Control System Designer basics

Introduction: 

The controlSystemDesigner tool is primarily used for designing single-input, single-output (SISO) controllers with feedback systems.

It consists of a Bode plot, root locus, and Nichols graphical editors for designing the controllers by modifying the gains, poles, and zeros.

Consider a second-order transfer function

G = 12 / (s2+15s+10)

To open the control system designer toolbox,

Enter the transfer function in the Matlab command window, followed by the Matlab command to open the control system designer app.

>> G = tf(12,[1 15 10]);

>> controlSystemDesigner(G)

This opens the Control system Designer toolbox with G as the default plant model

The characteristics of the step response can be found by right-clicking on the step response editor plot → Characteristics  and choosing the required characteristics

The system has a rise time of 1.35s and a steady-state value of 0.545.

We can adjust the compensator gains and the following compensator dynamics

• Real and complex pole, Integrator
• Real and complex zero, Differentiator
• Lead and Lag networks
• Notch filters

Adding Compensator gains:

The compensator gains can be adjusted by dragging the magnitude plot's response in the Bode editor or by Right-clicking the plot area → Edit compensator → Specify the compensator gain in the Compensator Editor Toolbox.

The compensator gains are used to:

• Reduce rise time
• Reduce steady-state error
• Increase the overshoot

For the above example changing the compensator gain by dragging the response has reduced the rise time from 1.35s to 0.018s and increased steady-state value from 0.545 to 0.998. But it has also increased the overshoot.

Adding Poles and zeros from the editor plot:

      Right-click on the plot area → Add Pole or Zero → Select the type of pole or zero

The editable poles and zeros are marked in red X, and O. The editable poles and zeros can be dragged to adjust by our design needs. The location of the pole or zero can also be specified.

Right-click on the plot area → Edit compensator → Select the pole or zero on the Dynamics table → Specify the location on the Edit Selected Dynamics section. The natural frequency and damping can also be specified for complex poles or zeros.

Delete Poles and zeros from the editor plot:

      Right-click on the plot area à Delete Pole or Zero. On the editor plot, click the pole or zero to delete.

Alternatively, Right click on the plot area → Edit compensator → Right-click the pole or zero on the Dynamics table → Delete Pole or Zero.

Poles or zeros that can be added to the compensator:

• Real pole or zero
• Complex poles or zeros
• Integrator - Add a pole at the origin to eliminate steady-state error for step inputs.
• Differentiator - Add a zero at the origin to improve the transient response.

Example: Consider the peak overshoot to be reduced to less than 5% for the above example

Add a real zero and drag the zero to adjust the overshoot to less than 5%

The overshoot has been reduced to less than 5%. There are multiple ways to achieve the design goals. The same transfer function can be designed if a specified gain and phase margin is needed by adding a compensator, integrator and lead /lag networks

 

Lead and lag networks: 

 

• Lead network
• Right click on Bode editor → Add Poles or Zeros → Lead
• A pole and zero on the negative real axis, with the zero having a lower natural frequency
• Increase stability margin
• Improve system bandwidth
• Reduce rise time 
• Lag network:
• Right-click on Bode editor → Add Poles or Zeros → Lag
• A pole and zero on the negative real axis, with the pole having a lower natural frequency 
• Increase phase margin
• Improve steady state accuracy
• Reduce high-frequency gain
• Lead-Lag: A combination of lead network and lag network
• Combined effects of lead and lag networks.
• Configuring the lead-lag network: 
• Specify pole and zero locations. Placing the pole and zero further apart increases the amount of phase angle change.
• Specify the maximum amount of phase angle change and the frequency at which this change occurs. The app automatically computes the pole and zero locations.

Consider the same transfer function with the following design requirements

Rise time: <0.5s

Overshoot: < 5 %

Gain margin: > 25 dB

Phase margin: > 60 deg

Steady-state error: < 1%

Adjust the compensator to 3 rad/s as crossover frequency by dragging the response in the magnitude plot. The steady-state value has been increased to 0.8, and the rise time is decreased to 0.43.

Then add the integrator by right clicking on the Bode editor plot and selecting Add Poles or Zeros → Integrator

The Steady state value is one which gives zero steady state error. But the overshoot is 65% and rise time is 0.6s which exceeds the design requirements. This can be reduced by adding a lead compensator. 

Add the lead compensator and drag the zeros and poles to attain the desired Gain and Phase margin. The Result will be reflected in the step response plot. The pole and zero can also be tuned in the Compensator editor toolbox. Click the Lead option in the Dynamics table and change the Real zero value in Edit Selected Dynamics section  as -1 and Real pole value as -60.

The system with the desired design values is obtained. The gain margin is >25 dB, Phase margin > 60 degrees, Overshoot < 5%, Rise time < 0.5s and Steady state error < 1%.

Notch Filter:  

• When a system has disturbances at a particular frequency, the notch filter is added  to attenuate the gain of the system at that frequency.
• Configuring Notch filter: Natural frequency, Notch depth, Notch width and Damping can be specified in the compensator editor dialog box.