Nyquist path

Nyquist Path: The Nyquist path is a crucial concept in stability analysis for feedback control systems. It provides a visual representation of the relations...

Nyquist Path:

The Nyquist path is a crucial concept in stability analysis for feedback control systems. It provides a visual representation of the relationship between the closed-loop system's stability and the bandwidth of its controllability.

The Nyquist path is defined as the locus of all feasible complex-valued transfer functions of the closed-loop system. A system is BIBO stable if its Nyquist path lies within the unit circle, indicating stable closed-loop behavior.

Key Points:

The Nyquist path is centered at the origin (0) and radiates outward.
A closed-loop system is asymptotically stable if its Nyquist path lies within the unit circle.
A closed-loop system is asymptotically stable if its Nyquist path lies outside the unit circle but not touching it.
A closed-loop system is marginally stable if its Nyquist path lies outside the unit circle but touches it at one endpoint.
The size of the Nyquist path corresponds to the bandwidth of the controllability of the system. A wider bandwidth implies a lower controllability.

Examples:

Consider a stable feedback system with a transfer function of:

$G(s) = \frac{1}{s + 1}$

The Nyquist path for this system would be a circle centered at the origin with a radius of 1, indicating a stable closed-loop system.

For another system with a transfer function of:

$G(s) = \frac{s}{s - 1}$

The Nyquist path would be a circle centered at the origin with a radius of 2, indicating an unstable closed-loop system.

The Nyquist path is a valuable tool for analyzing the stability and controllability of feedback control systems. By understanding the relationship between the Nyquist path and the stability of the system, engineers can design feedback controllers that achieve desired stability and performance characteristics