5.3. Nyquist Criterion

The Nyquist Criterion is a frequency-domain stability criterion used to determine the stability of a feedback system. It involves plotting the Nyquist plot, which is the plot of the system's frequency response G(jω) as ω varies from −∞ to +∞. The criterion helps determine if the closed-loop system will be stable based on the open-loop frequency response.

The Nyquist Criterion is a method used to assess the stability of feedback control systems. A feedback system is one where the output is fed back into the input to achieve desired performance. The Nyquist plot is a graphical representation that shows how the frequency response of the system changes as the frequency varies from negative infinity to positive infinity. By analyzing this plot, engineers can identify whether the system will remain stable under various conditions.

Think of the Nyquist Criterion like a balance scale. When you add weights (feedback) to one side, the balance point must remain stable. If the scale tips (instability) too much in one direction, it indicates that the system could collapse under certain conditions.

Steps to Construct the Nyquist Plot:

1. Calculate the frequency response G(jω) for various frequencies ω, where ω varies from −∞ to +∞.
2. Plot G(jω) in the complex plane.

To create the Nyquist plot, you first need to calculate the frequency response, G(jω), which involves substituting complex numbers into the transfer function of the system. This is done for a range of frequencies, allowing us to observe how the system behaves as the frequency changes. Once you have these values, you plot them on a complex plane where the x-axis represents the real part and the y-axis represents the imaginary part. This plot helps visualize how the system will respond to different frequencies.

Imagine plotting a graph of your shopping expenses over time. As you spend more (change frequencies), you want to see if you can stay within budget (stability). The points you plot show how your budget changes, similar to how the frequency response shows the system's behavior across different inputs.

Count the number of encirclements of the point −1 in the Nyquist plot:

• The system is stable if the Nyquist plot does not encircle the point −1 in the complex plane.
• The system is marginally stable if the plot encircles −1 once in a counter-clockwise direction.
• The system is unstable if the plot encircles −1 in a clockwise direction or does not encircle −1 as needed.

Once the Nyquist plot is constructed, you need to analyze how it interacts with the critical point at -1 in the complex plane. If the plot does not encircle this point at all, the system is considered stable. However, if it encircles -1 counter-clockwise once, it indicates marginal stability, which means the system can maintain a steady response but may not be robust under changes. In contrast, if it encircles -1 clockwise or fails to do so appropriately, it indicates that the system is unstable.

Think of the Nyquist plot like a racing track with a critical checkpoint (the -1 point) to determine if a car (the system) can safely navigate it. If the car smoothly passes the checkpoint without veering off track (no encirclements), it’s stable. If it just touches the checkpoint once (marginal stability), it may still manage. However, if the car spins out or runs into barriers (encircling -1 clockwise), it indicates a loss of control or instability.

For an open-loop transfer function G(s)=Ks(s+1)(s+2), calculate G(jω) and plot the Nyquist plot to analyze the stability of the closed-loop system. As ω → ∞, the plot approaches zero. Analyze the behavior of the Nyquist plot near the critical points, particularly −1, and determine if the system is stable.

To apply the Nyquist Criterion, you start with a specific open-loop transfer function, like G(s) = K/(s(s+1)(s+2)). First, determine the frequency response G(jω) by substituting jω into the transfer function. Then, you plot these values on the complex plane to construct the Nyquist plot. Observing how this plot behaves, especially in relation to the critical point of -1, will help you determine if the closed-loop system remains stable. Paying close attention to how the curve approaches zero as the frequency becomes very large also provides key insights into the system’s stability.

Consider tuning a radio to a specific station (analogous to the frequency response). Just like adjusting the dial helps you find the right signal, calculating G(jω) allows you to understand how the system behaves at different frequencies. Watching how the overall signal (Nyquist plot) interacts with noise and static (the point -1) determines if your favorite channel remains clear or becomes distorted (stable or unstable).

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