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8/15/2019 MODERN CONTROL SYS-LECTURE II.pdf
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MODERN CONTROL
SYSTEMS ENGINEERING
COURSE : CS421
INSTRUCTOR:
DR. RICHARD H. MGAYA
Date: October 25th, 2013
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Linear Difference Equation and Z-Transform
Linear difference equations and the z-transform
Techniques used in design and analysis of a digital control system.
• Example of systems that use digital computers:
Radar antenna positioning system
Airplane autopilot control
Chemical process control
Machine tool control
Goal : Representing a digital computer as a transfer function
similar to other subsystems.
Note: Control system design using analog, i.e., continuous time
devices, are still used and are valid
Dr. Richard H. Mgaya
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Linear Difference Equation and Z-Transform
Difference Equation
• Sequence of real values associated with temporal index k ,{k = 0, 1, 2, …}
• Digital control concerns in generating a sequence u(k ), i.e., controleffort, given a sequence y(k ), i.e., sampled data measurements
sequence The k th control effort is defined in terms of k th measurement or sample
Assumption : u(k ) is a linear combination of measurements and past control
efforts
Where: ai and bi are independent of k , i.e., time invariant
Task : Selection of ai and bi such that the control signal has the dynamic properties
if the desired controlDr. Richard H. Mgaya
)](,),2(),1(),(,),1(),([)( nk uk uk unk yk yk y f k u
)()1()()()1()( 0101 mk yak yak yank ubk ubk u mmn
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Linear Difference Equation and Z-Transform
Z-Transform of a Sequence
• Long-hand form:
Example: Sequence of finite series
Dr. Richard H. Mgaya
21 )2()1()0()()( z f z f f k f Z z F
0
)()()(k
k z k f k f Z z F
0
** )()()]([)(k
kTsekT f kT f Z t f L s F
0 )()( k k
z kT f z F
Let z = eTs
k z kT f z T f z T f f z F )()2()()0()( 21
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Linear Difference Equation and Z-Transform
Unit Step Sequence
• z-transform
• Geometric series convergence
Dr. Richard H. Mgaya
0 1
0 0)(
k
k kT u
0
)()]([)(k
k z kT f kT u Z z U
12 1
1
1
111
z z
z
z z
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Linear Difference Equation and Z-Transform
Unit Ramp Sequence
• z-transform
Multiply by z both sides
Subtract eqn. i and ii
Dr. Richard H. Mgaya
kT kT f )(
00
)()]([)(k
k
k
k kz T z kT f kT f Z z F
)32(321
z z z T
)4321()( 321 z z z T z zF
)1()()1()()(321
z z z T z F z z F z zF
21
11
1
1 z z
z
11
1)()1(
1
z
Tz
z T z F z
2)1()(
z
Tz z F
But
….ii
….i
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Linear Difference Equation and Z-Transform
Exponential Function Sequence
• z-transform
• Geometric series convergence
Dr. Richard H. Mgaya
0
0 0)(
k e
k kT f
akT
0
1)(][)(k
k aT akT z ee Z z F
aT e z
z
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Linear Difference Equation and Z-Transform
General Exponential function Sequence
•
z-transform
Dr. Richard H. Mgaya
k r k f )(
0
1 )(][)(k
k k z r f r Z z F
r r z
z z F
zfor)(
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Linear Difference Equation and Z-Transform
Discrete Impulse Function Sequence
• z-transform
Delayed impulse function sequence
• Z-transform
Dr. Richard H. Mgaya
0 1]1[)]([)( k k
z Z k Z z F
0 0
0 1)(
k
k k f
01
0)(
nk
nk k f
n Z nk Z z F )]([)(
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Linear Difference Equation and Z-Transform
Solving Linear Difference Equation with Z -Transform
• Delay and advance theorem
Theorem 1:
Theorem 2:
Note: s is associated with differentiation of a differential equation
z is associated with the shifting of the difference equation
Example: Consider a homogenous first-order differential equation
• Initial value x(0) = 1
Dr. Richard H. Mgaya
)()( z F z nk f Z n
0)(8.0)1(
k xk x
)1()0()()( n zf f z z F z nk f Z nn
0)(8.0)0()( z X zx z zX
8.0
)(
z
z z X k k x )8.0()(
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Linear Difference Equation and Z-Transform
Pulse Transfer Function
• Consider the following block diagram
• U *( s) – Sample input to G( s)
• X 0( s) – Continuous output
• X 0*( s) – Sampled output
• The figure is the pulse transfer function where U *( s) = U ( z )and X 0( z ) = X 0
*( s)
Dr. Richard H. Mgaya
)()(
)(0 z G z U
z X
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Linear Difference Equation and Z-Transform
Pulse Transfer Function
• Cascaded blocks
• Note: G1G2( z ) ≠ G1( z )G2( z )
Dr. Richard H. Mgaya
)()()(
)(21
0
z G z G z U
z X
)()()(
21
0
sGG Z z U
z X
)()(
)(21
0 z GG z U
z X
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Inverse Z-Transform
Long Division
Example: Find inverse sequence of the function:
• Multiply by z -2 in the numerator and denominator
Dr. Richard H. Mgaya
1)0( f
43)(
2
2
z z
z z z F
21
1
431
1)(
z z
z z F
21
121
841
1431
z z
z z z 21
431
z z 21 44 z z 321 16124 z z z 32 168
z z 432 32248 z z z 43
328 z z
Division Sequence
4)1( f 8)2( f
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Inverse Z-Transform
Partial Fraction Expansion
• Consider the following function:
• The limit of β implies the poles of the quadratic term arecomplex
• Partial fraction expansion:
• The 1st and 2nd terms are the z-transform of
Rk cosΩT and Rk sinΩT respectively, Q( z ) is the reminder
Dr. Richard H. Mgaya
Tcos
)2)((
)()(
2
22
T e R
R z R z z P
z N z F
)(2
sin
2
)(
)( 2222
2
z Q R z R z
T BzR
R z R z
zR z A
z F
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Inverse Z-Transform
• Correct form of the partial fraction expansion:
Dr. Richard H. Mgaya
787.4376.0
8.1
64.013.1
18.0)(
8.0
2
z z z
z
z
z z F C
8.064.013.1
)7079.0)(8.0(
64.013.1
))7063.0)(8.0(()(
22
2
z
Cz
z z
Bz
z z
z z A z F
8.064.013.1
)5663.0(
64.013.1
)565.0(22
2
z
Cz
z z
z B
z z
z z A
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Pole location on the z -Plane
Relationship Between Pole location and the nature of the
temporal sequence
• Consider the following function
• Partial fraction expansion:
• In polar notation the complex poles are written as follows:
Dr. Richard H. Mgaya
rootscomplexhasrdenominato
)()(
2 cbz z
z N z F
conjugatecomplexdenotes-*
)(*
*
p z
A
p z
Az z F
j j e pe p *
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Pole location on the z -Plane
Relationship Between Pole location and the nature of the
temporal sequence
• Pole location: z -plane
• Nature of the sequence
D Ri h d H M