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A method is presented to analyze the stability of feedback systems withneural network controllers. Two stability theorems are given to proveasymptotic stability and to compute an ellipsoidal inner-approximation to theregion of attraction (ROA). The first theorem addresses linear time-invariantsystems, and merges Lyapunov theory with local (sector) quadratic constraintsto bound the nonlinear activation functions in the neural network. The secondtheorem allows the system to include perturbations such as unmodeled dynamics,slope-restricted nonlinearities, and time delay, using integral quadraticconstraint (IQCs) to capture their input/output behavior. This in turn allowsfor off-by-one IQCs to refine the description of activation functions bycapturing their slope restrictions. Both results rely on semidefiniteprogramming to approximate the ROA. The method is illustrated on systems withneural networks trained to stabilize a nonlinear inverted pendulum as well asvehicle lateral dynamics with actuator uncertainty.

Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers