×

Robust \({\mathcal H}_{\infty}\) control of linear neutral systems. (English) Zbl 0988.93024

This paper investigates the problems of robust stability and robust \({\mathcal H}_\infty\) control for a class of uncertain neutral systems of the form: \[ \dot x(t)- D\dot x(t-\tau) =(A+\Delta A)x(t)+ (A_d+\Delta A_d) x(t-\tau) +\bigl[B+ \Delta B(t)\bigr] u(t) \]
\[ x(t_0+ \eta) =\varphi(\eta), \quad \forall\;\eta \in[- \tau,0], \] where \(u(t)\in \mathbb{R}^p\) is the control input, \(A\in \mathbb{R}^{n\times n}\) and \(A_d\in \mathbb{R}^{n \times n}\), \(B\in \mathbb{R}^{n\times p}\) are known real constant matrices, \(\tau>0\) is an unknown constant delay and \(\Delta A\in \mathbb{R}^{n\times n}\) \(\Delta A_d\in \mathbb{R}^{n \times n}\) are matrices of uncertain parameters, \(\Delta B(t)\) represents time-varying parametric uncertainties.
The class describes linear state models with norm-bounded uncertain system parameters and unknown constant state delay. First, a sufficient condition for robust stability independent of the delay is developed. Then, the author provides a sufficient condition that guarantees an \({\mathcal H}_\infty\)-norm bound constraint on the disturbance attenuation for all admissible uncertainties and unknown state delay. In both problems the results are expressed in the form of linear matrix inequalities.

MSC:

93B36 \(H^\infty\)-control
93C23 Control/observation systems governed by functional-differential equations
93D09 Robust stability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dugard, L., & Verriest, E. I. (Eds.). (1997). Stability and control of time-delay systems. New York: Springer.; Dugard, L., & Verriest, E. I. (Eds.). (1997). Stability and control of time-delay systems. New York: Springer. · Zbl 0901.00019
[2] Gorecki, H. S., Fuska, P., Grabowski, S., & Korytowski, A. (1989). Analysis and synthesis of time delay systems. New York: Wiley.; Gorecki, H. S., Fuska, P., Grabowski, S., & Korytowski, A. (1989). Analysis and synthesis of time delay systems. New York: Wiley. · Zbl 0695.93002
[3] Huang, S.; Ren, W., Longitudinal control with time-delay in platooning, Proceedings of IEE Control Theory and Applications, 148, 211-217 (1998)
[4] Kolomanovskii, V., & Myshkis, A. (1992). Applied theory of functional differential equations. New York: Kluwer Academic Publishers.; Kolomanovskii, V., & Myshkis, A. (1992). Applied theory of functional differential equations. New York: Kluwer Academic Publishers.
[5] Lee, C. S.; Leitmann, G., Continuous feedback guaranteeing uniform ultimate boundedness for uncertain linear delay systems: An application to river pollution control, Computer and Mathematical Modeling, 16, 929-938 (1988) · Zbl 0673.93052
[6] Li, X.; de Souza, C. E., Delay-dependent robust stability and stabilization of uncertain linear dealy systems: A linear matrix inequality, IEEE Transactions on Automatic Control, 42, 1144-1148 (1997) · Zbl 0889.93050
[7] Logemann, H.; Townley, S., The effect of small delays in the feedback loop on the stability of neutral systems, Systems and Control Letters, 27, 267-274 (1996) · Zbl 0866.93089
[8] Luo, J. S.; Van Den Bosch, P. P.J., Independent delay stability criteria for uncertain linear state space models, Automatica, 33, 171-179 (1997) · Zbl 0887.93048
[9] Mahmoud, M. S. (1994). Output feedback stabilization of uncertain systems with state delay. In: C. T. Leondes, Advances in theory and applications, San Diego: Academic, vol. 63, (pp. 197-257).; Mahmoud, M. S. (1994). Output feedback stabilization of uncertain systems with state delay. In: C. T. Leondes, Advances in theory and applications, San Diego: Academic, vol. 63, (pp. 197-257). · Zbl 0825.93623
[10] Mahmoud, M. S., Dynamic control of systems with variable state-delay, International Journal of Robust and Nonlinear Control, 6, 123-146 (1996) · Zbl 0853.93091
[11] Mahmoud, M. S., Robust stability and stabilization of a class of uncertain nonlinear systems with delays, Journal of Mathematical Problems in Engineering, 4, 165-185 (1998) · Zbl 0929.93033
[12] Mahmoud, M. S. (1999). Robust control and filtering for time-delay systems. New York: Marcel-Dekker.; Mahmoud, M. S. (1999). Robust control and filtering for time-delay systems. New York: Marcel-Dekker.
[13] Mahmoud, M. S.; Zribi, M., \(H∞\) controllers for time-delay systems using linear matrix inequalities, Journal of Optimization Theory and Applications, 100, 1, 89-122 (1999) · Zbl 0927.93024
[14] Niculescu, S. I., \(H∞\) memoryless control of with an \(α\)-stability constraint for time-delay systems: An LMI approach, IEEE Transactions on Automatic Control, 43, 739-743 (1998) · Zbl 0911.93031
[15] Niculescu, S. I.; de Souza, C. E.; Dugard, L.; Dion, J.-M., Robust exponential stability of uncertain systems with time-varying delays, IEEE Transactions on Automatic Control, 43, 743-748 (1998) · Zbl 0912.93053
[16] O’Conner, D.; Tarn, T. J., On the stabilization by state feedback for neutral differential difference equations, IEEE Transactions on Automatic Control, AC-28, 615-618 (1983) · Zbl 0527.93049
[17] Shen, J. C.; Chen, B. S.; Kung, F. C., Memoryless \(H∞\) stabilization of uncertain dynamic delay systems: Riccati equation approach, IEEE Transactions on Automatic Control, 36, 638-640 (1991)
[18] Slemrod, M.; Infante, E. F., Asymptotic stability criteria for linear systems of differential difference equations of neutral type and their discrete analogues, Journal of Mathematical Analysis and Applications, 38, 399-415 (1972) · Zbl 0202.10301
[19] Verriest, E. J., & Niculescu, S. I. (1997). Delay-independent stability of linear neutral systems: A Riccati equation approach. (Chapter 3) In: L. Dugard, & E. I. Verriest, Stability and control of time-delay systems (pp. 93-100). New York: Springer.; Verriest, E. J., & Niculescu, S. I. (1997). Delay-independent stability of linear neutral systems: A Riccati equation approach. (Chapter 3) In: L. Dugard, & E. I. Verriest, Stability and control of time-delay systems (pp. 93-100). New York: Springer. · Zbl 0923.93049
[20] Xie, L., & de Souza, C. E. (1993). Robust stabilization and disturbance attenuation for uncertain delay systems. Proceedings of the second European control conference, Groningen, (pp. 1145-1149).; Xie, L., & de Souza, C. E. (1993). Robust stabilization and disturbance attenuation for uncertain delay systems. Proceedings of the second European control conference, Groningen, (pp. 1145-1149).
[21] Zhou, K. (1998). Essentials of robust control. New York: Prentice-Hall.; Zhou, K. (1998). Essentials of robust control. New York: Prentice-Hall.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.