No abstract

A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal inner-approximation to the region of attraction. The first theorem addresses linear time-invariant plant dynamics, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the plant dynamics to include perturbations such as unmodeled dynamics, sloperestricted nonlinearities, and time delay, using integral quadratic constraint (IQCs) to capture their input/output behavior. Both results rely on semidefinite programming to compute Lyapunov functions and inner-estimates of the region of attraction. The method is illustrated on systems with neural networks trained to stabilize a nonlinear inverted pendulum as well as vehicle lateral dynamics with actuator uncertainty.

A method is presented to synthesize neural network (NN) controllers with stability and safety guarantees through imitation learning. Convex stability and safety conditions are derived for linear time-invariant plant dynamics with NN controllers. The proposed approach merges Lyapunov theory with local quadratic constraints to bound the nonlinear activation functions in the NN. The safe imitation learning problem is formulated as an optimization problem with the goal of minimizing the imitation learning loss, and maximizing volume of the region of attraction associated with the NN controller, while enforcing the stability and safety conditions. An alternating direction method of multipliers based algorithm is proposed to solve the optimization. The method is illustrated on an inverted pendulum system and aircraft longitudinal dynamics.

We present a safe-by-design approach to path planning and control for nonlinear systems. The planner uses a low fidelity model of the plant to compute reference trajectories by solving an MPC problem, while the plant being controlled utilizes a feedback control law that tracks those trajectories with an upper-bound on the tracking error. Our main goal is to allow for maximum permissiveness (that is, room for constraint feasibility) of the planner, while maintaining safety after accounting for the tracking error bound. We achieve this by parametrizing the state and input constraints imposed on the planner and deriving corresponding parametrized tracking control laws and tracking error bounds, which are computed offline through Sum-of-Squares programming. The parameters are then optimally chosen to maximize planner permissiveness, while guaranteeing safety.

We present a control design procedure for nonlinear control systems in which we represent a potentially high dimensional system with a low dimensional continuous-state abstraction. The abstraction generates a reference which the original system follows with a low level controller. We propose sum-of-squares programming as a tool to design this controller and to provide an upper bound on the relative error between the system and its abstraction. We compute the low level controller simultaneously with a simulation function that gives the boundedness guarantee for the relative error. arXiv:1909.06468v1 [math.OC]