Nyquist Stability Criterion

Definition

The Nyquist Stability Criterion is a graphical method used in control theory to determine the stability of a feedback control system based on the open-loop frequency response. It involves plotting the Nyquist plot of the system's open-loop transfer function and analyzing the encirclements of the critical point (-1, 0) in the complex plane, which helps to ascertain whether the closed-loop system will be stable or unstable based on the number of poles in the right half-plane.

5 Must Know Facts For Your Next Test

1. The Nyquist Stability Criterion is particularly useful for systems that cannot be easily analyzed using traditional methods, such as those with time delays.
2. To apply the Nyquist criterion, you first need to determine the open-loop transfer function of the control system and create its Nyquist plot.
3. The critical point (-1, 0) is pivotal for assessing stability; encirclements around this point indicate how many unstable poles exist in the closed-loop system.
4. A system is stable if the number of clockwise encirclements of (-1, 0) equals the number of open-loop poles in the right half-plane.
5. Understanding phase margin and gain margin is essential when applying the Nyquist Stability Criterion, as they provide insight into how close a system is to instability.

Review Questions

• How does the Nyquist Stability Criterion help in determining the stability of a feedback control system?
  • The Nyquist Stability Criterion assesses stability by analyzing the Nyquist plot of a system's open-loop transfer function. By observing how many times and in what direction the plot encircles the critical point (-1, 0) in the complex plane, we can infer whether the closed-loop system is stable. Specifically, if the number of clockwise encirclements equals the number of poles in the right half-plane, stability is confirmed. This method provides a visual and intuitive way to evaluate stability.
• Explain how you would use a Nyquist plot to analyze a control system that exhibits time delay.
  • To analyze a control system with time delay using a Nyquist plot, first derive the open-loop transfer function that includes both the system dynamics and any delay term. Construct the Nyquist plot by plotting the frequency response over a range of frequencies. When interpreting this plot, it’s important to pay close attention to how it approaches the critical point (-1, 0) since time delays can introduce additional phase lag. This phase lag can lead to potential instability, which can be detected through changes in encirclement patterns around (-1, 0), thus informing about stability under varying conditions.
• Critically analyze how variations in system parameters affect the application of the Nyquist Stability Criterion.
  • Variations in system parameters such as gain or time constants can significantly impact stability as determined by the Nyquist Stability Criterion. As these parameters change, they alter the shape and position of the Nyquist plot, potentially affecting how it encircles the critical point (-1, 0). Increased gain can lead to more encirclements around this critical point, possibly indicating instability. Furthermore, adjustments in parameters may require reevaluation of both phase margin and gain margin. Understanding these interdependencies allows for better design and tuning of control systems to achieve desired stability.

Related terms

Bode Plot: A graphical representation of a linear time-invariant system's frequency response, showing magnitude and phase as functions of frequency.

Root Locus: A graphical technique used to analyze how the roots of a system change with variations in a particular parameter, typically gain.

Feedback Control System: A system that uses feedback to automatically adjust its operation, ensuring that the output remains stable and within desired parameters.