Abstract
An approximation theory of optimal control for nonlinear stochastic dynamic systems has been established. Based on the generalized Hamilton-Jacobi-Bellman equation of the cost function of nonlinear stochastic systems, general iterative procedures for approximating the optimal control are developed by successively improving the performance of a feedback control law until a satisfactory suboptimal solution is achieved. A successive design scheme using upper and lower bounds of the exact cost function has been developed for the infinite-time stochastic regulator problem. The determination of the upper and lower bounds requires the solution of a partial differential inequality instead of equality. Therefore it provides a degree of flexibility in the design method over the exact design method. Stability of the infinite-time sub-optimal control problem was established under not very restrictive conditions, and stable sequences of controllers can be generated. Several examples are used to illustrate the applicati on of the proposed approximation theory to stochastic control. It has been shown that in the case of linear quadratic Gaussian problems, the approximation theory leads to the exact solution of optimal control.
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References
Al’brekht, E. G. (1961). On the optimal stabilization of nonlinear systems, J. Appl. Math. Mech. (ZPMM), 25,5, 1254–1266.
Aoki, M. (1967). Optimization of Stochastic Systems, Academic Press, N.Y.
Bellman, R. (1956). Dynamic Programming, Princeton University Press, Princeton, N.J.
Doob, J. L. (1953). Markov Processes. Wiley, N.Y.
Dynkin, E. B. (1953). Stochastic Processes. Academic Press, N.Y.
Itô, K. (1951). On Stochastic Differential Equations? Memn Amer. Math. Soc., 4.
Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 4, 106.
Kushner, H. (1971). Introduction to Stochastic Control, Holt, Reinhardt and Winston, NY.
Kwakernaak, H. and R. Sivan. (1972). Linear Optimal Control Systems, Wiley, N.Y.
Leake, R.J. and R.-W. Liu. (1967). Construction of suboptimal control sequences, SIAM J. Control, 5,1, 54–63.
Lukes, D. L. (1969). Optimal regulation of nonlinear dynamical systems. SIAM J. Control, 7,1, 75–100.
Nishikawa, Y., N. Sannomiya and H. Itakura. (1962). A method for suboptimal design of nonlinear feedback systems, Automatica, 7,6, 703–712.
Ohsumi, A. (1984). Stochastic control with searching a randomly moving target, Proc. Of American control Conference, San Diego, CA, 500–504.
Panossian, H. V. (1988). Algorithms and computational techniques in stochastic optimal control, C.T. Lenodes (ed.), Control and Dynamic Systems, 28,1
Rekasius, Z.V. (1964). Suboptimal design of intentionally nonlinear controllers. IEEE Trans. Automatic Control, AC-9,4, 380–386.
Sage, A. P. and C. C. White. (1977). Optimun Systems Control, Prentice-Hall, Englewood Cliffs, N.J.
Saridis, G. N. and Fei-Yue Wang. (1994). Suboptimal control of nonlinear stochastic systems, Control — Theory and Advanced Technology, Vol. 10,No. 4, Part l, pp.847–871.
Saridis, G. N. (1988). Entropy formulation for optimal and adaptive control. IEEE Trans. Automatic Control, AC-33,8, 713–721.
Saridis, G. N. and J. Balaram. (1986). Suboptimal control for nonlinear systems. Control-Thoery and Advanced Technology (C-TAT), 2.3, 547–562.
Saridis, G. N. and C. S. G. Lee. (1976). An approximation theory of optimal control for trainable manipulators. IEEE Trans. Systems, Man, and Cybernetics, SMC-9,3, 152–159.
Skorokhod, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.
Wang, Fei-Yue and G. N. Saridis. (1992). Suboptimal control for nonlinear stochastic systems, Proceedings of 31st Conference on Decision and control, Tucson, AZ, Dec.
Wonham, W. M. (1970). Random differential equations in control theory. A. T. BharuchaReid (ed.), Probabilistic Methods in Applied Mathematics, Academic Press, NY.
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Wang, FY., Saridis, G.N. (2002). On Successive Approximation of Optimal Control of Stochastic Dynamic Systems. In: Dror, M., L’Ecuyer, P., Szidarovszky, F. (eds) Modeling Uncertainty. International Series in Operations Research & Management Science, vol 46. Springer, New York, NY. https://doi.org/10.1007/0-306-48102-2_16
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DOI: https://doi.org/10.1007/0-306-48102-2_16
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