I. Setup and Assumptions:

    Feedback System:

    Figure 1.

    Closed Loop Transfer Function:

    Characteristic Equation:         1 + GcGvGpGm=0

    (Note: This equation is not a polynomial but a ratio of polynomials)

    Stability Condition: None of the zeros of ( 1 + GcGvGpGm )are 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.

II.  Objective : to determine the stability of a feedback control system using frequency response data.

III. Preliminaries:

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
    c) Nyquist path:
      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:

    Figure 2.

      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).
IV. Nyquist Stability 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.

Click here for sample applications of the stabilty criterion


This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 2/9/1999.
     Tomas B. Co
     Associate Professor
     Department of Chemical Engineering
     Michigan Technological University
     1400 Townsend Avenue
     Houghton, MI 49931-1295

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