In this dissertation we propose a control synthesis and analysis framework for nonlinear, and neural network (NN) controlled systems with robustness guarantees. We quantify systems' robustness against external disturbances and perturbations using the following measures: (i) the forward reachable set; (ii) the backward reachable set; (iii) the tracking error bound; (iv) the region of attraction. These measures are all characterized by sublevel sets of storage functions satisfying appropriate dissipation inequalities that account for external disturbances and perturbations. Integral quadratic constraints (IQCs) are used to describe perturbations, allowing for a variety of perturbations including parametric uncertainty, unmodeled dynamics, and nonlinear activation functions in NNs. We formulate sum-of-squares (SOS) constraints and Linear matrix inequality conditions by merging dissipation inequalities with IQCs to compute controllers and their associated robustness measures.
We start off by focusing on the finite time horizon robustness of uncertain nonlinear (polynomial) systems, which are modeled as interconnections of nominal polynomial systems and perturbations. We propose a method of outer-approximating the forward reachable set on finite horizons for the uncertain nonlinear systems with controllers given. Then we move from analysis to control synthesis, and present a method for synthesizing a polynomial control law that steers the system to the target set with the goal of maximizing inner-approximations to the backward reachable set. The approximations to both the forward reachable set and backward reachable set are characterized by time dependent polynomial storage functions, and are computed using SOS programming. IQCs with both hard and soft factorizations are used to describe perturbations.
Furthermore, we address robust trajectory planning and control design for nonlinear systems. A hierarchical trajectory planning and control framework is proposed, where a low-fidelity model is used to plan trajectories satisfying planning constraints, and a high-fidelity model is used for synthesizing tracking controllers guaranteeing the boundedness of the error state between the low- and high-fidelity models. We formulate SOS optimizations for computing the tracking controllers and their associated tracking error bound, with the goal of minimizing the volume of the tracking error bound. The tracking error bound is then used to redesign the planning constraints to guarantee safety of the system.
Finally, we move to NN controlled systems. We propose two theorems to prove local stability of NN controlled linear time invariant systems, and to compute inner-approximations to the region of attraction. The first theorem merges dissipation inequalities with local sector quadratic constraints (QCs) to bound the nonlinear activation functions in the NN. The second theorem includes IQCs to allow for perturbations, and further refine the description of activation functions by capturing their slope information. Then we move from analysis to control synthesis. Loop transformation is used to derive stability and safety conditions that are jointly convex in the Lyapunov function, weights of the NN controller, and the Lagrange multipliers for including QCs. These convex conditions are incorporated in the imitation learning process, which trades off between imitation learning accuracy, and size of the region of attraction inner-approximations, to learn robust NN controllers. We propose an alternating direction method of multipliers based algorithm to solve the constrained imitation learning problem.
