A BMI approach forH∞ gain scheduling of discrete time-varying systems

The problem of gain-scheduled state feedback control for discrete-time linear systems with time-varying parameters is considered in this paper. The time-varying parameters are assumed to belong to the unit simplex and to have bounded rates of variation, which depend on the values of the parameters and can vary from slow to arbitrarily fast. An augmented state vector is defined to take into account possible time-delayed inputs, allowing a simplified closed-loop analysis by means of parameter-dependent Lyapunov functions. A gain-scheduled state feedback controller that minimizes an upper bound to the H∞ performance of the closed-loop system is proposed. No grids in the parametric space are used. The design conditions are expressed in terms of bilinear matrix inequalities (BMIs) due to the use of extra variables introduced by the Finsler’s lemma. By fixing some of the extra variables, the BMIs reduce to a convex optimization problem, providing an alternate semi-definite programming algorithm to solve the problem. Robust controllers for time-invariant uncertain parameters, as well as gain-scheduled controllers for arbitrarily time-varying parameters, can be obtained as particular cases of the proposed conditions. As illustrated by numerical examples, the extra variables in the BMIs can provide better results in terms of the closed-loop H∞ performance. Copyright q 2009 John Wiley & Sons, Ltd.