Matrix input for Delayed Differential Equations

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d7366431 2018-12-25

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编辑： d7366431 2018-12-25

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Hello, could you please help me with solving system of Delayed Differential equations with matrix input?

Initially my equation is looks like

τ - constant delay

X - is vector of n elements

c - constant

- diag. matrix of n elements

- vector of zeros, only first element is equal to

I decided to choose DDE23 solver, but I'm not sure is it correct or not. Would be great to here some comments.

I put "lags" as constant = [1], I'm not sure is it correct or not, my idea was to get data from previous timestep which is in my case is 1 as I put in tspan vector.

Also I decrease order of differential equation as

Could you please tell me is it correct solver for this task? I noticed that nothing is changes with changing delay value.

How can I display movement of last element of my system (n th element). Is it correct way how I did below?

Below is my Matlab Code with some comments:

tspan = 1:1:5;
%%sol   = dde23(    ddefile,    lags,        history,           tspan);
  sol   = dde23(       @dde,    [1],        @history,           tspan);

ax1 = subplot(2,1,1);
plot(sol.x,sol.y);
grid on;
title('sol.x, sol.y')
    xlabel('time t');
    ylabel('x(t)');
 
ax2 = subplot(2,1,2);
grid on;
title('deval')
    xplot = 1:1:10;
    yplot = deval(sol,xplot,size(n));
    plot(xplot,yplot);
    xlabel('time t');
    ylabel('x(t)');

function xdot=dde(t,x,Z)                % x - state of system (position of each block)
n=3; %number of elements for simplicity put 3 for now       
v0=zeros(n,1);
v0(1)=1;
% c matrix
c=1;
C=zeros(n);
C(end,end)=c;
% D Matrix
d=1;
D=d*eye(n);
% M Matrix
m=1;                                   
M=m*eye(n);
%% The differential equations:
        xlag    = Z(1:n,1);
        x1      = zeros(n,1);
        x2      = zeros(n,1);
        x1dot   = zeros(n,1);
        x2dot   = zeros(n,1);
        xdot    = zeros(2*n,1);
        x1 = xdot(1:n);
        x2 = xdot(n+1:end);
        x1dot = x2;
        x2dot = v0*t-M*xlag-D*x1-C*x2;
        xdot = [x1dot; x2dot];
end
function s = history(t)
n=3
    s=zeros(2*n,1)
    for i=1:2*n
        s(i,1)=1;
    end
end