Non linear plant equation and design the PID controller

I have a set of equation which is non linear equation and I want to design a non a PID controller for the the equation. The equation and the condition are in thre attachments. Please give me any hints that might help me with equation. Where u is the PID controller equation.

Answer 1

[Sam Chak]

Hi @Jay Sutariya

The motion dynamics of your Z-axis robot can be expressed through the following equation:

Here, the combined effects of friction and gravity are represented as a lumped disturbance. While the friction force exhibits nonlinear behavior, resembling a mathematically-saturated function (not easily visualized in If-Else or Piecewise forms), the motion system without disturbance can be characterized as linear:

Upon taking the Laplace transform, the transfer function of the plant is derived as:

,

This clearly represents a Double Integrator system. With this understanding, a classical PID controller can be designed with a specific focus on Disturbance Rejection.

From the simulation result, the amplitude (max) of the output response (red curve) resulting from a unit step disturbance is marginally below 0.0025 m. With a friction force of 50 Nm, this disturbance will induce a maximum change in height of less than 0.125 m, dissipating within the subsequent 5 seconds.

%% Plant
t   = 0:0.01:8;
s   = tf('s');
m   = 20;           % robot mass
P   = 1/(m*s^2)     % Plant transfer function

P =
 
    1
  ------
  20 s^2
 
Continuous-time transfer function.

%% Classical PID Controller with a focus on Disturbance Rejection
wc  = 6;        % <-- Tune this parameter to make the response faster or slower
opt = pidtuneOptions('DesignFocus', 'Disturbance-Rejection');
[C, info] = pidtune(P, 'PID', wc, opt)

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

  with Kp = 360, Ki = 289, Kd = 112
 
Continuous-time PID controller in parallel form.

info = struct with fields:
                Stable: 1
    CrossoverFrequency: 6
           PhaseMargin: 60.0000

%% Closed-loop TF from Reference R(s) to Output Y(s)
Gcl = feedback(C*P, 1)

Gcl =
 
      112 s^2 + 360 s + 289.4
  --------------------------------
  20 s^3 + 112 s^2 + 360 s + 289.4
 
Continuous-time transfer function.

yc  = step(Gcl, t);

%% Closed-loop TF from Disturbance D(s) to Output Y(s)
Gdl = feedback(P, C)

Gdl =
 
                 s
  --------------------------------
  20 s^3 + 112 s^2 + 360 s + 289.4
 
Continuous-time transfer function.

yd  = step(Gdl, t);

%% Step responses
yyaxis left
plot(t, yc), 
ylabel('Height, z (m)')

yyaxis right
plot(t, yd), grid on
xlabel('Time, t (s)')
legend('Response to a Step Reference', 'Response to a Step Disturbance', 'location', 'East')
title('Step Responses')