Nyquist stability criterion

The Nyquist Stability Criterion is a unique and powerfull method for determining the stability of a closed-loop control system. The criterion was established by Harry Nyquist.

given a Transfer function $\mathcal{T}(s)$ it becomes necessary in control systems engineering to determine how many poles of a closed-loop feedback system can be found in the right-half of the complex $s$ -plane (laplace domain plane). See Laplace Transform.

Contents

1 Background

2 Terminology

3 Stability Concerns

4 The Principle of the Argument

5 The Nyquist Criterion

6 References

Background

any transfer function can be written in the form $\mathcal{T}(s) = \frac{\Sigma_i (s + Z_i)}{\Delta (s)}$ (Mason's Rule) where $Δ(s) = 0$ is known as the "Characteristic Equation." Solving the characteristic equation for $s$ yeilds the "Poles of the Closed-Loop Transfer Function." In a negative feedback loop, the characteristic equation $Δ(s)$ is equal to $\Delta (s) = 1 + \mathcal{F}(s)$ where $\mathcal{F}(s)$ is known as the "Loop Transfer function", or in situations where their is only a single feedback loop, it is known as the "Open-Loop Feedback Function."

Through further expansion, $\Delta (s) = 1 + \mathcal{F}(s) = 1 + \frac{N(s)}{D(s)} = \frac{D(s) + N(s)}{D(s)}$ (eq 1)

Terminology

Zero: Given an equation $F(s) = \frac{A(s)}{B(s)}$ , solving the equation $A (s) = 0$ for $s$ yeilds the Zeros of $F (s)$ . Literally, a Zero of a function of $s$ is a value for $s$ where the function returns $0$

Pole: Given the same equation $F(s) = \frac{A(s)}{B(s)}$ , solving $B (s) = 0$ for $s$ yeilds the Poles of $F (s)$ . Literally, a Pole $s = p$ is a value for which $\lim_{n \to p} \mathcal{F}(s) = \infty$

Stability Concerns

In the complex Laplace Domain, a system's Transfer Function may not have Poles in the right half of the plane, and remain stable. Through a careful examination of equation 1 (above), it can be seen that the Zeros of $Δ(s)$ are the Poles of $\mathcal{T}(s)$ . Therefore, by examining $Δ(s)$ , one can determine the overall stability of the system.

The Principle of the Argument

According to a theorem stated originally by Cauchy, a contour $Γ s$ drawn in the complex $s$ plane, that may encompass any number of non-analytic points but may not pass directly through any such points, can be mapped to another plane (the $F (s)$ plane) by a function $F (s)$ . A result of this mapping is that the resultant contour $Γ F (s)$ will encircle the origin of the $F (S)$ plane $N$ times, where $N = Z - P$ . $Z$ and $P$ are the number of Zeros and Poles of $F (s)$ , respectively.

The Nyquist Criterion

To Be finished - some notation conflicts A feedback system is stable if the path of the loop gain (A(s) B(s)) in the complex plane, (plotted for all real frequency) does not enclose the point (1 + j0), provided that A(s) and B(s) are themselves stable (and AB -> 0 as mod(s) -> infinity). Here A is the open-circuit amplifier response, and B is the feedback loop characteristic.

See Also:

References

Faulkner E A (1969) 'Introduction to the Theory of Linear Systems' Chapman & Hall, ISBN 412 09400 2

Pippard A B (1985) 'Response & Stability' Cambridge University Press, ISBN 0 521 31994 3