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Nyquist stability criterion

From Biocrawler, the free encyclopedia.

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

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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 $s$ yields 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 there 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)

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Terminology

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

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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.

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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.

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The Nyquist Criterion

The Nyquist Contour Nyquist proposed a contour in the $s$ plane that goes as such:

a path traveling up the $j ω$ axis, from $0 - j\infty$ to $0 + j\infty$ .
a semicircular arc, with radius $r \to \infty$ , that starts at $0 + j\infty$ and travels clock-wise to $0 - j\infty$

The Nyquist Criterion Given a Nyquist contour in the $s$ plane, $Γ s$ , the resultant contour in the $F (s)$ -plane, $Γ F (s)$ shall, for a stable feedback system, encircle the point (-1 + j0) a number of times $N$ such that $N = Z - P$ where $P$ is the number of poles of $F (s)$ encircled by $Γ s$ , and $Z$ is the number of zeros of $F (s)$ (and therefore the poles of $\mathcal{T}(s)$ ) enclosed by $Γ s$

Note: Some texts claim that any encirclements of the point (-1 + j0) causes the system to become unstable. This is not strictly accurate. Given the equation $N = Z - P$ , only $Z$ must be zero to ensure that the system is stable. A closed-loop feedback system may have a zero in the right half plane, without compromising stability. Such a system is called a "non-minumum phase system."

See Also:

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References

Faulkner, E.A. (1969): Introduction to the Theory of Linear Systems; Chapman & Hall; ISBN 0-412-09400-2
Pippard, A.B. (1985): Response & Stability; Cambridge University Press; ISBN 0-521-31994-3

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