## 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

### Citations:

Zbl 0416.93077; Zbl 0416.93076
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### References:

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