## Root Clustering in Parameter SpaceThis book deals with algebraic criteria for root clustering (inclusion) in general regions in the complex plane. It is based on the view that there are three approaches to root clustering: composite matrices and polynomial symmetric matrices, and rational mappings. The book presents two main results of potential benefit to the reader. First, given a linear dynamical system, it is possible to analyze its relative stability. Second, for that system it is possible to construct a dynamic controller of fixed order for closed loop relative stability. The reader is assumed to have some knowledge of linear algebra and the theory of complex variables. The book is aimed at systems and control researchers, mathematicians and physicists. |

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### Contents

REVIEW OF CLASSICAL RESULTS | 9 |

INTRODUCTION TO ROOT CLUSTERING | 26 |

Chapter TRANSFORMABLE REGIONS | 45 |

Copyright | |

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### Common terms and phrases

algebraic region asymptotic stability Bezoutian boundary Chapter characteristic polynomial Chojnowski cl(K closed loop characteristic Cnxn Coef coeff1cients coefficients Combining Theorems companion matrix compensator completes the proof complex plane composite polynomials consider Contr Control coprime Corollary criterion critical constraint Decision Algebra def1ned defined Definition diagonal double hyperbola eigenvalues Example Given a polynomial Gutman hand side Hermitian Hurwitz Hurwitz matrix IEEE Trans implies Kharitonov's theorem Kronecker product left half plane left hyperbola left sector Lemma Let A(X linear loop characteristic polynomial Lyapunov Lyapunov equation M-transformable region matrix equations minimization nonsingular Note obtain open left half P-transformable parameter space polynomial A(X polynomial inequalities polynomial version polynomial with roots q(Ti rational function real polynomial Res[a(X respect result Rnxn root clustering criteria root-clustering roots of A(X satisfying Section Sylvester matrix symmetric symmetric matrix Theorem 4.3 transformable unique solution unit disk x+iy zero