Introduce step signal after system response

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Hello,
sorry for the summary, I dont know how to resume my question in few words.
I have a second order system which I know how will respond without an input signal, like this:
x1'=x2;
x2'=ax1+bx2
using ODE 45 and I get this response
where the position (x1) acts like this
now I need to insert a step signal "u" at second 20. The system looks like this but I dont know how to do it. I have created this signal as a vector but I'm getting an error in the ODE 45 function. How can I introduce u into the equation in order to alterate this system in the second 20?
x1'=x2;
x2'=ax1+bx2 + u
this is my code so far:
%% INIT
clear variables;
close all;
clc;
%% Reference input
%STEP
t = (1:1:60)'; % entering U as a vector is not working
u=t>=20;
% figure()
% plot(t,u)
%% variable initialization
%coefs
a=-0.3;
b=-1.1;
%gains
% Kp=3;
% Ki=0.9;
% Kd=0.8;
%initial conditions
x1init= 2.5;
x2init= 2.5;
% time
time=100;
%% Space state equation
%define second order equation system
f = @(t,X) [X(2); a*X(1)+b*X(2)+u];
% define the area where the function will be analysed
[x1,x2] = meshgrid(-5:5);
x1mat = zeros(size(x1));
x2mat = zeros(size(x2));
%
t=0;
for i = 1:numel(x1)
Xprima = f(t,[x1(i); x2(i)]);
x1mat(i) = Xprima(1);
x2mat(i) = Xprima(2);
end
%% PLOT THE AREA
figure
quiver(x1,x2,x1mat,x2mat,'r'); figure(gcf)
xlabel('x1')
ylabel('x2')
axis tight equal;
title('Phase diagram ')
hold on
%% PLOT FUNCTION SOLUTION
%
%options=odeset('InitialStep',1e-3,'MaxStep',100);
for sol1 = [x1init ; x2init]
[ts,ys] = ode45(f,[0,time],sol1);
plot(ys(:,1),ys(:,2),'r')
plot(ys(1,1),ys(1,2),'bo') % starting point
plot(ys(end,1),ys(end,2),'ks') % ending point
end
hold off
figure
subplot(2,1,1)
plot(ts,ys(:,1),'k', 'LineWidth',2)
xlabel('time(s)','fontweight','bold','FontSize',12)
ylabel('position','fontweight','bold','FontSize',12)
subplot(2,1,2)
plot(ts,ys(:,2),'k', 'LineWidth',2)
xlabel('time(s)','fontweight','bold','FontSize',12)
ylabel('speed','fontweight','bold','FontSize',12)
title('function evolution')

Accepted Answer

Paul
Paul on 19 Jul 2022
Hi mikel,
No need to use ode45 for this particular problem. The delayed step response with initial conditions can be obtained as
a=-0.3;
b=-1.1;
sys = ss([0 1;a b],[0;1],eye(2),0);
%initial conditions
x1init= 2.5;
x2init= 2.5;
% time
time=100;
ts = 0:.1:time; % small time step, make sure a point in ts hits 20
ys = initial(sys,[x1init x2init],ts) + step(sys*tf(1,1,'InputDelay',20),ts);
% or use lsim
%ys = lsim(sys,ts>=20,ts,[x1init x2init]);
figure
subplot(2,1,1)
plot(ts,ys(:,1),'k', 'LineWidth',2)
xlabel('time(s)','fontweight','bold','FontSize',12)
ylabel('position','fontweight','bold','FontSize',12)
subplot(2,1,2)
plot(ts,ys(:,2),'k', 'LineWidth',2)
xlabel('time(s)','fontweight','bold','FontSize',12)
ylabel('speed','fontweight','bold','FontSize',12)
title('function evolution')
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