Numerical Methods for Roots of Polynomials - Part II: Chapter 14. Stability Considerations

Numerical Methods for Roots of Polynomials - Part II: Chapter 14. Stability Considerations
ISBN-10
0128077042
ISBN-13
9780128077047
Series
Numerical Methods for Roots of Polynomials - Part II
Category
Mathematics
Pages
728
Language
English
Published
2013-07-19
Publisher
Elsevier Inc. Chapters
Authors
J.M. McNamee, V.Y. Pan

Description

In considering the stability of mechanical systems we are led to the characteristic equation . Continuous-time systems are stable if all the roots of this equation are in the left half-plane (Hurwitz stability), while discrete-time systems require all (Schur stability). Hurwitz stability has been treated by the Cauchy index and Sturm sequences, leading to various determinantal criteria and Routh’s array, and several other methods. We also have to consider the question of robust stability, i.e. whethera system remains stable when its coefficients vary. In the Hurwitz case Kharitonov’s theorem reduces the answer to the consideration of 4 extreme polynomials, and other authors consider cases where the coefficients depend on parameters in various ways. Schur stability is notably dealt with by the Schur–Cohn algorithm, which constructs a sequence of polynomials and tests whether all their constant terms are negative. Methods are described which reduce overflow in this process. Robust Schur stability is harder to deal with than Hurwitz, but several partial solutions are described.

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