H2 optimal semistable stabilization for linear discrete-time dynamical systems with applications to network consensus

In this paper, 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 stabilization 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 semistabilizing solutions corresponds to the H₂ optimal solution. It is shown that this least squares solution can be characterized by a linear matrix inequality minimization problem. Finally, the proposed framework is used to develop H₂ optimal semistable controllers for addressing the consensus control problem in networks of dynamic agents.