On the bounds of the fastest admissible decay of generalized energy in controlled LTI systems subject to state and input constraints

This paper contributes to the field of Lyapunov stability theory where positive definite radially unbounded functions – which can be interpreted as generalized energy for the system of interest – play a central role. Given a discrete–time LTI system subject to hard constraints, which are affine in state and control variables, upper and lower barriers are developed which bound the fastest possible decay of generalized energy over the set of admissible control policies for which the closed–loop system is asymptotically stable on a compact polyhedral set including the origin. It is shown that the lower barrier is a greatest feasible lower bound of the fastest possible decay of generalized energy over any set of admissible control policies. A simple example is given to illustrate the main results.