I have a system (of first-order, homogeneous ODEs) which contains a time delay
$$b(t) = a(t-T).$$
Are there any standard methods for approximating $b$ by $\bar{b}$, where $\bar{b}$ derives from $a$ via a system of ODEs? I envision this system as a "black box" with inherent time delay $T$, rather than hard-coding the delay as in the original equation.
For example, we could try
$$\bar{b}^\prime(t) = k \left(a(t) -\bar{b}(t)\right)$$
where
$$k = \frac{2 \log 2}{T}$$
as a simple choice. If $a = a_0$ were constant, $\bar{b} \to a$ as $t \to \infty$, and moreover at $t = \frac12 T$, $\bar{b}(t) = \frac12 \left(b_0+a_0 \right)$: after half the delay, the system has moved half-way to the correct answer.
Are there more formal approximations available?
You might try using a Taylor polynomial $$ a(t-T) \approx \sum_{j=0}^d \frac{(-T)^j}{j!} a^{(j)}(t) $$