Abstract
This paper investigates the problem of stabilizing a single input uncertain linear system using linear state feedback control. The uncertain system is described by a linear state equation which contains uncertain parameters which are unknown but bounded. A quadratic Lyapunov function is used to establish the stability of the closed loop system.
A number of papers have appeared in recent years in which the desired stabilization of the system is achieved using nonlinear state feedback control. The main result of this paper demonstrates that for a class of single input uncertain linear systems, linear state feedback control can equally well be used to achieve this stabilization. That is, we describe a class of single input uncertain linear systems which have the following property: If the system can be stabilized using nonlinear state feedback control, then it can also be stabilized using linear state feedback control.
This work was supported by the National Science Foundation under Grant Number ECS-8108804.
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References
G. Leitmann, "Guaranteed Asymptotic Stability for Some Linear Systems with Bounded Uncertainty," ASME Journal of Dynamical Systems, Measurement and Control, Vol. 101, no. 3, 1979.
G. Leitmann, "On the Efficacy of Nonlinear Control in Uncertain Linear Systems," ASME Journal of Dynamical Systems, Measurement and Control, Vol. 103, no. 2, 1981.
S. Gutman and Z. Palmor, "Properties of Min-Max Controllers in Uncertain Dynamical Systems," SIAM Journal on Control and Optimization, Vol. 20, no. 6, 1982.
J.S. Thorp and B.R. Barmish, "On Guaranteed Stability of Uncertain Linear Systems via Linear Control," Journal of Optimization Theory and Applications, Vol. 35, no. 4, 1981.
B.R. Barmish, I.R. Petersen and A. Feuer, "Linear Ultimate Boundedness Control of Uncertain Dynamical Systems," Automatica, Vol. 19, no. 5, 1983.
A.M. Meilakhs, "Design of Stable Control Systems Subject to Parametric Perturbation," Automatika i Telemekhanika, no. 10, 1978.
C.V. Hollot and B.R. Barmish, "Optimal Quadratic Stabilizability of Uncertain Linear Systems," Proceedings of the 18th Allerton Conference on Communication, Control and Computing, University of Illinois, Monticello, 1980.
B.R. Barmish, "Necessary and Sufficient Conditions for Quadratic Stabilizability of an Uncertain Linear System," to appear in the Journal of Optimization Theory and Applications.
I.R. Petersen, "Investigation of Control Structure in the Stabilization of Uncertain Dynamical Systems," PhD dissertation, Department of Electrical Engineering, University of Rochester, Rochester, New York, 1983.
B.R. Barmish, "Fundamental Issues in Guidance and Control of Uncertain Systems," Proceedings of the American Control Conference, Arlington, Virginia, 1982.
B.R. Barmish, "Stabilization of Uncertain Systems via Linear Control," Proceedings of the IEEE Asilomar Conference on Computers, Circuits and Systems, Monterey, 1981; see also IEEE Transactions on Automatic Control, Vol. AC-28, no. 8, 1983.
W. Rudin, Principles of Mathematical Analysis, third edition, McGraw Hill, New York, 1976.
J.L. Willems and J.C. Willems, "Robust Stabilization of Uncertain Systems," SIAM Journal on Control and Optimization, Vol. 21, no. 3, 1983.
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© 1984 Springer-Verlag
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Petersen, I.R., Barmish, B.R. (1984). The stabilization of single input uncertain linear systems via linear control. In: Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems. Lecture Notes in Control and Information Sciences, vol 62. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0004946
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DOI: https://doi.org/10.1007/BFb0004946
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