Using event function in matlab ode45 for multi-dimensional state vector

Question

I have a set of odes written in matrix form as $X' = AX$; I also have a desired value of the states $X_des$. $X$ is a five dimensional vector. I want to stop the integration after all the states reach their desired values (or atleast close to them by $1e{-3}$). How do I use event function in matlab to do this? (All the help I have seen are about 1 dimensional states)

PS: I know for sure that all the states approach their desired values after long time. I just want to stop the integration once they are $1e{-3}$ within the desired values.

Answer 1

First, I presume that you're aware that you can use the matrix exponential (expm in Matlab) to solve your system of linear differential equations directly.

There are many ways to accomplish what you're trying to do. They all depend a bit on your system, how it behaves, and the particular event you want to capture. Here's a small example for a 2-by-2 system of linear differential equations:

function multipleeventsdemo
A = [-1 1;1 -2]; % Example A matrix
tspan = [0 50];  % Initial and final time
x0 = [1;1];      % Initial conditions
f = @(t,y)A*y;   % ODE function
thresh = 0;      % Threshold value
tol = 1e-3;      % Tolerance on threshold

opts = odeset('Events',@(t,y)events(t,y,thresh,tol)); % Create events function
[t,y] = ode45(f,tspan,x0,opts);                       % Integrate with options
figure;
plot(t,y);

function [value,isterminal,direction] = events(t,y,thresh,tol)
value = y-thresh-tol;
isterminal = all(y-thresh-tol<=0)+zeros(size(y)); % Change termination condition
direction = -1;

Integration is stopped when both states are within tol of thresh. This is accomplished by adjusting the isterminal output of the events function. Note that separate tolerance and threshold variables isn't really necessary – you simply need to define the crossing value.

If your system oscillates as it approaches it's steady state (if A has complex eigenvalues), then you'll need to do more work. But you questions doesn't indicate this. And again, numerical integration may not be the easiest/best way to solve your problem which such a system. Here is how you could use expm in conjunction with a bit of symbolic math:

A = [-1 1;1 -2];
x0 = [1;1];
tol = 1e-3;
syms t_sym
y = simplify(expm(A*t_sym)*x0)           % Y as a function of t
t0 = NaN(1,length(x0));
for i = 1:length(x0)
    sol = double(solve(y(i)==tol,t_sym)) % Solve for t when y(i) equal to tol
    if ~isempty(sol)                     % Could be no solution, then NaN
        t0(i) = max(sol);                % Or more than one solution, take largest
    end
end
f = matlabFunction(y);                   % Create vectorized function of t
t_vec = linspace(0,max(t0),1e2);         % Time vector
figure;
plot(t_vec,f(t_vec));

This will only work for fairly small A, however, because of the symbolic math. Numerical approaches using expm are also possible and likely more scalable.