# Proof of Forward Euler for discretizing a transfer function

In Levine book "The control handbook" it is shown that, for discretizing a transfer function $$\frac{1}{s}$$ using Forward Euler i simply have to replace s with $$\frac{z-1}{T}$$. How can extend the proof and show that the substitution works for every possible transfer function $$G(s)$$ ?

• What do you mean by "works"? Of course you can replace $s$ by that function and see what you get. It is just one out of several ways to discretize a continuous system. There are other transformations (such as backward Euler) that will guarantee that a stable system transforms to a stable system. That is not the case with forward Euler. – Matt L. Jan 12 '20 at 17:52
• I don't understand what's the proof for the subtitution, to me laplace and z-transform are completely different things, so i don't understand why is the correct to apply that substitution. In my textbook, the proof is derived from a particular case, and i would like to generalize it. (Levine - The Control Handbook. Volume 1 pag 283) – themagiciant95 Jan 12 '20 at 18:22
• Does my answer clarify things a bit? – Matt L. Jan 12 '20 at 19:05
• Read this answer to see that the Laplace and Z-transforms are very much related. The Z-transform is in the discrete domain what the Laplace transform is in the continuous domain. – Matt L. Jan 12 '20 at 19:08
• @MattL., maybe here my doubt is more clear .... dsp.stackexchange.com/questions/63229/… – themagiciant95 Jan 14 '20 at 10:22

Another way to see how the forward Euler method approximates a continuous-time system is by considering the "ideal" mapping of the $$s$$-plane to the $$z$$-plane (why?):
$$z=e^{sT}\tag{1}$$
For frequencies that are much smaller than the sampling frequency (i.e., $$|s|T\ll 1$$) we can approximate $$e^{sT}$$ by its first order Taylor series:
$$z\approx 1+sT\tag{2}$$
$$s=\frac{z-1}{T}\tag{3}$$