Abstract
The problem of estimation and control for discrete-time systems with multiplicative noise is examined. Such systems occur naturally in the modeling of stochastic systems with random or unknown coefficients and appear to be robust in contrast to LQG regulators which are sensitive to errors in the coefficients.
The statistics of the white sequences of the system are unknown. The problem of stochastic estimation and control of such a system is difficult not only because of the unknown statistics but also because the state is not Gaussian.
The approach of this work is to convert the stochastic problem to a deterministic game-theoretic one. We find the estimator and controller so as to minimize a suitable performance measure assuming the worst behavior of nature.
A set of necessary and sufficient conditions is developed for the existence of a saddle-point estimator. When both estimation and control are considered, two difficulties appear: the optimality conditions are only necessary and the separation principle collapses. As a result, the saddle-point conditions are only necessary. If the covariances belong to sets with maximal points, then the necessary conditions are satisfied at these points. If, on the other hand, they belong to convex and compact sets and the system has a steady state, then the estimation problem alone has always a saddle-point solution.
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References
Sagirow, P.,Stochastic Methods in the Dynamics of Satellites, Lecture Notes, CISM, Udine, Italy, 1970.
Haussmann, U. G.,Optimal Stationary Control with State and Control Dependent Noise, SIAM Journal on Control, Vol. 9, pp. 184–198, 1971.
Golubev, G. A.,Minimax Linear Filters of Coordinates of Dynamic Objects, Tehniceskaja Kibernetica, No. 3, pp. 155–162, 1978 (in Russian).
Golubev, G. A.,Minimax Linear Filtering of Dynamic Discrete-Time Processes, Automation and Remote Control, Vol. 2, pp. 203–211, 1984.
Poor, V., andLooze, D. P.,Minimax State Estimation for Linear Stochastic Systems with Noise Uncertainty, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 902–906, 1981.
Looze, D. P., Poor, V., Vastola, K., andDarragh, J. C.,Minimax Control of Linear Stochastic Systems with Noise Uncertainty, IEEE Transactions on Automatic Control, Vol. AC-29, pp. 882–888, 1983.
Phillis, Y. A.,Estimation and Control of Systems with Unknown Covariance and Multiplicative Noise, Proceedings of the 26th IEEE Conference on Decision and Control, Los Angeles, California, pp. 321–325, 1987.
Fan, K.,Minimax Theorems, Proceedings of the Academy of Sciences of the USA, Vol. 39, pp. 42–47, 1953.
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Communicated by C. T. Leondes
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Phillis, Y.A. Optimal estimation and control of discrete multiplicative systems with unknown second-order statistics. J Optim Theory Appl 64, 153–168 (1990). https://doi.org/10.1007/BF00940029
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DOI: https://doi.org/10.1007/BF00940029