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# A novel approach to the analysis and synthesis of controllers for parametrically uncertain systems

#### Abstract

This dissertation describes several new connections between a polynomial's coefficients and its zeros. The most important of these, the Finite Nyquist Theorem (FNT), states that one can prove a polynomial has all its roots in an open, bounded or unbounded, convex region ${\cal D}$ of the complex plane given only the polynomial's degree and its phase at finitely many points along ${\cal D}$'s boundary. An immediate and very useful corollary to FNT is the Finite Inclusions Theorem (FIT), with which one can prove a class of polynomials has all its zeros in ${\cal D}$ given only the polynomials' degree and approximate knowledge of the class's value set at finitely many points along ${\cal D}$'s boundary.^ From FIT a procedure we call FIT synthesis is developed for synthesizing robustly ${\cal D}$-stabilizing controllers for parametrically uncertain systems (note, all the systems considered here are assumed to be linear time-invariant and finite dimensional). This procedure uses FIT to directly search for robust controllers by way of solving a sequence of systems of linear inequalities. Two numerical examples of this procedure are given to show its effectiveness. In these examples the systems of inequalities are solved via the projection method which is an elegantly simple technique for solving (finite or infinite) systems of convex inequalities in an arbitrary Hilbert space. Since this method has yet to appear in standard textbooks on numerical methods, it is covered here in detail with the aim of better popularizing the method and, where possible, extending the known theory concerning its convergence. ^

#### Subject Area

Mathematics|Electrical engineering

#### Recommended Citation

Kaminsky, Richard David, "A novel approach to the analysis and synthesis of controllers for parametrically uncertain systems" (1993). Doctoral Dissertations Available from Proquest. AAI9408293.
https://scholarworks.umass.edu/dissertations/AAI9408293

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