Spectral Abscissa Minimization when Algebraic Control of Unstable LTI-TDS

Abstract

Optimal pole assignment minimizing the spectral abscissa when algebraic control of linear time-invariant time delay systems (LTI-TDS) is focused in this paper. We concentrate on algebraic controller design approach in the RMS ring resulting in delayed controllers as well. In the case of unstable delayed plants, the use a simple feedback loop results in a characteristic quasipolynomial instead of polynomial is obtained which means that the closed loop has an infinite spectrum. Thus, it is not possible to place all feedback poles to the prescribed positions exactly by a finite number of free controller parameters. The pole placement problem is translated to the minimization of the spectral abscissa which is a nonsmooth nonconvex function of free parameters in many cases. We initially solve the problem via standard quasi-continuous shifting algorithm followed by a comparative utilization of three iterative optimization algorithms; namely, Nelder-Mead algorithm, Extended Gradient Sampling Algorithm and Self-Organizing Migration Algorithm. Simulation control of an unstable LTI-TDS - the roller skater on the swaying bow - serves as an illustrative example for the algebraic control with the spectral abscissa minimization.