Learning-Based Stability Certification and System Identification of Nonlinear Dynamical Systems

Abstract

In recent decades, by taking advantage of the abundance of sensory measurements, learning-based methods have been prevalent and shown their effectiveness in tackling challenging or intractable problems for classical approaches in systems and control. For instance, many systems with complex nonlinearities, high-dimensional state spaces, or unknown dynamics cannot be effectively handled by classical mathematical tools, and computing stability certifications for such systems is often intractable. This thesis aims to construct systematic approaches to perform system identification tasks and learning-based Lyapunov functions for nonlinear dynamical systems, with some extensions to optimal control. The first aspect of this thesis is to develop an efficient method based on a special feedforward neural network structure, an extreme learning machine, to compute stability certificates for nonlinear systems by solving linear PDEs when the dynamics are accessible. Differing from the typical neural network-based approaches that require training on high-performance computing platforms, one only needs to solve a convex optimization problem. On top of that, the proposed method can also be used to efficiently solve the notable HJB equation via policy iteration to obtain optimal control policies for nonlinear systems. The second aspect of this research is to tackle these issues for nonlinear systems with (partially) unknown dynamics. We first show that with two feedforward neural networks, the unknown system and a Lyapunov-based stability certificate can be learned simultaneously. With the help of satisfiability modulo theories (SMT) solvers, the resulting Lyapunov function can be formally verified to provide stability certificates for the unknown nonlinear system. Alternatively, in the past two decades, the Koopman operator and its generator have demonstrated advantages in identifying discrete-time systems and continuous-time systems, respectively, requiring significantly less data while achieving better performance than most existing classical methods. For unknown continuous-time dynamical systems, we propose a novel resolvent operator-based learning framework to learn the Koopman generator, which is a linear operator that describes the infinitesimal evolution of the Koopman operator. The learned generator, thereafter, can be used to identify the vector field of the nonlinear systems. Moreover, with the learned high-accuracy Koopman generator, we can also construct a Lyapunov-based stability certificate for the unknown nonlinear system in the same function space. By formulating the linear PDEs as a linear least squares problem, Lyapunov functions can be computed efficiently. The learned Lyapunov functions can be formally verified using an SMT solver and provide less conservative estimates of the region of attraction, compared to existing methods. Taken together, these contributions provide a coherent pathway that begins with model-based stability certification computation and continues to fully data-driven system identification and thereafter computing Lyapunov-based stability certificates.