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:

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:

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)

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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

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

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.

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.