Stabilization of interconnected dynamical systems by online convex optimization

The problem of stabilizing networks of interconnected dynamical systems (NDS) in a scalable fashion is considered. As the first contribution, a generalized lemma and example network are provided to demonstrate that state-of-the-art, tractable dissipation-based NDS stabilization methods can fail even for simple unconstrained, linear, and time-invariant dynamics. Then, a solution to this issue is proposed, in which controller synthesis is decentralized via a set of parameterized storage functions. The corresponding stability conditions allow for max-type construction of a trajectory-specific Lyapunov function for the full closed-loop network, whereas neither of the local storage functions is required to be monotonically converging. The provided approach is indicated to be nonconservative in the sense that it can generate converging closed-loop trajectories for the motivating example network and a prescribed set of initial conditions. For input-affine NDS and quadratic parameterized storage functions, the synthesis scheme can be formulated as a set of low-complexity semidefinite programs that are solved online, in a receding horizon fashion. Moreover, for linear and time-invariant networks, an even simpler, explicit control scheme is derived by interpolating a collection of a priori generated converging state and control trajectories in a distributed fashion.