28-12-2012, 03:54 PM
Nyquist stability criterion
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The stability of a linear control system depends on the location of the characteristic equation roots in the s-plane.
characteristic equation should lie on the left side of the s-plane
Two general methods for determining the presence of unstable roots are :
Routh Stability Criterion
Works with the closed-loop system characteristic equation in an algebraic fashion.
Nyquist Stability Criterion
It is a semi-graphical technique based on open loop frequency response polar plot.
The Nyquist stability criterion can be used to judge the stability of a closed loop system using the frequency domain plot of its open loop transfer function.
The criterion is based on the complex analysis result known as Cauchy’s principle of argument. As the system transfer function is a complex function.
By applying Cauchy’s principle of argument to the open-loop system transfer function, L(s) = G(s)H(s) , information about stability of the closed-loop system transfer function can be achieved and arrive at the Nyquist stability criterion.
Principle of Argument
If a point or region is encircled by a closed path, a number ‘N’ can be assigned to indicate the number of times it is encircled by the closed path. For example point A in figure is encircled once by the closed path and point B is encircled twice by the closed path. N should be taken positive if the encirclement is in the clockwise direction and negative for counterclockwise.
Principle of Argument
Now instead of taking a discrete point if we take a close contour in s-plane such that the contour does not pass through any singular point, then for each value of s we can find a corresponding value of q(s) which can be plotted on the q(s)-plane.
It can be easily understood that for a closed contour in s-plane the corresponding plot in the q-plane will also form a closed contour.