Abstract
This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix Ω(q) (formed using the given data) is negative definite for allq ∈Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.
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References
Leitmann, G.,Guaranteed Asymptotic Stability for Some Linear Systems with Bounded Uncertainties, Journal of Dynamic Systems, Measurement, and Control, Vol. 101, No. 3, 1979.
Gutman, S., andLeitmann, G.,Stabilizing Control for Linear Systems with Bounded Parameter and Input Uncertainty, Proceedings of 7th IFIP Conference on Optimization Techniques, Springer-Verlag, Berlin, Germany, 1975.
Gutman, S., andLeitmann, G.,Stabilizing Feedback Control for Dynamical Systems with Bounded Uncertainty, Proceedings of the IEEE Conference on Decision and Control, Clearwater Beach, Florida, 1976.
Gutman, S.,Uncertain Dynamical Systems, A Lyapunov Min-Max Approach, IEEE Transactions on Automatic Control, Vol. AC-24, No. 3, 1979.
Leitmann, G.,On the Efficacy of Nonlinear Control in Uncertain Linear Systems, Proceedings of the 1980 Winter Meeting of ASME, Chicago, Illinois (to appear).
Leitmann, G.,Stabilization of Dynamical Systems under Bounded Input and Parameter Uncertainty, Proceedings of the 2nd Kingston Conference on Differential Games and Control Theory, Marcel Dekker, New York, New York, 1976.
Leitmann, G.,Guaranteed Ultimate Boundedness for a Class of Uncertain Linear Dynamical Systems, IEEE Transactions on Automatic Control, Vol. AC-23, No. 6, 1978.
Leitmann, G.,Guaranteed Ultimate Boundedness- for a Class of Uncertain Linear Dynamical Systems, Proceedings of 3rd Kingston Conference on Differential Games and Control Theory, Marcel Dekker, New York, New York, 1979.
Chang, S. S. L., andPeng, T. K. C.,Adaptive Guaranteed Cost Control of Systems with Uncertain Parameters, IEEE Transactions on Automatic Control Vol. AC-17, No. 4, 1972.
Vinkler, A., andWood, I. J.,A Comparison of Several Techniques for Designing Controllers of Uncertain Dynamic Systems, Proceedings of the IEEE Conference on Decision and Control, San Diego, California, 1979.
Vinkler, A., andWood, I. J.,Multistep Guaranteed Cost Control of Linear Systems with Uncertain Parameters, Journal of Guidance and Control, Vol. 2, No. 6, 1979.
Salmon, D. M.,Minimax Controller Design, IEEE Transactions on Automatic Control, Vol. AC-13, No. 4, 1968.
Ly, U., andCannon, R. H.,A Direct Method for Designing Robust Optimal Control Systems, Proceedings of the AIAA Guidance and Control Conference, Palo Alto, California, 1978.
Molander, P.,Stabilization of Uncertain Systems, Lund Institute of Technology, Report No. LUTFD2/(TFRT-1020)/1-111/(1979), 1979.
Luenberger, D.,Canonical Forms for Linear Multivariable Systems, IEEE Transactions on Automatic Control, Vol. AC-12, No. 3, 1967.
Hahn, W.,Theory and Application of Lyapunov's Direct Method, Prentice Hall, New Jersey, Englewood Cliffs, 1963.
Barmish, B. R., andLeitmann, G.,On Ultimate Boundedness Control of Uncertain Systems in the Absence of Matching Conditions (to appear).
Monopoli, R. V.,Synthesis Techniques Employing the Direct Method, IEEE Transactions on Automatic Control, Vol. AC-10, No. 3, 1965.
Monopoli, R. V.,Discussion of Two Theorems on the Second Method, IEEE Transactions on Automatic Control, Vol. AC-11, No. 3, 1966.
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Communicated by G. Leitmann
This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.
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Thorp, J.S., Barmish, B.R. On guaranteed stability of uncertain linear systems via linear control. J Optim Theory Appl 35, 559–579 (1981). https://doi.org/10.1007/BF00934932
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DOI: https://doi.org/10.1007/BF00934932