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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.

MSC:

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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References:

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