Switched LQR control

Abstract

This thesis studies the Switched Linear Quadratic Regulator (SLQR) problem, over a hybrid (continuous and discrete) dynamical model known as "switched system". The problem is defined as computing the optimal continuous and discrete switching control to minimise a quadratic cost function that weights the states and the continuous controls. The original SLQR problem does not handle constraints on states, continuous or discrete controls, and there is no probabilistic behaviour. This thesis focuses on the discrete dynamics in a SLQR problem. The first part of the thesis describes the SLQR problem with discrete constraints, whereas the second part is dedicated to probabilistic switching behaviour. The problem with discrete constraints is described as finding the optimal hybrid switching policy that minimises a quadratic cost function, weighting states and continuous controls, without violating the discrete constraints. The problem with probabilistic switches is defined as finding the optimal hybrid switching policy that minimises an expected value of a quadratic cost function, weighting states and continuous controls. For the SLQR problem with discrete constraints a general relaxation framework is developed to simplify the representations of the value functions and the corresponding control strategies. It is shown that the closed loop performance of the obtained solution with the relaxation framework can be made arbitrarily close to the optimal solution. For the SLQR problem with probabilistic switches it is shown that a relaxation framework can only be developed when there are no discrete constraints involved. Finally, the thesis concludes with a few case studies to illustrate how the optimal hybrid control sequence is computed.