Constructing a homogeneous LTI descriptor system with desired properties using perturbation theory

In the literature regarding linear systems and mathematical control theory, several different techniques have been developed for obtaining the solution of homogeneous linear time-invariant (LTI) descriptor differential systems. In this article, applying the complex Weierstrass canonical form, we investigate the conditions under which a descriptor system with a specific structure and desired properties is being constructed using perturbation theory. Our approach is very general, and as an example, a stable homogeneous LTI descriptor system is designed. Thus, a proportional and derivative controller can be used, such as the case where a family of perturbed pencils is defined and the solutions of the initial and the relative perturbed systems are ρM(t)-close with respect to a Frobenius distance. A Step-algorithm and an illustrative example are also presented to illustrate the results of this article.