I am trying to solve DDE using ode45 in Matlab. My question is about the way that I am solving this equation. I don't know if I am right or I am wrong and I should use dde23 instead. I have a following equation:
xdot(t)=Ax+BU(t-td)+E(t) & U(t-td)=Kx(t-td) & K=constant
Normally, when I don’t have delay on my equation, I solve this using ode45. Now with delay on my equation, again I am using ode45 to get the result. I have the exact amount of U(t-td) at each step and I replace its amount and solve the equation.
Is my solution correct or should I use dde23?
You have two problems here:
ode45 is a solver with adaptive step size. This means that your sampling steps are not necessarily equivalent to the actual integration steps. Instead, the integrator splits a sampling step into several integration steps as needed to achieve the desired accuracy (see this question on Scientific Computing for more information).
As a consequence, you may not provide correct delayed value of U at each step of the integration, even if you believe to do so.
However, if your sampling steps are sufficiently small, you will indeed have one time step per sampling step. The reason for this is that you effectively disable the adaptive integration by making your time step smaller than needed (and thus waste computation time).
Higher-order Runge–Kutta methods such as ode45 do not only make use of the value of the derivative at each integration step, but also evaluate it in-between (and no, they cannot provide a usable solution for this in-between time step).
For example, suppose that your delay and integration step are td=16. To make the integration step from t=32 to t=48, you need to evaluate U not only at t = 32−16 = 16 and t = 48−16 = 32, but also at t = 40−16 = 24. Now, you might say: Okay, let’s integrate such that we have an integration step at all those time points. But for these integration steps, you again need steps in the middle, e.g., if you want to integrate from t=16 to t=24, you need to evaluate U at t=0, t=4, and t=8. You get a never-ending cascades of smaller and smaller time steps.
Due to problem 2, it is impossible to provide the exact states from the past with any but a one-step integrator – using which is probably not a good idea in your case. For this reason, it is inevitable to use some sort of interpolation to obtain past values if you want to integrate DDEs with a multi-step integrator. dde23 does this in a sophisticated way using a good interpolation.
If you only provide U at the integration steps, you are essentially performing a piecewise-constant interpolation, which is the worst possible interpolation and therefore requires you to use very small integration steps. While you can do this if you really want to, dde23 with its more sophisticated piecewise cubic Hermite interpolation can work with much larger time steps and integrate adaptively, and therefore will be much faster. Also, it’s less likely that you somehow make a mistake. Finally, dde23 can deal with very small delays (smaller than the integration step), if you’re into that sort of thing.