Generally speaking, time delays are mean.

As a first step, time delay can be represented by an exponential function in the frequency domain when you take the Laplace transform, i.e.:

L(x(t-a))=X(s)e^(-as),

where L is the Laplace transform symbol, and a is the delay. This means that time delays introduce nonlinearities, in general. Then, your problem will be a nonlinear one. Many approaches can be used in this case, but one can think of "Pade approximation" to represent the exponential function as a rational function, usually one degree is enough (it depends on your system).

For modelling time delay systems in time domain, you can use the function dde23 in Matlab.

Finally,

To quote Hitch-hiker's Guide - the problem as posed is too broadly based. Asking how to handle a time delay is like asking how to handle an integral. There isn't any generic answer. The Laplace approach is mainly useful for linear systems, and in a practical sense, not all of them - but it is one thing to try. But, you said "time domain". The simplest answer to that is use $f(t-\Delta t)$, of course, and do the algebra, but you could also convolve the function with a shifted impulse. There are also time-domain differential expressions for a time delay - although in that case you should make sure that the functions involved are analytic (Taylor series converges to function). For things such as lag in gears, often a state-based approach is a good idea. Some indicator conditions are used to decide which function applies. This can have the problem of an exponential blow-out in the number of cases, but it can be a viable approach.

thanks all of you Mutaz Saif Eddin Tuffaha Bruce I. Mills

I am not sure this will help but from a modeling point of view, in general:

- Frequency domain: exp(-tau*s)

- Linear systems (Transf. Func. or State-Space): Trans.Func. approx (Padè Approximation, order to be decided on a case-by-case in order to match Freq. Resp. and/or time resp.)

- Simulation in continuous domain:

* Fortran / C / etc.: custom built functions

* Simulink / MATRIXx / Scicos: Time Delay Block or custom function

- Simulation in discrete domain with a single rate: a rounded number of unit delays that will represent the delay

- Simulation in discrete domain with multi-rate: a unit delay that corresponds to the delay

- Simulation of hybrid system (continuous+discrete): approach for continuous or for discrete.

thanks Pierluigi Capone , but my model will be in a time domain so that i can neither use Simulink nor frequency domain work. thanks anyway for your reply.