The **Nyquist Stability Criterion** is a powerful analytical tool used in control systems to determine the stability of a closed-loop system based on the open-loop frequency response. Introduced by Harry Nyquist in 1932, this criterion provides a graphical method to assess whether a system will remain stable under feedback conditions. The open-loop frequency response refers to the system's behavior when feedback is removed, and it can be determined through calculations or experiments.
This method is particularly useful because it allows engineers to visualize the system's behavior without solving complex equations directly. The Nyquist criterion applies only to linear, time-invariant (LTI) systems. In classical control theory, it is primarily used for single-input, single-output (SISO) systems, forming a core part of frequency domain analysis. Over time, its application has been extended to multi-variable systems as well.
To understand the Nyquist criterion, one must first grasp the concept of the **argument principle** from complex analysis. This principle relates the number of zeros and poles of a function within a closed contour. Applying this to control systems, the key idea is to examine the trajectory of the open-loop transfer function G(s)H(s) in the complex plane as s traverses a specific path—typically the imaginary axis combined with a large semicircle in the right half-plane.
The critical point to monitor is the point (-1, j0). If the Nyquist plot passes through this point, the system is **critically stable**. Otherwise, the stability is determined by the relationship between the number of encirclements around this point and the number of open-loop poles in the right half-plane.
The formula Z = P - 2N helps quantify this:
- **Z** represents the number of closed-loop poles in the right half-plane (i.e., unstable poles).
- **P** is the number of open-loop poles in the right half-plane.
- **N** is the number of clockwise encirclements of the (-1, j0) point by the Nyquist plot.
If Z = 0, the system is stable. If Z > 0, the system is unstable. This approach allows engineers to predict system behavior without explicitly solving characteristic equations, making it both efficient and insightful.
The Nyquist plot is symmetric about the real axis if the system is real, meaning we only need to analyze the upper half of the imaginary axis. The lower half is automatically inferred due to this symmetry. Additionally, special care must be taken when there are poles at the origin, which require an arc to avoid them during the contour traversal.
By plotting the open-loop transfer function’s frequency response, engineers can gain deep insights into the stability margins, phase margins, and overall robustness of the system. This makes the Nyquist criterion an essential tool in both theoretical analysis and practical design.
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