How does the Nyquist criterion determine the stability of feedback control systems?
1 Answer
The Nyquist criterion is a graphical tool used to analyze the stability of feedback control systems. It is based on the Nyquist plot, which is a plot of the frequency response of the open-loop transfer function of the system. The Nyquist criterion provides valuable insights into stability and robustness of feedback control systems.
Here's how the Nyquist criterion determines the stability of a feedback control system:
Open-loop transfer function: The first step is to obtain the open-loop transfer function of the control system. This transfer function represents the relationship between the output and input of the system in the absence of feedback.
Complex frequency analysis: The Nyquist criterion operates in the frequency domain. It involves analyzing the open-loop transfer function at various points in the complex plane (i.e., for different complex frequencies s = σ + jω, where σ is the real part and ω is the angular frequency).
Nyquist plot: For each value of s in the complex plane, evaluate the open-loop transfer function G(s). Then plot the complex values of G(s) in the complex plane. The plot is known as the Nyquist plot.
Encirclement of critical point (-1, 0): The critical point (-1, 0) represents the point on the Nyquist plot corresponding to the frequency ω = 0 dB (unity gain) and 180-degree phase shift. The stability of the feedback system is determined by checking if the Nyquist plot encircles the critical point in the clockwise direction. The number of encirclements of the critical point is equal to the number of unstable poles of the closed-loop system.
Stability assessment: Based on the Nyquist plot's behavior, the system stability can be determined:
a. If the Nyquist plot does not encircle the critical point (-1, 0) at all, the system is stable.
b. If the Nyquist plot encircles the critical point in the clockwise direction a finite number of times, the system has that many unstable poles.
c. If the Nyquist plot encircles the critical point an infinite number of times (goes around the critical point in a loop), the system is marginally stable or unstable.
The Nyquist criterion is particularly useful for determining the stability of systems with complex transfer functions and multiple poles. By examining the Nyquist plot, control system engineers can assess the system's stability and make design decisions to ensure a stable and robust feedback control system. It also aids in identifying the necessary modifications to improve system stability, such as adjusting controller gains or adding compensators.
Here's how the Nyquist criterion determines the stability of a feedback control system:
Open-loop transfer function: The first step is to obtain the open-loop transfer function of the control system. This transfer function represents the relationship between the output and input of the system in the absence of feedback.
Complex frequency analysis: The Nyquist criterion operates in the frequency domain. It involves analyzing the open-loop transfer function at various points in the complex plane (i.e., for different complex frequencies s = σ + jω, where σ is the real part and ω is the angular frequency).
Nyquist plot: For each value of s in the complex plane, evaluate the open-loop transfer function G(s). Then plot the complex values of G(s) in the complex plane. The plot is known as the Nyquist plot.
Encirclement of critical point (-1, 0): The critical point (-1, 0) represents the point on the Nyquist plot corresponding to the frequency ω = 0 dB (unity gain) and 180-degree phase shift. The stability of the feedback system is determined by checking if the Nyquist plot encircles the critical point in the clockwise direction. The number of encirclements of the critical point is equal to the number of unstable poles of the closed-loop system.
Stability assessment: Based on the Nyquist plot's behavior, the system stability can be determined:
a. If the Nyquist plot does not encircle the critical point (-1, 0) at all, the system is stable.
b. If the Nyquist plot encircles the critical point in the clockwise direction a finite number of times, the system has that many unstable poles.
c. If the Nyquist plot encircles the critical point an infinite number of times (goes around the critical point in a loop), the system is marginally stable or unstable.
The Nyquist criterion is particularly useful for determining the stability of systems with complex transfer functions and multiple poles. By examining the Nyquist plot, control system engineers can assess the system's stability and make design decisions to ensure a stable and robust feedback control system. It also aids in identifying the necessary modifications to improve system stability, such as adjusting controller gains or adding compensators.