Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. (English) Zbl 0549.93045

Consider an uncertain system (\(\Sigma)\) described by the equation \(\dot x(t)=A(r(t))x(t)+B(s(t))u(t),\) where \(x(t)\in R^ n\) is the state, \(u(t)\in R^ m\) is the control, r(t)\(\in {\mathcal R}\subset R^ p\) represents the model parameter uncertainty, and s(t)\(\in {\mathcal S}\subset R^{\ell}\) represents the input connection parameter uncertainty. The matrix functions \(A(\cdot)\), \(B(\cdot)\) are assumed to be continuous and the restraint sets \({\mathcal R}\), \({\mathcal S}\) are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds \({\mathcal R}\), \({\mathcal S}\). Much of the previous literature has concentrated on a fundamental question: Under what conditions on \(A(\cdot)\), \(B(\cdot)\), \({\mathcal R}\), \({\mathcal S}\) can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (\(\Sigma)\) satisfies certain matching conditions, then quadratic stabilizability is indeed assured [see G. Leitmann, J. Dyn. Syst. Meas. Control 101, 212-216 (1979; Zbl 0416.93077) and S. Gutmann, IEEE Trans. Autom. Control AC-24, 437-443 (1979; Zbl 0416.93076)]. Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to completely characterize the class of systems for which quadratic stabilizability can be guaranteed.


93D15 Stabilization of systems by feedback
93B35 Sensitivity (robustness)
93C05 Linear systems in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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