How to check the linearity of a system by using a Transfer function which is consisted of PID Parameters and other two constants?

I have derived a transfer function which consisted of a PID Controller, Motor driver+actuator. And I need to check the linearity of the system by using that transfer function. But I am confused with the values that I need to substitute for those PID Constants and Constants related to motor driver + actuator.

here, Kp, Ki, Kd are PID parameters and Ka = Constant for motor driver + actuator and Ks = constant for Position sensor.

[Aquatris]

You drived the transfer function so I assume you already did some linearization. What do you mean when you say you are trying to check the linearity of the system?

[Raveesha Bandara]

I want to check the stability of this transfer function. But I am confused with those PID parameters and other constants.

Answer 1

[Sam Chak]

Hi @Raveesha Bandara

If your models of the actuator and sensor are fairly accurate, then consider using these formulas to calculate the PI gains. Yes, a PI controller is sufficient for the task since the actuator is a 1st-order system. Additionally, if the sensor introduces a non-unity factor in the feedback path, the setpoint needs to be scaled by a factor of . Here are the formulas:

, where is the desired Time that settles within 2% of the final value.

By the way, take note that the formula of your closed-loop transfer function is incorrect.

%% Parameters
Ka  = 5;    % Gain parameter in Actuator
Ta  = 3;    % Time constant  in Actuator
Ks  = 2;    % Gain parameter in Sensor

%% Actuator
Ga  = tf(Ka, [Ta 1])

Ga =
 
     5
  -------
  3 s + 1
 
Continuous-time transfer function.

%% Sensor
Gs  = tf(Ks)

Gs =
 
  2
 
Static gain.

%% PI Controller
Ts  = 1;        % desired settling time
Ki  = -log(0.02)/(Ka*Ks*Ts);
Kp  = Ki*Ta;
Gpi = pid(Kp, Ki)

Gpi =
 
             1 
  Kp + Ki * ---
             s 

  with Kp = 1.17, Ki = 0.391
 
Continuous-time PI controller in parallel form.

%% Closed-loop transfer function
Gcl = feedback(Gpi*Ga, Gs)

Gcl =
 
      5.868 s + 1.956
  -----------------------
  3 s^2 + 12.74 s + 3.912
 
Continuous-time transfer function.

%% Scale the Setpoint by a factor of Ks
G   = Ks*Gcl

G =
 
      11.74 s + 3.912
  -----------------------
  3 s^2 + 12.74 s + 3.912
 
Continuous-time transfer function.

stepinfo(G), 

ans = struct with fields:
         RiseTime: 0.5616
    TransientTime: 1.0000
     SettlingTime: 1.0000
      SettlingMin: 0.9045
      SettlingMax: 0.9993
        Overshoot: 0
       Undershoot: 0
             Peak: 0.9993
         PeakTime: 1.8717

step(G), grid on

[Raveesha Bandara]

I derived the transfer function by considering a PID controller. It is because my project is implementing a PID controller.

[Raveesha Bandara]

@Sam Chak eventhough a PI is sufficient for this system, could you please help me with PID as well?

[Sam Chak]

@Raveesha Bandara, Adding the derivative action might be excessive for the system. Nevertheless, if you're insistent, go ahead and add the two lines for Kd and Tf, then update the pid() command. For simplicity, adjust the single parameter Kd until you achieve the desired response.

Let me know if it works.

%% PID Controller
Ts  = 1;        % desired settling time no longer applies
Ki  = -log(0.02)/(Ka*Ks*Ts);    % no change unless you pidtune()
Kp  = Ki*Ta;                    % no change unless you pidtune()
Kd  = 1;        % <-- adjust this value 
Tf  = 1/100;    % <-- add this line
Gpid= pid(Kp, Ki, Kd, Tf)

[Raveesha Bandara]

@Sam Chak could you please explain the way that you choose the values for Ka, Ks, Ta and Ts?

[Sam Chak]

Hi @Raveesha Bandara

The values for the non-control parameters {Ka, Ks, Ta} should come from you. I randomly selected some integers. The desired settling time Ts, on which the PI control gains depend, is determined by me (the Control Designer). If I aim for a fast response, allowing the solar tracker to follow the input signal within 1 second, then I choose Ts = 1. Regardless of the values of {Ka, Ks, Ta}, the solar tracking response always converges to 1 second, thanks to the formulas used to determine the PI gain.

Answer 2

[Raveesha Bandara]