Control of linear discrete-time systems over a finite-time interval

In this paper we deal with some finite-time control problems for discrete-time linear systems. In particular, we consider a system subject to an exogenous output and provide a sufficient condition for finite-time boundedness, which guarantees, provided a certain Riccati difference inequality is satisfied, that the system state does not exceed some prespecified bounds. Finite-time boundedness translates into the classical concept of finite-time stability for autonomous systems. In this case necessary and sufficient conditions are proved; these conditions require either the computation of the state transition matrix of the system or the solution of a certain difference Lyapunov equation (or inequality). The design problem, i.e. the problem of finding a state feedback controller which stabilizes the closed loop system in the finite-time sense, is then addressed. The way these conditions can be solved numerically is finally considered and a design example is presented.