A PD-type iterative learning control algorithm for singular discrete systems
- Senping Tian^{1}Email author,
- Qian Liu^{1},
- Xisheng Dai^{2} and
- Jianxiang Zhang^{2}
https://doi.org/10.1186/s13662-016-1047-4
© The Author(s) 2016
Received: 10 March 2016
Accepted: 29 September 2016
Published: 9 December 2016
Abstract
Based on a specific decomposition of discrete singular systems, in this paper, we study the problem of state tracking control by using PD-type algorithm of iterative learning control. The convergence conditions and theoretical analysis of the PD-type algorithm are presented in detail. An illustrative example supporting the theoretical results and the effectiveness of the PD-type iterative learning control algorithm for discrete singular systems is shown at the end of the paper.
Keywords
singular discrete systems PD-type iterative learning control convergence analysis1 Introduction
Iterative learning control (ILC) is an effective control scheme in handling a system that repetitively perform the same task with a view to sequentially improving the accuracy on a finite interval. The research of ILC is of great significance for dynamic systems with complex modeling, strong nonlinear coupling effects, and uncertainty [1–5]. The aim of ILC is to look for a proper learning control algorithm of the controlled systems so that the output state can track the given desired trajectory over a finite interval time and in the meantime the constructed learning control sequences can uniformly converge to the desired control. Since Arimito proposed the concept of iterative learning control in 1984, the research of ILC has become a topic of focus in the field of control and fruitful research progress has been made in theory and application [6–12].
Singular systems have been a subject of interest over the last two decades due to their many practical applications. For instance, we have engineering systems, social systems, economic systems, biological systems, network analysis, time-series analysis, etc. [13, 14]. Such systems describe a wider class of systems, including physical models and non-dynamic constraints. Currently, ILC of singular system has also attracted the attentions of many scholars. The existing ILC methods of singular discrete systems use either a P-type algorithm or a D-type algorithm to track the desired output trajectory. Compared with these existing ILC methods, we use a PD-type algorithm related to the current error and the following error to improve the accuracy. Considering the equivalence of two norms, we use the λ norm in the paper to prove the convergence of the PD-type algorithm. Reference [15] studied the state tracking problem of the singular system with time-delay and proved that the iterative learning algorithm is convergent under certain conditions. In [16], the convergence of the P-type iterative learning control algorithm for a fast subsystem of a linear singular system is proved under a certain sufficient condition. Reference [17] proposes the convergence results for a continue linear time-invariant singular system by the close-loop PD-type iterative learning control algorithm. In [18], a new iterative learning control algorithm to study the state tracking for a class of singular systems is proposed and the convergence of the algorithm is completely analyzed.
As a result, the PD-type iterative learning control algorithm is applied to study the state tracking for a class of discrete singular systems. And then the convergence conditions are proposed and analyzed from the theoretical perspective. Finally, the numerical simulation results, showing that the given ILC algorithm for the state tracking of singular system is effective, are presented.
2 Description of singular discrete system
- (1)
The singular discrete system is regular, controllable, and observable, \(A_{22}\) is invertible.
- (2)
The system (2) satisfies the initial conditions: \(x_{k}(0)=x_{d}(0)\), \(k=0,1,\ldots\) .
- (3)For a given desired target \(x_{d}(i)\), there always exists a corresponding control input \(u_{d}(i)\) over the finite interval \([0, T]\), such that$$ \left \{ \textstyle\begin{array}{l} x_{d}^{(1)}(i+1) = A_{11}x_{d}^{(1)}(i)+A_{12}x_{d}^{(2)}(i)+B_{1}u_{d}(i), \\ 0 = A_{21}x_{d}^{(1)}(i)+A_{22}x_{d}^{(2)}(i)+B_{2}u_{d}(i). \end{array}\displaystyle \right . $$(4)
3 Convergence analysis of PD-type iterative learning control algorithm
Throughout this paper, we will use the following notation.
Theorem 1
Proof
4 Simulation of the new algorithm
Figure 1 shows the tracking process of the desired trajectory \(x_{d}^{(1)}(i)\) of the discrete singular system by using iterative learning control algorithm (3) at 18th and 25th iterations. According to the definition of complete tracking and the simulation data we can derive that the algorithm can track the desired trajectory completely at the 30th iteration.
The simulation examples illustrate the effectiveness of PD-type iterative learning control algorithm for discrete singular systems. It shows that the research on discrete iterative learning control problem of a class of singular systems has obtained good results in this paper.
5 Conclusions
In this paper, an iterative learning control problem for a class of discrete singular systems is studied by using the dynamic decomposition standard of singular systems. The PD-type iterative learning control algorithm and the sufficient condition are designed, and we have proved in theory the algorithm can guarantee that the output can track the desired trajectory completely on a finite time interval. The simulation example shows the effectiveness of PD-type iterative learning control algorithm for the discrete singular system.
Declarations
Acknowledgements
First and foremost, we would like to show our deepest gratitude to the editors and the reviewers for giving the chance for our paper to be published. The comments we received were all valuable and very helpful for revising and improving our paper, as well as being of significance and important as a guide to our researches. This work was supported by the National Natural Science Foundation of China (Nos. 61374104, 61364006) and the Natural Science Foundation of Guangdong Province, China (No. 2016A030313505).
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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