Ph.D Thesis

Ph.D StudentDolgin Yuri
SubjectUncertain Systems: Identification and Approximation
DepartmentDepartment of Electrical and Computer Engineering
Supervisor PROFESSOR EMERITUS Ezra Zeheb
Full Thesis textFull thesis text - English Version


This research focuses on the treatment of uncertain systems in control problems. Important aspects of order reduction and identification of systems in the presence of uncertainty are investigated. The stability problem of the reduced-order system and various related aspects are examined in this research. The notion of stability used in this research is broader than in regular stability problems and encompasses problems of robustness, sensitivity, performance measures and others. Namely, considering a polynomial, by ``stability'' we mean that all zeros of the polynomial are in a given region in the complex plane. The following results were obtained in this research.

-          The Finite Nyquist and Finite Inclusions Theorems were generalized to the case of disjoint stability regions. The main advantage of these theorems is the ability to formulate the stability constraints as a set of simple checks on a finite number of frequencies. The generalization of the above theorems to the case of disjoint stability regions allows

o   better treatment of systems with separate dynamics and systems with complex robust performance constraints

o   simple treatment of model reduction problems of unstable systems

o   performing non-fragility analysis

-          It was shown that the recent results on Linear Matrix Inequality based characterization of stability domain in the coefficients space of a polynomial can be well approximated by a linear programming problem characterization. The linear programming characterization has the apparent advantage of being simpler to handle. It was shown in the research that in the case of interval polynomials, the linear programming characterization is particularly advantageous, requiring only O(N) the number of constraints used in the fixed-coefficients case, compared to at least O(N^2) required by LMI characterization.

-          Other new results were obtained in stability domain characterization in the coefficients space of a polynomial. The results are based on a novel approach using Rouche theorem and are expressed in the form of LMI, providing convex approximation of generally non-convex stability domain in coefficients space of a polynomial. The new results allow treatment of very general stability regions including non-convex and disjoint stability regions.

-          New result was obtained for the problem of design for multivariate positivity. This result is useful in very many control problems, including the problem of stability.

-          New results were obtained for the problem of stable-antistable factorization of polynomials and the problem of zeroth order model reduction of time delay systems, where we considered the delay-dependent case.