Quadratic optimization for controller initialization in multivariable switching systems

This paper deals with switching between several linear time-invariant controllers to be used in feedback with a MIMO linear time-invariant (not necessarily stable) plant to be controlled. We explore the fact that, aside from selecting the controllers´ transfer functions and deciding which controller to use in the feedback loop at each time instant, the system designer has two additional degrees of freedom that are seldom used: the choice of the realizations for the controllers´ transfer functions and selection of the initial condition that is assigned to the controller when it is switching into the feedback loop. We show that by selecting appropriate controller realizations and finding optimal values for the initial controller state, one can achieve significantly better transients. We also show that the closed-loop system remains input-to-state stable by adding a mild constraint to the optimization and to select the initial controller state. We illustrate the use of our results in the control of a flexible beam actuated in torque. This system is unstable with two poles at the origin and contains several lightly damped modes, which can be easily excited by controller switching.