How to integrate a control system by ode45 with PID control input ?

I want to simulate a control system using ode45 function. The system model can be expressed as , where the control input u is the PID control which can be expressed as . The is the desired state. My question is how to integrate the error in the ode model as illustrated in the following code example, since we only obtain the current time state X when the f function is called

function dX = f(t,X,P)

% ...

% ...

u = P.Kp * (P.Xd - X) + P.Ki * (P.Xd - X) + ... ; % How to integrate the error Xd - X ?

end

Answer 1

[Sam Chak]

Hi @Chuguang Pan

Perhaps you can try this approach.

G   = tf(1, [1, 0, 0]);
opt = pidtuneOptions('DesignFocus', 'disturbance-rejection');
wc  = 5;
C   = pidtune(G, 'PID', wc, opt);

% Parameters
P.Xd    = 3;        % desired state for X(1)
P.Kp    = C.Kp;     % proportional gain
P.Ki    = C.Ki;     % integral gain
P.Kd    = C.Kd;     % derivative gain

% ODE function
function dX = f(t, X, P)
    % error
    e       = P.Xd - X(1);
    
    % PID controller
    u       = P.Kp*e + P.Ki*X(3) - P.Kd*X(2);
    
    % disturbance
    d       = 1;
    
    % original 2nd-order system
    dX(1)   = X(2);
    dX(2)   = d + u;
    
    % X(3) is ∫ e dt
    dX(3)   = e;
    
    % system dynamics
    dX      = [dX(1)
               dX(2)
               dX(3)];
end

% Solve and plot
[t, X]  = ode45(@(t, X) f(t, X, P), [0, 10], zeros(3, 1));
plot(t, X(:,1)), grid on
title('Time response for state x_{1}')
xlabel('t')
ylabel('x_{1}')

●

[Chuguang Pan]

@Sam Chak. Thanks a lot for your answer, your solution works well.

Answer 2

[Torsten]

Add a second differential equation

dV/dt = P.Xd - X 

with initial condition

V(0) = 0

Then

V(t) = integral_{0}^{t} (P.Xd - X) dt