Global stabilization of nonlinear systems with a class of unmatched uncertainties.(English)Zbl 0759.93060

The paper studies global stability properties of dynamical systems described by $\dot x(t)=f(x,t)+\tilde\xi(x,t)+B(x,t)[\xi(x,t)+u],\;x(t_ 0)=x_ 0,$ where $$x(t)\in\mathbb{R}^ n$$ denotes the state, $$u(t)\in\mathbb{R}^ n$$ the control, $$\tilde\xi(x,t)$$ the unmatched constraints and $$\xi(x,t)$$ the matched constraints. Under the assumption of global asymptotic stability of $$\dot x(t)=f(x,t)$$ and under certain unboundedness assumptions on the uncertainties of the constraints as well as of $$f$$ and $$B$$, the author constructs a state feedback controller $$u(x,t)$$ which stabilizes the closed loop system in the large. The techniques make use of the fact that Lyapunov functions are in general not unique.

MSC:

 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C10 Nonlinear systems in control theory 70K20 Stability for nonlinear problems in mechanics 93C15 Control/observation systems governed by ordinary differential equations 93D09 Robust stability 93D30 Lyapunov and storage functions
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References:

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