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Linear systems subject to input saturation and time delay: Global asymptotic and \(L^p\)-stabilization. (English. Abridged French version) Zbl 1076.93038

The problem of stabilizing the input delay system \[ \dot{x}(t) = Ax(t) + Bu(t-h) \] subject to bounded inputs of the form \(\| u\| \leq r\leq 1\) is considered. Two aspects are tackled: global asymptotic stabilization and \(L^p\) finite-gain stabilizability. In the first case the authors construct a nested nonlinear control leading to the system \[ \dot{x}(t) = Ax(t) - B\sigma(F_h(x(t-h))) \] where \(\sigma\) is a multivariable saturation function and \(F_h:R^n \mapsto R^m\) is a globally Lipschitz function which is synthesized. The second problem is to find a matrix \(F_h\) such that the system \[ \dot{x}(t) = Ax(t) - rB\sigma(F_h^Tx(t-h)+u_1(t-h) + ru_2(t-h)) \] is finite-gain \(L^p\)-stable for every integer \(p>0\). The authors prove the existence of such an \(F_h\). All results are “delay-dependent”, i.e. they are valid for \(0\leq h\leq h_*\).

MSC:

93D15 Stabilization of systems by feedback
93C23 Control/observation systems governed by functional-differential equations
93C10 Nonlinear systems in control theory
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