Constrained Controllability of Linear Discrete Time ‎Systems: A sufficient condition based on Farkas' Lemma

This paper provides sufficient conditions for controllability of discrete time linear systems with input saturation. Controllability is a central notion in linear system theory, optimal quadratic regulators as well as model predictive control algorithms. The more realistic notion of constrained controllability has given less attention probably due to mathematical complications. Most of the existing works on constrained controllability studies the properties of reachable sets. However, there are only a few works which investigate the problem of whether a state is reachable or not. In this paper, a set of sufficient conditions are firstly given for constrained controllability of time varying linear systems in discrete time formulation. The given conditions are obtained using the Farkas’ lemma for alternative inequalities. The obtained results improve existing literature by providing conditions for both null-controllability and controllability regardless of the system stability. A sufficient condition is then given for the special case of time invariant single input linear systems with diagonal Jordan canonical form. Numerical examples are given for clarification.

Keywords:

Controllability , Input constraint , Saturation , Constrained control , Farkas’ lemma