Linear behavior and dynamic behavior of a sensor

[Aleoa]

I want to model a sensor with the static behavior:

$y(t)=a+by_{0}(t)$

using a first order lag:

$G(s)=\frac{K}{1+Ts}$

However, if i try to convert this order lag in time domain and set the derivative as 0, what i get as static response is:
$y(t)=Ky_{0}(t)=by_{0}(t)$

And the a constant has disappeared, what can i do ?

 

[berkeman]

And the a constant has disappeared, what can i do ?

See if section 3 of this PDF helps. It deals with DC offsets in the transfer functions of first order systems...

http://engineering.ju.edu.jo/Laboratories/04 - First Order System.pdf