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.

Keywords:

• semistability
• ℋ₂ optimal semistable control
• semicontrollability
• semiobservability
• linear matrix inequalities
• multiagent systems
• discrete-time systems

Acknowledgement

This research was supported in part by the Air Force Office of Scientific Research under Grant FA9550-06-1-0240.