The **Nyquist Stability Criterion** is a powerful method used in control systems to assess the stability of a closed-loop system based on its open-loop frequency response. Introduced by Harry Nyquist in 1932, this criterion provides a graphical approach to determine whether a system will remain stable under feedback conditions. The open-loop frequency response refers to the system's behavior when feedback is disconnected, and it can be obtained through calculations or experiments. This makes the Nyquist criterion both practical and intuitive for engineers.
The criterion is particularly useful for linear time-invariant (LTI) systems. In classical control theory, it’s primarily applied to single-input, single-output (SISO) systems, where the frequency response method plays a central role in analysis and design. Since the 1970s, the concept has been extended to multi-variable systems, leading to more advanced frequency-domain techniques.
To understand the Nyquist criterion, we must consider the **argument principle** from complex analysis. The key idea involves mapping the open-loop transfer function G(s)H(s) as s traverses a specific contour in the complex plane. This contour typically includes the imaginary axis and a large semicircle in the right half-plane. The number of encirclements around the critical point (-1, j0) determines the stability of the system.
The criterion states that if the Nyquist plot does not pass through the (-1, j0) point, the system is stable if Z = P - 2N = 0. Here:
- **Z** is the number of zeros of F(s) = 1 + G(s)H(s) in the right half-plane.
- **P** is the number of poles of the open-loop transfer function in the right half-plane.
- **N** is the number of clockwise encirclements of the (-1, j0) point by the Nyquist plot.
If the plot passes through (-1, j0), the system becomes critically stable. Understanding this requires a solid grasp of complex variable functions and the argument principle. The direction of traversal (clockwise or counterclockwise) also affects the sign of N, which is crucial in determining stability.
In practice, the Nyquist plot is constructed by evaluating G(jω)H(jω) over a range of frequencies from 0 to ∞. The resulting curve helps visualize how the system responds to different inputs and whether it tends to oscillate or diverge. Additionally, special cases like poles at the origin require careful handling, often involving small arcs to avoid singularities.
By analyzing the Nyquist plot, engineers can make informed decisions about system design, ensuring that the closed-loop system remains stable even in the presence of uncertainties or changes in operating conditions. This makes the Nyquist criterion an essential tool in modern control engineering.
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