MATH319 Slides 141 Nyquist’s locus 143 Argument principle

142 Nyquist’s criterion for stability of $T$

Nyquist’s criterion

Let $R$ be the transfer function of a plant such that $R$ is stable. Suppose that the contour $R(i\omega)$ $(-\infty\leq\omega\leq\infty)$ does not pass through or wind around $-1$ . Then $T=R/(1+R)$ is also stable, so the feedback system with constant feedback $-1$ is also stable.

We let $c=\lim_{s\rightarrow\infty}R(s)$ where $c\neq-1$ by assumption. Hence we can write $R(s)=c+p(s)/q(s)$ where $p(s)$ and $q(s)$ are polynomials, and degree $p(s)$ is less than the degree of $q(s)$ . Then

T(s)={{R(s)}\over{1+R(s)}}={{c+p(s)/q(s)}\over{c+1+p(s)/q(s)}}={{cq(s)+p(s)}% \over{(1+c)q(s)+p(s)}}

and the degree of $(1+c)q(s)+p(s)$ equals the degree of $q(s)$ , hence $T(s)$ is proper.