Nyquist Stability Criterion

The Nyquist Stability Criterion is one of the most powerful tools in control systems for determining closed-loop stability directly from the open-loop frequency response. Unlike the Routh-Hurwitz criterion that works in the s-domain, Nyquist operates in the frequency domain, making it especially useful when the open-loop transfer function is known experimentally or contains time delays.

Core Concept of the Nyquist Criterion

The Nyquist criterion is based on the principle of argument from complex variable theory. When a closed contour in the s-plane (the Nyquist contour, which encloses the entire right-half plane) is mapped through the open-loop transfer function G(s)H(s), the resulting curve in the GH-plane tells us how many times the mapped contour encircles the critical point -1+j0. This encirclement count directly reveals closed-loop stability.

The key relationship is expressed as Z = P - N, where Z is the number of closed-loop poles in the right-half s-plane (RHP), P is the number of open-loop poles in the RHP, and N is the number of clockwise (CW) encirclements of the -1+j0 point by the Nyquist plot. For a stable closed-loop system, Z must equal zero, meaning N must equal P.

If the open-loop system is itself stable (P = 0), then stability requires N = 0, meaning the Nyquist plot must not encircle the -1+j0 point at all. If the open-loop system has P unstable poles, then the Nyquist plot must encircle -1+j0 exactly P times in the counter-clockwise direction to ensure Z = 0.

An important note on sign convention: some textbooks define N as counter-clockwise (CCW) encirclements, leading to the form N = Z - P. The GATE exam typically follows the convention where N is CW encirclements, giving Z = P - N. Always confirm the sign convention being used.

Mathematical Expression and Derivation Basis

The characteristic equation of a unity feedback system is 1 + G(s)H(s) = 0. The closed-loop poles are the zeros of F(s) = 1 + G(s)H(s). By the principle of argument, the net change in angle of F(s) as s traverses the Nyquist contour equals 2pi times (Z - P), where Z and P are zeros and poles of F(s) in the RHP. Since the zeros of F(s) are closed-loop poles and the poles of F(s) are open-loop poles, this gives the encirclement relationship directly.

The Nyquist contour in the s-plane consists of the imaginary axis from -j-infinity to +j-infinity and a large semicircle of radius R approaching infinity closing the RHP. Poles on the imaginary axis are handled by small indentations. The mapping of this contour through G(s)H(s) generates the Nyquist plot.

Gain Margin and Phase Margin from the Nyquist Plot

The Nyquist plot provides a direct graphical measure of stability margins. The gain margin (GM) is the reciprocal of the magnitude of G(jw)H(jw) at the phase crossover frequency, where the phase angle is -180 degrees. On the Nyquist plot, this is the reciprocal of the distance from the origin to where the plot crosses the negative real axis.

The phase margin (PM) is 180 degrees plus the phase of G(jw)H(jw) at the gain crossover frequency where the magnitude equals unity. On the Nyquist plot, the gain crossover point lies on the unit circle, and phase margin is the angle between that point and the negative real axis. A system with positive GM and PM is stable.

Practical Significance

The Nyquist criterion is preferred over Routh-Hurwitz in frequency domain design because it handles time-delay systems naturally. A pure time delay e^(-sT) adds a phase lag of wT radians, which distorts the Nyquist plot in a way that Routh-Hurwitz cannot capture. In process control, instrumentation, and communication feedback systems where transport delays are present, the Nyquist approach gives a complete picture of stability.

It is also useful for systems defined by experimental frequency response data rather than an explicit transfer function. If Bode plots are measured in the lab, the Nyquist plot can be constructed and stability assessed without ever deriving a transfer function.

Solved Numerical Example

Consider an open-loop transfer function G(s) = K / (s(s+2)(s+4)). The Nyquist plot is drawn for the positive frequency axis (w from 0 to infinity) and mirrored. We need to find the range of K for closed-loop stability. The characteristic equation approach using Routh will confirm the Nyquist result.

Given:
G(s) = K / [s(s+2)(s+4)], H(s) = 1, unity feedback
P = 0 (no open-loop RHP poles)

Why this formula applies:
For stability: Z = P - N = 0 => N = 0
Nyquist plot must NOT encircle -1+j0.
This means |G(jw)| at phase = -180 deg must be < 1/K threshold.

Formula:
Phase crossover: angle of G(jw) = -180 deg
G(jw) = K / [jw(jw+2)(jw+4)]

Substitution:
Angle = -90 - arctan(w/2) - arctan(w/4) = -180
arctan(w/2) + arctan(w/4) = 90 deg

Calculation:
tan(arctan(w/2) + arctan(w/4)) = tan(90) => infinity
Using tan addition: [w/2 + w/4] / [1 - (w^2/8)] = infinity
=> 1 - w^2/8 = 0 => w^2 = 8 => w_pc = 2√2 rad/s

|G(j*2√2)| = K / [2√2 * √(8+4) * √(8+16)]
= K / [2√2 * √12 * √24]
= K / [2√2 * 2√3 * 2√6]
= K / [8√36] = K / 48

For stability: K/48 < 1 => K < 48

Final Answer: System is stable for 0 < K < 48

Exam Tip: In GATE problems, if the open-loop system is stable (P=0) and the Nyquist plot does not encircle -1+j0, the closed-loop is stable. The number of CW encirclements minus P gives the number of unstable closed-loop poles. Also remember: gain margin in dB = 20 log(1/|G(jwpc)|).

Mechanism Summary

• Construct the Nyquist plot by substituting s = jw in G(s)H(s) for w from 0 to infinity, then mirror it about the real axis.
• Handle imaginary-axis open-loop poles by indenting the Nyquist contour with small semicircles into the RHP.
• Count N as the number of clockwise encirclements of the -1+j0 point. CCW encirclements count as negative N.
• Apply Z = P - N. If Z = 0, the closed-loop system is stable regardless of how many open-loop RHP poles exist.
• Gain margin is read directly as the inverse of the real-axis crossing magnitude; phase margin is the angle at unit-circle crossing.

Quick Revision

• Nyquist criterion relates closed-loop stability to open-loop frequency response via Z = P - N.
• Z = number of closed-loop RHP poles (must be 0 for stability). P = open-loop RHP poles. N = CW encirclements of -1+j0.
• For stable open-loop systems (P=0): stable closed-loop requires zero encirclements of -1+j0.
• Gain margin = 20 log(1/|G(jwpc)|) dB. Phase margin = 180 + angle of G(jw) at gain crossover.
• Nyquist handles time delays and experimentally obtained frequency responses where Routh-Hurwitz cannot be applied.
• Common trap: confusing CW with CCW convention. GATE uses Z = P - N with N as CW encirclements.
• For the system G(s) = K/[s(s+a)(s+b)], critical gain K = ab(a+b) and phase crossover frequency w = sqrt(ab).