Nyquist Stability Criterion

1. Definition and Purpose of the Nyquist Criterion

Definition and Purpose of the Nyquist Criterion

The Nyquist Stability Criterion is a graphical technique used to determine the stability of a closed-loop control system by analyzing the frequency response of its open-loop transfer function. Developed by Harry Nyquist in 1932, this criterion provides a way to assess stability without explicitly solving for the poles of the closed-loop system.

Mathematical Foundation

The criterion is rooted in complex analysis, specifically the argument principle, which relates the number of zeros and poles of a meromorphic function to a contour integral of its logarithmic derivative. For a system with open-loop transfer function \( L(s) \), the Nyquist plot is a polar plot of \( L(j\omega) \) for \( \omega \) ranging from \( -\infty \) to \( +\infty \).

$$ N = Z - P $$

Here, \( N \) is the number of encirclements of the point \( (-1, 0) \) in the complex plane, \( Z \) is the number of unstable closed-loop poles, and \( P \) is the number of unstable open-loop poles. The system is stable if \( Z = 0 \), which implies \( N = -P \).

Practical Application

In control engineering, the Nyquist Criterion is particularly useful for systems with time delays or non-rational transfer functions, where root locus methods may be cumbersome. It also provides insights into gain and phase margins, which quantify how far the system is from instability.

Visual Interpretation

The Nyquist plot is a parametric curve where the real and imaginary parts of \( L(j\omega) \) are plotted as \( \omega \) varies. The plot must not encircle the \( (-1, 0) \) point in a clockwise direction for a stable system, assuming \( P = 0 \). For systems with \( P \neq 0 \), the number of counter-clockwise encirclements must equal \( P \).

(-1,0)

Historical Context

Nyquist's work was initially motivated by the stability analysis of telephone amplifiers, which were prone to oscillations due to feedback. His criterion provided a robust method to ensure stable operation, laying the groundwork for modern control theory.

Nyquist Plot Stability Analysis A polar plot showing the Nyquist curve with encirclements around the (-1,0) point in the complex plane, illustrating stability conditions. Re Im (-1,0) ω → +∞ ω → -∞ N = 1
Diagram Description: The diagram would physically show the Nyquist plot with encirclements around the (-1,0) point in the complex plane, illustrating stability conditions.

1.2 Key Assumptions and System Requirements

Linear Time-Invariant (LTI) System Assumption

The Nyquist Stability Criterion applies strictly to Linear Time-Invariant (LTI) systems. The system's transfer function G(s) must satisfy:

$$ G(s) = \frac{N(s)}{D(s)} $$

where N(s) and D(s) are polynomials in the Laplace variable s, and the system must exhibit superposition and time-invariance. Nonlinearities or time-varying parameters invalidate the criterion unless linearized around an operating point.

Proper Transfer Function

The transfer function must be proper (degree of numerator ≤ degree of denominator) or strictly proper (degree of numerator < degree of denominator). This ensures the Nyquist plot remains finite as s → ∞.

No Poles on the Imaginary Axis

The open-loop system must have no poles lying exactly on the imaginary axis (s = jω). If such poles exist, the Nyquist contour must be modified with infinitesimal semicircular indentations to avoid them, introducing additional phase contributions.

Closed-Loop Stability Analysis

The criterion evaluates stability of the closed-loop system:

$$ T(s) = \frac{G(s)}{1 + G(s)H(s)} $$

where H(s) is the feedback path transfer function. The open-loop transfer function L(s) = G(s)H(s) must be known precisely.

Nyquist Contour Requirements

The analysis uses a contour in the complex plane that:

Frequency Response Data

Accurate knowledge of the system's frequency response L(jω) is essential. This is typically obtained through:

Relative Degree Consideration

The difference between the denominator and numerator degrees (relative degree) determines the Nyquist plot's behavior at infinite frequency:

$$ \lim_{\omega \to \infty} L(j\omega) = \begin{cases} 0 & \text{if strictly proper} \\ K & \text{if exactly proper (non-zero constant)} \\ \infty & \text{(invalid for Nyquist)} \end{cases} $$

Minimum-Phase vs. Non-Minimum Phase Systems

While the criterion applies to both cases, non-minimum phase systems (with RHP zeros) require special attention as they introduce additional phase lag that affects stability margins.

Relationship to the Principle of Argument

The Nyquist Stability Criterion is fundamentally rooted in the Principle of Argument, a key result from complex analysis. This principle states that for a meromorphic function \( F(s) \), the number of zeros \( Z \) and poles \( P \) inside a closed contour \( \Gamma \) in the complex plane is related to the net change in the argument (phase) of \( F(s) \) as \( s \) traverses \( \Gamma \). Mathematically, this is expressed as:

$$ N = Z - P $$

where \( N \) is the number of clockwise encirclements of the origin by \( F(s) \) as \( s \) moves along \( \Gamma \). The Nyquist Criterion applies this principle to the open-loop transfer function \( L(s) = G(s)H(s) \), mapping the right-half-plane (RHP) contour \( \Gamma \) via \( 1 + L(s) \).

Mapping the Nyquist Contour

The Nyquist contour \( \Gamma \) is constructed to enclose the entire right-half-plane, typically consisting of:

When \( \Gamma \) is mapped through \( L(s) \), the resulting Nyquist plot reveals stability by examining encirclements of the critical point \( (-1, 0) \). The argument principle ensures that:

$$ N = P_{cl} - P_{ol} $$

where \( P_{cl} \) is the number of unstable closed-loop poles, and \( P_{ol} \) is the number of unstable open-loop poles. For stability, \( P_{cl} = 0 \), requiring \( N = -P_{ol} \).

Practical Implications

In control system design, this relationship allows engineers to:

For example, in a system with \( P_{ol} = 2 \), the Nyquist plot must encircle \( (-1, 0) \) twice counterclockwise to ensure \( P_{cl} = 0 \). Violations of this condition directly indicate unstable closed-loop poles.

Mathematical Derivation

Let \( F(s) = 1 + L(s) \). The zeros of \( F(s) \) correspond to closed-loop poles, while its poles match the open-loop poles of \( L(s) \). Applying the argument principle:

$$ \frac{1}{2\pi} \Delta_{\Gamma} \arg F(s) = Z - P $$

where \( \Delta_{\Gamma} \arg F(s) \) is the net phase change. Since \( F(s) \) encircles the origin whenever \( L(s) \) encircles \( (-1, 0) \), the Nyquist criterion follows by substituting \( Z = P_{cl} \) and \( P = P_{ol} \).

This derivation underscores why the Nyquist plot's encirclements of \( (-1, 0) \)—not the origin—are the focus of stability analysis.

Nyquist Contour and Mapping Diagram showing the Nyquist contour Γ in the complex plane and its mapping through L(s) to illustrate encirclements of the critical point (-1, 0). Im Re RHP Γ Im Re (-1, 0) L(s)
Diagram Description: The diagram would show the Nyquist contour Γ in the complex plane and its mapping through L(s) to illustrate encirclements of the critical point (-1, 0).

2. Mapping the Open-Loop Transfer Function

2.1 Mapping the Open-Loop Transfer Function

The Nyquist Stability Criterion evaluates closed-loop system stability by analyzing the open-loop transfer function L(s). The process begins by mapping the contour of L(s) in the complex plane as s traverses the Nyquist contour—a semicircular path enclosing the right-half plane (RHP).

Constructing the Nyquist Contour

The Nyquist contour is defined in the s-plane as follows:

For a system with open-loop transfer function L(s) = G(s)H(s), the Nyquist plot is the image of this contour under L(s).

Mathematical Formulation

Consider L(s) in factored form:

$$ L(s) = K \frac{\prod_{i=1}^m (s - z_i)}{\prod_{j=1}^n (s - p_j)} $$

As s traverses the Nyquist contour, the phase and magnitude of L(s) evolve. The critical observation is the encirclement condition:

$$ N = Z - P $$

where:

Practical Mapping Procedure

  1. Identify singularities: Locate poles of L(s) on the imaginary axis, which require detours via semicircular indentations.
  2. Evaluate L(s) along Segment 1: Compute L(jω) for ω ∈ (−∞, ∞). Due to symmetry, L(−jω) is the complex conjugate of L(jω).
  3. Evaluate L(s) along Segment 2: For |s| → ∞, L(s) typically converges to zero or a constant gain, simplifying the plot.

Example: First-Order System

Let L(s) = \frac{K}{s + a} (a > 0). The Nyquist plot is a semicircle:

No encirclements of (−1, 0) occur, and since P = 0, the closed-loop system is stable for all K > 0.

Visual Interpretation

The Nyquist plot’s geometry reveals stability:

For higher-order systems, numerical tools (e.g., MATLAB’s nyquist function) automate this mapping, but manual analysis remains essential for interpreting edge cases.

Nyquist Contour and Mapping A diagram showing the Nyquist contour in the s-plane (left) and its mapping to the Nyquist plot in the complex plane (right), illustrating the encirclement condition around the point (-1, 0). Re Im RHP Nyquist Contour Re Im (-1, 0) Encirclements Nyquist Plot (L(s)) Mapping
Diagram Description: The diagram would physically show the Nyquist contour in the s-plane and its mapping to the Nyquist plot in the complex plane, illustrating the encirclement condition.

Handling Poles and Zeros at the Origin

When applying the Nyquist stability criterion, systems with open-loop poles or zeros at the origin (s = 0) require special consideration due to their impact on the Nyquist contour and encirclement interpretation. These singularities introduce phase discontinuities and influence the stability analysis.

Modification of the Nyquist Contour

To avoid passing through the origin, the Nyquist contour is adjusted with an infinitesimal semicircular detour of radius r → 0 in the right-half plane (RHP). This exclusion ensures the contour remains analytic while encircling the pole/zero:

$$ C = \lim_{r \to 0} \left( \text{Semicircle around } s = 0 \text{ defined by } s = re^{j heta}, \quad heta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \right) $$

Phase Contribution Analysis

For an open-loop transfer function L(s) with N poles at the origin (L(s) = s^{-N}G(s)), the phase shift introduced by the detour is:

$$ \Delta \angle L(s) = -N \cdot \frac{\pi}{2} \quad \text{as } s \text{ traverses the semicircle from } \omega = 0^- \text{ to } \omega = 0^+ $$

This results in an instantaneous phase drop of −Nπ/2 radians, which must be accounted for when plotting the Nyquist diagram.

Practical Implications

Example: Type-1 System

Consider L(s) = K/(s(s+1)). The Nyquist contour’s detour around s = 0 introduces a phase shift of −π/2. The low-frequency asymptote of the Nyquist plot thus begins at −90° and magnitude |L(j\omega)| → ∞ as \omega → 0.

$$ \lim_{\omega \to 0} L(j\omega) = \frac{K}{j\omega} \Rightarrow \angle L(j\omega) = -\frac{\pi}{2}, \quad |L(j\omega)| \to \infty $$

Visualizing the Effect

The Nyquist plot for systems with poles at the origin exhibits an infinite semicircular arc in the clockwise direction, corresponding to the detour’s phase contribution. The number of such arcs equals the multiplicity of poles at s = 0.

Critical Point (−1, 0)
Nyquist Contour Detour and Phase Shift A schematic diagram showing the Nyquist contour with a semicircular detour around the origin and its corresponding Nyquist plot, illustrating the phase shift and critical point. s = 0 r → 0 jω → +∞ jω → -∞ Re Im -Nπ/2 (-1, 0) Re Im Nyquist Contour Detour and Phase Shift Nyquist Contour Nyquist Plot
Diagram Description: The diagram would physically show the modified Nyquist contour with the infinitesimal semicircular detour around the origin and its impact on the Nyquist plot's phase and magnitude.

2.3 Dealing with Non-Minimum Phase Systems

Non-minimum phase (NMP) systems are characterized by transfer functions with zeros in the right-half plane (RHP). These systems exhibit counterintuitive behavior, such as initial undershoot in their step response, complicating stability analysis under the Nyquist criterion. Unlike minimum-phase systems, where phase and magnitude responses are uniquely linked, NMP systems introduce additional phase lag without affecting the magnitude response.

Impact on Nyquist Stability Analysis

The Nyquist stability criterion relies on the principle of argument applied to the open-loop transfer function L(s). For NMP systems, the presence of RHP zeros modifies the encirclement condition. Specifically, if L(s) has N RHP poles and M RHP zeros, the Nyquist plot must encircle the critical point −1 + j0 N − M times counterclockwise for closed-loop stability.

$$ Z = N - P $$

where Z is the number of RHP closed-loop poles, N is the number of clockwise encirclements of −1, and P is the number of RHP poles of L(s). For NMP systems, P must account for both RHP poles and zeros.

Practical Example: Aircraft Control Systems

In aircraft dynamics, altitude control systems often exhibit NMP behavior due to the effect of elevator deflection. A sudden upward elevator command initially causes a downward pitch moment before the aircraft climbs. The Nyquist plot for such a system shows an unexpected phase drop near the crossover frequency, necessitating careful gain margin adjustment to avoid instability.

−1

Compensation Techniques

Stabilizing NMP systems requires:

$$ C(s) = K \frac{s + z}{s + p} \quad (z < p) $$

where C(s) is the compensator transfer function, and z, p are chosen to shift the Nyquist contour away from the critical point.

Robustness Considerations

NMP systems are inherently less robust to parameter variations. A small perturbation in the zero location can drastically alter the Nyquist plot’s shape. Sensitivity analysis using the H∞ norm is recommended to ensure stability margins are maintained under uncertainty.

Nyquist Plot for NMP System with RHP Zero A Nyquist plot illustrating the frequency response of a non-minimum phase (NMP) system with a right-half-plane (RHP) zero. The plot shows the Nyquist contour wrapping around the critical point (-1 + j0) in the complex plane. Re Im -1 + j0 ω
Diagram Description: The section includes a Nyquist plot example for an NMP system, which is a highly visual concept showing the relationship between phase and magnitude in the complex plane.

3. Counting Encirclements of the Critical Point (-1, 0)

3.1 Counting Encirclements of the Critical Point (-1, 0)

The Nyquist Stability Criterion assesses closed-loop stability by examining the encirclements of the critical point (-1, 0) in the complex plane by the Nyquist plot of the open-loop transfer function L(s). The number and direction of these encirclements determine the stability of the closed-loop system.

Mathematical Foundation

The argument principle from complex analysis underpins this criterion. For a closed contour Γ in the s-plane, the change in the argument of L(s) as s traverses Γ is related to the number of zeros and poles of 1 + L(s) enclosed by Γ:

$$ N = Z - P $$

where:

Determining Encirclement Direction

Encirclements are counted as:

The direction corresponds to the phase increase of L(s) as the frequency increases from -∞ to +∞.

Practical Counting Method

To systematically count encirclements:

  1. Draw a vector from (-1, 0) to a point on the Nyquist plot
  2. Track the angle change of this vector as ω sweeps from -∞ to +∞
  3. Each full 360° rotation constitutes one encirclement
  4. Sum the net rotations (clockwise positive, counter-clockwise negative)

Stability Condition

For a stable closed-loop system:

$$ Z = N + P = 0 $$

This means the number of clockwise encirclements must exactly cancel the open-loop RHP poles. If L(s) has no RHP poles (P = 0), the Nyquist plot must not encircle (-1, 0) at all for stability.

Visual Interpretation

Consider a Nyquist plot that:

The phase and gain margins can be directly observed from how close the Nyquist plot approaches (-1, 0).

Example Case

For a system with P = 2 RHP poles:

$$ N = -2 $$

Requires two counter-clockwise encirclements of (-1, 0) for stability (Z = 0).

Nyquist Plot Encirclements of (-1, 0) A Nyquist plot on the complex plane showing encirclements of the critical point (-1, 0) with directional arrows indicating frequency sweep. Re Im (-1, 0) ω = +∞ ω = -∞ Clockwise Counter-Clockwise
Diagram Description: The section describes spatial relationships (encirclements of (-1, 0)) and directional tracking of Nyquist plots, which are inherently visual concepts.

3.2 Determining Closed-Loop Stability from Open-Loop Data

The Nyquist Stability Criterion provides a powerful method to assess the stability of a closed-loop system by analyzing the frequency response of its open-loop transfer function L(s). The key insight lies in mapping the open-loop Nyquist plot and counting encirclements of the critical point (-1, 0) in the complex plane.

Mathematical Foundation

Consider a closed-loop system with open-loop transfer function L(s) = G(s)H(s). The characteristic equation is given by:

$$ 1 + L(s) = 0 $$

The Nyquist criterion relates the number of right-half-plane (RHP) poles P of L(s) to the number of encirclements N of (-1, 0) by:

$$ Z = N + P $$

where Z is the number of RHP roots of the characteristic equation (unstable closed-loop poles). For stability, we require Z = 0.

Step-by-Step Stability Assessment

  1. Count RHP poles of L(s): Determine P from the open-loop transfer function.
  2. Generate Nyquist plot: Plot L(jω) for ω from -∞ to +∞, including the infinite semicircle.
  3. Count encirclements: Observe how many times the plot encircles (-1, 0) clockwise.
  4. Apply the criterion: Calculate Z = N + P. The system is stable if Z = 0.

Practical Considerations

In real-world applications:

Example: Third-Order System

Consider L(s) = k/(s(s+1)(s+2)):

$$ P = 0 \text{ (no RHP poles)} $$ $$ \text{For } k = 6: N = -1 \Rightarrow Z = -1 \text{ (unstable)} $$ $$ \text{For } k = 2: N = 0 \Rightarrow Z = 0 \text{ (stable)} $$
(-1,0)

The plot shows two Nyquist contours for different gains, demonstrating how stability depends on the encirclement count.

Extensions to Non-Minimum Phase Systems

When P > 0, the stability condition becomes N = -P. This frequently occurs in:

Nyquist Plot with Encirclements A Nyquist plot illustrating encirclements of the (-1, 0) point on the complex plane, with trajectories for stable and unstable cases. Re Im (-1, 0) Stable Unstable ω₁ ω₂ Stability Region Stable Unstable
Diagram Description: The Nyquist plot and encirclements of the (-1, 0) point are inherently spatial concepts that require visualization.

3.3 Special Cases: Systems with Open-Loop Poles on the Imaginary Axis

When applying the Nyquist stability criterion, a complication arises if the open-loop transfer function L(s) has poles on the imaginary axis. These poles introduce singularities in the mapping, requiring a modified Nyquist contour to avoid them while preserving the stability analysis.

Modified Nyquist Contour

To handle poles at s = jω₀, the standard Nyquist contour is adjusted with infinitesimal semicircular detours into the right-half plane (RHP) around each pole. The radius r → 0 ensures the contour does not pass through the singularity. For a pole at s = jω₀, the detour is parameterized as:

$$ s = jω₀ + re^{jθ}, \quad θ \in \left[-\frac{π}{2}, \frac{π}{2}\right] $$

As r → 0, the contribution of this detour to the Nyquist plot becomes a semicircular arc of infinite radius in the L(s)-plane. The direction depends on the pole order:

Practical Implications

Systems with integrators (L(s) = 1/s) or harmonic oscillators (L(s) = 1/(s² + ω₀²)) exhibit this behavior. For example, a motor control loop with an integrator:

$$ L(s) = \frac{K}{s(s+1)} $$

Here, the Nyquist contour must detour around s = 0. The mapping of this detour results in an infinite-radius arc rotating -180° (due to the 1/s term) in the L(s)-plane.

Stability Criterion Adjustment

The Nyquist criterion is still applied as Z = P - N, where:

For the motor control example above (P = 0), if the Nyquist plot encircles (-1, 0) once clockwise (N = -1), then Z = 1, indicating instability.

Case Study: Phase-Locked Loop (PLL)

In PLLs, the open-loop transfer function often includes a pole at the origin (integrator) and complex poles near the imaginary axis. The Nyquist plot must account for detours around these poles, with careful attention to the phase contribution of the infinite-radius arcs. Incorrect handling can lead to false stability conclusions.

s = jω₀ Infinite-radius arc
Nyquist Contour Detour for Imaginary-Axis Pole A schematic diagram showing the modified Nyquist contour detouring around the imaginary-axis pole at s = jω₀ in the s-plane (left) and the resulting infinite-radius arc in the L(s)-plane (right). Re Im s = jω₀ r → 0 +180° Nyquist Contour Re Im (-1, 0) L(s)-plane Infinite-radius arc Nyquist Contour Detour for Imaginary-Axis Pole s-plane L(s)-plane
Diagram Description: The diagram would physically show the modified Nyquist contour detouring around the imaginary-axis pole and the resulting infinite-radius arc in the L(s)-plane.

4. Nyquist Analysis for Simple Feedback Systems

Nyquist Analysis for Simple Feedback Systems

Fundamentals of the Nyquist Criterion

The Nyquist Stability Criterion evaluates the stability of a closed-loop feedback system by analyzing the open-loop transfer function L(s). For a system with forward path gain G(s) and feedback path H(s), the open-loop transfer function is:

$$ L(s) = G(s)H(s) $$

The criterion relates the number of encirclements of the point (−1, 0) in the complex plane by the Nyquist plot of L(jω) to the number of unstable poles of the closed-loop system. The Nyquist path is a contour in the s-plane that encloses the right half-plane (RHP).

Mathematical Derivation

Consider the characteristic equation of the closed-loop system:

$$ 1 + L(s) = 0 $$

The argument principle states that the number of zeros N of 1 + L(s) in the RHP is given by:

$$ N = Z - P $$

where:

For stability, N = 0 (no zeros of 1 + L(s) in the RHP), requiring:

$$ Z = P $$

Nyquist Plot Construction

The Nyquist plot is generated by evaluating L(jω) for ω ∈ (−∞, ∞) and plotting the imaginary part against the real part. Key features include:

Stability Interpretation

The closed-loop system is stable if and only if the number of counter-clockwise encirclements of (−1, 0) equals the number of RHP poles of L(s). For minimum-phase systems (P = 0), stability requires no encirclements.

Practical Example: First-Order System

Consider a unity feedback system with:

$$ G(s) = \frac{K}{s + a} $$

The Nyquist plot is a semicircle in the lower half-plane with radius K/(2a) centered at (K/(2a), 0). For K > a, the plot does not encircle (−1, 0), confirming stability.

Gain and Phase Margins

The Nyquist plot directly reveals:

These margins quantify robustness against parameter variations.

Non-Minimum Phase Systems

For systems with RHP zeros or poles, the Nyquist criterion remains valid but requires careful interpretation of encirclements. The plot may exhibit unexpected behavior due to phase non-monotonicity.

Multivariable Extensions

The generalized Nyquist criterion extends to MIMO systems using the determinant of the return difference matrix:

$$ \det(I + L(jω)) $$

where L(jω) is the open-loop transfer matrix. The number of encirclements of the origin by the characteristic loci determines stability.

Nyquist Plot with Encirclements A Nyquist plot on the complex plane showing encirclements of the critical point (-1, 0) to illustrate stability analysis. Re Im (-1, 0) ω = 0 ω → ∞ N = 2
Diagram Description: The Nyquist plot and encirclements of the point (−1, 0) are inherently spatial concepts that require visualization to understand the relationship between the plot and stability.

Stability Margins: Gain and Phase Margins from Nyquist Plots

The Nyquist stability criterion provides a graphical method to assess the stability of a closed-loop system by examining the open-loop transfer function's Nyquist plot. Two critical metrics derived from this plot are the gain margin (GM) and phase margin (PM), which quantify the system's robustness against instability due to gain variations or phase delays.

Gain Margin (GM)

The gain margin is defined as the amount by which the open-loop gain can be increased before the system reaches the verge of instability. Mathematically, it is determined at the phase crossover frequency (ω180), where the phase of the open-loop transfer function equals −180°.

$$ \text{GM} = \frac{1}{|G(j\omega_{180})H(j\omega_{180})|} $$

In decibels (dB), the gain margin is expressed as:

$$ \text{GM}_{\text{dB}} = -20 \log_{10} |G(j\omega_{180})H(j\omega_{180})| $$

A positive gain margin (GM > 1 or GMdB > 0) indicates stability, while a negative margin implies instability. For robust designs, engineers typically aim for GM ≥ 6 dB.

Phase Margin (PM)

The phase margin is the additional phase lag required at the gain crossover frequency (ω1), where the magnitude of the open-loop transfer function equals unity (0 dB), to bring the system to the brink of instability. It is calculated as:

$$ \text{PM} = 180° + \angle G(j\omega_1)H(j\omega_1) $$

A positive phase margin (PM > 0°) ensures stability, with typical design targets ranging from 30° to 60° for adequate robustness.

Interpreting Nyquist Plots for Stability Margins

On the Nyquist plot:

(−1,0) PM

Practical Implications

Stability margins are crucial in control system design:

For example, in aerospace systems, insufficient phase margins can lead to oscillatory instabilities, while inadequate gain margins may cause saturation or failure under high-gain conditions.

Nyquist Plot with Stability Margins A polar plot showing the Nyquist curve intersecting the unit circle and negative real axis, with annotations for gain margin (GM), phase margin (PM), and critical frequencies (ω₁, ω₁₈₀). Unit Circle (-1, 0) Nyquist Curve ω₁ ω₁₈₀ GM PM Re Im
Diagram Description: The diagram would physically show the Nyquist plot with the unit circle, the −1 point, and the angular relationship representing phase margin.

4.3 Case Study: Nyquist Criterion in Control System Design

Consider a feedback control system with an open-loop transfer function given by:

$$ G(s)H(s) = \frac{K}{s(s+1)(s+2)} $$

To assess stability using the Nyquist criterion, we first analyze the open-loop poles. The poles are at s = 0, -1, -2, all in the left half-plane (LHP) except for the origin. The Nyquist path must encircle the entire right half-plane (RHP), avoiding the pole at the origin with an infinitesimal semicircle.

Constructing the Nyquist Plot

We evaluate G(jω)H(jω) along three critical segments:

  1. Segment 1: As s moves along the positive imaginary axis (s = jω, ω: 0⁺ → +∞), the magnitude and phase are:
    $$ |G(jω)H(jω)| = \frac{K}{ω\sqrt{ω^2 + 1}\sqrt{ω^2 + 4}} $$ $$ \angle G(jω)H(jω) = -90° - \tan^{-1}(ω) - \tan^{-1}(ω/2) $$
  2. Segment 2: The infinite semicircle maps to the origin in the G(s)H(s) plane.
  3. Segment 3: The negative imaginary axis (s = jω, ω: -∞ → 0⁻) mirrors Segment 1.

Stability Analysis

The Nyquist plot’s encirclements of the critical point (-1, 0) determine stability. For K = 6:

Practical Implications

In motor control systems, this analysis prevents oscillations by ensuring gain margins are respected. A Bode plot derived from the Nyquist data would show:

$$ \text{Gain Margin} = 20\log_{10}\left(\frac{6}{K}\right) $$

Engineers use this to tune K while maintaining stability, particularly in aerospace and robotics where overshoot can be catastrophic.

Nyquist plot for G(s)H(s) = K/[s(s+1)(s+2)] (-1,0)
Nyquist Plot for K/[s(s+1)(s+2)] Polar plot showing Nyquist curve for K=6 (marginal stability) and K>6 (unstable case), with critical point (-1,0), ω=√2, and direction of increasing ω. Re Im (-1,0) ω=√2 ω↑ |G|=1 |G|=1 K=6 (Marginal) K>6 (Unstable)
Diagram Description: The Nyquist plot's encirclements of the critical point (-1,0) and its phase/magnitude variations are inherently spatial relationships.

5. Key Textbooks and Papers on Nyquist Stability

5.1 Key Textbooks and Papers on Nyquist Stability

5.2 Online Resources and Interactive Tools

5.3 Advanced Topics and Extensions of the Nyquist Criterion