NYQUIST STABILITY CRITERION
I. Setup and Assumptions:
II. Objective : to determine the stability of a feedback
control system using frequency response data.
Closed Loop Transfer Function:
1 + GcGvGpGm=0
(Note: This equation is not a polynomial but a ratio of polynomials)
Stability Condition: None of the zeros of ( 1 + GcGvGpGm
in the right half plane.
Assumption: Gc, Gv, Gp and Gm
are all stable, i.e. stable (open-loop) process, controller, actuator and
sensor. This means (1 + GcGvGpGm
has no poles in the right half plane.
a) A transfer function G(s) is said to be proper if
and strictly proper if
(Note: All physically realizable systems are either proper
or strictly proper.)
b) If G(s) is a ratio of polynomial functions, then
G(s) is proper if the order of the numerator polynomial is less than or
equal than the order of the denominator polynomial
G(s) is strictly proper if the order of the numerator polynomial is less
than the order of the denominator polynomial
c) Nyquist path:
IV. Nyquist Stability Criterion:
In the s-plane, the clockwise path A-B-C-A is a simple closed
semi-circle contour in the right half of the complex plane as shown below:
As R approaches infinity, the contour will encompass all the
points in the right half plane.
d) Complex mapping using the Nyquist path:
Let G(s) be proper, then
- The mapping of G(s) along the path A-B is just G(iw)
for w=0 to R. This information is obtained from
frequency response experiments.
- The mapping of G(s) along the path B-C is just a finite constant
as R approaches infinity.
- The mapping of G(s) along the path C-A is G(-iw)
with w= - R to 0, i.e. the mirror image of G(iw).
Click here for
sample applications of the stabilty criterion
Assume Gc, Gv, Gp and Gm
are all stable. If the Nyquist plot of GcGvGpGm
(s) does not encircle the point -1+0i in the complex plane in the
clockwise direction, then the system is asymptotically stable.
Proof: According to the complex mapping theorem, the map
of (1+GcGvGpGm) along the Nyquist
path will encircle the origin N times, where N=Z-P. Since Gc,
Gv, Gp and Gm are all stable, P=0. Thus
N=Z. This means the number of encirclements of the origin indicates the
number of roots of the characteristic equation in the right half plane.
Thus for stability, we must have N=Z=0.
Finally, the map of GcGvGpGm
(s) is the map of 1+ GcGvGpGm
(s) shifted one unit to the left.
This page is maintained by Tomas B. Co (firstname.lastname@example.org).
Last revised 2/9/1999.
Tomas B. Co
Department of Chemical Engineering
Michigan Technological University
1400 Townsend Avenue
Houghton, MI 49931-1295
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