A stabilization algorithm for a class of uncertain linear systems. (English) Zbl 0618.93056

Necessary and sufficient conditions are given for the quadratic stabilization of the uncertain control system \[ \dot x(t)=(A+DF(t)E)x(t)+Bu(t),\quad F^ T(t)F(t)\leq I, \] \(x\in {\mathfrak R}^ n\), \(u\in {\mathfrak R}^ m\), and \(F\in {\mathfrak R}^{p\times q}\). The form of the stabilizing feedback control is \(u^*(t)=-R^{-1}B^ TPx(t)\), where R and P are positive definite matrices of appropriate dimension, and P solves an algebraic Riccati equation. It is demonstrated that with the notion of ”overbounding”, it is possible to use the control law \(u^*(t)\) to stabilize larger classes of uncertain linear systems.
Reviewer: J.Gayek


93D15 Stabilization of systems by feedback
15A24 Matrix equations and identities
93C05 Linear systems in control theory
34D20 Stability of solutions to ordinary differential equations
65K10 Numerical optimization and variational techniques
93B40 Computational methods in systems theory (MSC2010)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
Full Text: DOI


[1] Chang, S.S.L.; Peng, T.K.C., Adaptive guaranteed cost control of systems with uncertain parameters, IEEE trans. automat. control, 17, 474, (1972) · Zbl 0259.93018
[2] Noldus, E., Design of robust state feedback laws, Internat. J. control, 35, 935, (1982) · Zbl 0491.93050
[3] Barmish, B.R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain linear system, J. optim. theory appl., 46, 399, (1985) · Zbl 0549.93045
[4] Petersen, I.R.; Hollot, C.V., A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 22, (July 1986)
[5] Vidysagar, M., Nonlinear systems analysis, (1978), Prentice-Hall Engelwood Cliffs, NJ
[6] Jacobson, D.H., Extensions of linear-quadratic control, optimization and matrix theory, (1977), Academic Press New York · Zbl 0359.90085
[7] Van Dooren, P., A generalized eigenvalue approach for solving Riccati equations, SIAM J. sci. statist. comput., 2, 121, (1981) · Zbl 0463.65024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.