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Lecture 8System Modeling and Identification
Dr.-Ing. Erwin SitompulPresident University
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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:
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With
Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
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)
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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
1 1 ( )( ) (1 ) G sG z zs
Z L
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Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
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
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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: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
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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:
( )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
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Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
Backward Difference ApproximationThe exact integration operation will now be approximated using Backward Difference Approximation.
This method follows the equation given as:
Approximation of Z-Transform
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
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Chapter 5 Discrete-Time Process Models
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:
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
Approximation of Z-Transform
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Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
Example
ZOH
FDA TA
BDA
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Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
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
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Chapter 5 Discrete-Time Process Models
Example: Discretization of Single-Tank System
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Chapter 5 Discrete-Time Process Models
Example: Discretization of Single-Tank System
: Linearized model: Discretized linearized model
i 7 liters s10 ss
qT
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Chapter 5 Discrete-Time Process Models
Example: Discretization of Single-Tank System
: Linearized model: Discretized linearized model
i 7 liters s2 ss
qT
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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
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.
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Chapter 5 Discrete-Time Process Models
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.