5 Design with Transform Methods

5.4 Direct Design in the z-PlaneAll Activities

A discrete tranfer function of the antenna plant, with a sampling time $T=2$ sec, is

$$G(z)=\frac{0.1873(z+0.9355)}{(z-1)(z-0.8187))}$$

The figure below shows the z-plane root locus for proportional control or for a Pole/Zero compensator of the form

$$D(z)=K\frac{z-b}{z-a}$$

where the zero at $b$ is place at $z=0.8187$ to cancel the plant's pole at $z=0.8187$

The unit step response of the closed-loop system is shown in the figure. The transient response is what we expected from the design.

root locus step response
  

Figure 1: Root Locus and corresponding step response Solid lines are drawn for clarity. They do not represent the intersample response. of antenna system using a pole/zero compensator. $K$=1.08 . Acceptable design region is shown in grey on the root locus.

Use the figure above to select the controller type and change the controller parameters. The corresponding step response is updated automatically.

For proportional control, drag on the gain location (■) to change the gain value.
For Pole/Zero compensator, drag on the gain location (■) or compensator pole location (×) to change their values.

  1. For protional control, what value of $K$ results in an unstable system?
  2. For proportional control, what happens to the response as $K$ approaches zero. Note that the graph only shows the results to 20 seconds.
  3. When $a$=0.2 in the pole/zero compensator, what value of $K$ results in an unstable system?
  4. How does changing the value of $a$ in the pole/zero compensator affect the system response?
  5. For the pole/zero compensator, what range of $a$ and $K$ values satisfy the design criteria?