Linear matrix inequality characterisation for ℋ∞ and ℋ2 guaranteed cost gain-scheduling quadratic stabilisation of linear time-varying polytopic systems

Abstract

Necessary and sufficient linear matrix inequality (LMI) conditions are provided to compute parameter-dependent state feedback control gains that ensure closed-loop quadratic stability for linear systems affected by arbitrarily fast time-varying parameters inside a polytope. The proposed conditions, based on an extension of Pólya's theorem and on the systematic construction of homogeneous polynomial solutions for parameter-dependent LMIs, are written as a sequence of progressively less and less conservative LMI conditions. Necessity is attained as the level of relaxation increases, providing a parameter-dependent state feedback gain that quadratically stabilises the system whenever such a gain exists. Moreover, parameter-dependent gains of arbitrary degree assuring quadratic stability with ℋ∞ and ℋ2 guaranteed costs are also provided. The convergence to the minimum values of the attainable ℋ∞ and ℋ2 guaranteed costs under closed-loop quadratic stability occurs as the degree of the polynomially parameter-dependent gain increases. Numerical results illustrate the efficiency of the proposed conditions.