Discrete-Time Transfer Functions

2. 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. 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. 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 • TheZ-transform of Continuous-time Transfer Function g(t) • TheZ-transform of Input Signal u(t)

5. 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. 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. 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. 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. 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. 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. 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. 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 BDA TA FDA

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

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

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

20. 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 (Attendance List No.1-4) • Backward Difference (Attendance List No.5-8) (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.