Nyquist Stability Theorem

This is a key theorem in Control Theory which relates the stability of a closed-loop system to that of its open-loop frequency response (represented as the polar locus or a Bode plot) that can be theoretically or experimentally determined.

The definition of the Generalised Nyquist Stability Criterion is: A closed-loop system is stable if and only if the number of clockwise encirclements of the critical (−1) point by the polar locus, as $-\infty \to \omega \to \infty$ is equal to the number of right-half-plane open-loop poles.

For Control 3 we only consider systems with no right-half-plane open-loop poles, therefore the Nyquist Stability Criterion reduces to: A closed-loop system is stable if and only if the polar locus, as $-\infty \to \omega \to \infty$ does not encircle the critical (−1) point.

Core Principle

The important points to note are:

1. We only sketch $G(j\omega )$ for positive $\omega$
2. Stability is determined by the behaviour of the polar locus near the critical (−1) point
3. First and second order systems are stable for all values of gain (i.e. they have infinite Gain Margin).
4. If an open-loop system has n poles and m zeros such that $n-m>2$ (relative order greater than two), then the closed-loop will be unstable for large enough gain.