__I. Setup and Assumptions:__

Figure 1.

**Closed Loop Transfer Function:**

**Characteristic Equation:**
1 + G_{c}G_{v}G_{p}G_{m}=0

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

**Stability Condition:** None of the zeros of ( 1 + G_{c}G_{v}G_{p}G_{m
})are
in the right half plane.

**Assumption:** G_{c}, G_{v}, G_{p} and G_{m}
are all stable, i.e. stable (open-loop) process, controller, actuator and
sensor. This means (1 + G_{c}G_{v}G_{p}G_{m
})
has no poles in the right half plane.

__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.)

- 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

- 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).

- Assume G

Thus for stability, we must have N=Z=0.

Finally, the map of G_{c}G_{v}G_{p}G_{m}
(s) is the map of 1+ G_{c}G_{v}G_{p}G_{m}
(s) shifted one unit to the left.

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