The frequency response of the system after introducing the above compensator is shown in Figure 4, which shows that the desired performance criteria are met.

$\includegraphics[width=5.0in]{m5l8fig4.eps}$

Figure 4: Frequency response of the system in Example 1 with a lag-lead compensator

Example 2:

Now let us consider that the system as described in the previous example is subject to a sampled data control system with sampling time sec. We would use MATLAB to derive the plant transfer function -plane.

Use the below commands.

>> s=tf('s');
>> gc=1/(s*(1+0.1*s)*(1+0.2*s));
>> gz=c2d(gc,0.1,'zoh');

You would get

$\displaystyle G_z(z) = \frac{{0.005824 z^2 + 0.01629 z + 0.002753}}{{z^3 - 1.974 z^2 + 1.198 z - 0.2231}}$

The bi-linear transformation

$\displaystyle z = \frac{{1+ wT/2}}{{1-wT/2}}= \frac{{(1 + 0.05w)}}{{(1 - 0.05w)}}$

will transfer

into

-plane. Use the below commands

>> aug=[0.1,1];
>> gwss = bilin(ss(gz),-1,'S_Tust',aug)
>> gw=tf(gwss)

to find out the transfer function in

-plane, as

$\displaystyle G_w(w)= \frac{0.001756 w^3 - 0.06306 w^2 - 1.705 w + 45.27}{w^3 + 14.14 w^2 + 45.27 w - 5.629\times 10^{-13}}$