A New Method To H2 Robust Filter Design

In this paper a new method to H2 robust filter design is proposed. Both continuous and discrete time settings are considered for systems subject to polytopic parameter uncertainty. Lower and upper bounds of the true cost are determined in order to evaluate the degree of sub-optimality of the proposed robust filter. The design method is based on the parametrization of all robust filters as a convex combination of Kalman filters associated to each vertex of the uncertainty domain. Among all feasible filters, the one minimizing a guaranteed H2 cost of the estimation error is determined by a pure convex programming problem, expressed in terms of linear matrix inequalities (LMIs). The order of the robust filter is generally greater than the order of the plant, a fact that contributes to reduce conservatism. The proposed design technique is compared with other methods available in the literature. In several examples solved the proposed method outperforms all other designs. © 2008 Elsevier Inc. All rights reserved.4301145154Anderson, B.D.O., Moore, J.B., (1979) Optimal Filtering, , Prentice-Hall, Englewood Cliffs, NJBarbosa, K.A., de Souza, C.E., Trofino, A., Robust H2 filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions (2005) Syst. Contr. Lett., 54 (3), pp. 251-262Boyd, S.P., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , SIAM, PhiladelphiaColaneri, P., Geromel, J.C., Locatelli, A., (1997) Control Theory and Design - An RH2 and RH∞ Viewpoint, , Academic Pressde Oliveira, M.C., Geromel, J.C., Hsu, L., LMI characterization of structural and robust stability: the discrete-time case (1999) Linear Algebra Appl., 296, pp. 27-38Geromel, J.C., Regis, L.A.V., H2 Optimal robust filtering (2006) Eur. J. Control, 12 (1), pp. 30-39Geromel, J.C., de Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM J. Control Optim., 41 (3), pp. 700-711Geromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Trans. Signal Process., 47 (1), pp. 168-175Geromel, J.C., de Oliveira, M.C., Hsu, L., LMI characterization of structural and robust stability (1998) Linear Algebra Appl., 285, pp. 69-80Jain, B.N., Guaranteed error estimation in uncertain systems (1975) IEEE Trans. Automat. Contr., 20, pp. 230-232Li, L., Luo, Z.Q., Davidson, T.N., Wong, K.M., Bossé, E., Robust filtering via semidefinite programming with applications to target tracking (2002) SIAM J. Optim., 12 (3), pp. 740-755Marcos, S., A network of adaptive Kalman filters for data channel equalization (2000) IEEE Trans. Signal Process., 48 (9), pp. 2620-2627Martin, C.J., Mintz, M., Robust filtering and prediction for linear systems with uncertain dynamics: a game-theoretic approach (1983) IEEE Trans. Automat. Contr., 28 (9), pp. 888-896Poor, H.V., On robust Wiener filtering (1980) IEEE Trans. Automat. Contr., 25 (3), pp. 531-536Rockafellar, R., (1970) Convex Analysis, , Princeton PressShaked, U., Xie, L., Soh, Y.C., New approaches to robust minimum variance filter design (2001) IEEE Trans. Signal Process., 49 (11), pp. 2620-2629de Souza, C.E., Trofino, A., An LMI approach to the design of robust H2 filters (1999) Recent Advances on Linear Matrix Inequalities Methods in Control, , El Ghaoui L., and Niculescu S.-I. (Eds), SIAM, Englewood Cliffs, NJTheodor, Y., Shaked, U., Robust discrete-time minimum-variance filtering (1996) IEEE Trans. Signal Process., 44, pp. 181-189Tuan, H.D., Apkarian, P., Nguyen, T.Q., Robust and reduce-order filtering: new LMI-based characterizations and methods (2001) IEEE Trans. Signal Process., 49 (12), pp. 2975-2984L.H. Xie, Y.C. Soh, C. Du, Robust H2 estimation and control, Sch. Elect. Electron Eng., Nanyang Technol. Univ., Singapore, Tech. Rep., 1999Xie, L.H., Soh, Y.C., Robust Kalman filtering for uncertain systems (1994) Syst. Contr. Lett., 22, pp. 123-129Xie, L.H., Soh, Y.C., de Souza, C.E., Robust Kalman filtering for uncertain discrete-time systems (1994) IEEE Trans. Automat. Contr., 39, pp. 1310-131

H2 Optimal Robust Filtering

In this paper, a design procedure for H2 optimal robust filtering is presented. The robust filter is determined from the equilibrium solution of a minimax programming problem where the H2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all full order, linear, rational and causal filters. As the main contribution, it is shown that for the class of parameter uncertainty considered, the equilibrium solution of the aforementioned minimax problem can be exactly determined. In contrast to the design methods available in the literature to date, dealing with norm bounded or convex bounded parameter uncertainty, which naturally provide suboptimal solutions to the robust filtering problem, the one presented in this paper does not present any degree of conservatism due to the particular parameter uncertainty model taken into consideration. However, the price to be payed to reach robust optimality is that the order of the optimal robust filter is, in general, very high. The classical static linear approximation problem as well as the filter design problem corresponding to continuous and discrete time linear systems are considered. In the last section, an illustrative example is presented to make clear the main features of the reported results. © 2006 EUCA.1213039Anderson, B.D.O., Moore, J.B., (1979) Optimal Filtering, Englewood Cliffs, , NJ, Prentice HallBen-Tal, A., Nemirovski, A., Roos, C., Robust solutions to uncertain quadratic and conic-quadratic optimization problems (2002) SIAM J Optim, 13 (2), pp. 535-560Bertsekas, D.P., (1995) Nonlinear Programming, , Athena Scientific, USABoyd, S.P., El, G.L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities In System and Control Theory, , SIAM, PhiladelphiaColaneri, P., Geromel, J.C., Locatelli, A., (1997) Control Theory and Design - An RH2 and RH1 Viewpoint, , Academic Press, London, UKGeromel, J.C., de Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM J Control Optim, 41 (3), pp. 700-711Geromel, J.C., Bernussou, J., Garcia, G., de Oliveira, M.C., H2 and H1 robust filtering for discrete-time linear systems (2000) SIAM J Control Optim, 38 (5), pp. 1353-1368Geromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Trans Signal Process, 47 (1), pp. 168-175Hindi, H.A., Boyd, S.P., (1998) Robust Solutions to '1, '2 and '1 Uncertain Linear Approximation Problems Using Convex Optimization, pp. 3487-3490. , In: Proceedings of the American Control Conference, Philadelphia, PA, USAJain, B.N., Guaranteed error estimation in uncertain systems (1975) IEEE Trans Autom Control, 20, pp. 230-232Kassam, S.A., Poor, H.V., Robust techniques for signal processing: A Survey (1985) Proc. IEEE, 73 (3), pp. 433-481Li, L., Luo, Z.Q., Davidson, T.N., Wong, K.M., Bossé, E., Robust filtering via semidefinite programming with applications to target tracking (2002) SIAM J On Optim, 12 (3), pp. 740-755Liu, J., Wang, J.L., Yang, G.-H., Reliable Guaranteed Variance Filtering Against Sensor Failures (2003) IEEE Trans Signal Process, 51 (5), pp. 1403-1411Martin, C.J., Mintz, M., Robust filtering and prediction for linear systems with uncertain dynamics: A gametheoretic approach (1983) IEEE Trans Autom Control, 28 (9), pp. 888-896Poor, H.V., On robust Wiener filtering (1980) IEEE Trans Autom Control, 25 (3), pp. 531-536Rockafellar, R.T., (1970) Convex Analysis, , Princeton Press, Princeton, NJ, USAShaked, U., de Souza, C.E., Robust minimum variance filtering (1995) IEEE Trans Signal Process, 43 (11), pp. 2474-2483Shaked, U., Xie, L., Soh, Y.C., New approaches to robust minimum variance filer design (2001) IEEE Trans Signal Process, 49 (11), pp. 2620-2629Tuan, H.D., Apkarian, P., Nguyen, T.Q., Robust and reduce-order filtering: New LMI-based characterizations and methods (2001) IEEE Trans Signal Process, 49 (12), pp. 2975-2984Xie, L., Soh, Y.C., Robust Kalman filtering for uncertain systems (1994) Syst. Control Lett, 22, pp. 123-129Wang, F., Balakrishnan, V., Robust steady-state filtering for systems with deterministic and stochastic uncertainties (2003) IEEE Trans Signal Process, 51 (10), pp. 2550-255

H2 Robust Filter Design With Performance Certificate Via Convex Programming

In this paper a new approach to H2 robust filter design is proposed. Both continuous- and discrete-time invariant systems subject to polytopic parameter uncertainty are considered. After a brief discussion on some of the most expressive methods available for H2 robust filter design, a new one based on a performance certificate calculation is presented. The performance certificate is given in terms of the gap produced by the robust filter between lower and upper bounds of a minimax programming problem where the H2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all linear, rational and causal filters. The calculations are performed through convex programming methods developed to deal with linear matrix inequality (LMI). Many examples borrowed from the literature to date are solved and it is shown that the proposed method outperforms all other designs. © 2007 Elsevier Ltd. All rights reserved.444937948Anderson, B.D.O., Moore, J.B., (1979) Optimal filtering, , Prentice Hall, Englewood Cliffs, NJBarbosa, K.A., de Souza, C.E., Trofino, A., Robust H2 filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions (2005) Systems & Control Letters, 54 (3), pp. 251-262Boyd, S.P., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear matrix inequalities in system and control theory, , SIAM, PhiladelphiaColaneri, J., Geromel, C., Locatelli, A., (1997) Control theory and design-An RH2 and RH∞ viewpoint, , Academic Press, New YorkGeromel, J.C., Convex analysis and global optimization of joint actuator location and control problems (1989) IEEE Transactions on Automatic Control, 34 (7), pp. 711-720Geromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Transactions on Signal Processing, 47 (1), pp. 168-175Geromel, J.C., de Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM Journal on Control Optimization, 41 (3), pp. 700-711Geromel, J.C., de Oliveira, M.C., Hsu, L., LMI characterization of structural and robust stability (1998) Linear Algebra and Its Applications, 285, pp. 69-80Geromel, J.C., Regis, L.A.V., H2 Optimal robust filtering (2006) European Journal of Control, 12 (1), pp. 30-39Hoang, N.T., Tuan, H.D., Apkarian, P., Hosoe, S., Robust filtering for discrete nonlinear fractional transformation systems (2004) IEEE Transactions Circuit and Systems II-Express Briefs, 51, pp. 587-592Jain, B.N., Guaranteed error estimation in uncertain systems (1975) IEEE Transactions on Automatic Control, 20, pp. 230-232Li, L., Luo, Z.Q., Davidson, T.N., Wong, K.M., Bossé, E., Robust filtering via semidefinite programming with applications to target tracking (2002) SIAM Journal on Optimization, 12 (3), pp. 740-755Martin, C.J., Mintz, M., Robust filtering and prediction for linear systems with uncertain dynamics: A game-theoretic approach (1983) IEEE Transactions on Automatic Control, 28 (9), pp. 888-896Meirovitch, L., Baruh, H., Oz, H., A comparison of control techniques for large flexible systems (1983) J. Guidance, 6 (4), pp. 302-310de Oliveira, M.C., Geromel, J.C., Hsu, L., LMI characterization of structural and robust stability: The discrete-time case (1999) Linear Algebra and Its Applications, 296, pp. 27-38Poor, H.V., On robust wiener filtering (1980) IEEE Transactions on Automatic Control, 25 (3), pp. 531-536Rockafellar, R., (1970) Convex analysis, , Princeton PressScherer, C. W., Köse, I. E. (2006). Robust H2 estimation with dynamic IQCs: A convex solution. In IEEE Conference on Decision and Control, USAShaked, U., Xie, L., Soh, Y.C., New approaches to robust minimum variance filter design (2001) IEEE Transactions on Signal Processing, 49 (11), pp. 2620-2629de Souza, C.E., Trofino, A., An LMI approach to the design of robust H2 filters (1999) Recent Advances on linear matrix inequalities methods in control, , El Ghaoui L., and Niculescu S.-I. (Eds), SIAM, Englewood Cliffs, NJTheodor, Y., Shaked, U., Robust discrete-time minimum variance filtering (1996) IEEE Transactions on Signal Processing, 44, pp. 181-189Tuan, H.D., Apkarian, P., Nguyen, T.Q., Robust and reduce-order filtering : New LMI-based characterizations and methods (2001) IEEE Transactions on Signal Processing, 49 (12), pp. 2975-2984Xie, L.H., Soh, Y.C., Robust Kalman filtering for uncertain systems (1994) Systems Control Letters, 22, pp. 123-129Xie, L.H., Soh, Y.C., de Souza, C.E., Robust Kalman filtering for uncertain discrete-time systems (1994) IEEE Transactions on Automatic Control, 39, pp. 1310-1314Xie, L. H., Soh, Y. C., Du, C. (1999). RobustH2 estimation and control. School of Electrical and Electronic Engineering Nanyang Technological University, Singapore, Technical Repor