On the synthesis of linear \(H_\infty\) filters for polynomial systems. (English) Zbl 1250.93119
Summary: This paper is concerned with the \(H_{\infty }\)-filtering problem for polynomial systems. By means of Lyapunov theory and matrix inequality techniques, sufficient conditions are first obtained to ensure that the filtering error system is asymptotically stable and satisfies \(H_{\infty }\) performance constraint. Then, a sufficient condition for the existence of desired filters is established with a free matrix introduced, which will greatly facilitate the design of filter matrices. By virtue of Sum-Of-Squares (SOS) approaches, a convergent iterative algorithm is developed to tackle the polynomial \(H_{\infty }\) filtering problem. Note that the approach can be efficiently implemented by means of recently developed SOS decomposition techniques, and the filter matrices can be designed explicitly. Finally, a numerical example is given to illustrate the main results of this paper.
MSC:
| 93E10 | Estimation and detection in stochastic control theory |
| 93E11 | Filtering in stochastic control theory |
| 93B50 | Synthesis problems |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C10 | Nonlinear systems in control theory |
Keywords:
fixed-order filtering; \(H_{\infty }\)-filtering; iterative algorithm; polynomial systems; sum of squares (SOS)References:
| [1] | Anderson, B. D.O.; Moore, J. B., Optimal Filtering (2005), Dover Publications: Dover Publications New York · Zbl 0758.93070 |
| [2] | Lewis, F.; Xie, L.; Popa, D., Optimal and Robust Estimation: with an Introduction to Stochastic Control Theory (2008), CRC press · Zbl 1133.93040 |
| [3] | Nagpal, K. M.; Khargonekar, P. P., Filtering and smoothing in an \(H_\infty\) setting, IEEE Trans. Automat. Control, 36, 2, 152-166 (1991) · Zbl 0758.93074 |
| [4] | Gao, H.; Wang, C., Delay-dependent robust \(H_\infty\) and \(L_2-L_\infty\) filtering for a class of uncertain nonlinear time-delay systems, IEEE Trans. Automat. Control, 48, 9, 1661-1666 (2003) · Zbl 1364.93210 |
| [5] | Xu, S.; Chen, T., Reduced-order \(H_\infty\) filtering for stochastic systems, IEEE Trans. Signal Process., 50, 12, 2998-3007 (2002) · Zbl 1369.94325 |
| [6] | Wang, Z.; Yang, F.; Ho, D. W.C.; Liu, X., Robust \(H_\infty\) filtering for stochastic time-delay systems with missing measurements, IEEE Trans. Signal Process., 54, 7, 2579-2587 (2006) · Zbl 1373.94729 |
| [7] | Shen, B.; Wang, Z.; Shu, H.; Wei, G., \(H_\infty\) filtering for nonlinear discrete-time stochastic systems with randomly varying sensor delays, Automatica, 45, 4, 1032-1037 (2009) · Zbl 1162.93039 |
| [8] | Nguang, S. K.; Fu, M., Robust nonlinear \(H_\infty\) filtering, Automatica, 32, 8, 1195-1199 (1996) · Zbl 0855.93086 |
| [9] | de Souza, C. E.; Xie, L.; Wang, Y., \(H_\infty\) filtering for a class of uncertain nonlinear-systems, Systems Control Lett., 20, 6, 419-426 (1993) · Zbl 0784.93095 |
| [10] | Wang, Z.; Liu, Y.; Liu, X., \(H_\infty\) filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica, 44, 5, 1268-1277 (2008) · Zbl 1283.93284 |
| [11] | Abbaszadeh, M.; Marquez, H. J., LMI optimization approach to robust \(H_\infty\) observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems, Internat. J. Robust Nonlinear Control, 19, 3, 313-340 (2009) · Zbl 1163.93027 |
| [12] | Basin, M.; Shi, P.; Calderon-Alvarez, D., Central suboptimal \(H_\infty\) filter design for nonlinear polynomial systems, Internat. J. Adapt. Control Signal Process., 23, 10, 926-939 (2009) · Zbl 1298.93148 |
| [13] | Basin, M.; Soto, P.; Calderon-Alvarez, D., Central suboptimal \(H_\infty\) controller design for linear time-varying systems with unknown parameters, Internat. J. Systems Sci., 42, 5, 709-716 (2011) · Zbl 1233.93030 |
| [14] | Chesi, G., LMI techniques for optimization over polynomials in control: a survey, IEEE Trans. Automat. Control, 55, 11, 2500-2510 (2010) · Zbl 1368.93496 |
| [15] | Prajna, S.; Parrilo, P.; Rantzer, A., Nonlinear control synthesis by convex optimization, IEEE Trans. Automat. Control, 49, 2, 310-314 (2004) · Zbl 1365.93164 |
| [16] | Henrion, D.; Garulli, A., Positive Polynomials in Control (2005), Springer: Springer Berlin · Zbl 1059.93002 |
| [17] | Henrion, D.; Lasserre, J. B., Convergent relaxations of polynomial matrix inequalities and static output feedback, IEEE Trans. Automat. Control, 51, 2, 192-202 (2006) · Zbl 1366.93180 |
| [18] | C. Hol, C. Scherer, Computing optimal fixed order \(H_\infty \); C. Hol, C. Scherer, Computing optimal fixed order \(H_\infty \) |
| [19] | Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach, IEEE Trans. Automat. Control, 50, 3, 365-370 (2005) · Zbl 1365.93365 |
| [20] | Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems (2009), Springer: Springer Berlin · Zbl 1218.93002 |
| [21] | Prajna, S.; Papachristodoulou, A.; Wu, F., Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach, Proc. Asian Control Conf., 157-165 (2004) |
| [22] | S. Prajna, A. Papachristodoulou, P. Seiler, P.A. Parrilo, SOSTOOLS: sum of squares optimization toolbox for MATLAB, Available from http://www.cds.caltech.edu/sostoolshttp://www.mit.edu/ parrilo/sostools; S. Prajna, A. Papachristodoulou, P. Seiler, P.A. Parrilo, SOSTOOLS: sum of squares optimization toolbox for MATLAB, Available from http://www.cds.caltech.edu/sostoolshttp://www.mit.edu/ parrilo/sostools |
| [23] | J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in: Proceedings of the CACSD Conference, Taipei, Taiwan, pp. 284-289, 2004.; J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in: Proceedings of the CACSD Conference, Taipei, Taiwan, pp. 284-289, 2004. |
| [24] | Henrion, D.; Lasserre, J. B.; Löfberg, J., GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw., 24, 761-779 (2009) · Zbl 1178.90277 |
| [25] | Xu, J.; Xie, L.; Wang, Y., Simultaneous stabilization and robust control of polynomial nonlinear systems using SOS techniques, IEEE Trans. Automat. Control, 54, 8, 1892-1897 (2009) · Zbl 1367.93547 |
| [26] | Ma, H.; Yang, G., Fault-tolerant control synthesis for a class of nonlinear systems: Sum of squares optimization approach, Internat. J. Robust Nonlinear Control, 19, 5, 591-610 (2009) · Zbl 1160.93332 |
| [27] | Toker, O.; Ozbay, H., On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback, Proc. Amer. Control Conference, 4, 2525-2526 (1995) |
| [28] | Sturm, J. F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11, 1, 625-653 (1999) · Zbl 0973.90526 |
| [29] | Chesi, G., On the gap between positive polynomials and SOS of polynomials, IEEE Trans. Automat. Control, 52, 6, 1066-1072 (2007) · Zbl 1366.93174 |
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.
