Polynomial Controllers for Optimal Trajectory Matching with Stability Guarantees

Abstract

We formulate a trajectory matching problem in which a set of reference trajectories for a plant is given, and a control law that causes the plant’s trajectories to be as close as possible to the reference trajectories is desired. These trajectories might be generated by an implicit controller such as a model predictive control (MPC) algorithm or manually chosen by a user. This thesis presents a nonconvex optimization approach for solving the trajectory matching problem that generates explicit polynomial controllers. The value of this approach is that the explicit control laws it generates are simpler to implement, and can be used for stability analysis. Additionally, the method presented in this thesis guarantees local stability of the generated controller by ensuring local contractivity towards the generated trajectories. This thesis presents several theoretical results that justify the method described here. Firstly, a proof that the local contractivity constraints used to ensure local stability can be expressed as a set of matrix inequalities is presented, which turns an infinite set of constraints into a finite one. Secondly, a theorem that describes how symmetries in the trajectory matching problem correspond to symmetries in its solution is presented and proven, which enables a reduction in the control design problem size and resulting solution. Finally, this thesis demonstrates the method it describes on two example problems motivated by real-world applications. The first of these is stabilization and disturbance recovery for a single-machine infinite-bus (SMIB) power system, and the second is a lane change manoeuvre for Dubin’s vehicle, a simple vehicle model. In each case, the reference trajectories are generated by MPC.