SCO colloquium: Paul Wijnbergen

Title: Optimal control of switched differential algebraic equations and more…

 

Abstract: The first part of this presentation will be concerned with linear quadratic regulator (LQR) problem for switched differential algebraic equations (sDAEs). Due to the linearity of the dynamics and the quadratic cost functional it can be shown that the optimal cost is a quadratic function of the initial value and the optimal controller is linear in the state. Taking dynamic programming approach it will be shown that the LQR problem for sDAEs can be reduced to a repeated finite horizon LQR problem for non-switched DAEs. However, as this LQR problem is considered in the context of sDAEs additional constraints on the terminal state and the terminal cost matrix have to be imposed. As a consequence of these constraints, there generally does not exist an optimal solution for all initial values, but only for initial values contained in a certain subspace. As a main result, it is shown how to characterize this subspace and how to compute an optimal control if it exists. This part of the presentation will be concluded with some simulations to illustrate the results.

In the second part of the presentation, the platooning problem will briefly be addressed. It will be shown that if a leader-follower structure is adopted, the problem of finding a decentralized controller can be posed as a disturbance decoupling problem combined with output stabilization. Within this geometrical context, string stability follows from the spacing policy. Necessary and sufficient conditions on the existence of a decentralized controller are given and the results are illustrated through simulation results.