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Affine parameter-dependent Lyapunov functions and real parametric uncertainty. (English) Zbl 0854.93113
Consider the system \[ \dot x = A \bigl( \theta (t) \bigr) x \] with \(A (\theta (t)) = A_0 + \theta_1 (t) A_1 + \cdots + \theta_k (t) A_k\), \(\theta_i \in [\underline{\theta}_i, \overline \theta_i]\), \(\dot \theta_i \in [\underline{\nu}_i, \overline \nu_i]\). This system is called affinely quadratically stable if there exists an affine quadratic Lyapunov function \(V(x, \theta) = x^* P (\theta) x\), \(P (\theta) = P_0 + \theta_1 P_1 + \cdots + \theta_k\) such that \(V(x, \theta) > 0\), \(dv/dt < 0\) along all admissible parameter trajectories and for all initial conditions \(x_0\). The problem of constructing such a Lyapunov function is reduced to solving some linear matrix inequalities. Other control problems connected with quadratic Lyapunov functions (e.g. \(H_\infty\) performance) are studied for the same type of parametric uncertainties. Numerical implementations are suggested.

93D09 Robust stability
93D30 Lyapunov and storage functions
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