Design of H∞ Gain-Scheduled Controllers for Linear Time-Varying Systems by means of Polynomial Lyapunov Functions

This paper proposes convex conditions to design parameter-dependent (i.e. gain-scheduled) state feedback controllers that ensure closed-loop stability with H_∞, performance for linear systems affected by time-varying parameters that belong to a polytope and have bounded time-derivatives. The conditions, based on homogeneous polynomially parameter-dependent Lyapunov functions of arbitrary degree, are expressed as a set of linear matrix inequalities written in terms of the vertices of the polytope, the bounds on the rates of parametric variations and the degree of the Lyapunov function. Differently from other gain-scheduled strategies in the literature, all the system matrices here are supposed to be affected by time-varying parameters and the control gains are obtained from a finite dimension convex optimization problem, which avoids to cope with nonlinear matrix inequalities and does not require the use of gridding procedures in the space of the parameters. Numerical results illustrate the efficiency of the proposed conditions, providing progressively less conservative results as the degree of the Lyapunov function increases