Implement PID in ode45 code

So, I have a ode45 function and I have an error defined in it which changes at different iterations of the solver. I was able to define the derivative part by just differentiating the error formula but I am unable to implement the integral part. The error expression is:

, where q represents position and v represents velocity. Is it possible to get the PID part in the ode function.

Answer 1

[Sam Chak]

Hi @Kashish Pilyal

If the error is defined as

then

P part is

I part is

D part is

and you can arrange then in the state-space form. For example, a Double Integrator system

can be rewritten in state-space as:

.

The PID has 3 terms, and the state-space is in differential form. So you have no issue with the P and the D part, because they are part of the state variables. The I part is in integral form, so you have to create an additional state variable. See Example below:

[t, x] = ode45(@DIsystem, [0 20], [0; 0; 0]);
plot(t, x(:,1), 'linewidth', 1.5)
grid on, xlabel('t'), ylabel('y(t)'), % ylim([-0.2 1.2])

function dxdt = DIsystem(t, x)
    dxdt    = zeros(3, 1);
    % construction of PID 
    r       = 1;                            % reference signal
    e       = x(1) - r;                     % error signal
    Kp      = 1 + sqrt(2);                  % proportional gain
    Ki      = 1;                            % integral gain
    Kd      = 1 + sqrt(2);                  % derivative gain
    u       = - Kp*e - Ki*x(3) - Kd*x(2);   % the PID thing
    
    % the dynamics
    A       = [0 1; 0 0];                   % state matrix
    B       = [0; 1];                       % input matrix
    dxdt(1:2) = A*[x(1); x(2)] + B*u;       % the Double Integrator system
    dxdt(3) = e;                            % for integral action in PID
end

[Kashish Pilyal]

Um... what do you mean by state-space form? How do I get the integral part during the ode45 running? As at every iteration when the solver is called, the position and the velocity changes and so does the error. I want to integrate at every time step which also means that every error during all the previous iterations need to be added. The code for error is:

e_1=x(1)-(  x(4)+(h_i*x(5))  );

As you can see it will change at every instance so how do I integrate it at every time step when the ode45 is running?

[Sam Chak]

Hi @Kashish Pilyal

I have provided an example in the Answer using a Double Integrator system:

You will get the idea of how to implement that. Hope it works out for your system.

[Kashish Pilyal]

Thanks I got the idea, in my case I have the error:

e_1=x(1)-(  x(4)+(h_i*x(5))  );

So, If I am right I need to introduce a new state which I will be equal to this error at the end of every iteration like you did with dxdt=x(3) and in the next iteration use this state as the integral of the error.