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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. Harry Nyquist (February 7, 1889 - April 4, 1976) was an important contributor to information theory. ...
Given a Transfer function 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. In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by: This integral transform has a number of properties that make it useful for analysing linear dynamical systems. ...
Background
Any transfer function can be written in the form (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 where 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, (eq 1)
Terminology - Zero: Given an equation , 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 , solving B(s) = 0 for s yeilds the Poles of F(s). Literally, a Pole s = p is a value for which
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 . Therefore, by examining Δ(s), one can determine the overall stability of the system. A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...
The Principle of the Argument According to a theorem stated originally by Cauchy, a contour Γs 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 ΓF(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. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...
The Nyquist Criterion The Nyquist Contour Nyquist proposed a contour in the s plane that goes as such: - a path traveling up the jω axis, from to .
- a semicircular arc, with radius , that starts at and travels clock-wise to
The Nyquist Criterion Given a Nyquist contour in the s plane, Γs, the resultant contour in the F(s)-plane, ΓF(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 Γs, and Z is the number of zeros of F(s) (and therefore the poles of ) enclosed by Γs
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:
A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ...
The Routh Hurwitz Stability Criterion is a necessary, and frequently sufficient method to establish the stability of a Single-Input, Single-Output, Linear Time Invarient (LTI) control system. ...
A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...
Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ...
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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