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CONTROL ICONTROL IELEN3016
Discrete Signals & Systemsandand
Z-Transform(Lecture 22)
OverviewOverview
• First Things First!• Signals and Samplingg p g• Z-Transforms• Systems and SamplingSyste s a d Sa p g• Digital Controller Design
(Classical Methods, Numerical Methods,(Classical Methods, Numerical Methods, Analytical Design, Optimum Response Digital Design)
• Tutorial Exercises & Homework
First Things First!First Things First!
• Tut on Thursday 20 September 2012• Lab assessment booking sheets are• Lab assessment booking sheets are
available at reception.
Signals and SamplingSignals and Sampling
• Signal Sampling
)(tf )(* tfT
T2 T3 T4 T5 T6T0 tT2 T3 T4 T5 T6T0 t
Signals and SamplingSignals and Sampling
• Signal Sampling cont’d
)(tf )(* tfT
T2
T3 T4
T5 T6T0 tT2 T5 T6T0 t
Signals and SamplingSignals and Sampling
• Signal Sampling cont’d
)(tf )(* tfT
T2
T3 T4
T5 T6T0 tT2 T5 T6T0 t
Signals and SamplingSignals and Sampling
• Signal Sampling cont’dSampled signal as a convolution sumSampled signal as a convolution sum
0
)()()( nTtnTftf
Taking the Laplace transform yields0n
T
Z T f
0)()( )(
n
snTenTfsF tfL
ff )()()( LZZ-Transform: ze TstftfzF
)()()( LZ
Z-TransformZ Transform
• Some Important Z-Transforms
1 zzaaZ kakkn
11.
2
az
zaaZza
kz
k
n
100
)( azZZZ ndndn2.
3
2)()(
azaZaaaZnaZ n
dadn
dadn
zZZZ nn3. 211 )1(
z
anZanZnZ an
an
nnj zZZZ 4.
jean
eannj
ezzaZaZeZ jj
Z-TransformZ Transform
• Some Important Z-Transforms cont’d
5 sin1i znjnj eeZZ5.
6
1cos2
sin21
2...sin
zz
zjjj eeZnZ
1lili)( nn ZZZ 6.
7
1limlim)(00
n
an
aaZaZnZ
)( zZZZ nn7.
C l i f f ll f h f h
1
)( 11
z
zaZaZnuZ an
an
Conclusion: Many transforms follow from that of the standard exponential function in 1.) above.
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 1
)(*FT)(sF )(Y)(sG)(sFT)(sF )(sY
)(* sYT
For the above system we have
(Signals & Systems)
)()()( kTtgkTfty
T
(Signals & Systems)
and
0
)()()(k
kTtgkTfty
0
)()()(n
nTtnTyty
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 1 cont’d
T ki th L l t f i ld
0 0
)()()()(n k
nTtkTnTgkTfty
Taking the Laplace transform yields
)()()( snTekTnTgkTfsY
0 0
)()(
)()()(
snT
n k
ekTnTgkTf
gf
0 0
)()(k n
ekTnTgkTf
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 1 cont’d
)()()(
0esGkTfsY
k
skT
)()(0
ekTfsGk
skT
)()( sFsG
TLetting yields .zesT )()()( zFzGzY
Systems and SamplingSystems and Sampling
• Block Diagram Algebra cont’d
From the above we obtain the followingFrom the above we obtain the following algebraic rules:
1 )()( GG 1.
2. i.e.
)()( sGsG
)()( tyty LL )()( sYsY
3.
)()()()()()( sGsFsGsFsGsF
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 2
)(*FT)(sF )(Y)(1 sG
)(sFT)(sF )(sY)(* sY
T
)(2 sG
From the above system we haveT
.)()()()( 21 sFsGsGsY
Applying the star-operator yields
)()()()()()()( FGGFGGY
)()()()( 21 GG
.)()()()()()()( 2121 sFsGsGsFsGsGsY
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 2 cont’d
For ease of reference we defineFor ease of reference we define
hence )()(:)( 2121 sGsGsGG
hence )()()( 21 sFsGGsY
TLetting yields .)()()( 21 zFzGGzY zesT
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 3)(* sE)(sR )(sC)(sE
)(sG)(sE
T)(sR )(sC
)(* sCT
)(sE
)(sB T)(sH
)(sB
For the above system we have
(1).)()()()()()( sEsGsCsEsGsC ( ))()()()()()(
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 3 cont’d
FurthermoreFurthermore
)()()()()()(
sCsHsRsBsRsE
.)()()()(
)()()(
sEsHsGsR
sCsHsR
Applying the star-operator yields
.)()()()( sEsGHsRsE )()()()(
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 3 cont’d
Solving for gives)(sESolving for gives
. (2))(1
)()(GH
sRsE
)(sE
Substituting (2) into (1) yields
)(1 sGH
.)()(1
)()( sRsGH
sGsC
)(
Systems and SamplingSystems and Sampling
• Block Diagram Algebra – Example 3 cont’d
Letting yieldszesT Letting yields
)()( sCzCzesT
ze
)()(1
)( sRsGH
sG
.)()(
)(1
zRzG
sGH zesT
)()(1 zGH
Tutorial Exercises & HomeworkTutorial Exercises & Homework
• Tutorial Exercises
– Prove that in general.)()()( 2121 zGzGzGG Prove that in general.
H k
)()()( 2121 zGzGzGG
• Homework
– None
ConclusionConclusion
• Z-transforms• Signals & Samplingg p g• Tutorial Exercises & Homework
• Digital Control System Design (Burns, Chapter 7)( , p )
...
Thank you yfor your interest!
