A simple stabilization and algebraic control of unstable delayed first-order systems using meromorphic functions

Abstract

The paper is focused on control of unstable delayed systems. The control design is performed in the R(MS) ring of (retarded quasipolynomial) meromorphic functions. The unstable systems are modeled in anisochronic philosophy as a ratio of quasipolynomials where also denominator contains delay terms. The goal is to find a suitable stable quasipolynomial as a common denominator of R(MS) terms. This task is equivalent to the stabilization of a plant by a proportional feedback loop. Then, the appropriate controller can be found. In this paper, three algebraic methods are suggested, two of them are based on the solution of the Bezout equation with Youla-Kucera parameterization. The third one utilizes modified internal model control (IMC) structure with an affine parameterization. All methods offer a real positive real parameter m(0) which defines closed loop poles placement. The modified "equalization method" for determining of m(0) can be applied. An example illustrates the proposed methodology, properties and benchmarking of all principles.