National Instruments 370753C-01

Chapter 1 Introduction

One approach to stabilizing this system is to try to cancel the pole at

–0.656513 by adding a compensator, K1(s), with a zero at –0.656513.

Note It is important to understand that this is primarily an academic exercise. Accurate

pole-zero cancellations are impracticable in the real world, and the mode corresponding to

that pole still exists internally.

This compensator must have a pole for realizability, so you add one at –10,

which is far enough away that its effect on dynamic response will be small

compared to that of the system’s other modes. In addition, you need to add

a zero to the left of the positive (and unstable) poles to pull the closed-loop

system roots into the left half plane. Choose s= 0 for the zero location and,

again, select a corresponding pole at –10. Call this second compensator

K2(s). To create these two compensators:

K1s=polynomial(ol_poles(3),"s")/...

polynomial(-10,"s");

K2s=polynomial(0,"s")/polynomial(-10,"s");

You then can cascade them in series with the original system Gs (or ssys)

and examine the locus of closed-loop roots for varying total compensator

gain Kc. The poles out at –10 have a smaller effect on the system dynamics

than do the poles closer to the origin, so you can use the optional

rlocus( ) keywords to zoom in on the part of the locus nearer the origin.

rlocus(K2s*K1s*Gs, {xmin = -2, xmax = 2})

The single input syntax activates interactive mode. A user interface lets

you change the gain through the Feedback Loop Gain slider and button.

The graphics window shows the closed-loop locus as a solid line, with the

open-loop poles shown as large xs and the open-loop zeros shown as Os.

Increase the gain by moving the slider; notice the asterisks (*) denoting

closed-loop pole location moving along the locus. The system is maximally

stable with total compensator gain Kc= 2 as shown in Figure 1-2. In this

figure, smal l xs denote the pole location for Kc= 2, and the root locus gain

window show s settings.