A convex approach for robust state feedback control of discrete-time systems with state delay

Uncertain discrete-time systems with state delay are investigated. The uncertainty is supposed to belong to a known convex polytope. Linear matrix inequality conditions are given for the robust stability of the system, encompassing quadratic stability based results. Then, convex conditions assuring the existence of a robust state feedback gain are derived, assuring the delay independent quadratic stability of the closed-loop system (thus allowing to deal with time-varying uncertain systems) or, in the time-invariant case, guaranteeing the robust stability irrespective of the value of the delay. Moreover, the feedback control law can also include a term depending on the delayed state which, if the value of the delay is known, can be used to improve the control design. Numerical examples illustrate the effectiveness of the proposed techniques.