2 optimal semistable stabilisation for linear discrete-time dynamical systems with applications to network consensus

Abstract

In this article, we develop ℋ2 semistability theory for linear discrete-time dynamical systems. Using this theory, we design ℋ2 optimal semistable controllers for linear dynamical systems. Unlike the standard ℋ2 optimal control problem, a complicating feature of the ℋ2 optimal semistable stabilisation problem is that the closed-loop Lyapunov equation guaranteeing semistability can admit multiple solutions. An interesting feature of the proposed approach, however, is that a least squares solution over all possible semistabilising solutions corresponds to the ℋ2 optimal solution. It is shown that this least squares solution can be characterised by a linear matrix inequality minimisation problem. Finally, the proposed framework is used to develop ℋ2 optimal semistable controllers for addressing the consensus control problem in networks of dynamic agents.