Nyquist Criterion with an Unstable Pole

Example: Calculating winding and poles

Consider the system with open-loop response L(s) = K/(s - 1) and K = 2. We can determine its stability as follows:

1. The open-loop system has a pole at s = 1, so P = 1 open right-half-plane pole.
2. The closed-loop pole is the root of 1 + L(s) = 0, which means 1 + K/(s - 1) = 0, or s = 1 - K. For K = 2, the closed-loop pole is at 1 - 2 = -1.
3. Since the closed-loop pole is at -1 (in the left-half plane), the closed-loop system is stable. The number of closed right-half-plane poles is Z = 0.
4. According to the Nyquist Criterion, Z = P - W. Therefore, 0 = 1 - W, which implies the counterclockwise winding W must be 1.
5. At frequency w = 1, the loop frequency response is L(i) = -2/(1 + 1) - i*2*1/(1 + 1) = -1 - i. The real part is -1.0 and imaginary part is -1.0.

Exploring the effect of Gain K

You can use the interactive module to see how the gain affects the Nyquist plot and stability.

1. Open the Nyquist Criterion with an Unstable Pole interactive module.
2. Set Gain K to 2. Observe the Nyquist plot encircling the critical point -1 counterclockwise once. The system status is Stable.
3. Change Gain K to 0.5. The radius of the Nyquist plot decreases and it no longer encircles -1. The winding becomes 0, and the status changes to Unstable.
4. Change Frequency to -1 with K = 2. Observe the selected point moving to Loop real -1.0 and Loop imaginary 1.0.

Model limitations and boundaries

This model uses a simplified loop transfer function L(s) = K/(s - 1) which describes a simple plant with a single unstable open-loop pole. It assumes a strictly negative feedback configuration.

When K = 1, the Nyquist plot passes exactly through the critical point -1. In this boundary case, the winding number is undefined ("Unavailable") and the closed-loop pole is at 0 (on the imaginary axis). The system is marginally stable but not asymptotically stable.

The visual representation uses a normalized scale to fit the Nyquist contour within the viewport regardless of the selected gain, which allows focusing on the topological encirclement rather than absolute magnitudes.

References

The foundational theory of the Nyquist stability criterion, particularly for systems with open-loop unstable poles, is a standard topic in classical control theory.

• University of Washington, Feedback Control Systems, Lecture 24: Nyquist Criterion: Explains the application of the Cauchy argument principle and the P-Z criterion for right-half-plane poles.