Finding Kp,Ti,Td with PID tuner

Hello, I'm trying to find and built a PID system.But it gets error.

P=4;
bode(P)
I=tf([1],[5]);
bode(I);
D=9;
bode(D);

sys=tf([270 50 1],[225 5 0])
[C_pi,info]=pidtune(sys,'PI')
[C_p,info]=pidtune(sys,'P')

bode(sys)
nyquist(sys)

[C_PD,info]=pidtune(syz,'PID')

I have to find,reponse time,steady stae error,overshot from Kp,Ki,Td.Function I have to work with is (1/s^2+10s+20).

Tune controller is in this sequence;

1.Only P element

2.Enable D, and returne P

3.Enable I and retune P,D.

And I have to plot, P,I,D,PI,PD,PID, bode and nyquist graphs.

Answer 1

[Sam Chak]

Hi @Esin Derin,

Kind of remember seeing a similar question few days ago. Probably your classmate. Anyway, this is how you can plot the step response of a closed-loop system.

Question 1:

% Plant
Gp = tf(1, [1 10 20])

Gp =
 
         1
  ---------------
  s^2 + 10 s + 20
 
Continuous-time transfer function.

% Question 1
[Gc1, info] = pidtune(Gp, 'P')

Gc1 =
 
  Kp = 94.9
 
P-only controller.

info = struct with fields:
                Stable: 1
    CrossoverFrequency: 8.2132
           PhaseMargin: 60.0248

% Closed-loop system
Gcl1 = minreal(feedback(Gc1*Gp, 1))

Gcl1 =
 
        94.86
  ------------------
  s^2 + 10 s + 114.9
 
Continuous-time transfer function.

step(Gcl1, 3)   % steady-state error between 0.8 and 1.0 is significant.

% bode(Gcl1)
% nyquist(Gcl1)

Question 2:

% Question 2
[Gc2, info] = pidtune(Gp, 'PD')

Gc2 =
 
             
  Kp + Kd * s
             

  with Kp = 1.22e+03, Kd = 30.6
 
Continuous-time PD controller in parallel form.

info = struct with fields:
                Stable: 1
    CrossoverFrequency: 41.6138
           PhaseMargin: 60.0000

% Closed-loop system
Gcl2 = minreal(feedback(Gc2*Gp, 1))

Gcl2 =
 
     30.62 s + 1216
  --------------------
  s^2 + 40.62 s + 1236
 
Continuous-time transfer function.

step(Gcl2, 3)   % response is super fast and steady-state error is reduced substantially 1 - 1216/1236, but overshoot is relatively high

% bode(Gcl2)
% nyquist(Gcl2)

Question 3:

% Question 3
[Gc3, info] = pidtune(Gp, 'PID')

Gc3 =
 
             1          
  Kp + Ki * --- + Kd * s
             s          

  with Kp = 39.3, Ki = 116, Kd = 3.33
 
Continuous-time PID controller in parallel form.

info = struct with fields:
                Stable: 1
    CrossoverFrequency: 4.1614
           PhaseMargin: 74.0404

% Closed-loop system
Gcl3 = minreal(feedback(Gc3*Gp, 1))

Gcl3 =
 
     3.326 s^2 + 39.27 s + 115.9
  ---------------------------------
  s^3 + 13.33 s^2 + 59.27 s + 115.9
 
Continuous-time transfer function.

step(Gcl3, 3)   % no steady-state error and the overshoot is reduced. Response settles under 2 seconds.

% bode(Gcl3)
% nyquist(Gcl3)