Understanding the Nyquist Criterion
The Nyquist criterion is a fundamental tool in control systems engineering used to determine the stability of a closed-loop system based on the open-loop transfer function. It's particularly useful for systems with time delays or complex pole-zero configurations where root locus or Bode plots might be less intuitive.
Core Concepts of Nyquist Stability
The Nyquist criterion relates the number of poles and zeros of the open-loop transfer function that lie in the right-half of the s-plane to the encirclements of the critical point (-1, 0) by the Nyquist locus (the frequency response of the open-loop system).
Stability is determined by how the Nyquist plot encircles the critical point (-1, 0).
The Nyquist criterion states that for a closed-loop system to be stable, the number of clockwise encirclements of the critical point (-1, 0) by the Nyquist locus must equal the number of open-loop poles in the right-half s-plane (P).
Mathematically, the criterion is expressed as N = P - Z, where N is the number of clockwise encirclements of the critical point (-1, 0) by the Nyquist locus, P is the number of open-loop poles in the right-half s-plane, and Z is the number of closed-loop poles in the right-half s-plane. For stability, Z must be zero. Therefore, N must equal P. If the open-loop system is stable (P=0), then N must be 0 for the closed-loop system to be stable.
Constructing the Nyquist Locus
To apply the Nyquist criterion, we first need to plot the Nyquist locus. This involves evaluating the open-loop transfer function G(s)H(s) for s = jω, where ω ranges from -∞ to +∞. Special attention is paid to contours around poles on the jω-axis.
Steps for Plotting
- Determine Open-Loop Poles: Identify all poles of the open-loop transfer function G(s)H(s).
- Frequency Response: Substitute s = jω into G(s)H(s) to get G(jω)H(jω).
- Plot for ω from 0 to ∞: Calculate the magnitude |G(jω)H(jω)| and phase ∠G(jω)H(jω) for ω ≥ 0. Plot these values in the complex plane.
- Plot for ω from -∞ to 0: The locus for ω < 0 is the mirror image of the locus for ω > 0 with respect to the real axis.
- Handle Poles on jω-axis: If there are poles on the jω-axis (e.g., at s=0), the contour must be indented around them. A small semi-circular arc in the right-half plane is used to bypass these poles, corresponding to a phase shift of ±180° as ω approaches the pole from the right.
Applying the Criterion for Stability
Once the Nyquist locus is plotted, we count the number of clockwise encirclements (N) of the critical point (-1, 0). We also determine the number of open-loop poles in the right-half s-plane (P).
| Condition | Closed-Loop Stability |
|---|---|
| N = P | Stable |
| N > P | Unstable |
| N < P | Unstable |
Remember: The critical point is (-1, 0). A clockwise encirclement contributes +1 to N. An anti-clockwise encirclement contributes -1 to N. If the Nyquist locus passes through the critical point, the system is marginally stable.
Gain Margin and Phase Margin
The Nyquist plot also provides insights into the relative stability of the system through gain and phase margins. The phase margin is the amount of additional phase lag required to make the system unstable at the gain crossover frequency (where the magnitude is 1). The gain margin is the amount of additional gain that can be added before the system becomes unstable at the phase crossover frequency (where the phase is -180°).
The Nyquist criterion is visualized by plotting the frequency response of the open-loop system G(jω)H(jω) in the complex plane. This plot, known as the Nyquist locus, is then analyzed for encirclements of the critical point (-1, 0). The number of clockwise encirclements (N) must equal the number of open-loop poles in the right-half s-plane (P) for the closed-loop system to be stable (Z=0). If P=0 (open-loop stable), then N must be 0 for closed-loop stability. The critical point is the origin of the transformed s-plane when using the bilinear transformation, or simply the point (-1,0) in the G(jω)H(jω) plane.
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Example Scenario
Consider an open-loop transfer function G(s)H(s) = 1 / (s+1). This system has one pole at s = -1 (in the left-half plane, so P=0). The Nyquist locus for this system is a semi-circle in the complex plane. Since P=0, for the closed-loop system to be stable, N must also be 0. This means the Nyquist locus should not encircle the critical point (-1, 0).
The critical point is (-1, 0) in the complex plane.
N represents the number of clockwise encirclements of the critical point (-1, 0) by the Nyquist locus.
Learning Resources
Provides a clear explanation of the Nyquist criterion, including the steps for plotting and applying the stability rules.
A detailed article covering the Nyquist criterion, its graphical interpretation, and examples for stability analysis.
A community discussion offering practical insights and common questions about the Nyquist criterion.
A video tutorial demonstrating how to plot a Nyquist locus and apply the criterion for stability analysis.
The Wikipedia page offers a comprehensive overview of the Nyquist criterion, its mathematical basis, and historical context.
Another excellent video explaining the Nyquist criterion with clear examples and visual aids.
A PDF document from NPTEL providing a rigorous treatment of the Nyquist criterion within a broader control systems course.
Explains the Nyquist criterion with a focus on its application in determining system stability and relative stability.
Official MATLAB documentation on the `nyquistplot` function, useful for understanding how to implement the criterion computationally.
A video focusing on the practical application of the Nyquist criterion for stability analysis of control systems.