# Calculating the discrete-time system transfer function from continious transfer function.

#### Elbobboo

Joined Mar 24, 2023
4
Ok so I am required to develop proportional controller from a given continuous transfer function, however first I have been asked to find the discrete-time system transfer function. From my understanding this requires the z-transform to be taken from the continuous transfer function G(s).

$$G(s)=\frac{5e^{-\tau s}}{s+5}$$, where $$\tau = 1$$ and $$e^{-\tau s}$$ is the pure time delay in the system.

I have given the hint: $$Z[e^{−\tau s}G(s)]=z^{−k}Z[(G(s)]$$, where $$k=\frac{\tau}{T}$$
I have already calculated the value of T, using bode plots, finding the cut off frequency and from there the sample interval.

Finding the Z-transform of G(s)
$$Z[G(s)]=Z[\frac{5}{s+5}]=\frac{5Z}{Z−1}\frac{1}{1−(−e^{−T})}=\frac{5Z}{(Z−1)(1+e^{−T})}$$

Substituting this into the equation for,
$$Z[e^{−τs}G(s)]$$

$$Z[e^{−τs}G(s)]=Z^{-k}Z[G(s)]=Z^{-k}\frac{5Z}{(Z-1)(1+e^{-T})}$$

Substituting
$$k=\frac{\tau}{T}$$

$$G_{HP}(Z)=Z[e^{−τs}G(s)]=Z^{-\frac{\tau}{T}} \frac{5Z}{(Z−1)(1+e^{T})}=\frac{5Z1-\frac{\tau}{T}}{(z-1)(1+e-T)}$$

So the discrete-time system transfer function is:

$$G_{HP}(Z)=\frac{5Z1-\frac{\tau}{T}}{(z-1)(1+e-T)}$$

Am I on the right page?

#### RoofSheep

Joined Mar 7, 2023
36
Ok so I am required to develop proportional controller from a given continuous transfer function, however first I have been asked to find the discrete-time system transfer function. From my understanding this requires the z-transform to be taken from the continuous transfer function G(s).

$$G(s)=\frac{5e^{-\tau s}}{s+5}$$, where $$\tau = 1$$ and $$e^{-\tau s}$$ is the pure time delay in the system.

I have given the hint: $$Z[e^{−\tau s}G(s)]=z^{−k}Z[(G(s)]$$, where $$k=\frac{\tau}{T}$$
I have already calculated the value of T, using bode plots, finding the cut off frequency and from there the sample interval.

Finding the Z-transform of G(s)
$$Z[G(s)]=Z[\frac{5}{s+5}]=\frac{5Z}{Z−1}\frac{1}{1−(−e^{−T})}=\frac{5Z}{(Z−1)(1+e^{−T})}$$

Substituting this into the equation for,
$$Z[e^{−τs}G(s)]$$

$$Z[e^{−τs}G(s)]=Z^{-k}Z[G(s)]=Z^{-k}\frac{5Z}{(Z-1)(1+e^{-T})}$$

Substituting
$$k=\frac{\tau}{T}$$

$$G_{HP}(Z)=Z[e^{−τs}G(s)]=Z^{-\frac{\tau}{T}} \frac{5Z}{(Z−1)(1+e^{T})}=\frac{5Z1-\frac{\tau}{T}}{(z-1)(1+e-T)}$$

So the discrete-time system transfer function is:

$$G_{HP}(Z)=\frac{5Z1-\frac{\tau}{T}}{(z-1)(1+e-T)}$$

Am I on the right page?
There are various methods for converting continuous-time transfer functions to discrete-time transfer functions. Examples are the zero-order-hold method and the impulse-invariance method. Choosing the right method depends on how your digital processor is connected to the analogue world.