Robust Stability of Neural-Network Controlled Nonlinear Systems with Parametric Variability

Stability certification and identification of the stabilizable operating region of a dynamical system are two important concerns to ensure its operational safety/security and robustness. With the advent of machine-learning tools, these issues are especially important for systems with machine-learned components in the feedback loop. Here, in presence of unknown discrete variation (DV) of its parameters within a bounded range, a system controlled by a static feedback controller in which the closed-loop (CL) equilibria are subject to variation-induced drift is equivalently represented using a class of time-invariant systems, each with the same control policy. To develop a general theory for stability and stabilizability of such a class of neural-network (NN) controlled nonlinear systems, a Lyapunov-based convex stability certificate is proposed and is further used to devise an estimate of a local Lipschitz upper bound for the NN and a corresponding operating domain in the state space containing an initialization set, starting from where the CL local asymptotic stability of each system in the class is guaranteed, while the trajectory of the original system remains confined to the domain if the DV of the parameters satisfies a certain quasi-stationarity condition. To compute such a robustly stabilizing NN controller, a stability-guaranteed training (SGT) algorithm is also proposed. The effectiveness of the proposed framework is demonstrated using illustrative examples.