Accelerating the pace of engineering and science

# Documentation

## Nichols Plot Design

### What Is Nichols Plot Design?

An alternative method for designing compensators is to use the Nichols plot, which combines gain and phase information in a single plot. The combination of the two is useful when you are designing to gain and phase margin specifications.

You can design compensators with the SISO Design Tool by using Nichols plot techniques. This topic repeats the DC motor compensator design presented in Bode Diagram Design for DC Motor, only this time the focus is on Nichols plot techniques. The design strategy, however, is the same.

1. Adjust the compensator gain to improve the rise time.

3. Add a lead network to further improve the rise time while minimizing overshoot.

### Nichols Plot Design for DC Motor

From SISO Example: The DC Motor, the transfer function of the DC motor is

```Transfer function:
1.5
------------------
s^2 + 14 s + 40.02
```

This example uses the design criteria specified in Design Specifications:

• Rise time of less than 0.5 second

• Steady-state error of less than 5%

• Overshoot of less than 10%

• Gain margin greater than 20 dB

• Phase margin greater than 40 degrees

### Opening a Nichols Plot

To open the SISO Design Tool with a Bode diagram and a Nichols plot, use these commands:

```load ltiexamples
sisotool({'bode','nichols'},sys_dc)
```

The SISO Design Task node on the Control and Estimation Tools Manager opens and the Graphical Tuning window with sys_dc opens.

Graphical Tuning Window with a Bode Diagram and a Nichols Plot

You can adjust the compensator gain by entering a value in the Compensator Editor page.

1. Click the Compensator Editor tab to open the Compensator Editor page.

2. Select C from the compensator selection list.

3. In the text box to the right of the equal sign in the Compensator area, enter the gain amount and press Enter.

Adjusting Compensator Gain in the Compensator Editor Page

In this example, the new gain is 112.

You can also adjust the compensator gain in the Graphical Tuning window by moving the Nichols curve up and down with your mouse. To do this, place your mouse over the curve. The cursor turns into a hand. Left-click and move the curve up to increase the gain. When you adjust the gain in this manner, the compensator gain is automatically updated in the Compensator Editor page.

Click the Analysis Plots tab to set the analysis plots. Select Plot Type Step for Plot 1, and then select plot 1 for Closed-Loop r to y, as shown in the following figure, to open a linked LTI Viewer with the closed-loop step response from reference signal r to output signal y.

Analysis Plots Loop Response Selection

LTI Viewer Step Response for Compensator Gain = 112

The rise time is quite fast, about 0.15 second, but the overshoot is 18.4% and the steady-state is about 0.82.

One approach to eliminating the steady-state error is to add an integrator.

1. Click the Compensator Editor tab to open the Compensator Editor page.

2. Right-click in the Dynamics table and select Add Pole/Zero > Integrator.

This figure shows the process.

Adding an Integrator in the Dynamics Table

You can also add an integrator by selecting Add Pole/Zero > Integrator from the right-click menu in the Graphical Tuning window. When you add the integrator in this manner, it is automatically added to the Dynamics table on the Compensator Editor page.

Adding an integrator changes the gain margin from infinity to 10.5 dB. Since the gain and phase margins are now both finite, the Nichols plot shows a vertical line for the gain margin and a horizontal line for the phase margin.

Step Response for a Compensator Consisting of a Gain and an Integrator

The steady-state value is now 1, which means the steady-state error has been eliminated, but the overshoot is 34% and the rise time is about 0.7 second. You must do more work to create a good design.

Improving the rise time requires that you increase the compensator gain, but increasing the gain further deteriorates the gain and phase margins while increasing the overshoot. You need to add a lead network to selectively raise the gain about the frequency crossover. To add the lead network:

1. Click the Compensator Editor tab to open the Compensator Editor page.

2. Right-click in the Dynamics table and select Add Pole/Zero > Lead.

This figure shows the process of adding a lead network on the Compensator Editor page.

You can track the pole's movement in the status bar at the bottom of the Graphical Tuning window. The status bar tells you the current location of the pole.

Using the Compensator Editor page, move the lead network pole to -28 and the zero to -4.3 for this design. The zero should be almost on top of the right-most pole of the plant (the DC motor model). Adjust the compensator gain to 84. This gives the final design.

Final Nichols Plot Design for the DC Motor Compensator

The gain and phase margins are 21.9 dB and 65.7 degrees, respectively. Inspect the closed-loop step response to see if the rise time and overshoot requirements are met.

Closed-Loop Step Response for the Final Compensator Design

As this figure shows, the rise time is 0.448 second, and the overshoot is a little over 3%. This satisfies the rest of the design requirements.