Positive stabilization of discrete-time systems with unknown delay and bounded controls

Abstract

This paper solves the stabilization problem for a class of linear discrete-time systems with delay, for which the states must not take nonnegative values whenever the initial conditions are nonnegative. In particular, the synthesis of state-feedback controllers is first solved in terms of Linear Programming for the unbounded controls case. This result is then extended to stabilization by nonnegative controls, and stabilization by bounded controls. The proposed conditions are necessary and sufficient and do not involve the value of the delay (which is assumed to be fixed and constant but maybe unknown): only the multiplicative matrix on the delayed states is needed.