## Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays.(English)Zbl 1170.34054

Appl. Math. Comput. 208, No. 1, 58-68 (2009); erratum ibid. 215, No. 1, 427-430 (2009).
Linear continuous-time control systems with variable coefficients and variable point delay in the state variables are considered. It is generally assumed that the system matrices are known with some uncertainties. Using Lyapunov functionals and linear matrix inequality, sufficient conditions for feedback stabilizability are formulated and proved. Simple numerical examples, which illustrate theoretical considerations are presented. Moreover, many remarks and comments on stabilization problems for delayed control systems are given. The relationships to the results existing in the literature are mentioned and discussed. Finally, it should be pointed out, that similar stabilization problems have been recently considered in the papers [J. H. Park and O. Kwon, Appl. Math. Comput. 162, No. 2, 627–637 (2005; Zbl 1077.34075)] and [P. G. Park and J. W. Ko, Stability and robust stability for systems with a time-varying delay. Automatica 43, No. 10, 1855–1858 (2007; Zbl 1120.93043)].

### MSC:

 34K35 Control problems for functional-differential equations 93C05 Linear systems in control theory 93C23 Control/observation systems governed by functional-differential equations 93D09 Robust stability 93D15 Stabilization of systems by feedback 34K06 Linear functional-differential equations

### Citations:

Zbl 1077.34075; Zbl 1120.93043

LMI toolbox
Full Text:

### References:

 [1] Hale, J.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York [2] Niculescu, S.I., Delay effects on stability: a robust approach, Lecture notes in control and information sciences, vol. 2, (2002), Springer-Verlag New York [3] Richard, J.P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302 [4] Park, Ju H.; Kwon, O.M., On new stability criterion for delay-differential systems of neutral type, Applied mathematics and computations, 162, 627-637, (2005) · Zbl 1077.34075 [5] Xu, S.; Lam, J.; Zou, Y., Further results on delay-dependent robust stability conditions of uncertain neutral systems, International journal of robust and nonlinear control, 15, 233-246, (2005) · Zbl 1078.93055 [6] T. Li, L. Guo, Y. Zhang, Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay, International Journal of Robust and Nonlinear Control doi:10.1002/rnc.1280, 2007. · Zbl 1298.93263 [7] Park, P.G.; Ko, J.W., Stability and robust stability for systems with a time-varying delay, Automatica, 43, 738-743, (2007) [8] Park, P.G., A delay-dependent stability criterion for systems with uncertain linear state-delayed systems0, IEEE transactions on automatic control, 35, 876-877, (1999) · Zbl 0957.34069 [9] Moon, Y.S.; Park, P.G.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, International journal of control, 74, 1447-1455, (2001) · Zbl 1023.93055 [10] Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE transactions on automatic control, 47, 1931-1937, (2002) · Zbl 1364.93564 [11] Fridman, E.; Shaked, U., Delay dependent stability and $$H_\infty$$ control: constant and time-varying delays, International journal of control, 76, 48-60, (2003) · Zbl 1023.93032 [12] Yue, D.; Won, S.; Kwon, O., Delay dependent stability of neutral systems with time delay: an LMI approach, IEEE Proceedings – control theory and applications, 150, 23-27, (2003) [13] Kwon, O.M.; Park, J.H., On improved delay-dependent robust control for uncertain time-delay systems, IEEE transactions on automatic control, 49, 1991-1995, (2004) · Zbl 1365.93370 [14] Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE transactions on automatic control, 50, 384-387, (2005) · Zbl 1365.93376 [15] Parlakļi, M.N.A., Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems, International journal of robust and nonlinear control, 16, 599-636, (2006) · Zbl 1128.93380 [16] Parlakļi, M.N.A., Robust stability of uncertain time-varying state-delayed systems, IEE Proceedings – control theory and applications, 153, 469-477, (2006) [17] O.M. Kwon, Ju H. Park, S.M. Lee, On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays, Applied Mathematics and Computation 197 (2008) 864-873. · Zbl 1144.34052 [18] Knospe, C.R.; Roozbehani, M., Stability of linear systems with interval time delay excluding zero, IEEE transactions on automatic control, 51, 1271-1288, (2006) · Zbl 1366.34099 [19] Yu, K.W.; Lien, C.H., Stability criteria for uncertain neutral systems with interval time-varying delay, Chaos solitons & fractals, 36, 920-927, (2008) · Zbl 1139.93354 [20] Yue, D.; Peng, C.; Tang, G.Y., Guaranteed cost control of linear systems over networks with state and input quantisation, IEEE Proceedings – control theory and applications, 153, 658-664, (2006) [21] Boyd, S.; Ghaoui, L.; Feron, E.; Balakrishanan, V., Linear matrix inequalities in system and control theory, Studies in applied mathematics, vol. 15, (1994), SIAM Philadelphia, Pennsylvania [22] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of 39th IEEE Conference on Decision Control, Sydney, Australia, 2000, pp. 2805-2810. [23] Gahinet, P.; Nemirovski, A.; Laub, A.J.; Chilali, M., LMI control toolbox, (1995), The Mathworks Natik, Massachusetts [24] Park, H.S.; Kim, Y.H.; Kim, D.S.; Kwon, W.H., A scheduling method for networked based control systems, IEEE transactions on control systems technology, 10, 318-330, (2002) [25] Kim, D.S.; Lee, Y.S.; Kwon, W.H.; Park, H.S., Maximum allowable delay bounds of networked control systems, Control engineering practice, 11, 1301-1313, (2003) [26] Yue, D.; Han, Q.L.; Peng, C., State feedback controller design of networked control systems, IEEE transactions on circuits and systems-II: express briefs, 51, 640-644, (2004)
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.