Abstract
This paper considers the adaptive control of discrete-time hybrid stochastic systems with unknown randomly jumping parameters described by a finite-state hidden Markov chain. An intuitive yet longstanding conjecture in this area is that such hybrid systems can be adaptively stabilized whenever the rate of transition of the hidden Markov chain is small enough. This paper provides a rigorous positive answer to this conjecture by establishing the global stability of a gradient-algorithm-based adaptive linear-quadratic control.
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References
Elliott, R. J., Aggoun, L., Moore, J. B., Hidden Markov Models, Estimation and Control, New York: Springer-Verlag, 1995.
Krasovskii, N. N., Lidskii, E. A., Analytic design of controller in systems with random attributes, Part I–III, Automat. Remote Contr., 1961, 22: 1021; 1141; 1289.
Sworder, D. D., Feedback control of a class of linear systems with jump parameters, IEEE Trans. Automatic Contr., 1969, AC-14: 9.
Wonham, W. M., Random differential equations in control theory, Probabilistic Methods in Applied Mathematics, Vol II, (ed. Bharucha, A. T.), New York: Academic, 1971, 131–213.
Blair, W. P. Jr., Sworder, D. D., Feedback control of linear discrete systems with jump parameters and quadratic criteria, Int. J. Contr., 1975, 21(5): 833.
Chizeck, H. J., Willsky, A. S., Castanon, D., Discreet-time Markovian-jump linear quadratic optimal control, Int. J. Contr., 1986, 43(1): 231.
Ji, Y. D., Chizeck, H. J., Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automat. Control, 1990, AC-35(7): 777.
Ji, Y. D., Chizeck, H. J., Jump linear quadratic Gaussian control: steady-state solution and testable conditions, Control Theory and Advanced Technology, 1990, 6(3): 289.
Ji, Y. D., Chizeck, H. J., Controllability, observability, and discrete-time Markovian jump linear quadratic control, Int. J. Contr., 1988, 48(2): 481.
Mariton, M., Bertrand, P., Output feedback for a class of linear systems with stochastic parameters, IEEE Trans. Automat. Control, 1985, AC-30(9): 898.
Caines, P. E., Chen, H. F., Optimal adaptive LQG control for systems with finite state process parameters, IEEE Trans. Automat. Control, 1985, AC-30: 185.
Caines, P. E., Zhang, J. F., On the adaptive control of jump parameter systems via nonlinear filtering, SIAM J. Control and Optimization, 1995, 33(6): 1758.
Dufour, F., Bertrand, P., Stabilizing control law for hybrid models, IEEE Transactions on Automatic Control, 1994, 39 (11): 2354.
Sworder, D. D., Hybrid adaptive control, Appl. Math. Comp., 1991, 45: 173.
Everdij, M. H. C., Blom, H. A. P., Embedding adaptive JLQG into LQ martingale control with a complete observable stochastic control matrix, IEEE Trans. Automat. Control, 1996, AC-41: 424.
Dufour, F., Elliott, R. J., Adaptive control of linear systems with Markov perturbations, IEEE Trans. Automat. Control, 1998, AC-43: 351.
Guo, L., On adaptive stabilization of time-varying stochastic systems, SIAM J. Control and Optimization, 1990, 28(6): 1432.
Ford, J. J., Moore, J. B., Adaptive estimation of HMM transition probability, IEEE Trans. Signal Processing, 1998, SP-46: 1374.
Guo, L., Ljung, L., Wang, G. J., Necessary and sufficient condition for stability of LMS, IEEE Trans. Automat. Control, 1997: 761.
Ioannou, P. A., Sun, J., Robust Adaptive Control, Upper Saddle River, NJ: Prenticce-Hall, 1996.
Naik, S. M., Kumar, P. R., Robust indirect adaptive control of time-varying plants with unmodeled dynamics and disturbances, SIAM J. Control and Optimization, 1994, 32(6): 1696.
Delchamps, D., Analytic feedback control and the algebraic Riccati equation, IEEE Trans. Automat. Control, 1984, AC-29: 1031.
Doob, J. L., Stochastic Processes, New York: Wiley, 1953.
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Huang, M., Guo, L. Stabilization of stochastic systems with hidden Markovian jumps. Sci China Ser F 44, 104–118 (2001). https://doi.org/10.1007/BF02713969
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DOI: https://doi.org/10.1007/BF02713969