# Convert z to e^(-sT) in transfer function

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Malte Eggers on 9 Nov 2023
Edited: Les Beckham on 9 Nov 2023
I know I can simply write
exp(-s*T)
But I want to convert an entire transfer function from the z domain to the s domain by simply replacing z with said term. I've tried extracting denominator and numerator coefficients and multiplying each with an appropriate exponential, but any way I try to combine them to a single transfer function would throw an error. A loop would probably do the trick, but there must be a better way.
Ideally, it would work for MIMO systems as well

Les Beckham on 9 Nov 2023
Edited: Les Beckham on 9 Nov 2023
If you have the Control System Toolbox you can use the d2c function to convert from the z domain to the s domain.
Note that this should work for your existing state space model without first converting to transfer functions.

Harald on 9 Nov 2023
Hi Malte,
you could use Symbolic Math Toolbox to accomplish this:
syms z s T
f = (z-1)/(z^2 + 2*z + 3) % an example
f =
f2 = subs(f, z, exp(-s*T))
f2 =
Be sure to use Live Editor to see nicely formatted formulas. :)
Best wishes,
Harald
Malte Eggers on 9 Nov 2023
Hi Harald,
I'm starting off with a state space system actually. Ideally I'd be converting that, but I initially tried converting a transfer function as that seemed more straightforward. I do I have a MISO system that I could split up into several transfer functions if need be.
The issue with your suggestion is that it requires a symbolic expression to start off whereas I would be having a state space system (or a transfer function) as a result of previous steps. I'd have to convert that to a symbolic expression, do the substitution and then convert back to a transfer function, which would have me face some of the same issues and some new ones.
Cheers,
Malte
Harald on 9 Nov 2023
Hi Malte,
in general, please consider including some code example as this may be clearer and easier to follow.
While it's always possible that there is an easier way to do this, you could convert the transfer function to a symbolic expression as shown here, perform the substitution, and transform back if needed. If the main concern is the code starting to look ugly, this could be moved into a function.
Best wishes,
Harald

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