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Non-fragile $H_\infty$ filter design for linear continuous-time systems. (English) Zbl 1152.93365
Summary: This paper studies the problem of non-fragile $H_\infty$ filter design for linear continuous-time systems. The filter to be designed is assumed to include additive gain variations, which result from filter implementations. A notion of structured vertex separator is proposed to approach the problem, and exploited to develop sufficient conditions for the non-fragile $H_\infty$ filter design in terms of solutions to a set of Linear Matrix Inequalities (LMIs). The designs guarantee the asymptotic stability of the estimation errors, and the $H_\infty$ performance of the system from the exogenous signals to the estimation errors below a prescribed level. A numerical example is given to illustrate the effect of the proposed method.

##### MSC:
 93B36 $H^\infty$-control 93C05 Linear control systems 93D20 Asymptotic stability of control systems
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##### References:
 [1] Dorato, P.: Non-fragile controller design, an overview. Proceedings of American control conference 5, 2829-2831 (1998) [2] Famularo, D.; Dorato, P.; Abdallah, C. T.; Haddad, W. M.; Jadbabais, A.: Robust non-fragile LQ controllers: the static state feedback case. International journal of control 73, No. 2, 159-165 (2000) · Zbl 1006.93514 [3] Fu, M.; De Souza, C. E.; Xie, L.: H$\infty$estimation for uncertain systems. International journal of robust nonlinear control 2, 87-105 (1992) · Zbl 0765.93032 [4] Gao, H. J.; Lam, J.; Wang, C. H.: Induced L2 and generalized H2 filtering for systems with repeated scalar nonlinearities. IEEE transactions on signal processing 53, No. 11, 4215-4226 (2005) [5] Geromel, J. C.; De Oliviera, M. C.: H2 and H$\infty$robust filtering for convex bounded uncertain systems. IEEE transactions on automatic control 46, No. 1, 100-107 (2001) · Zbl 1056.93628 [6] Haddad, W. M.; Corrado, J. R.: Robust resilient dynamic controllers for systems with parametric uncertainty and controller gain variations. International journal of control 73, No. 15, 1405-1423 (2000) · Zbl 1062.93503 [7] Ho, M. -T.; Datta, A.; Bhatacharyya, S. P.: Robust and non-fragile PID controller design. International journal of robust and nonlinear control 11, No. 7, 681-708 (2001) · Zbl 0993.93009 [8] Iwasaki, T.; Shibata, G.: LPV system analysis via quadratic separator for uncertain implicit systems. IEEE transactions on automatic control 46, 1195-1208 (2001) · Zbl 1006.93053 [9] Jadbabaie, A.; Abdallah, C. T.; Famularo, D.; Dorato, P.: Robust, non-fragile and optimal controller design via linear matrix inequalities. Proceedings of American control conference 5, 2842-2846 (1998) [10] Keel, L. H.; Bhatacharyya, S. P.: Robust, fragile, or optimal?. IEEE transactions on automatic control 42, No. 8, 1098-1105 (1997) · Zbl 0900.93075 [11] Li, H.; Fu, M.: A linear matrix inequality approach to robust H$\infty$filtering. IEEE transaction on signal processing 45, 2338-2350 (1997) [12] Li, G.: On the structure of digital controller with finite word length consideration. IEEE transaction on automatic control 43, 689-693 (1998) · Zbl 0989.93535 [13] Mahmoud, M. S.: Resilient linear filtering of uncertain systems. Automatica 40, 1797-1802 (2004) · Zbl 1162.93403 [14] Mahmoud, M. S.: Resilient L2-L1 filtering of polytopic systems with state delays. IET control theory and applications 1, No. 1, 141-154 (2007) [15] Nagpal, K. M.; Khargonekar, P. P.: Filtering and smoothing in an H$\infty$setting. IEEE transaction on automatic control 36, No. 2, 152-166 (1991) · Zbl 0758.93074 [16] De Oliveira, M. C.; Geromel, J. C.: H2 and H$\infty$filtering design subject to implementation uncertainty. SIAM journal on control and optimization 44, No. 2, 515-530 (2006) [17] Palhares, R. M.; Peres, P. L. D.: Robust H$\infty$filtering design with pole placement constraint via lmis. Journal of optimization theory and applications 102, No. 2, 239-261 (1999) · Zbl 0941.93018 [18] Petersen, I. R.: A stabilization algorithm for a class of uncertain linear systems. System & control letters 8, No. 5, 351-357 (1987) · Zbl 0618.93056 [19] Peaucelle, D., Arzelier, D., & Farges, C. (2004). LMI results for resilient state-feedback with H\infty performance. In Proceedings of the 43th IEEE conference on decision and control (pp. 400-404) [20] Scherer, C. W. (1997). A full block S-procedure with applications. In Proceedings of the 36th IEEE conference on decision and control (pp. 2602-2607) [21] Takahashi, R. H. C., Dutra, D. A., Palhares, R. M., Peres, P. L. D. (2000). On robust non-fragile static state-feedback controller synthesis. In Proceedings of the 39th IEEE conference on decision and control, (pp. 4909-04914) [22] Yang, G. -H.; Wang, J. L.; Lin, C.: Non-fragile H$\infty$control for linear systems with additive controller gain variations. International journal of control 73, No. 16, 1500-1506 (2000) · Zbl 1009.93020 [23] Yang, G. -H.; Wang, J. L.: Robust nonfragile Kalman filtering for uncertain linear systems with estimation gain uncertainty. IEEE transaction on automatic control 46, No. 2, 343-348 (2001) · Zbl 1056.93635 [24] Yang, G. -H.; Wang, J. L.: Non-fragile H$\infty$control for linear systems with multiplicative controller gain variations. Automatica 37, No. 5, 727-737 (2001) · Zbl 0990.93031 [25] Yang, G. -H.; Wang, J. L.: Non-fragile H$\infty$output feedback controller design for linear systems. Journal of dynamic systems measurement and control transactions of the ASME 125, No. 1, 117-123 (2003) [26] Yaz, E. E.; Jeong, C. S.; Yaz, Y. I.: An LMI approach to discrete-time observer design with stochastic resilience. Journal of computational and applied mathematics 188, No. 2, 246-255 (2006) · Zbl 1108.93026 [27] Yaesh, I.; Shaked, U.: Game theory approach to optimal linear estimation in the minimum H$\infty$norm sense. IEEE transaction on automatic control 37, No. 6, 828-831 (1992) · Zbl 0769.90087 [28] Zhou, K.; Doyle, J. C.; Glover, K.: Robust and optimal control. (1996) · Zbl 0999.49500