To perform interactive compensator design functions, consider a system with the open-loop transfer function:

You are required to design a lag compensator for this
system such that a phase margin of 45^{o} is maintained. Assume that
*K _{c}* = 1.

The angle θ* _{GH}* is computed from the
equation:

θ* _{GH}* = -180

**To interactively design a lag
compensator**

1. Create and simulate the following system:

2.
Set the polynomial coefficients for the **transferFunction** block to the
following values:

Numerator: 1

Denominator: 1 3 2 0

**Note:** Always leave spaces
between coefficient values.

3.
Select the **transferFunction** block.

4.
Choose **Analyze** **>** **Frequency** **Response **to display
the Bode magnitude and phase plots.

5. Move and resize them for better viewing.

As can be seen from the Bode
phase plot, the frequency corresponding to a phase of -130^{o} is
approximately 0.48 rad. Therefore, ω_{1} = 0.48. The corresponding
magnitude of the open-loop transfer function, from the Bode magnitude plot, is
0.853.

The lag zero frequency is
ω_{0} = 0.1 ω_{1} = 0.048. The lag pole frequency is given
by:

6.
Select the **transferFunction** block.

7.
Choose **Analyze** **>** **Compensator** **Design.**

8.
In the **Compensator Design** dialog with the plant poles and zeros
displayed, do the following:

• In the **Compensator** **Zeros** box, enter (-0.048,0) and click
**Add**.

A 0 is displayed in the Compensator Zeros list. The entry corresponds to a 0 in the complex plane, with a real part of -0.048 and an imaginary part of 0.

• In the **Compensator** **Poles** box, enter (-0.056,0) and click
**Add**.

This pole is displayed in the
**Compensator** **Poles** list.

• In the **Gain** box, enter **1.167**.

• Click **Replot**.

The Bode magnitude and phase
plots of the Compensator-Plant system, *C*(*s*) *GH*(*s*)
are displayed. The Bode plots indicate that the phase margin is approximately
48^{o}.

9.
Click **OK** in the **Compensator** **Design** dialog.

10. Place
the new **Compensator** block before the plant transfer function
*GH*(*s*), and connect it as shown below. The Compensator-Plant system
is now ready for use or further analysis.