PD controller (Time domain)

Question

Hidd_1 2021-3-11

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/769507-pd-controller-time-domain

回答： Mathieu NOE 2021-3-12

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!

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Mathieu NOE 2021-3-11

hello

have yu already started some code ?

Hidd_1 2021-3-11

编辑：Hidd_1 2021-3-11

在 MATLAB Online 中打开

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)

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Answer 1

Mathieu NOE 2021-3-12

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/769507-pd-controller-time-domain#answer_645843

在 MATLAB Online 中打开

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

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Answer 2

Mathieu NOE 2021-3-11

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/769507-pd-controller-time-domain#answer_645222

在 MATLAB Online 中打开

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;

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Hidd_1 2021-3-11

编辑：Hidd_1 2021-3-11

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)

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