Robust \(H_{\infty}\) control for singular systems with state delay and parameter uncertainty
- Yeping Sun^{1}Email author and
- Yuxiao Kang^{1}
https://doi.org/10.1186/s13662-015-0433-7
© Sun and Kang; licensee Springer. 2015
Received: 1 December 2014
Accepted: 2 March 2015
Published: 15 March 2015
Abstract
This paper considers the problem of robust \(H_{\infty}\) control for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. By using the linear matrix inequality (LMI) approach, a sufficient condition is presented for a prescribed uncertain singular system with time-delay to have generalized quadratic stability and \(H_{\infty}\) performance. Furthermore, the design methods of state feedback controllers are considered such that the resulting closed-loop system has generalized quadratic stability with \(H_{\infty}\) performance. By means of matrix inequalities, sufficient conditions are derived for the existence of memory-less and memorial static state feedback controllers. The controllers are obtained by the solutions of matrix inequalities.
Keywords
1 Introduction
It is well known that a real system inevitably contains some uncertain parameters because of work environment change, measure error, model approximation and so on. The uncertain parameters perhaps change the system structure and even destroy the system. For a practical system, the uncertain parameters should be considered, otherwise, one cannot obtain their desired goals. Recently, robust \(H_{\infty}\) (sub) optimal control has become one of the most important notions in the field of automatic control theory, it has drawn considerable attention from many researchers. Although robust \(H_{\infty}\) control theory has been perfectly developed over the last decade, most of the results were developed based on uncertain linear systems [1–4]. Besides, some physical phenomena, like impulse and hysterics, which are important in circuit theory, cannot be treated in the linear system models. It is well known that time delay is frequently encountered in a variety of industrial and engineering systems, and it has become one of the main sources for causing instability and poor performance of the network system [5, 6].
Singular systems are also referred to as generalized systems, descriptor systems, differential-algebraic systems, or implicit systems, which are also a natural representation of dynamic systems and describe a larger family of systems than the normal linear systems [7]. A singular system provides a suitable way to handle such problems, the robust control theory based on singular system models has been widely developed for many years. Dai first gave some notions of controllability, observability and duality in singular systems [8], some excellent results on disk pole constraints [9] and robust control [10, 11]. Problem of control and stabilization for uncertain dynamical systems with deviating argument is modern now. For example, robust stability and \(H_{\infty}\) were studied for uncertain systems with impulsive perturbations [12]. Moreover, robust \(H_{\infty}\) synchronization was studied for chaotic systems with input saturation and time varying delay [13]. In [14, 15], stabilization and perturbation estimation were studied in neutral type direct control systems. In [16], stabilization was studied for Lur’e-type nonlinear control systems by using Lyapunov-Krasovskii functionals. In addition, dissipativity was studied for singular systems with Markovian jump parameters and mode-dependent mixed time-delays [7]. However, for the singular system, robust \(H_{\infty}\) control problem has been little considered with uncertainties and time-delay recently.
In this paper, by means of linear matrix inequalities (LMIs), we present sufficient conditions for the existence of memory-less and memorial linear state feedback controllers such that the closed-loop system not only has \(H_{\infty}\) performance, but it also is generalized quadratically stable; moreover, the design methods for such controllers are also provided.
2 System description and preliminaries
ΔA, \(\Delta A_{d}\), \(\Delta B_{1}\), \(\Delta B_{2}\), \(\Delta C_{1}\), \(\Delta C_{1d}\), \(\Delta D_{11}\) and \(\Delta D_{12}\) are said to be admissible if both (2) and (3) hold.
Lemma 1
[11]
Suppose that the pair \((E,A)\) is regular and impulse free, then the solution to (4) exists and is impulse free and unique on \([0, \infty)\).
Definition 1
- (1)
The singular delay system (4) is said to be regular and impulse free if the pair \((E,A)\) is regular and impulse free.
- (2)
The singular delay system (4) is said to be asymptotically stable if for any \(\epsilon>0\), there exists a scalar \(\delta(\epsilon)>0\) such that, for any compatible initial conditions \(\phi(t)\) satisfying \(\sup_{-d\leq t\leq0}\|\phi(t)\|\leq\delta(\epsilon)\), the solution \(x(t)\) of system (4) satisfies \(\|x(t)\|\leq\epsilon\) for \(t\geq 0\). Furthermore, \(x(t)\rightarrow0\), \(t\rightarrow\infty\).
Definition 2
[17]
The uncertain singular delay system (1) is said to be robust stable if system (1) with \(u(t)\equiv0\) and \(\omega(t)\equiv0\) is regular, impulse free and asymptotically stable for all admissible uncertainties ΔA, \(\Delta A_{d}\).
Definition 3
[11]
Lemma 2
[11]
If the uncertain singular delay system (1) is generalized quadratically stable, then it is robustly stable.
Lemma 3
[18]
3 Main results
Theorem 1
Proof
Then we prove that the system has \(H_{\infty}\) performance γ.
Therefore the system has \(H_{\infty}\) performance γ from dissipative theory. □
Based on Theorem 1, we will further discuss the robust \(H_{\infty}\) control problem via state feedback for system (1).
From Theorem 1, we can easily obtain the following theorem.
Theorem 2
Remark 1
Obviously, (14) is a strict LMI about matrix P and a scalar \(\epsilon>0\), which can be solved numerically very efficiently by using the LMI Toolbox of Matlab.
Substituting the matrix P and the scalar \(\varepsilon>0\) obtained by solving (14) into (13), we can get the strict LMI about K, so the gain matrix can be obtained.
Similar to Theorem 2, we have the following conclusion.
Theorem 3
Remark 2
Obviously, (18) is a strict LMI about matrix P, \(K_{2}\) and a scalar \(\varepsilon>0\).
Substituting the matrix P, \(K_{2}\) and the scalar \(\varepsilon>0\) obtained by solving (18) into (17), we can get the strict LMI about \(K_{1}\), so the two gain matrices can be obtained.
The robust \(H_{\infty}\) control problem for singular time delay system with norm-bounded parametric uncertainties is considered in this paper. All the coefficient matrices except the matrix E include uncertainties. The authors derive sufficient conditions about the generalized quadratic stability and \(H_{\infty}\) performance of the closed-loop systems. The control laws proposed by using strict LMI approaches can guarantee that the resultant closed-loop systems are generalized quadratic stable for all admissible uncertainties.
4 Numerical examples
In this section, we present an example to illustrate the application of the proposed theoretical method given in this paper.
Example
5 Conclusions
A positive solution matrix was proposed for the problem of robust \(H_{\infty}\) control via state feedback for a class of uncertain continuous-time singular systems with state delay. The solution provides sufficient conditions in the form of linear matrix inequalities. It was shown by the numerical example that the proposed method can solve generalized quadratic stability with \(H_{\infty}\) performance for the parameter uncertain continuous-time singular systems with state delay.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 10571114 and the Henan Province Natural Science Foundation under Grant 0511012000.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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