- Open Access
Robust stability and -control of uncertain systems with impulsive perturbations
© Hu and Wang; licensee Springer. 2014
- Received: 3 January 2014
- Accepted: 24 February 2014
- Published: 7 March 2014
In this paper, the problems of robust stability, stabilization, and -control for uncertain systems with impulsive perturbations are investigated. The parametric uncertainties are assumed to be time-varying and norm-bounded. The sufficient conditions for the above problems are developed in terms of linear matrix inequalities. Numerical examples are given which illustrate the applicability of the theoretical results.
MSC:34H05, 34H15, 34K45.
- uncertain impulsive system
- robust stability
- robust stabilization
- linear matrix inequality (LMI)
Many evolutionary processes are subject to short temporary perturbations that are negligible compared to the process duration. Thus the perturbations act instantaneously in the form of impulses. For example, biological phenomena involving thresholds, bursting rhythm models in pathology, optimal control of economic systems, frequency-modulated signal processing systems do exhibit impulse effects. Impulsive differential systems provide a natural description of observed evolutionary processes with impulse effects.
Problems with qualitative analysis of impulsive systems has been extensively studied in the literature, we refer to [1–7] and the references therein. Also, the control of impulsive or nonlinear systems received more recently researchers’ special attention due to their applications; see, for example [8–11]. In , Guan et al. studied the control problem for impulsive systems. In terms of the solutions to an algebraic Riccati equation, they obtained sufficient conditions for the existence of state feedback controllers guaranteeing asymptotic stability and prescribed performance of the closed-loop system. But the result in  is based on the assumption that the state jumping at the impulsive time instant has a special form. This assumption is not satisfied for most impulsive systems. Therefore, the results in  are less applicable. Furthermore, the parameter uncertainties of impulsive systems were not considered in .
The goal of this paper is to study the robust stability, stabilization, and -control of uncertain impulsive systems under more general assumption on state jumping. Sufficient conditions for the existence of the solutions to the above problems are derived. Moreover, these sufficient conditions are all in linear matrix inequality (LMI) formalism, which makes their resolution easy.
The rest of this paper is organized as follows: Section 2 describes the system model; Section 3 addresses the robust stability and stabilization problems; Section 4 studies the robust problem; Section 5 provides two examples to demonstrate the applicability of the proposed approach.
In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation is used to denote a symmetric positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. and represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. denotes the Euclidean norm for vectors or the spectral norm of matrices.
where , , , are known real constant matrices and is an unknown matrix functions satisfying . It is assumed that the elements of are Lebesgue measurable.
Throughout this paper, we shall use the following concepts of robust stability and robust performance for system (2.1).
Definition 2.1 System (2.1) with and is said to be robustly stable if the trivial solution of (2.1) with and is asymptotically stable for all admissible uncertainties satisfying (2.2).
The following lemma is essential for the developments in the next sections.
Lemma 2.1 (see )
if , ;
if , .
First, we present some sufficient conditions for robust stability of system (3.1) with .
then system (3.1) with is robustly asymptotically stable.
It follows that system (2.1) with is robustly asymptotically stable. The proof is completed. □
for some matrix and some scalar . Condition (3.12) is exactly the sufficient condition for robust stability of continuous-time linear systems with norm-bounded uncertainty, for example, see .
where , . It follows that under the conditions of Theorem 3.1, system (3.1) with is robustly exponentially stable with decay rate . For prescribed decay rate δ, we can choose to find the feasible solution to LMIs (3.2) and (3.3) by tuning parameter .
to stabilize system (3.1), where is a constant gain to be designed.
then the controller (3.13) with robustly stabilizes system (3.1).
where with .
then (3.17) holds.
which combined with Schur complement leads to (3.15).
which combined with Schur complement leads to (3.16). The proof is completed. □
This section is devoted to studying the robust -control problem for system (2.1).
where , , then system (2.1) has robust stabilization with disturbance attenuation γ. Moreover, the controller (3.13) with robustly stabilizes system (2.1).
By and , it is easy to see that . So, if , then by (3.15) and (3.16) and Theorem 3.2, we can conclude that system (4.2) has robust stabilization. Next, we proceed to prove that system (4.2) verifies noise attenuation γ. To this end, we assume the zero initial condition, that is, , for .
so the proof will be completed.
Pre- and post-multiplying (4.8) by and using Schur complement, it is easy to prove that (4.1) is equivalent to (4.8). The proof is completed. □
Remark 4.1 In , under the assumption that and , sufficient condition for the existence of state feedback controller was derived in terms of the Riccati equation. As compared to , our results can be used for a wider class of impulsive system. Moreover, Theorem 4.1 cast the existence problem of state feedback controller into the feasibility problem of the LMIs (3.15), (3.16), and (4.1), the latter can be efficiently solved by the developed interior-point algorithm .
In this section, we shall give two numerical examples to demonstrate the effectiveness of the proposed results.
If we select decay rate , then by Theorem 3.1, choosing and , the obtained maximum value of c such that the above system is robustly exponentially stable is . If we select the decay rate , by choosing the same values of and , the corresponding maximum value of c is .
First we assume that , . By Theorem 4.1, choosing and , it has been found that the smallest value of γ for the above system to have robust stabilization with disturbance attenuation γ is . The corresponding stabilizing control law is given by .
Next we assume that , . By Theorem 4.1, choosing and , it has been found that the smallest value of γ is and the corresponding stabilizing control law is given by .
Three problems for uncertain impulsive systems have been studied, namely, robust stability, robust stabilization, and robust -control. In each case, the sufficient conditions in terms of linear matrix inequalities have been established. Moreover the method to design a state feedback controller is provided. Our method is helpful to improve the existing technologies used in the analysis and control for uncertain impulsive systems. Numerical examples have been provided to demonstrate the effectiveness and applicability of the proposed approach.
This work was supported by the National Natural Sciences Foundation of People’s Republic of China (Tianyuan Fund for Mathematics, Grant No. 11326113).
- Song X, Li A: Stability and boundedness criteria of nonlinear impulsive systems employing perturbing Lyapunov functions. Appl. Math. Comput. 2011, 217(24):10166–10174. 10.1016/j.amc.2011.05.011MathSciNetView ArticleMATHGoogle Scholar
- Dashkovskiy S, Kosmykov M, Mironchenko A, Naujok L: Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods. Nonlinear Anal. Hybrid Syst. 2012, 6(3):899–915. 10.1016/j.nahs.2012.02.001MathSciNetView ArticleMATHGoogle Scholar
- Wu S, Li C, Liao X, Duan S: Exponential stability of impulsive discrete systems with time delay and applications in stochastic neural networks: a Razumikhin approach. Neurocomputing 2012, 82: 29–36.View ArticleGoogle Scholar
- Fu X, Li X: Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems. J. Comput. Appl. Math. 2009, 224(1):1–10. 10.1016/j.cam.2008.03.042MathSciNetView ArticleMATHGoogle Scholar
- Perestyuk NA, Plotnikov VA, Samoilenko AM, Skripnik NV: Differential Equations with Impulsive Effects: Multivalued Right-Hand Sides with Discontinuities. de Gruyter, Berlin; 2011.View ArticleMATHGoogle Scholar
- Perestyuk MO, Chernikova OS: Some modern aspects of the theory of impulsive differential equations. Ukr. Math. J. 2008, 60(1):91–104. 10.1007/s11253-008-0044-5MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham V, Bainov D, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleMATHGoogle Scholar
- Liu X, Zhong S, Ding X: Robust exponential stability of impulsive switched systems with switching delays: a Razumikhin approach. Commun. Nonlinear Sci. Numer. Simul. 2012, 17(4):1805–1812. 10.1016/j.cnsns.2011.09.013MathSciNetView ArticleMATHGoogle Scholar
- Antunes D, Hespanha J, Silvestre C: Stability of networked control systems with asynchronous renewal links: an impulsive systems approach. Automatica 2013, 49(2):402–413. 10.1016/j.automatica.2012.11.033MathSciNetView ArticleMATHGoogle Scholar
- Shatyrko AV, Khusainov DY, Diblik J, Bastinec J, Rivolova A: Estimates of perturbation of nonlinear indirect interval control system of neutral type. J. Autom. Inf. Sci. 2011, 43: 13–28. 10.1615/JAutomatInfScien.v43.i1.20View ArticleGoogle Scholar
- Shatyrko A, Diblik J, Khusainov D, Ruzickova M: Stabilization of Lur’e-type nonlinear control systems by Lyapunov-Krasovski functionals. Adv. Differ. Equ. 2012., 2012: Article ID 229Google Scholar
- Guan Z, Liao J, Liao R:Robust -control of uncertain impulsive systems. Control Theory Appl. 2002, 19(4):623–626.MathSciNetMATHGoogle Scholar
- Wang Y, Xie L, Souza C: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 1992, 19: 139–149. 10.1016/0167-6911(92)90097-CView ArticleMathSciNetMATHGoogle Scholar
- Gu K, Kharitonov V, Chen J: Stability of Time-Delay Systems. Birkhäuser, Boston; 2003.View ArticleMATHGoogle Scholar
- Boyd S, Ghaoui L, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia; 1994.View ArticleMATHGoogle Scholar
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