Discrete-time min-max tracking

The problem of optimal state estimation of linear discrete-time systems with measured outputs that are corrupted by additive white noise is addressed. Such estimation is often encountered in problems of target tracking where the target dynamics is driven by finite energy signals, whereas the measurement noise is approximated by white noise. The relevant cost function for such tracking problems is the expected value of the standard H/sub /spl infin// performance index, with respect to the measurement noise statistics. The estimator, serving as a tracking filter, tries to minimize the mean-square estimation error, and the exogenous disturbance, which may represent the target maneuvers, tries to maximize this error while being penalized for its energy. The solution, which is obtained by completing the cost function to squares, is shown to satisfy also the matrix version of the maximum principle. The solution is derived in terms of two coupled Riccati difference equations from which the filter gains are derived. In the case where an infinite penalty is imposed on the energy of the exogenous disturbance, the celebrated discrete-time Kalman filter is recovered. A local iterations scheme which is based on linear matrix inequalities is proposed to solve these equations. An illustrative example is given where the velocity of a maneuvering target has to be estimated utilizing noisy measurements of the target position.

[1]  U. Shaked,et al.  A transfer function approach to the problems of discrete-time systems: H/sub infinity /-optimal linear control and filtering , 1991 .

[2]  H. Rotstein,et al.  An exact solution to general four-block discrete-time mixed H2/H∞ problems via convex optimization , 1998, IEEE Trans. Autom. Control..

[3]  Thia Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software , 2001 .

[4]  I. Yaesh,et al.  Transfer Function Approach to the Problems of Discrete-time Systems : H_∞-optimal Linear Control and Filtering , 1991 .

[5]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[6]  M. Sznaier,et al.  An exact solution to general 4-blocks discrete-time mixed H/sub 2//H/sub /spl infin// problems via convex optimization , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[7]  H. W. Sorenson,et al.  Kalman filtering : theory and application , 1985 .

[8]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[9]  Paul Zarchan,et al.  Tactical and strategic missile guidance , 1990 .

[10]  E. Tse,et al.  A direct derivation of the optimal linear filter using the maximum principle , 1967, IEEE Transactions on Automatic Control.

[11]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[12]  Isaac Yaesh,et al.  Min-max Kalman filtering , 2004, Syst. Control. Lett..

[13]  U. Shaked,et al.  Game theory approach to optimal linear estimation in the minimum H/sup infinity /-norm sense , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[14]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[15]  H. Kwakernaak.  Minimax frequency domain performance and robustness optimization of linear feedback systems , 1985 .

[16]  G. Zames.  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .