile, there is some work which has been reported on the ILC for singular systems, but most of it has focused mainly on the continue-time singular systems (see [14–17]). For instance, reference [14] analyzed the convergence of D-type and PD-type closed-loop learning algorithms for linear singular systems in the sense of the Frobenius norm. Based on the Weierstrass canonical form of singular systems, reference [15] proposed a P-type ILC algorithm for the fast subsystems with impulse. In [16], the ILC technique was applied to a class of singular systems with state delay, then the convergence of the algorithm and the possibility of the state tracking were analyzed. Based on the nonsingular transformation method, a PD-type algorithm was designed in [17] to study the state tracking problem for a class of singular systems. Very recently, reference [18] applied the ILC strategy to a class of discrete singular systems, then the convergence analysis of the algorithm was given in detail by using λ-norm.
On the other hand, it should be pointed out that most of the singular systems studied in the above-mentioned works are based on the assumption that the matrix \(A_{22}\) is nonsingular (see [16–18]), which implies that the systems are impulse-free (for continue-time singular systems) or causal (for discrete-time singular systems). However, in many practical singular system models, the matrix \(A_{22}\) may be singular. Motivated by the aforementioned discussions, the ILC problem for a class of discrete-time singular systems will be further considered in this paper. According to the characteristics of the systems, a closed-loop PD-type learning algorithm is proposed and the convergence condition of the algorithm is established. It is worth pointing out that the algorithm presented in this paper has the ability to eliminate the non-causality of discrete-time singular systems. Under the action of the algorithm, the uniform convergence of the output tracking error is guaranteed with the aid of λ-norm. Furthermore, the result is extended to discrete-time singular systems with state delay. In the end, two numerical examples are given to support the theoretical analysis.
Throughout this paper, I denotes the identity matrix with appropriate dimensions. For a given vector or matrix X, \(\Vert X \Vert \) denotes its Euclidean norm. For a discrete system, \(t \in [0,T]\) denotes the integer sequence \(t = 0,1,2,\ldots, T\) . For a function h: \([0,T] \to {R} ^{n}\) and a real number \(0<\lambda < 1\), \({\Vert h \Vert _{\lambda }}\) denotes the λ-norm defined by \({\Vert h \Vert _{\lambda }} = \sup_{t \in [0,T]} \{ {{\lambda^{t}}\Vert {h(t)} \Vert } \} \).
2 Problem description
Definition 1
([9])
The system (1) is said to be regular if there exists a constant complex \({s_{0}}\) such that \(\det ({s_{0}}E - A) \ne 0\).
Before giving our ILC law, basic assumptions for the system (1) are first given as follows.
Assumption 1
Assumption 2
Given a desired output trajectory \(y_{d}{(t)}\), the target of this paper is to design an appropriate learning algorithm and generate the control sequence \(u_{k}{(t)}\), such that the system output \(y_{k}{(t)}\) can track the desired trajectory \(y_{d}{(t)}\) as the iteration number increases.
3 Convergence analysis of the algorithm
Theorem 1
Proof
4 Extension to systems with state delay
Basic assumptions for the system (11) are given for further analysis.
Assumption 4
Assumption 5
Assumption 6
The system (11) is regular, controllable and observable.
Theorem 2
Consider the system (11) satisfying Assumptions 4-6. If there exists the gain matrix \({\Gamma } \in {{R}^{{m} \times {r}}}\) such that the matrix \(E + B\Gamma C\) is nonsingular and the convergence condition (3) holds, then the system output \({y_{k}}(t)\) can converge to the desired trajectory \({y_{d}}(t)\) on the time interval \([0,T+1]\) by using the algorithm (2), i.e., \(\mathop{\lim } _{k \to \infty } {y_{k}}(t) = {y_{d}}(t), t\in [0,T+1]\).
Proof
5 Numerical examples
In this section, two numerical examples are constructed to demonstrate the validity of the presented closed-loop PD-type learning algorithm.
Example 1
Example 2
6 Conclusion
In this paper, the problem of iterative learning control is investigated for a class of discrete-time singular systems. Then a closed-loop PD-type learning algorithm is adopted for such singular systems, and the convergence condition of the algorithm is established. We show that the algorithm can ensure the output tracking error converges to zero on the whole time interval. The corresponding result is further extended to discrete-time singular systems with state delay. In the end, two numerical examples are constructed to illustrate the effectiveness of the presented algorithm.
Declarations
Acknowledgements
The authors would like to express their gratitude to the anonymous reviewers for their valuable suggestions that have improved the quality of this paper. This work was supported by the National Natural Science Foundation of China (Nos. 61374104, 61773170) and the Natural Science Foundation of Guangdong Province of China (No. 2016A030313505).
Authors’ contributions
All the authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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