Understanding the Nyquist Plot

The Nyquist plot is a fundamental tool in control engineering, used to analyze the stability of feedback systems. Named after Harry Nyquist, this graphical representation depicts the frequency response of a system by plotting the complex transfer function G(jω) as the angular frequency ω varies from 0 to ∞. The plot is drawn in the complex plane, with the real part of G(jω) on the horizontal axis and the imaginary part on the vertical axis. Its primary utility lies in assessing closed-loop stability from the open-loop frequency response, without explicitly computing closed-loop poles. The Nyquist plot also provides insight into gain and phase margins, making it indispensable for robust controller design.

The mathematical underpinnings of Nyquist plots require a thorough understanding of complex analysis, particularly the evaluation of transfer functions on the imaginary axis and the geometric interpretation of contour mapping. Engineers must grasp the concept of the Nyquist contour, a closed path in the complex plane that encloses the entire right half-plane (RHP), and how this contour maps through the open-loop transfer function G(s)H(s) to produce the Nyquist path. The resulting plot's winding around the critical point −1 + 0j directly indicates system stability, following the Nyquist stability criterion. This article expands on these mathematical foundations, walking through the construction logic, the principle of argument, and practical pitfalls.

Complex Transfer Function Evaluation

The open-loop transfer function G(s)H(s) is a rational function of the complex variable s = σ + jω. For frequency response analysis, we set σ = 0, so s = jω. The resulting expression G(jω)H(jω) is a complex number that depends on ω:

G(jω)H(jω) = Re(ω) + j Im(ω) = M(ω) e^{jφ(ω)}

Here, M(ω) is the magnitude (also called gain) and φ(ω) is the phase angle. Constructing the Nyquist plot involves evaluating G(jω)H(jω) at many frequencies and plotting the complex points. For a typical system, the plot starts at ω = 0 (often at a point on the real axis) and ends at ω → ∞ (usually at the origin, unless the system has pure integrators). The shape of the curve reveals key dynamic properties: for example, how the phase decreases with increasing frequency, or how the magnitude rolls off.

To compute G(jω)H(jω) mathematically, consider a general transfer function factored into poles and zeros:

G(s)H(s) = K \frac{\prod_{i=1}^{m} (s - z_i)}{\prod_{k=1}^{n} (s - p_k)}

When s = jω, each term becomes a complex vector from the zero or pole location to the point jω on the imaginary axis. The total magnitude is the product of the lengths of the zero vectors divided by the product of the lengths of the pole vectors, multiplied by |K|. The total phase is the sum of the angles of the zero vectors minus the sum of the angles of the pole vectors, plus the phase of K (which is 0 if K is real and positive). This geometric interpretation is central to understanding how the Nyquist curve behaves, especially near frequencies corresponding to the system's natural frequencies.

Mathematical Construction Steps

Constructing a Nyquist plot by hand (or using software) involves systematic evaluation of G(jω)H(jω) over a frequency range. The key steps are:

  1. Identity the transfer function and write it in Bode form (poles and zeros). For systems with time delays, include the e^{-jωT} factor.
  2. Compute the magnitude and phase at a set of frequencies. Typically, low frequencies (near 0) and high frequencies (near ∞) are chosen, with intermediate points near break frequencies (where pole or zero contributions change sign).
  3. Start at ω = 0: The point depends on the system type. For Type 0 (no integrators), the plot starts at K (the DC gain) on the positive real axis. For Type 1 (one integrator), the plot starts at an infinite magnitude with a phase of −90°. For Type 2 (two integrators), the phase is −180° at ω = 0.
  4. Plot points at increasing frequencies, connecting them smoothly. As frequency increases, the curve generally moves toward the origin (since magnitude decreases). The phase may cross −180° at the phase crossover frequency where the magnitude is critical for stability analysis.
  5. Mirror for negative frequencies (if using the full Nyquist contour). The plot for ω from −∞ to 0 is the complex conjugate of the plot for positive ω, reflected across the real axis. Combined with the positive frequency branch (and the infinite semicircle if the system has net pole-zero excess), the complete Nyquist path is obtained.
  6. Identify the number of encirclements of −1 + 0j to apply the stability criterion.

Mathematically, the Nyquist curve for positive frequencies can be parameterized as ω ∈ [0,∞). For systems with relative degree (n − m) ≥ 1, the curve ends at the origin, approaching it with a phase of −(n−m)×90°. For relative degree 0 (same number of poles and zeros), the final point is a finite non-zero complex number at ω → ∞.

The Nyquist Stability Criterion in Detail

The Nyquist stability criterion is a powerful method to determine closed-loop stability from the open-loop frequency response. It leverages Cauchy's argument principle, also known as the principle of the argument, from complex analysis. This principle states that if a closed contour Γ in the s-plane is mapped through a function F(s) to a closed contour ΓF in the F-plane, then the number of times ΓF encircles the origin equals (Z − P), where Z is the number of zeros of F(s) inside Γ and P is the number of poles of F(s) inside Γ. For control systems, we choose the function F(s) = 1 + G(s)H(s), and the contour Γ is the Nyquist contour that encloses the entire right half-plane (RHP) by traversing the imaginary axis from −j∞ to +j∞ and then a semicircle of infinite radius to the right.

Because F(s) is related to the open-loop transfer function by F(s) = 1 + G(s)H(s), the map of F(s) through the Nyquist contour is simply the Nyquist plot of G(s)H(s) shifted to the right by 1. Equivalently, encirclements of the origin in the F-plane correspond exactly to encirclements of the point −1 + 0j in the G(s)H(s)-plane. The Nyquist stability criterion then reads:

N = Z − P

where:

  • N = number of encirclements of −1 + 0j by the Nyquist plot (counted in a clockwise direction as positive, counterclockwise as negative).
  • Z = number of closed-loop poles in the RHP (zeros of 1 + G(s)H(s) in the RHP).
  • P = number of open-loop poles in the RHP.

For a stable closed-loop system, we require Z = 0. Thus, the closed-loop system is stable if and only if the Nyquist plot of G(s)H(s) encircle the point −1 + 0j exactly −P times (i.e., counterclockwise encirclements equal P if we count clockwise as positive). If P = 0 (open-loop stable), then the Nyquist plot must not encircle −1 at all for closed-loop stability.

Applying the Criterion: Encirclement Counting

Counting encirclements rigorously involves analyzing the Nyquist path over the complete contour, including the mapping of the infinite semicircle. For systems with relative degree ≥ 1, the infinite semicircle maps to a small circle around the origin in the G(s)H(s) plane (or to a fixed point if relative degree zero). The practical approach for engineers is to:

  1. Draw the Nyquist plot for ω from 0 to ∞.
  2. Reflect it across the real axis for ω from 0 to −∞.
  3. If the system has poles on the imaginary axis (e.g., integrators), indent the Nyquist contour with small semicircles to the right to avoid these poles, and map those indentations to large arcs in the G(s)H(s) plane.
  4. Count the net number of clockwise encirclements of −1 by tracing the complete closed curve (including the infinite semicircle mapping).

Mathematically, the encirclement count N can be computed via the winding number integral:

N = \frac{1}{2πj} \oint_{Nyquist \, contour} \frac{G'(s)H'(s)}{1+G(s)H(s)} \, ds

but this is rarely evaluated analytically; engineers rely on geometric inspection or software tools. The key insight is that encirclements occur when the phase of G(jω)H(jω) crosses −180° (i.e., the point −1 on the real axis) while the magnitude is ≥ 1. This condition defines the gain margin and phase margin.

Practical Construction Examples

To solidify understanding, we consider a simple first-order system and a second-order system with damping.

First-Order System: G(s) = 1/(s+1)

Here G(jω) = 1/(jω+1). The magnitude is M(ω) = 1/√(1+ω²) and phase is φ(ω) = −tan⁻¹(ω). At ω = 0, the point is (1, 0). As ω → ∞, the point approaches (0, 0) from an angle of −90°. The Nyquist curve is a semicircle in the lower half-plane (since phase is always negative) spanning from the positive real axis to the origin. This curve does not encircle −1, and since P = 0, the closed-loop system is stable for any positive gain K if we replace G(s) with KG(s). However, for very large K, the curve expands radially; it may still not encircle −1 because its shape remains a semicircle. The first-order system is never destabilized by increasing gain—consistent with intuition.

Second-Order System with Damping: G(s) = ω_n²/(s² + 2ζω_n s + ω_n²)

For a second-order system with natural frequency ωn and damping ratio ζ, the Nyquist plot begins at (1, 0) for ω = 0 and ends at (0, 0) with a phase of −180° as ω → ∞. The shape depends critically on ζ. For underdamped systems (ζ < 1), the curve may exhibit a loop that crosses the negative real axis at a frequency ωpc where the phase is −180°. At that frequency, the magnitude is Mpc. If Mpc > 1, then the Nyquist plot encircles −1 and the closed-loop system is unstable. This corresponds to a gain margin less than 1 (0 dB). Conversely, if ζ is large enough such that Mpc < 1, then no encirclement occurs and the system is stable. The critical damping ratio ζc for which Mpc = 1 can be found analytically: the phase crossover occurs at ωn for a standard second-order system? Actually, for G(s) = ω_n²/(s² + 2ζω_n s + ω_n²), the phase at ω = ωn is −90°, not −180°. The −180° crossing occurs at infinite frequency (asymptotically) unless the system has additional dynamics. Wait, that's an important nuance: a pure second-order system without zeros has a phase that asymptotically approaches −180°, but never reaches −180° at any finite frequency. Therefore, its Nyquist plot never crosses the negative real axis (except at the origin). Thus it never encircles −1, and the closed-loop is always stable for any gain—contradicting typical second-order behavior with damping? Actually, a standard second-order system is stable for all positive gains because its open-loop phase never exceeds −180° in the finite frequency range; it only reaches −180° at infinite frequency with zero magnitude. So no encirclement. This illustrates that the Nyquist criterion reveals stability for more complex systems like those with time delays or non-minimum phase zeros, where the phase can exceed −180° at finite frequencies.

To see a proper phase crossover, consider a system with an additional pole at low frequency: G(s) = 1/((s+1)(s² + 2ζ s + 1)). Here the phase can cross −180° at some finite ω, and the plot may encircle −1 if the gain is sufficiently high.

Connecting Nyquist, Bode, and Nichols Plots

The Nyquist plot is intimately related to the Bode plot (magnitude and phase) and the Nichols chart. The Bode plot provides magnitude in dB and phase in degrees as functions of log frequency. The Nyquist plot is a parametric combination of the magnitude and phase: each point on the Nyquist curve corresponds to a magnitude and phase at a given ω. The gain margin is the factor by which the gain can be increased before the magnitude at the phase crossover frequency becomes 1 (i.e., the distance from −1 to the Nyquist curve along the negative real axis). The phase margin is the additional phase lag required at the gain crossover frequency (where magnitude = 1) to make the phase −180°. These margins are directly readable from the Nyquist plot: the gain margin is 1/|G(jωpc)| (the reciprocal of the distance from 0 to the point where the curve crosses the negative real axis), and the phase margin is the angle between the negative real axis and the line from 0 to the point where the magnitude = 1.

The Nichols chart transforms the Nyquist plot into a Cartesian plot of magnitude (dB) vs. phase (deg). This transformation is useful for designing controllers because gain and phase margins are easily seen, and closed-loop response can be read using M-circles. All three representations are equivalent mathematically; choosing one depends on the context and engineer's preference.

Limitations and Advanced Considerations

While powerful, the Nyquist plot has limitations that engineers must acknowledge. First, for systems with time delays, the Nyquist curve becomes infinite and highly curved, requiring careful handling of the infinite semicircle mapping. The delay adds a phase shift −ωT that accumulates linearly, causing the Nyquist plot to spiral inward or outward depending on the delay. Stability analysis then requires the modified Nyquist criterion that accounts for the delay using the concept of the Nyquist contour that excludes the delay pole at infinity.

Second, the Nyquist plot as described is for continuous-time systems. For discrete-time systems (z-domain), the Nyquist contour becomes the unit circle, and the stability criterion examines encirclements of −1 in the w-plane or the use of the bilinear transform to map to the s-plane. The mathematical foundations remain similar, but the contour changes.

Third, the Nyquist criterion assumes that the closed-loop system is stable if and only if the open-loop Nyquist plot encircle −1 appropriately. However, this requires that the character of the open-loop poles in the RHP is known a priori. If the open-loop transfer function has poles on the imaginary axis (including pure integrators), the Nyquist contour must be indented to avoid these poles, and the mapping of those indentations produces infinite arcs that must be included in the encirclement count. This can be tricky for engineers new to the method.

Finally, the Nyquist plot is a frequency-domain tool; it provides necessary and sufficient conditions for stability only for linear time-invariant (LTI) systems. For nonlinear or time-varying systems, the Nyquist criterion does not directly apply, though extensions like the circle criterion exist for certain classes of nonlinearities.

Conclusion

The construction of a Nyquist plot is deeply rooted in complex analysis, particularly the mapping of the Nyquist contour through the open-loop transfer function and the principle of the argument. By mastering the mathematical steps—evaluating G(jω)H(jω), plotting the complex points, and counting encirclements of −1—engineers gain a powerful method for assessing closed-loop stability and designing robust controllers. The Nyquist criterion provides intuitive insight into gain and phase margins, and its connection to Bode and Nichols plots makes it a versatile tool in the control engineer's toolkit. Understanding these foundations enables engineers not only to apply the Nyquist plot correctly but also to interpret its results with confidence in real-world applications, from aerospace systems to industrial process control.

For further reading, refer to standard textbooks such as Wikipedia's article on the Nyquist stability criterion, Control Tutorials for MATLAB and Simulink, or the classic text Modern Control Engineering by Ogata. These resources provide additional examples and deeper mathematical derivations.