An algorithm for the quadratic stabilization of uncertain systems with structured uncertainty of the one-block type (Q1915797)

The authors present a new result in the so-called quadratic stabilization of uncertain systems [see e.g. \textit{B. R. Barmish}, IEEE Trans. Autom. Control AC--28, 848-850 (1983; Zbl 0554.93054)]. The main idea of this approach is to use a quadratic Lyapunov function in order to find a linear controller which stabilizes the family of the systems defined by state equations with uncertain parameters taking values within known bounded sets. The main contribution of the paper consists in a new sufficient condition for the quadratic stabilizability via linear control for a class of systems with structured uncertainty in the case when the number of unmatched uncertain parameters is not greater than the number of control inputs. Following this result an algorithm is constructed based on the solution of a Lyapunov equation. This solution is found to be a linear function of scaling parameters used to represent the uncertainty structure. This feature enables to treat the quadratic stabilization problem by solving a linear matrix inequality problem with the number of variables equal to the number of uncertain parameters. The authors present an illustrative example consisting of an uncertain system with both matched and unmatched uncertainties.