how can add PI to ode45?

hi

I have an system controled by PI control in simulink i can design it but in code when add PI control I can't do it

where my system is shown in next figure

any help

Answer 1

[Sam Chak]

Hi @karim_karimo

Here are the mathematical preliminaries for constructing the PI in ODE form.

[t, x] = ode45(@system, [0 10], [0; 0]);
plot(t, x(:,1), 'linewidth', 1.5)
grid on, xlabel('t'), ylabel('y(t)'), ylim([-0.2 1.2])

function dxdt = system(t, x)
    dxdt    = zeros(2, 1);
    % construction of PI 
    r       = 1;                % reference signal
    e       = r - x(1);         % error signal (based on Simulink)
    Kp      = 1/3;              % proportional gain (user's Kp is 1)
    Ki      = - 1/3;            % integral gain     (user's Ki is 2)
    u       = Kp*e + Ki*x(2);   % the PI thing      (user's PI gains cause overshoot)
    
    % the dynamics
    A       = 1;
    B       = 3;
    dxdt(1) = A*x(1) + B*u;     % the system
    dxdt(2) = e;                % for integral action
end

[Sam Chak]

User's selection of PI gains cause 50% overshoot, which is normally undesirable in most situations.

The proposed PI gains in the Answer drive the system to have almost the same convergence time but without overshoot.

[t, x] = ode45(@system, [0 10], [0; 0]);
plot(t, x(:,1), 'linewidth', 1.5)
grid on, xlabel('t'), ylabel('y(t)')

function dxdt = system(t, x)
    dxdt    = zeros(2, 1);
    % construction of PI 
    r       = 1;                % reference signal
    e       = r - x(1);         % error signal (based on Simulink)
    Kp      = 1;                % user's proportional gain
    Ki      = 2;                % user's integral gain
    u       = Kp*e + Ki*x(2);   % the PI thing
    
    % the dynamics
    A       = 1;
    B       = 3;
    dxdt(1) = A*x(1) + B*u;     % the system
    dxdt(2) = e;                % for integral action
end

[karim_karimo]

thank you

can I use integral instead using the second state

[Sam Chak]

The integral command requires the mathematical function.