Browsing by Author "Saadaoui, Karim"

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Open Access
Field of values of matrix polytopes
(Anadolu Üniversitesi, 2000) Özgüler, A. Bülent; Saadaoui, Karim
The tool of field of values (also known as the classical numerical range) is used to recover most results available in the literature and to obtain some new one s concerning Hurwitz and Schur stability of matrix polytopes. Some facts obtained by an application of the elementary properties of field of values are as follows. If the vertex matrices have polygonal field of values, then the matrix polytope is Hurwitz and Schur stable if and only if the vertex matrices are Hurwitz and Schur stable, respectively. If the polytope is nonnegative and the symmetric part of each vertex matrix is Schur stable, then the polytope is Schur stable. For polytopes with spectral vertex matrices, Schur stability of vertices is necessaryand sufficient for the Schur stability of the polytope.
Open Access
Fixed order controller design via parametric methods
(2003) Saadaoui, Karim
In this thesis, the problem of parameterizing stabilizing fixed-order controllers for linear time-invariant single-input single-output systems is studied. Using a generalization of the Hermite-Biehler theorem, a new algorithm is given for the determination of stabilizing gains for linear time-invariant systems. This algorithm requires a test of the sign pattern of a rational function at the real roots of a polynomial. By applying this constant gain stabilization algorithm to three subsidiary plants, the set of all stabilizing first-order controllers can be determined. The method given is applicable to both continuous and discrete time systems. It is also applicable to plants with interval type uncertainty. Generalization of this method to high-order controller is outlined. The problem of determining all stabilizing first-order controllers that places the poles of the closed-loop system in a desired stability region is then solved. The algorithm given relies on a generalization of the Hermite-Biehler theorem to polynomials with complex coefficients. Finally, the concept of local convex directions is studied. A necessary and sufficient condition for a polynomial to be a local convex direction of another Hurwitz stable polynomial is derived. The condition given constitutes a generalization of Rantzer’s phase growth condition for global convex directions. It is used to determine convex directions for certain subsets of Hurwitz stable polynomials.
Open Access
Local convex directions
(IEEE, 2001) Özgüler, Arif Bülent; Saadaoui, Karim
A proof of a strengthened version of the phase growth condition for Hurwitz stable polynomials is given. Based on this result, a necessary and sufficient condition for a polynomial p(s) to be a local convex direction for a Hurwitz stable polynomial q(s) is obtained. The condition is in terms of polynomials associated with the even and odd parts of p(s) and q(s).
Open Access
On the set of all stabilizing first-order controllers
(IEEE, 2003) Saadaoui, Karim; Özgüler, Arif Bülent
A computational method is given for determining the set of all stabilizing proper first-order controllers for finite dimensional, linear, time invariant, scalar plants. The method is based on a generalized Hermite-Biehler theorem.
Open Access
Stability robustness of linear systems: a field of values approach
(1997) Saadaoui, Karim
One active area of research in stability robustness of linear time invariant systems is concerned with stability of matrix polytopes. Various structured real parametric uncertainties can be modeled by a family of matrices consisting of a convex hull of a finite number of known matrices, the matrix poly tope. An interval matrix family consisting of matrices whose entries can assume any values in given intervals are special types of matrix polytopes and it models a commonly encountered parametric uncertainty. Results that allow the inference of the stability of the whole polytope from stability of a finite number of elements of the polytope are of interest. Deriving such results is known to be difficult and few results of sufficient generality exist. In this thesis, a survey of results pertaining to robust Hurwitz and Schur stability of matrix polytopes and interval matrices are given. A seemingly new tool, the field of values, and its elementary properties are used to recover most results available in the literature and to obtain some new results. Some easily obtained facts through the field of values approach are as follows. Poly topes with normal vertex matrices turn out to be Hurwitz and Schur stable if and only if the vertex matrices are Hurwitz and Schur stable, respectively. If the polytope contains the transpose of each vertex matrix, Hurwitz stability of the symmetric part of the vertices is necessary and sufficient for the Hurwiz stability of the polytope. If the polytope is nonnegative and the symmetric part of each vertex matrix is Schur stable, then the polytope is also stable. For polytopes with spectral vertex matrices, Schur stability of vertices is necessary and sufficient for the Schur stability of the polytope.