Proportional control
Proportional-Derivative control
Proportional-Integral-Derivative control

In the Pitch Controller Modeling page, the transfer function was derived as

The input (elevator deflection angle, delta e) will be 0.2 rad (11 degrees), and the output is the pitch angle (theta).

The design requirements are

• Overshoot: Less than 10%
• Rise time: Less than 2 seconds
• Settling time: Less than 10 seconds
• Steady-state error: Less than 2%

Recall from the PID Tutorial page, the transfer function of a PID controller is:

We will implement combinations of proportional (Kp), integral (Ki), and derivative (Kd) controllers in an unity feedback system shown below to study the system output.

Proportional control

Kp=[1];	%Enter any numerical value for the proportional gain
num=[1.151 0.1774];
num1=conv(Kp,num);
den1=[1 0.739 0.921 0];

[numc,denc]=cloop (num1,den1)

For now, let the proportional gain (Kp) equal 2 and observe the system behavior. We will use the closed-loop transfer function derived by hand. Enter the following commands into a new m-file and run it in the Matlab command window. You should obtain the step response similar to the one shown below:

de=0.2;
Kp=2;

numc=Kp*[1.151 0.1774];
denc=[1 0.739 1.151*Kp+0.921 0.1774*Kp];

t=0:0.01:30;
step (de*numc,denc,t)

As you see, both the overshoot and the settling time need some improvement.

PD control

Using the commands shown below and with several trial-and-error runs, a proportional gain (Kp) of 9 and a derivative gain (Kd) of 4 provided the reasonable response. To comfirm this, change your m-file to the following and run it in the Matlab command window. You should obtain the step response similar to the one shown below:

de=0.2;
Kp=9;
Kd=4;

numc=[1.151*Kd 1.151*Kp+0.1774*Kd 0.1774*Kp];
denc=[1 0.739+1.151*Kd 0.921+1.151*Kp+0.1774*Kd 0.1774*Kp];

t=0:0.01:10;
step (de*numc,denc,t)

This step response shows the rise time of less than 2 seconds, the overshoot of less than 10%, the settling time of less than 10 seconds, and the steady-state error of less than 2%. All design requirements are satisfied.

PID Control

Even though all design requirements were satisfied with the PD controller, the integral controller (Ki) can be added to reduce the sharp peak and obtain smoother response. After several trial-and-error runs, the proportional gain (Kp) of 2, the integral gain (Ki) of 4, and the derivative gain (Kd) of 3 provided smoother step response that still satisfies all design requirements. To cofirm this, enter the following commands to an m-file and run it in the command window. You should obtain the step response shown below:

de=0.2;
Kp=2;
Kd=3;
Ki=4;
       
numo=[1.151 0.1774];
deno=[1 0.739 0.921 0];
numpid=[Kd Kp Ki];
denpid=[1 0];

num1=conv(numo,numpid);
den1=conv(deno,denpid);

[numc,denc] = cloop(num1,den1);
t=0:0.01:10;      
step (de*numc,denc,t)

Comments

1. To find appropriate gains (Kp, Kd, and Ki), you can use the table shown in PID Tutorial page; however, please keep in your mind that changing one gain might change the effect of the other two. As a result, you may need to change other two gains as you change one gain.

2. Our system with a PI controller do not provide the desired response; thus, a PI Control section was omitted from this page. You may confirm this by using the m-file shown in PID Control section, and set Kd equals to zero.

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