PD controller (Time domain)

I would like to design a PD controller, to control a moving mass (=1Kg) along the x-axis so it moves from rest (point A) to another point B, like this:

I would appreciate any help!

[Mathieu NOE]

hello

have yu already started some code ?

[Hidd_1]

Yes here is it:

clf
clc

%Time:
t_ges = 2;
dt = 0.1;
t_vect = 0 : dt : t_ges;
%Initial position, Initial velocity, Initial acceleration:
q_0=0;
dq_0=0;
ddq_0=0;
%The position, velocity, acceleration:
q = q_0;
dq = dq_0;
ddq = dq_0;
%PD constants
Kp=2;
Td=2;
%Matrices to store the calculate values:
n_times = length(t_vect);
q_Matrix = zeros(1, n_times);
dq_Matrix = zeros(1, n_times);
ddq_Matrix = zeros(1, n_times);
F_Matrix = zeros(1, n_times);

for t = t_vect
    %position calculation:
     q = q + dt * dq + dt^2/2 * ddq;
     q_Matrix(:, 1) = q;
     %Velocity calculation:
     dq = dq + dt * ddq;
     dq_Matrix(:, 1) = dq;
     %PD controller calculation: (closed-loop, with the reference is 1 meter (Point B))
     F = Kp*(1-q) + Td*(0-dq);
     F_Matrix(:, 1) = F;
     %acceleration calculation:
     ddq = F;
     ddq_Matrix(:, 1) = ddq;

end

And for the rectangle mass simulation I am planning to use the following function: (I still don't know how to implement it for an array thou)

function ac_plot(~, marker, color, size)
plot(rectangle('Position',[0 0 0.2 0.2],'EdgeColor','b','LineWidth',2),...
    rectangle('Position',[0 0 0.2 0.2],'EdgeColor','b','LineWidth',2),...
    marker, 'Color', color, 'MarkerSize', size)

Answer 1

[Mathieu NOE]

hello again

so this is a modified code :

• the integration process computes the velocity first (from acceleration) , then the position (from the velocity)
• force can be limited , so you can have large gains from the controller (faster response) but you can then saturate the controller output to any given Fmax value. Results will look pretty much like a bang bang controller
• plot update by using handles (faster than re plotting the entire plot at each step)
• still need to figure out how to add the moving rectangle at the top of the figure

hope it helps

clf
clc
%Time:
t_ges = 2.5;
dt = 0.025;
t_vect = 0 : dt : t_ges;
%Initial position, Initial velocity, Initial acceleration:
q_0=0;
dq_0=0;
ddq_0=0;
%The position, velocity, acceleration:
q = q_0;
dq = dq_0;
ddq = dq_0;
%PD constants
Kp=10;
Kd=Kp/2;

% force limitation to Fmax (limitation of output of PD controller)
Fmax = 1;

%Matrices to store the calculate values:
n_times = length(t_vect);
q_Matrix = zeros(1, n_times);
dq_Matrix = zeros(1, n_times);
ddq_Matrix = zeros(1, n_times);
F_Matrix = zeros(1, n_times);

%// initiallize plot. Get a handle to graphic object

 f = figure(1);
    s1 = subplot(3,1,1);
    p1 = plot(NaN,NaN,'b');
    title('Position')
    ylabel('m');
    axis([0 t_ges -0.25 1.25]);%// freeze axes to their final size, to prevent Matlab from rescaling them dynamically
    s2 = subplot(3,1,2);
    p2 = plot(NaN,NaN,'b');
    title('Velocity');
    ylabel('m/s');
    axis([0 t_ges -1 2]);%// freeze axes to their final size, to prevent Matlab from rescaling them dynamically
    s3 = subplot(3,1,3);
    p3 = plot(NaN,NaN,'b');
    title('Force');
    ylabel('N');
    axis([0 t_ges -Fmax*1.25 Fmax*1.25]);%// freeze axes to their final size, to prevent Matlab from rescaling them dynamically
    

for ci = 1:n_times
     %Velocity calculation:
     dq = dq + dt * ddq;
     dq_Matrix(1,ci) = dq;
    
     %position calculation:
     q = q + dt * dq;
     q_Matrix(1,ci) = q;
     
     %PD controller calculation: (closed-loop, with the reference is 1 meter (Point B))
     F = Kp*(1-q) + Kd*(0-dq);
     
     % force limitation (optionnal)
     if abs(F)> Fmax
         F = Fmax*sign(F);
     end
     F_Matrix(1,ci) = F;
     
     %acceleration calculation:
     ddq = F;
     ddq_Matrix(1,ci) = ddq;
     

    % update the plot
    ind = (1:ci);  
    t_plot = t_vect(ind);
    pause(0.01)
    set(p1, 'XData', t_plot, 'YData', q_Matrix(ind));
    set(p2, 'XData', t_plot, 'YData', dq_Matrix(ind));
    set(p3, 'XData', t_plot, 'YData', F_Matrix(ind));
     
end

Answer 2

[Mathieu NOE]

a very simple and straigthforward simulation without much science behind :

NB : controller and plant in series in the main path, unitary feedback

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% original plant
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
s = tf('s');
%  Gs = 1.44e09/(s^2+5333*s+9.6e07);
 Gs = 1/(s^2+0*s+0);
 
%PID 
P= 10;
I= 0.0;
D= 1*P;
Ct = P+I/s+D*s/(1e-6*s+1);

figure()
Gscl = feedback(series(Ct,Gs),1); 
step(Gscl);
legend('closed loop');
grid on;

[Hidd_1]

Thank you very much for your reply!

I would like to make the simulation, that's why I tried to discretized the time, do you have any idea how to make a similar simulation like the one above.

The porpuse for my problem is to examine the controller which is going to take the shortest amount of time between the points A and B, and taking in consideration limits on the force F. ==> I think I need dynamic Programming to solve that! (Bang-bang controller)