How to keep track of accumulated error for integral term in a PID controller function?

This is my PI controller function:

function [ control ] = my_fastcontrol(t, v)
    Kp         = 20;
    K_i        = 10;
    v_desired  = 20;
    error      = v_desired - v ;
    control(2) = Kp*error + Ki(int(error, 0, t));
end

It is called by this line in another file:

[t1, y1] = ode45(@(tau, y1) my_dynamicvehicle(tau, y1, @(tau) my_fastcontrol(tau, y1(2)), friction_factors(j)), [t0 tf], z0);

where, y1(2) is the velocity. my_dynamicvehicle() is also a separate file that contains the differential equations that models the behaviors of a car.

My question is I am trying to take the integral of all the error from 0 to t. The proportional term has no problem as it only cares about the velocity that gets passed to the my_fastcontrol() function at the present instant, but the integral terms need a history of all the velocities at the previous times. How does my_fastcontrol() gets access to that history?

Answer 1

[Sam Chak]

Hi @David

This simple example shows how to obtain the integral error signal by adding an auxiliary state in the my_Dynamics() function. The example uses a first-order system and is given by . Two systems are used in the simulation for comparison purposes. The first one is approached using a transfer function, while the second one employs the ode45 method.

%% System 1
Gp  = tf(1, [1 1]);         % 1st-order plant
Gc  = pid(1, 1)             % PI controller

Gc =
 
             1 
  Kp + Ki * ---
             s 

  with Kp = 1, Ki = 1
 
Continuous-time PI controller in parallel form.

Gcl = feedback(Gc*Gp, 1);   % closed-loop feedback system
subplot(211)
step(Gcl, 10), grid on      % step response

%% System 2
tspan = [0 10];             % integration time interval
v0    = [0; 0];             % initial values, incl the auxiliary state
[t, v] = ode45(@my_Dynamics, tspan, v0);

% Plot result
subplot(212)
plot(t, v(:,2)), grid on    % plot the original system state, v(2)
yticks([0 0.5 1])
xlabel('t'), ylabel('v'), title('Result from ode45')

% Main function
function dvdt = my_Dynamics(t, v)
    dvdt      = zeros(2, 1);
    v_desired = 1;
    error     = v_desired - v(2);
    dvdt(1)   = error;    % add an auxiliary state that returns v(1) = int(error)
    dvdt(2)   = - v(2) + my_PIcontrol(v);   % <-- calling PI control signal from my_PIcontrol()
end

% Local function (to be used in the Main function)
function control = my_PIcontrol(v)
    Kp        = 1;                  % proportional gain
    Ki        = 1;                  % integral gain
    v_desired = 1;                  % desired steady-state value for v(2)
    error     = v_desired - v(2);
    control   = Kp*error + Ki*v(1); % remember that v(1) = int(error)
end

[David]

This works. Thank you.

[Sam Chak]

@David, I'm glad that it works for you. If you find the solution helpful, please consider clicking 'Accept' ✔ on the answer and voting 👍 for it.