Stability Analysis for State Feedback Control Systems Established as Neural Networks with Input Constraints

Considerable progress in deep learning has also lead to an increasing interest in using deep neural networks (DNN) for state feedback in closed-loop control systems. In contrast to other purposes of DNN, it is insufficient to consider them only as black box models in control, in particular, when used for safety-critical applications. This paper provides an approach allowing to use the well-established indirect method of Lyapunov for time-invariant continuous time nonlinear systems with neural networks as state feedback controllers in the loop. A key element hereto is the derivation of a closed-form expression for the partial derivative of the neural network controller with respect to its input. By using activation functions of the type of sigmoid functions in the output layer, the consideration of box-constrained inputs is further ensured. The proposed approach does not only allow to verify the asymptotic stability, but also to find Lyapunov functions which can be used to search for positively invariant sets and estimates for the region of attraction.