Exponential stability of uncertain dynamic systems including state delay. (English) Zbl 1220.34095

Kwon, O. M.; Park, Ju H.

Appl. Math. Lett. 19, No. 9, 901-907 (2006).

Systems with uncertainties and delays, described by equations of the form \[ \begin{gathered} \dot x(t)= (A+\Delta A) x(t)+ (A_1+\Delta A_1) x(t- h),\\ x(s)= \phi(s),\quad s\in [-h,0]\end{gathered} \] are considered. It is assumed that the uncertain terms can be written as \(\Delta A= DF(t)E\), \(\Delta A_1= D_1F_1(t)E_1\), where the matrices \(D\), \(E\), \(D_1\), \(E_1\) are known, and the matrices \(F\), \(F_1\) are small in same sense. The main result is a sufficient condition for exponential stability, expressed in terms of a LMI where the delay \(h\) appears explicitly. The result is illustrated by means of two examples.

Reviewer: Andrea Bacciotti (Torino)

MSC:

34K20	Stability theory of functional-differential equations
34D10	Perturbations of ordinary differential equations
93D20	Asymptotic stability in control theory

Keywords:

time-delay system; exponential stability; Lyapunov method; convex optimization

Software:

LMI toolbox

References:

[1]	Hale, J.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[2]	Malek-Zavarei, M.; Jamshidi, M., Time-delay systems: analysis, optimization and applications, (1987), North-Holland Amsterdam · Zbl 0658.93001
[3]	Lien, C.H.; Yu, K.W.; Hsieh, J.G., Stability conditions for a class of neutral systems with multiple time delays, Journal of mathematical analysis and applications, 245, 20-27, (2000) · Zbl 0973.34066
[4]	Park, J.H., Robust stabilization for dynamic systems with multiple time-varying delays and nonlinear uncertainties, Journal of optimization theory and applications, 108, 155-174, (2001) · Zbl 0981.93069
[5]	Park, J.H.; Jung, H.Y., On the exponential stability of a class of nonlinear systems including delayed perturbations, Journal of computational and applied mathematics, 159, 467-471, (2003) · Zbl 1033.93055
[6]	Park, J.H.; Jung, H.Y., On the design of nonfragile guaranteed cost controller for a class of uncertain dynamic systems with state delays, Applied mathematics and computation, 150, 245-257, (2004) · Zbl 1036.93054
[7]	Park, J.H., Delay-dependent criterion for asymptotic stability of a class of neutral equations, Applied mathematics letters, 17, 1203-1206, (2004) · Zbl 1122.34339
[8]	Kharitomov, V.L.; Melchor-Aguilar, D., On delay-dependent stability conditions for time-varying systems, Systems and control letters, 46, 173-180, (2002) · Zbl 0994.93022
[9]	Cao, J.; Wang, J., Delay-dependent robust stability of uncertain nonlinear systems with time delay, Applied mathematics and computations, 154, 289-297, (2004) · Zbl 1060.34041
[10]	Liu, P.L., Exponential stability for linear time-delay systems with delay dependence, Journal of the franklin institute, 340, 481-488, (2003) · Zbl 1035.93060
[11]	Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., ()
[12]	K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of 39th IEEE CDC, December 2000, Sydney, Australia, 2000, pp. 2805-2810
[13]	Gahinet, P.; Nemirovski, A.; Laub, A.; Chilali, M., LMI control toolbox user’s guide, (1995), The Mathworks Natick, MA
[14]	Yue, D.; Won, S., Delay-dependent robust stability of stochastic systems with time delay and nonlinear uncertainties, Electronics letters, 37, 992-993, (2001) · Zbl 1190.93095