How I can plot input signal in ode45

I want to plot the control signal (u) when we solve the equation with ode45. how I can plot control signal (u) in the following code.

tspan = 0:1:10000;
y0 = [0.2,0.3];
[t,y] = ode45(@(t,y) odefcn(t,y), tspan, y0);
% 

function dx = odefcn(t,x)
dx = zeros(2,1);
u=[2 -2]*x(1:2)+1;
dx(1)=x(1)-2*x(2)
dx(2)=-x(2)+u
end

Answer 1

[Star Strider]

Create a second output for ‘u’ (ode45 will ignore it during the integration) and then return it using a for loop after the integration finishes.

Try this —

tspan = 0:0.001:15;
y0 = [0.2,0.3];
[t,y] = ode45(@(t,y) odefcn(t,y), tspan, y0);

for k = 1:numel(t)
    [dx,u(k)] = odefcn(t(k),y(k,:).');
end

figure
plot(t,y(:,1), 'DisplayName','y_1(t)')
hold on
plot(t,y(:,2), 'DisplayName','y_2(t)')
plot(t,u, 'DisplayName','u(t)')
hold off
grid
legend('Location','best')

% set(gca,'XScale','log')

function [dx,u] = odefcn(t,x)
dx = zeros(2,1);
u=[2 -2]*x(1:2)+1;
dx(1)=x(1)-2*x(2);
dx(2)=-x(2)+u;
end

I changed ‘tspan’ because with a long vector, the details of the initial transient disappear from the plot.

.

[Dyuman Joshi]

Nice answer, @Star Strider.

Just a small suggestion to preallocate u and negate the output dx if it is needed.

Answer 2

[Sam Chak]

Hi @hossen hassanzadth

Your initial query has already been addressed. To achieve step profile tracking using your designed gain matrix, you should multiply the reference input (xref) by a scaling factor (sf). In doing so, the blue curve will consistently attain its steady-state position .

tspan  = 0:0.001:10;
x0     = [0.2, 0.3];
[t, x] = ode45(@odefcn, tspan, x0);

% Computing the control signal u from the ode solution
u = [2 -2]*x.' + (-0.5)*(1);    % sf = -0.5; xref = 1;

plot(t, x(:,1), 'DisplayName', 'x_1(t)'), hold on
plot(t, x(:,2), 'DisplayName', 'x_2(t)')
plot(t, u,      'DisplayName', 'u(t)'), hold off, grid on
xlabel('Time, (seconds)')
title('Step Response')
legend('location', 'SE', 'fontsize', 12)

% Dynamics
function [dx, u] = odefcn(t, x)
    dx    = zeros(2,1);
    
    xref  =  1;                 % reference input
    sf    = -0.5;               % scaling factor
    u     = [2 -2]*x + sf*xref; % control signal

    dx(1) = x(1) - 2*x(2);      % x'
    dx(2) = - x(2) + u;         % x"
end