Robust exponential stabilization for large-scale uncertain impulsive systems with coupling time-delays. (English) Zbl 1154.34041

Robust exponential stability of a large-scale uncertain impulsive systems with coupling time-delays is investigated. The robust and exponential stability criteria are established for the following system \[ \dot{x}_i(t)=A_ix_i(t)+f_i(t,x_i(t))+ \sum\limits_{j=1}^{N}B_{ij}x_j(t-\tau _j(t))+u_{ci}(t), \;t\in (t_k,t_{k+1}], \]
\[ \Delta x_i(t)=(C_{ik}-I)x_i(t)+u_{di}(t), \;t=t_k, \;k\in \mathbb{N}, i=1,2,\ldots N; \] where \(x_i=(x_{i1},x_{i1},\ldots, x_{in})^T \in \mathbb{R}^n\), represents the state vector of the \(i\)–th subsystem; \(\Delta x_i(t_k)=x(t_k^+)-x(t_k)\); \(f_i : \mathbb{R}_+\times \mathbb{R}^n \rightarrow \mathbb{R}^n\) is a smooth nonlinear vector function with \(f_i(t,0)\equiv 0\). By utilizing Lyapunov’s method and Razumikhin technique, the feeback hybrid controllers in terms of linear matrix inequalities are proved under which the robust exponential stability is achieved for a closed-loop large-scale uncertain impulsive systems with coupling time-delays. These criteria can be easily used for the design of a feedback controller. For illustration of this result one example is given. The numerical simulation procedure is coded and executed in the MATLAB environment.


34K35 Control problems for functional-differential equations
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
93C23 Control/observation systems governed by functional-differential equations
93D15 Stabilization of systems by feedback


Full Text: DOI Link


[1] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulse differential equations, (1989), Singapore World Scientific · Zbl 0719.34002
[2] Bainov, D.D.; Simeonov, P.S., Systems with impulsive effects: stability theory and applications, (1989), Halsted Press New York · Zbl 0676.34035
[3] Lakshmikantham, V.; Liu, X., Stability analysis in terms of two measures, (1993), Singapore World Scientific · Zbl 0797.34056
[4] Liu, X., Stability results for impulsive differential systems with applications to population growth models, Dynam. stability syst., 9, 2, 163-174, (1994) · Zbl 0808.34056
[5] Ye, H.; Michel, A.N.; Hou, L., Stability analysis of systems with impulsive effects, IEEE trans. automat. control, 43, 12, 1719-1723, (1998) · Zbl 0957.34051
[6] Ballinger, G.; Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dcdis, 5, 579-591, (1999) · Zbl 0955.34068
[7] Liu, X.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Comput. math. appl., 41, 903-915, (2001) · Zbl 0989.34061
[8] Shen, J.H.; Yan, J., Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear anal., 33, 519-531, (1998) · Zbl 0933.34083
[9] Liu, B.; Liu, X.; Liao, X., Robust stability analysis of uncertain impulsive systems, J. math. anal. appl., 290, 519-533, (2004) · Zbl 1051.93077
[10] Liu, B.; Liu, X.; Liao, X., Stability and robust stability of quasi-linear impulsive hybrid dynamical systems, J. math. anal. appl., 283, 416-430, (2003) · Zbl 1047.93041
[11] Liu, B.; Liu, X., Robust global exponential stability and estimate of decay rate of uncertain linear impulsive systems with time-delay, Rocky moutain J. math., 36, 2, 615-636, (2006) · Zbl 1145.34045
[12] Liu, B.; Liu, X.; Teo, K.L., Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA J. appl. math., 71, 47-61, (2006) · Zbl 1128.34047
[13] Moon, Y.M.; Park, P.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, Int. J. control, 74, 14, 1447-1455, (2001) · Zbl 1023.93055
[14] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequality in systems and control theory, (1994), SIAM Philadelphia
[15] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002
[16] Petersen, I.R., A stabilization algorithm for a class of uncertain systems, Syst. control lett., 8, 181-188, (1987)
[17] Chen, G., Controlling chaos and bifurcations in engineering systems, (2000), CRC Press Boca Raton, FL · Zbl 0929.00012
[18] Liu, B.; Liu, X.; Chen, G.; Wang, H., Robust impulsive synchronization of uncertain dynamical networks, IEEE trans. circuits syst.—I. regular paper, 52, 7, 1431-1441, (2005) · Zbl 1374.82016
[19] Gavel, D.T.; Siljak, D.D., Decentralized adaptive control: structural conditions for stability, IEEE trans. automat. control, 34, 413-426, (1989) · Zbl 0681.93001
[20] Bakule, L.; Rodellar, J., Decentralized control and overlapping decomposition of mechanical systems, Int. J. control, 61, 559-587, (1995) · Zbl 0825.93026
[21] Mahmoud, M.S.; Bingulac, S., Robust design of stabilizing controllers for interconnected time-delay systems, Automatica, 34, 795-800, (1998) · Zbl 0936.93044
[22] Gao, H.J.; Lam, J.; Wang, C.H.; Wang, Q., Hankel norm approximation of linear systems with time-varying delay: continuous and discrete cases, Int. J. control, 77, 17, 1520-1530, (2004) · Zbl 1067.93008
[23] Mao, X.R.; Lam, J.; Xu, S.Y.; Gao, H.J., Razumikhin method and exponential stability of hybrid stochastic delay interval systems, J. math. anal. appl., 283, 416-430, (2006)
[24] Li, Z.G.; Soh, C.B.; Xu, X.H., Controllability and observability of impulsive hybrid dynamic systems, IMA J. math. control inform., 16, 315-334, (1999) · Zbl 0943.93015
[25] Li, Z.G.; Soh, Y.C.; Wen, C.Y., Robust stability of a class of hybrid nonlinear systems, IEEE trans. automat. control, 46, 6, 897-903, (2001) · Zbl 1001.93069
[26] Li, Z.G.; Soh, Y.C.; Wen, C.Y., Switched and impulsive systems: analysis, design and application, ISBN: 3-540-23952-9, (2005), Springer-Verlag · Zbl 1001.93068
[27] Li, Z.G.; Wen, C.Y.; Soh, Y.C., Analysis and design of impulsive control systems, IEEE trans. automat. control, 46, 894-899, (2001) · Zbl 1001.93068
[28] Zhang, Y.; Sun, J.T., Stability of impulsive delay differential equations with impulses at variable times, Dynam. syst.—an international journal, 20, 3, 323-331, (2005) · Zbl 1088.34069
[29] Zhang, Y.; Sun, J.T., Stability of impulsive linear differential equations with time delay, IEEE trans. circuits syst. II—express briefs, 52, 701-705, (2005)
[30] Liu, X.; Shen, J.H., Stability theory of hybrid dynamical systems with time delays, IEEE trans. automat. control, 51, 4, 620-625, (2006) · Zbl 1366.93441
[31] Pepe, P., On the asymptotic stability of coupled delay differential and continuous time difference equations, Automatica, 41, 107-112, (2005) · Zbl 1155.93373
[32] Phat, V.U., Robust stability and stabilizability of uncertain linear hybrid systems with state delays, IEEE trans. circuits syst. II—express briefs, 52, 2, 94-98, (2005)
[33] B. Liu, Robust stability of uncertain discrete impulsive hybrid systems, IEEE Trans. Circuits Syst. II—Express Briefs, 2007 (in press)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.