Semistability-based robust and optimal control design for network systems

In this paper, we present a new Linear-Quadratic Semistabilizers (LQS) theory for linear network systems. This new semistable ℌ₂ control framework is developed to address the robust and optimal semistable control issues of network systems while preserving network topology subject to white noise. Two new notions of semistabilizability and semicontrollability are introduced as a means to connecting semistability with the Lyapunov equation based technique.With these new notions, we first develop a semistable ℌ₂ control theory for network systems by exploiting the properties of semistability. A new series of necessary and sufficient conditions for semistability of the closed-loop system have been derived in terms of the Lyapunov equation. Based on these results, we propose a constrained optimization technique to solve the semistable ℌ₂ network-topology-preserving control design for network systems. Then optimization analysis and the development of numerical algorithms for the obtained constrained optimization problem are conducted. We establish the existence of optimal solutions for the obtained nonconvex optimization problem. Next, we propose a swarm optimization based numerical algorithm towards efficiently solving this nonconvex, nonlinear optimization problem due to the strong resemblance between swarm behaviors in nature and the notion of semistability. Finally, several numerical examples will be provided to illustrate the effectiveness of the proposed method.