Talk on Physics-Informed Stability Control for Nonlinear Systems at CAIMS 2024, Kingston, Ontario, Canada.

A talk on `Physics-Informed Characterization and Control of Stability for Nonlinear Systems` was presented at the CAIMS (Canadian Applied and Industrial Mathematics Society) 2024 meeting in the subsession 'Recent Progress on the Intersections of Nonlinear Dynamics, Control, Learning, and Optimization'. It was truly a great experience meeting all the experts and sharing opinions with them in this area. See the abstract below.

Abstract

Stability analysis of nonlinear dynamical systems is crucial in many fields, where understanding the domain of attraction for an asymptotically stable equilibrium point helps determine if systems can stabilize after experiencing faults. Since Lyapunov's landmark paper more than a hundred years ago, Lyapunov functions have been key in nonlinear stability analysis, and extensive research has been conducted on constructing Lyapunov functions through computational methods. In this research series, we systematically investigate the use of physics-informed neural networks to compute Lyapunov functions, encoding the Lyapunov conditions into the Zubov partial differential equation (PDE) for training neural networks. We demonstrate that using the Zubov equation can closely approximate regions of attraction to their true domains. We also examine approximation errors and the convergence to Zubov’s unique viscosity solution, then provide verifiable conditions for neural Lyapunov functions using satisfiability modulo theories (SMT) solvers for formal verification. The results demonstrate that this computational tool surpasses traditional sums-of-squares (SOS) methods that employ semidefinite programming (SDP). Furthermore, we expand the tool's functionalities to support the formulation and solving of optimal control problems for control-affine systems using physics-informed neural network policy iteration (PINN-PI). We outline the methodology that enables the learning and verification of PINN for optimal stabilization tasks. Demonstrating with classical control examples, we show that the learned optimal controller significantly improves performance and provides verifiable regions of attraction, especially in high-dimensional nonlinear systems. Lastly, we address the scenario of unknown systems by proposing a lifting approach that maps observable data into an infinite-dimensional function space, thus generating a flow governed by the proposed `Zubov-Koopman' operators. By learning such an operator over a fixed time interval, we can indirectly approximate the solution to Zubov’s equation through iterative applications of the learned operator on the identity function. Our physics-informed algorithm, underpinned by a comprehensive investigation of the regularities of Zubov-Koopman operators and their associated quantities, demonstrates strong convergence to the true operator, reduces the amount of required data, and yields desirable estimation results.