Robust iterative learning control design for linear systems with time-varying delays and packet dropouts
- Bu Xuhui^{1}Email author,
- Hou Zhanwei^{2},
- Hou Zhongsheng^{3} and
- Yang Junqi^{1}
https://doi.org/10.1186/s13662-017-1140-3
© The Author(s) 2017
Received: 5 November 2016
Accepted: 10 March 2017
Published: 20 March 2017
Abstract
This paper proposes the robust iterative learning control (ILC) design for uncertain linear systems with time-varying delays and random packet dropouts. The packet dropout is modeled by an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, which renders the ILC system to be stochastic instead of a deterministic one. The main idea of this paper is to transform the ILC design into robust stability for a two-dimensional (2D) stochastic system described by the Roesser model with a delay varying in a range. A delay-dependent stability condition, which can guarantee mean-square asymptotic stability of such a 2D stochastic system, is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. An example for the injection molding is given to demonstrate the effectiveness of the proposed ILC method.
Keywords
1 Introduction
Iterative learning control (ILC) is an effective technique for systems that could perform the same task over a finite time interval repetitively. It updates the control input signal only depending on I/O data of previous iteration, and the tracking performance can become better and better. Due to its simplicity and effectiveness, ILC has been widely applied in many practical systems such as robotics, chemical batch processes, hard disk drives and urban traffic systems [1–8].
Time delay, which is a source of instability and poor performance, often appears in practical systems due to the finite speed of signal transmission and information processing. As a result, studies on batch process control have attracted considerable attention [9–13]. Recently, ILC has been introduced to systems with time delays to improve the tracking performance. In [14], a new ILC method is proposed for a class of linear systems with time delay using a holding mechanism. In [15], an ILC algorithm integrated with Smith predictor for batch processes with fixed time delay is proposed and analyzed in the frequency domain; itcan obtain perfect tracking performance under certain conditions. In [16], an ILC scheme is proposed for systems with time delay and model uncertainties based on the internal model control principle. In [17], a two-dimensional (2D) model based on ILC methods for systems both with state delays and with input delays is presented; necessary and sufficient conditions for the stability of ILC are also provided. In [18], the problem of ILC design for time-delay systems in the presence of initial shifts is considered, and the 2D system theory is employed to develop a convergence condition for both asymptotic stability and monotonic convergence of ILC.
In these existing studies [14–18], the time delay considered is known and fixed constants. As we know, many practical systems suffer from time-varying delays, which are less conservative than constant delays. It is a challenge to design ILC for systems with time-varying delays. There have already been a few results on this issue. In [19], a robust state feedback integrated with ILC scheme is proposed for batch processes with interval time-varying delay. The design is considered using a 2D Roesser model based on the general 2D system theory [20]. In [21], a robust closed-loop ILC scheme is proposed for batch processes with state delay and time-varying uncertainties, and the batch processes are described as a 2D FM model. In [22], a robust output feedback incorporated with ILC scheme is proposed for a kind of batch process with uncertainties and interval time-varying delay. The batch process is transformed into a 2D FM model with a delay varying in a range, and the design is cast as a robust H∞ control for uncertain 2D systems. It is noticed that almost all available results on ILC systems with time delays are based on an implicit assumption that sensor output measurement is perfect. However, the assumption is often not true in most cases in practice. The main reason is that sensors may suffer from probabilistic signal missing especially in a networked environment [23–25].
Actually, the problem of ILC for networked control systems (NCSs) with packet dropouts has received some attention in the research field. The stability of ILC for linear and nonlinear systems with intermittent measurement is investigated in [26, 27]. Some robust ILC designs are proposed for NCSs to suppress the effect of data dropouts in [28–33]. However, to our best knowledge, no work considering ILC systems with data dropouts and time delay simultaneously has been done up to now.
This paper proposes a robust ILC design scheme for uncertain linear systems with time-varying delays and random packet dropouts. Here the considered systems are implemented in a network environment, where data packet may be missed during transmission. For convenience, only the measurement packet dropout is taken into account. The packet dropout is modeled by an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, which renders the ILC system to be stochastic instead of a deterministic one. Then, a 2D stochastic Roesser model with a delay varying in a range is established to describe the entire dynamics. Based on 2D system theory, the ILC law is designed to guarantee mean-square asymptotic stability of the considered 2D stochastic systems. Afterwards, a delay-dependent stability condition is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. Finally, an example for the injection molding is given to demonstrate the effectiveness of the proposed ILC method.
Throughout the present paper, the following notations are used. The superscript ‘T’ denotes the matrix transposition, I denotes the identity matrix, 0 denotes the zero vector or matrix with the required dimensions. \(\operatorname{diag} \{ \bullet \}\) denotes the standard (block) diagonal matrix whose off-diagonal elements are zero. In symmetric block matrices, an asterisk ∗ is used to denote the term that is induced by symmetry. The notation \(\Vert \bullet \Vert \) refers to the Euclidean vector norm, \(E \{ x \},E \{ x|y \}\) mean the expectation of x and the expectation of x conditional on y, respectively. Matrices, if the dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
2 Problem formulation and 2D system representation
Remark 1
It is worth pointing out that 2D system (8) is a stochastic system due to the introduction of the stochastic variable \(\alpha (i,j)\). It differs from the deterministic 2D or ILC systems with time delay in recent works such as [17–22]. Here, the ILC design should be discussed under the framework of stochastic stability. To this end, we need to give the following definition of stochastic stability for 2D systems.
3 Stability analysis and controller design
3.1 Stability analysis
In this section, we focus on the problem of robust stability and robust stabilization for 2D stochastic system (8) using an LMI technique. The following lemma is needed in the proof of our main result.
Lemma 1
[34]
Now, we can give our main result.
Theorem 1
Proof
This completes the proof of Theorem 1. □
Remark 2
Theorem 1 provides a sufficient condition for 2D uncertain systems with a delay varying in a range and random packet dropouts. If the communication link existing between the plant and the controller is perfect, that is, there is no packet dropout during their transmission, then \(\alpha = 1\) and \(\theta = 0\). In this case, the condition in Theorem 1 becomes the condition obtained in [19, 20] for a 2D deterministic system with time delay. From this point of view, Theorem 1 can be seen as an extension to 2D time-delay systems with packet dropout.
3.2 Controller design
Theorem 1 gives a mean-square asymptotic stability condition where the controller gain matrix K is known. However, our eventual purpose is to determine a suitable K by system matrices \(A,A_{d},B,E,F_{1},F_{2}\) and parameter α.
Lemma 2
[35]
Now, we can give the following result.
Theorem 2
Proof
Remark 3
Notethat the condition in Theorem 2 is no longer LMIs owing to the term \(LX^{ - 1}L\) in \(\tilde{\psi}_{1}\). The available LMI tools cannot be used directly to obtain a feasible solution. However, we can use the idea of iterative algorithms in combination with LMI convex optimization problems as is done in [36, 37]. Then an available control law can be obtained.
Remark 4
It isnoticed thatthe data dropoutmay occur on both system output and control input sides in NCSs. In this paper, we only consider output measurement missing for the sake of convenience, as is also done in most existing works. However, the result in this paper can be extended to the control input signal dropouts.
Remark 5
Since the system performs the same task repetitively, computation complexity is an important issue for ILC systems. The large number of iterations leads to high accuracy but heavy computational burden. How to keep the balance between iteration number and tracking accuracy is an important problem for practical ILC systems. Some efforts can be made to address this issue. Firstly, we can pre-calculate an acceptable tracking error, which satisfies the accuracy requirement of the control objective. If the tracking error reaches the given value, then the iteration process is stopped. Secondly, we can design optimal ILC with monotonic convergence and fast convergence speed such that learning transient behavior is reduced.
4 Illustrative example
In simulation, the initial states are given as \(x_{1}(0,k) = x_{2}(0,k) = 0\) for all k, and the control input is selected as \(u(t,0) = 0\) for all t. The time-varying delay \(d(t)\) changes randomly with \(d(t) \in [0.5\ 4]\). The uncertain parameters \(\xi_{1,2}\) are assumed to vary randomly within \([0\ 1]\) along with both time and iteration direction.
5 Conclusions
The robust ILC design is considered for uncertain linear systems with time-varying delays and random packet dropouts. By modeling the packet dropout as an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, the considered system can be transformed into a 2D stochastic system described by the Roesser model with a delay varying in a range. Then, a delay-dependent stability condition is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. The results on injection velocity control have illustrated the feasibility and effectiveness of the proposed design.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 61573129, 61433002, 61573130), the Program for Science & Technology Innovation Talents in Universities of Henan Province (16HASTIT046), the Fundamental Research Funds for the Universities of Henan Province, the program of Key Young Teacher of Higher Education of Henan Province (2014GGJS-041), the program of Key Young Teacher of Henan Polytechnic University, the Science and Technology Innovation Talents Project of Henan Province (164100510004) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (CXTD2016054).
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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