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Nichols plot design is an interactive graphical method of modifying a compensator to
achieve a specific open-loop response (loop shaping). Unlike Bode Diagram Design, Nichols plot design uses Nichols plots to view the
open-loop frequency response. Nichols plots combine gain and phase information into a
single plot, which is useful when you are designing to gain and phase margin
specifications. You can also use the Nichols plot grid lines to estimate the closed-loop
response (see `ngrid`

). For more information on
Nichols plots, see `nicholsplot`

.

This example shows how to design a compensator for a DC motor using Nichols plot graphical tuning techniques.

**Plant Model and Requirements**

The transfer function of the DC motor plant, as described in SISO Example: The DC Motor, is:

$$G=\frac{1.5}{{s}^{2}+14s+40.02}$$

For this example, the design requirements are:

Rise time of less than 0.5 seconds

Steady-state error of less than 5%

Overshoot of less than 10%

Gain margin greater than 20 dB

Phase margin greater than 40 degrees

**Open Control System Designer**

At the MATLAB^{®} command line, create a transfer function model of the plant,
and open **Control System Designer** in the **Nichols
Editor** configuration.

```
G = tf(1.5,[1 14 40.02]);
controlSystemDesigner('nichols',G);
```

The app opens and imports `G`

as the plant model for the
default control architecture, **Configuration 1**.

In the app, the following response plots open:

Open-loop

**Nichols Editor**for the`LoopTransfer_C`

response. This response is the open-loop transfer function*GC*, where*C*is the compensator and*G*is the plant.**Step Response**for the`IOTransfer_r2y`

response. This response is the input-output transfer function for the overall closed-loop system.

To open the open-loop **Nichols Editor** when
**Control System Designer** is already open, on the
**Control System** tab, in the **Tuning
Methods** drop-down list, select **Nichols
Editor**. In the Select Response to Edit dialog box, select
an existing response to plot, or create a ```
New Open-Loop
Response
```

.

To view the open-loop frequency response and closed-loop step response
simultaneously, on the **Views** tab, click
**Left/Right**.

The app displays the **Nichols Editor** and
**Step Response** plots side-by-side.

**Adjust Bandwidth**

Since the design requires a rise time less than 0.5 seconds, set the open-loop DC crossover frequency to about 3 rad/s. To a first-order approximation, this crossover frequency corresponds to a time constant of 0.33 seconds.

To adjust the crossover frequency increase the compensator gain. In the
**Nichols Editor**, drag the response upward. Doing so
increases the gain of the compensator.

As you drag the Nichols plot, the app computes the compensator gain and updates the response plots.

Drag the Nichols plot upward until the crossover frequency is about 3 rad/s.

**View Step Response Characteristics**

To add the rise time to the **Step Response** plot,
right-click the plot area, and select **Characteristics** > **Rise Time**.

To view the rise time, move the cursor over the rise time indicator.

The rise time is around 0.23 seconds, which satisfies the design requirements.

Similarly, to add the peak response to the **Step
Response** plot, right-click the plot area, and select **Characteristics** > **Peak Response**.

The peak overshoot is around 3.5%.

**Add Integrator To Compensator**

To meet the 5% steady-state error requirement, eliminate steady-state
error from the closed-loop step response by adding an integrator to your
compensator. In the **Nichols Editor** right-click in the
plot area, and select **Add Pole/Zero** > **Integrator**.

Adding an integrator produces zero steady-state error. However, changing the compensator dynamics also changes the crossover frequency, increasing the rise time. To reduce the rise time, increase the crossover frequency to around 3 rad/s.

**Adjust Compensator Gain**

To return the crossover frequency to around 3 rad/s, increase the
compensator gain further. Right-click the **Nichols
Editor** plot area, and select **Edit
Compensator**.

In the Compensator Editor dialog box, in the
**Compensator** section, specify a gain of
`99`

, and press **Enter**.

The response plots update automatically.

The rise time is around 0.4 seconds, which satisfies the design requirements. However, the peak overshoot is around 32%. A compensator consisting of a gain and an integrator is not sufficient to meet the design requirements. Therefore, the compensator requires additional dynamics.

**Add Lead Network to Compensator**

In the **Nichols Editor**, review the gain margin and
phase margin for the current compensator design. The design requires a gain
margin greater than 20 dB and phase margin greater than 40 degrees. The
current design does not meet either of these requirements.

To increase the stability margins, add a lead network to the compensator.

In the **Nichols Editor**, right-click and select **Add Pole/Zero** > **Lead**.

To specify the location of the lead network pole, click on the magnitude
response. The app adds a real pole (red `X`

) and real zero
(red `O`

) to the compensator and to the **Nichols
Editor** plot.

In the **Nichols Editor**, drag the pole and zero to
change their locations. As you drag them, the app updates the pole/zero
values and updates the response plots.

To decrease the magnitude of a pole or zero, drag it towards the left. Since the pole and zero are on the negative real axis, dragging them to the left moves them closer to the origin in the complex plane.

As you drag a pole or zero, the app displays the new value in the status bar, on the right side.

As an initial estimate, drag the zero to a location around
`-7`

and the pole to a location around
`-11`

.

The phase margin meets the design requirements; however, the gain margin is still too low.

**Edit Lead Network Pole and Zero**

To improve the controller performance, tune the lead network parameters.

In the Compensator Editor dialog box, in the **Dynamics**
section, click the **Lead** row.

In the **Edit Selected Dynamics** section, in the
**Real Zero** text box, specify a location of
`-4.3`

, and press **Enter**. This value is
near the slowest (left-most) pole of the DC motor plant.

In the **Real Pole** text box, specify a value of
`-28`

, and press **Enter**.

When you modify a lead network parameters, the
**Compensator** and response plots update
automatically.

In the app, in the **Nichols Editor**, the gain margin of
`20.5`

just meets the design requirement.

To add robustness to the system, in the Compensator Editor dialog box,
decrease the compensator gain to `84.5`

, and press
**Enter**. The gain margin increases to
`21.8`

, and the response plots update.

In **Control System Designer**, in the response plots, compare the
system performance to the design requirements. The system performance
characteristics are:

Rise time is 0.445 seconds.

Steady-state error is zero.

Overshoot is 3.39%.

Gain margin is 21.8 dB.

Phase margin is 65.6 degrees.

The system response meets all of the design requirements.

Control System Designer | `nicholsplot`