Nyquist stability criterion

The **Nyquist stability criterion** is a graphical method for determining the stability of a closed-loop feedback system from its open-loop frequency response. Developed by Harry Nyquist in 1932, it states that the number of unstable closed-loop poles equals the number of unstable open-loop poles plus the number of clockwise encirclements of the point \(-1 + j0\) by the Nyquist plot of the open-loop transfer function.

The criterion's power lies in its avoidance of closed-loop pole computation. Rather than solving for the roots of the closed-loop characteristic equation — a numerically difficult problem for high-order systems — the engineer plots the open-loop frequency response and counts encirclements. The plot can be obtained analytically from the transfer function or experimentally by measuring the system's response to sinusoidal inputs.

The Nyquist criterion reveals not only whether a system is stable but how close it is to instability. The distance from the Nyquist curve to the \(-1\) point defines the stability margins: the reciprocal of the minimum distance is the maximum sensitivity, and the phase and gain margins are read directly from the plot's intersections with the negative real axis and the unit circle.

The criterion generalizes to multivariable systems through the multivariable Nyquist theorem, which examines the characteristic loci of the open-loop transfer function matrix. However, the multivariable version is less used in practice than the scalar version; mu analysis has largely supplanted it for robust multivariable stability assessment.

The Nyquist criterion is beautiful mathematics dressed as engineering. The encirclement theorem is a consequence of the argument principle from complex analysis — a theorem about meromorphic functions on the complex plane. That this abstract result tells us whether a steam turbine will overspeed or a spacecraft will tumble is one of the great unexplained miracles of applied mathematics. We use it constantly. We do not understand why it works so well.