## APPENDIX B

## PROOF OF THE NYQUISTSTABILITY CRITERION

The Nyquist stability criterion can be derived from Cauchy’s residue theorem, which states that

Let us replace *g*(*s*) by *f′*(*s*)/*f*(*s*), where *f*(*s*) is a function of *s* which is single valued on and within the closed contour *C* and analytic on *C*. Observe that the singularities *f′*(*s*)/*f*(*s*) occur only at the zeros and poles of *f*(*s*). The residue may be found at each singularity with multiplicity of the order of zeros and poles taken into account. The residues in the zeros of *f*(*s*) are positive and the residues in the poles of *f*(*s*) are negative. Therefore, if *f*(*s*) is not equal to zero along *C*, and if there are not at most a finite number of singular points that are all poles within the contour *C*, then

where *Z* = number of zeros of *f*(*s*) within *C*, with due regrd for their multiplicity of order, and *P* = number of poles of *f*(*s*) within *C*, with due regard for their multiplicity of order. The left-hand side of Eq. (B2) may be written as

In general, *f*(*s*) will have both real and imaginary parts along the contour *C*. Therefore, its logarithm can be written as

If we assume that *f*(*s*) is not zero anywhere on the contour *C*, the integration of Eq. (B3) results ...

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