Robust state feedback control for discrete-time linear systems via LMIs with a scalar parameter

This paper proposes an improved approach to ℋ
₂
and ℋ
_∞
robust state feedback control design for discrete-time polytopic time-invariant linear systems based on Linear Matrix Inequalities (LMIs) with a scalar parameter. The synthesis conditions, that depend on a real parameter lying in the interval (-1,1), become LMIs for fixed values of the scalar, reducing to standard conditions in the literature when the scalar is equal to zero. At the price of line searches combined with LMIs, less conservative results for robust state feedback control are obtained. The closed-loop stability and the ℋ
₂
and ℋ
_∞
guaranteed costs are certified by means of an affine parameter-dependent Lyapunov function. The validity and the efficiency of the method are illustrated by means of examples and exhaustive numerical comparisons.