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2 is just one formulation of Rouch e 's Theorem C. j j j on C implies that neither f (z ) nor f (z )+ g(z ) Note that the condition may have a zero on but it is sucient for our purposes. The next theorem is a simple application of Rouch e's Theorem. It is however most useful since it applies to polynomials. 6) and consider a circle k , of radius rk , centered at sk which is a root of P (s) of multiplicity tk . Let rk be xed in such a way that, 0 < rk < min sk sj ; for j = 1; 2; ; k 1; k + 1; ; m: (1:7) Then, there exists a positive number , such that i ; for i = 0; 1; ; n, implies that Q(s) has precisely tk zeros inside the circle k .

It turns out that this result on exposed edges also follows from a more powerful result, namely the Edge Theorem, which is established in Chapter 6. Here we show that this stability testing property of the exposed edges carries over to complex polynomials as well as to quasipolynomials which arise in control systems containing time-delay. A computationally ecient solution to testing the stability of general polytopic families is given by the Bounded Phase Lemma, which reduces the problem to checking the maximal phase dierence over the vertex set, evaluated along the boundary of the stability region.

In [52] Brasch and Pearson showed that arbitrary pole placement could be achieved in the closed loop system by a controller of order no higher than the controllability index or the observability index. In the late 1960's and early 1970's the interest of control theorists turned to the servomechanism problem. Tracking and disturbance rejection problems with persistent signals such as steps, ramps and sinusoids could not be solved in an obvious way by the existing methods of optimal control. The reason is that unless the proper signals are included in the performance index the cost function usually turns out to be unbounded.