By Edmond A. Jonckheere

During this ebook, probably unrelated fields -- algebraic topology and powerful keep watch over -- are introduced jointly. The e-book develops algebraic/differential topology from an application-oriented viewpoint. The ebook takes the reader on a course ranging from a well-motivated powerful balance challenge, exhibiting the relevance of the simplicial approximation theorem and the way it may be successfully applied utilizing computational geometry. The simplicial approximation theorem serves as a primer to extra severe topological concerns corresponding to the obstruction to extending the Nyquist map, K-theory of strong stabilization, and finally the differential topology of the Nyquist map, culminating within the rationalization of the shortcoming of continuity of the steadiness margin relative to rounding blunders. The ebook is appropriate for graduate scholars in engineering and/or utilized arithmetic, educational researchers and governmental laboratories.

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487 485 Fig. 3 Crossover X region and its projection on the space of uncertain parameters. Fig. 4 The largest square that can be inserted within the stable region for a < a ; the numbers indicate the number of LHP closed-loop poles. Largest square that can be inserted in stable region for a=a . Fig. 5 489 489 490 LIST OF SYMBOLS a°, a1, ... , zn] ai ai adj b0, b1, ... , 0)T, where 1 is in the ith position total space of a bundle the fiber above p in a bundle the module indexed by ( , ) at the rth step of a (homology) spectral sequence the ( , )-module at the rth step of a (cohomology) spectral sequence Nyquist mapping simplicial approximation to Nyquist mapping fPL piecewise-linear extension of vertex transformation ai f fixed-frequency Nyquist mapping chain map space of Fredholm operators bi LIST OF SYMBOLS xli F fiber of a bundle or a fibration Fi.

In case of a two-channel feedback with an SU(2,C) perturbation, the nominal fixed-frequency Nyquist map is fw : SU(2, C) C. Writing it explicitly requires choosing charts for the manifold SU(2, C). The problem can be simplified by defining the Nyquist map as fw o : T3 C. The 14 ROBUST MULTIVARIABLE NYQUIST CRITERION problem is that the latter is a Nyquist map that is more "many-to-one" than it really has to be. To further motivate D = U(ni), let us proceed from the premise that the fundamental multivariable robustness problem is whether a system can sustain multiplicative perturbation of its loop matrix by a nonsingular matrix A.

In the SISO gain margin problem, the uncertainty is the half-interval [0, ), which is not compact. Leaving the domain noncompact raises some questions as to what is "happening" at infinity. To clarify the behavior of the map at infinity, it is convenient to compactify the domain. We defer the compactification issue to the next section, where we will do it explicitly on the imaginary axis (—j , +j ). One way to define the multivariable gain margin is to introduce a positive definite Hermitian multiplicative uncertainty matrix somewhere along the feedback path.