Introduce step signal after system response

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')

Answer 1

[Paul]

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')

[mikel lasa]

This is much better that was doing! thanks

My goal is to input different signals like step, sinusoidal and impulse (with and without time delays) and see the system response.

Just a question, I dont understand what initial function does, and matlab help does not cleared my doubts.

initial(sys,[x1init x2init],ts)

[Paul]

initial computes the response of a state space system to initial conditions on the state variables with zero input. I'm not sure I can describe it any differently than the doc page. Note that in the state space model sys I selected C as eye(2) so that the outputs of sys are the state variables.

To explore the response to different inputs, lsim is most likely the way to go (see commented line above) with apprropriate selection of the "method" argument, except for an impulse input which would have to be handled as a special case with lsim.

[mikel lasa]

thank you! understood!