Giorgio VALMORBIDA (Associate Professor, CentraleSupélec, Laboratory of signals and systems (L2S) - Directeur de thèse
Antoine GIRARD (Directeur de Recherche, CNRS, Laboratory of signals and systems (L2S)) - Co-encadrant.
Application deadline: 23 May 2022
Funding: Université Paris-Saclay
Keywords: Optimisation, Learning Systems, Piece-wise affine systems, Safety
The increasing use of data and learning algorithms in dynamical systems introduces Neural Networks (NN) in feedback loops. The NN in these loops introduces nonlinear elements that may vary in time. These nonlinear terms and their variation should be studied to guarantee stability and satisfaction of safety constraints. Safety constraints appear, for instance, in autonomous vehicles since physical limits related to the speed or the road should be respected. Thus, the challenge is to guarantee that the vehicle, in feedback with the elements generated by the algorithm, respects the constraints.
On the other hand, a successful framework to incorporate actuator and state constraints in nonlinear systems is Model Predictive Control (MPC), which results in nonlinear piecewise affine (PWA) control laws. These seemingly different approaches to control, namely learning algorithms and MPC, may be connected by using nonlinear elements to represent the underlying function, namely a piecewise affine function. By exploiting the common features of these systems, we can formulate a unified strategy for stability and safety verification.
Importantly, the piecewise functions describing MPC are, in general, implicit, and the solution requires thus the solution of an implicit function. The impact of errors in its solutions should then be assessed. In this context, we will study the Input-to-State Stability of the PWA systems subject to disturbances. A challenge is to transform the evaluation of the PWA functions into evaluating explicit functions to reduce these computational errors. Feedforward neural networks present this convenient explicit structure, therefore making it easy to evaluate them. Therefore, for implementation purposes, it is desirable to obtain an explicit rather than implicit formulation for the PWAfunction stemming from MPC. One objective will be to develop a direct conversion from an implicit PWA model into an explicit one, similar to NN, and compare it with existing strategies for MPC.