By R. Wittkowski

ISBN-10: 1566760062

ISBN-13: 9781566760065

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Extra resources for Capillary Gas Chromatography in Food Control And Research

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For a given zero at s = q of a transfer matrix H, there may exist more than one input (or output) direction. In fact, there exist as many input (or output) directions as the drop in rank of the matrix H(q). This deficiency in rank of the matrix H(s) at s = q is called the geometric multiplicity of the zero at frequency q. The poles of a transfer matrix H are the eigenvalues of the evolution matrix of any minimal realization of H. We will assume that the sets of ORHP zeros and poles of H are disjoint.

3 ZS ZT {s ∈ {s ∈ + : S(s) = 0} , ✂ + : T (s) = 0} . 3 then establishes the fact that ZS is the set of ORHP poles of L and ZT is the set of ORHP zeros of L. In other words, we have translated the open-loop characteristics of instability and “nonminimum phaseness” into properties that the functions S and T must satisfy in the CRHP. , L = GK, then the constraints imposed by CRHP poles and zeros of G hold irrespective of the choice of K, provided that L has no unstable zero-pole cancelation. 2. 2 Recall from Chapter 2 that L is free of unstable hidden modes if there are no cancelations of CRHP zeros and poles between the plant and controller whose cascade connection forms L.

As a further illustration of these constraints, we have studied the effect on the closed-loop step response of pure integrators and ORHP zeros and poles of the open-loop system. We have seen, inter-alia, that a stable unity feedback system with a real ORHP open-loop zero must have undershoot in its step response. A similar conclusion holds with ORHP open-loop poles and overshoot in the step response. These limitations are obviously worse for plants having both nonminimum phase zeros and unstable poles; the inverted pendulum example illustrates these complications.

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