2. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
Now let us calculate the transient response of a combined discrete-time and continuous-time system, as shown below.
The input to the continuous-time system G(s) is the signal:
The system response is given by the convolution integral:

3. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
With
For 0 ≤ τ ≤ t,
We assume that the output sampler is ideally synchronized with the input sampler.
The output sampler gives the signal y*(t) whose values are the same as y(t) in every sampling instant t = jTs.
Applying the Z-transform yields:

4. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
Taking i = j – k, then:
For zero initial conditions, g(iTs) = 0, i < 0, thus:
where
Discrete-time Transfer Function
The Z-transform of Continuous-time Transfer Function g(t)
The Z-transform of Input Signal u(t)

5. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
Y(z) only indicates information about y(t) in sampling times, since G(z) does not relate input and output signals at times between sampling times.
When the sample-and-hold device is assumed to be a zero-order hold, then the relation between G(s) and G(z) is

6. Chapter 5
Discrete-Time Process Models
Example
Find the discrete-time transfer function of a continuous system given by:
where

7. Input-Output Discrete-Time Models
Chapter 5
Discrete-Time Process Models
Input-Output Discrete-Time Models
A general discrete-time linear model can be written in time domain as:
where m and n are the order of numerator and denominator, k denotes the time instant, and d is the time delay.
Defining a shift operator q–1, where:
Then, the first equation can be rewritten as: or

8. Input-Output Discrete-Time Models
Chapter 5
Discrete-Time Process Models
Input-Output Discrete-Time Models
The polynomials A(q–1) and B(q–1) are in descending order of q–1, completely written as follows:
The last equation on the previous page can also be written as:
Hence, we can define a function:
Identical, with the difference only in the use of notation for shift operator, q-1 or z–1

9. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Previous example shows how the Z-transform of a function written in s-Domain can be so complicated and tedious.
Now, several methods that can be used to approximate the Z-transform will be presented.
Consider the integrator block as shown below:
The integration result for one sampling period of Ts is:

10. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Forward Difference Approximation (Euler Approximation)
The exact integration operation presented before will now be approximated using Forward Difference Approximation.
This method follows the equation given as:
Taking the Z-transform of the above equation:
while
Thus, the Forward Difference Approximation is done by taking or

11. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Backward Difference Approximation
The exact integration operation will now be approximated using Backward Difference Approximation.
This method follows the equation given as:
Taking the Z-transform of the above equation:
while
Thus, the Backward Difference Approximation is done by taking or

12. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Trapezoidal Approximation (Tustin Approximation, Bilinear Approximation)
The exact integration operation will now be approximated using Backward Difference Approximation.
This method follows the equation given as:
Taking the Z-transform,
while
Thus, the Trapezoidal Approximation is done by taking or

13. Example Find the discrete-time transfer function of
Chapter 5
Discrete-Time Process Models
Example
Find the discrete-time transfer function of
for the sampling time of Ts = 0.5 s, by using (a) ZOH, (b) FDA, (c) BDA, (d) TA.
(a) ZOH
(b) FDA

14. Chapter 5
Discrete-Time Process Models
Example
(c) BDA
(d) TA

15. Chapter 5
Discrete-Time Process Models
Example
ZOH
FDA TA
BDA

16. Example: Discretization of Single-Tank System
Chapter 5
Discrete-Time Process Models
Example: Discretization of Single-Tank System
Retrieve the linearized model of the single-tank system. Discretize the model using trapezoidal approximation, with Ts = 10 s.
Laplace Domain
Z-Domain

17. Example: Discretization of Single-Tank System
Chapter 5
Discrete-Time Process Models
Example: Discretization of Single-Tank System

18. Example: Discretization of Single-Tank System
Chapter 5
Discrete-Time Process Models
Example: Discretization of Single-Tank System

19. Example: Discretization of Single-Tank System
Chapter 5
Discrete-Time Process Models
Example: Discretization of Single-Tank System
: Linearized model
: Discretized linearized model

20. Example: Discretization of Single-Tank System
Chapter 5
Discrete-Time Process Models
Example: Discretization of Single-Tank System
: Linearized model
: Discretized linearized model

21. Chapter 5
Discrete-Time Process Models
Homework 8
(a) Find the discrete-time transfer functions of the following continuous-time transfer function, for Ts = 0.25 s and Ts = 1 s. Use the Forward Difference Approximation
(b) Calculate the step response of both discrete transfer functions for 0 ≤ t ≤ 5 s.
(c) Compare the step response of both transfer functions with the step response of the continuous-time transfer function G(s) in one plot/scope for 0 ≤ t ≤ 0.5 s.

22. Chapter 5
Discrete-Time Process Models
Homework 8A
Find the discrete-time transfer functions of the following continuous-time transfer function, for Ts = 0.1 s and Ts = 0.05 s. Use the following approximation:
Forward Difference (Fiedel, Prananda)
Backward Difference (Yasin)
(b) Calculate the step response of both discrete transfer functions for 0 ≤ t ≤ 0.5 s. The calculation for t = kTs, k = 0 until k = 5 in each case must be done manually. The rest may be done by the help of Matlab Simulink.
(c) Compare the step response of both discrete transfer functions with the step response of the continuous-time transfer function G(s) in one plot/scope for 0 ≤ t ≤ 0.5 s.