Discrete-Time Transfer Functions

President University Erwin Sitompul SMI 8/1

Lecture 8System Modeling and Identification

Dr.-Ing. Erwin SitompulPresident University

http://zitompul.wordpress.com2 0 1 4


Now let us calculate the transient response of a combined discrete-time and continuous-time system, as shown below.

Chapter 5 Discrete-Time Process Models

Discrete-Time Transfer Functions

The input to the continuous-time system G(s) is the signal:

s s00

( ) ( ) ( ) ( )t

k

y t g t u kT kT d

*s s

0

( ) ( ) ( )k

u u kT kT

The system response is given by the convolution integral:


With


Discrete-Time Transfer Functions 1( ) ( )g t G sL

For 0 ≤ τ ≤ t, s s

0

( ) ( ) ( )k

y t g t kT u kT

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.

s s s s0

( ) ( ) ( )k

y jT g jT kT u kT

Applying the Z-transform yields:

s s s0 0

( ) ( ) ( ) j

j k

Y z g jT kT u kT z



Discrete-Time Transfer Functions Taking i = j – k, then:

( )s s

0

( ) ( ) ( ) i k

i k k

Y z g iT u kT z

For zero initial conditions, g(iTs) = 0, i < 0, thus:s s

0 0

( ) ( ) ( )i k

i k

Y z g iT z u kT z

( ) ( ) ( )Y z G z U z

where s

0

( ) ( )( ) i

i

G z g iT zg t

Z

s0

( ) ( )( ) k

k

U z u kT zu t

Z

• 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 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

1 1 ( )( ) (1 ) G sG z zs

Z L



ExampleFind the discrete-time transfer function of a continuous system given by:

1 2( ) ( ) ( )G s G s G s

wheres

1 21( ) , ( )

1

sTe KG s G ss s

1 1( ) (1 )( 1)K

G z zs s

Z L

1 1(1 )1

K Kz

s s

Z L

s( )

11 T

z z zKz z z e

s( )

11 T

zKz e

s

s

( )

( )

1( )T

T

eG z Kz e

s

s

1( )

( ) 111

T

T

zeKe z

11

111

b za z

s( )1 1 Tb K e

s( )1

Ta e



Input-Output Discrete-Time Models A general discrete-time linear model can be written in time

domain as:

1 1

( ) ( ) ( )n m

i ii i

y k a y k i bu k d i

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:1 ( ) ( 1)q y k y k

Then, the first equation can be rewritten as:( )

1 1

( ) ( ) ( )n m

i d ii i

i i

y k a q y k b q u k

or

1 1( ) ( ) ( ) ( )dA q y k q B q u k



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:1 1

1( ) 1 nnA q a q a q

1 11( ) m

mB q b q b q

1

1

( ) ( )( ) ( )

dy k B qqu k A q

The last equation on the previous page can also be written as:

11

1

( )( )( )

d B qG q qA q

Hence, we can define a function:1

11

( )( )( )

d B zG z zA z

• Identical, with the difference only in the use of notation for shift operator, q-1 or z–1



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:

( )Y s( )U s 1s

0

( ) (0) ( )t

y t y u d The integration result for one

sampling period of Ts is:s s

s

s s s( ) ( ) ( )kT T

kT

y kT T y kT u d

skT

s( )u kT

s s( )u kT T

s skT T

s s

s

s s sArea ( ) ( )

( )kT T

kT

y kT T y kT

u d



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:

Approximation of Z-Transform

s s s s s( ) ( ) ( )y kT T y kT T u kT

skT

s( )u kT

s s( )u kT T

s skT T

s sArea ( )T u kT Taking the Z-transform of the above equation:

s( ) ( ) ( )zY z Y z TU z

s

( )( )( ) ( )k

F zf tf t kT z F z

ZZ

s( ) ,( ) 1

TY zU z z

while ( ) 1( )Y sU s s

Thus, the Forward Difference Approximation is done by taking

s11

Ts z

s

1zsT

or



Backward Difference ApproximationThe exact integration operation will now be approximated using Backward Difference Approximation.



s s s s s s( ) ( ) ( )y kT T y kT T u kT T

skT

s( )u kT

s s( )u kT T

s skT T

s s sArea ( )T u kT T Taking the Z-transform of the above equation:

s( ) ( ) ( )zY z Y z T zU z

s

( )( )( ) ( )k

F zf tf t kT z F z

ZZ

s( ) ,( ) 1

T zY zU z z

while ( ) 1( )Y sU s s

Thus, the Backward Difference Approximation is done by taking

s11

T zs z

s

1zsT z

or



Trapezoidal Approximation(Tustin Approximation, Bilinear Approximation)The exact integration operation will now be approximated using Backward Difference Approximation.


s s ss s s s

( ) ( )( ) ( )

2u kT u kT T

y kT T y kT T

skT

s( )u kT

s s( )u kT T

s skT T

s s ss

( ) ( )Area

2u kT u kT T

T

Taking the Z-transform, s

1( ) ( ) ( ) ( )2

zY z Y z T U z zU z

s

( )( )( ) ( )k

F zf tf t kT z F z

ZZ

s( ) 1,( ) 2 1

TY z zU z z

while ( ) 1

( )Y sU s s

Thus, the Trapezoidal Approximation is done by taking

s1 12 1T z

s z

s

2 11

zsT z

or




ExampleFind 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.

5( )2 1

G ss

(a) ZOH

(b) FDA

11

11

( )1b zG za z

s( )1 1 Tb K e

s( )1

Ta e (0.5 2)e 0.779

(0.5 2)5 1 e 1.106

1

1

1.1061 0.779

zz

s

1zsT

s

5( )12 1

G zzT

512 1

0.5z

54 3z

1

1

1.251 0.75

zz



Example(c) BDA

(d) TA

s

1zsT z

s

5( )12 1

G zzT z

512 1

0.5zz

55 4zz

1

11 0.8z

s

2 11

zsT z

s

5( )2 12 1

1

G zz

T z

52 12 1

0.5 1zz

5 59 7zz

1

1

0.556 0.5561 0.778

zz



Example

ZOH

FDA TA

BDA



Example: Discretization of Single-Tank SystemRetrieve the linearized model of the single-tank system. Discretize the model using trapezoidal approximation, with Ts = 10 s.

1i

0

1 2 12a gh h qA h A

y h

i( ) ( ) ( )s H s H s Q s

is

2 1 ( ) ( ) ( )1

z h k h k q kT z

1

0 s

1 2 1 2 1, ,2 1a g zsA h A T z

i2( 1) ( ) ( 1) ( ) ( )z h k Ts z h k q k

i2( 1) ( 1) ( ) ( 1) ( )s sz T z h k T z q k

• Laplace Domain

• Z-Domain



Example: Discretization of Single-Tank System i2( 1) ( 1) ( ) ( 1) ( )s sz T z h k T z q k

i

( 1)( )( ) 2( 1) ( 1)

s

s

T zh kq k z T z

i

( )( ) 2 2

s s

s s

T z Th kq k T z T



Example: Discretization of Single-Tank System




: Linearized model: Discretized linearized model

i 7 liters s10 ss

qT




: Linearized model: Discretized linearized model

i 7 liters s2 ss

qT



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

2

10( )2 10

G ss s

(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.



Homework 8A(a) 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:1. Forward Difference (Attendance List No.1-4)2. Backward Difference (Attendance List No.5-8)

2

2( )5

G ss

(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.

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Discrete-Time Transfer Functions

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