Inversion of Separable Kernel Operator and Its Application in Control Synthesis

Abstract

In this chapter, we show how the problem of controller synthesis can be posed as a form of convex optimization in an operator-theoretic framework. Furthermore, we show how to: (a) Parameterize the integral and multiplier operator-valued decision variables using finite-dimensional vectors; (b) Verify and enforce positivity and negativity of multiplier and integral operators using positive matrices; (c) Invert positive integral and multiplier operators through the use of a new formula based on algebraic manipulation. Finally, we show how these 3 parts can be combined into a computational procedure for finding a stabilizing state-feedback controller for systems defined by differential-difference equations—a class which includes differential systems with discrete delays. Finally, a numerical example is used to illustrate the form of the resulting stabilizing controller.