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.
. 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
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
