Abstract
This paper solves the exact pole assignment problem for the single-input stochastic systems with unknown coefficients under the controllability assumption which is necessary and sufficient for the arbitrary pole assignment for systems with known coefficients. The system noise is required to be mutually independent with zero mean and bounded second moment. Two approaches to solving the problem are proposed: One is the iterative learning approach which can be applied when the state at a fixed time can be repeatedly observed with different feedback gains; the other is the adaptive control approach which works when the trajectories satisfy a nondegeneracy condition. Both methods are essentially based on stochastic approximation, and the feedback gains are recursively given without invoking the certainty-equivalency-principle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wonham, W. M., Linear Multivariable Control, New York: Springer-Verlag, 1974.
Kailath, T., Linear Systems, Cliffs, Englewood, Prentice Hall, 1980.
Lozano, R., Zhao, X. H., Adaptive pale placement without excitation probe signals, IEEE Trans. Automat. Contr., 1994, 39: 47.
Guo, L., Self-convergence of weighted least-squares with applications to stochastic adaptive control, IEEE Trans. Automat. Control., 1996, 41: 7889.
Nassiri-Toussi, K., Ren, W., Indirect adaptive pole-placement control of MIMO stochastic systems: self-tuning results, IEEE Trans. Autom. Control, 1997, 42: 38.
Chen, H. F., Cao, X. R., Controllability is not necessary for adaptive pole placement control, IEEE Trans. Automat. Contr., 1997, 42: 1222.
Chen, H. F., Zhang, J. F., Adaptive stabilization of unstable and nonminimum phase stochastic systems, Syst. Contr. Lett., 1993, 20: 27.
Vidyasagar, M., A Theory of Learning and Generalization, With Applications to Neural Networks and Control Systems, London: Springer-Verlag, 1997.
Sontag, E. D., A learning result for continuous time recurrent neural networks, Syst. Contr. Lett., 1998, 34: 151.
Chen, Y. Q., Gong, Z. M., Wou, C. Y., Analysis of a high-order iterative learning control algorithm for uncertain nonlinear systems with state delays, Automatica, 1998, 34(3): 345.
Xu, J. X., Direct learning of control efforts for trajectories with different time scales, IEEE Trans. Automat. Contr., 1998, 43: 1027.
Chen, H. F., Stochastic approximation and its new applications in Proceedings of 1994 Hong Kong International Workshop on New Directions of Control and Manufacturing, Hong Kong, 1994, 2–12.
Chen, H. F., Duncan, T., Pasik-Duncan, B., A Kiefer-Wolfowitz algorithm with randomized differences, IEEE Trans. Automat. Contr., 1999, 4: 442.
Chen, H. F., Zhu, Y. M., Stochastic Approximation, Shanghai: Shanghai Scientific & Technical Publishers, 1996.
Kushner, H. J., Clark, D. S., Stochastic Approximation for Constrained and Unconstrained Systems, New York: Springer-Verlag, 1978.
Benveniste, A., Métivier, M., Priouret, P., Adaptive Algorithms and Stochastic Approximations, Berlin: Springer-Verlag, 1990.
Kushner, H. J., Yin, G., Stochastic Approximation Algorithms and Applications, New York: Springer-Verlag, 1997.
Sontag, E. D., Mathematical Control Theory, New York: Springer-Verlag, 1990.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, H., Cao, X. Pole assignment for stochastic systems with unknown coefficients. Sci. China Ser. E-Technol. Sci. 43, 313–323 (2000). https://doi.org/10.1007/BF02916836
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02916836