Introduce step signal after system response

Question

mikel lasa 2022-7-19

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1763690-introduce-step-signal-after-system-response

评论： mikel lasa 2022-7-21

采纳的回答： Paul

在 MATLAB Online 中打开

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')

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Answer 1

Paul 2022-7-19

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1763690-introduce-step-signal-after-system-response#answer_1011225

在 MATLAB Online 中打开

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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Paul 2022-7-20

编辑：Paul 2022-7-20

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 2022-7-21