How to use matlab to solve ordinary differential equations with coupled intermediate variables inside?

Dear all:

I have a problem which can be simplified to the statement below (the real equations and functions are much more complicated):

System: dot_X1 = -A*X1; dot_X2 = B*(X2-5) + u;

dot_ means differentiate with time t, X1 and X2 are system states respectively, A and B are both intermediate variables, u is the system control input which can be a ramp or step input defined by myself.

I know how to use ode to solve normal similar problems when A and B and not coupled, for example, when A and B are only functions of X1, X2 and u respectively. But now in my problem, A and B are coupled, like:

A = f(X1, X2, u, B), B = g(X1, X2, u, A), f and g are very complicated functions.

Now in every time interval when solving the ode problem, for example, [0s, 0.01s], I have to calculate both A and B in order to use ode function in matlab. But since A and B are coupled and f and g are very complicated, I don't know how to deal with this.

Could you give me some help? Thanks a lot!

Answer 1

[Torsten]

Add A and B as algebraic variables to your ODE system.

To determine A and B, add the equations

A-f(X1,X2,u,B)=0
B-g(X1,X2,u,A)=0

to your system.

Then you solve for four unknowns: A,B,X1,X2.

Before calling the ODE integrator, define the mass matrix M as

M=[1 0 0 0
   0 1 0 0
   0 0 0 0 
   0 0 0 0].

Best wishes

Torsten.

[Tao]

Hi, Torsten:

Actually my A and B are twisted together and the function f and function g are implicit, which means I cannot write the equations A-f(X1,X2,u,B)=0 B-g(X1,X2,u,A)=0.

[Torsten]

But that's exactly the way you wrote your equations in your previous request...

I doesn't matter that it can not be explicitly solved for A and B - the ODE solvers are capable to solve for implicitly defined variables.

Best wishes

Torsten.