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Multivariable Control Multivariable Control SystemsSystems
Ali KarimpourAssistant ProfessorAssistant Professor
Ferdowsi University of Mashhad
Chapter 2Chapter 2Introduction to Multivariable ControlIntroduction to Multivariable Control
Topics to be covered include:
Multivariable Connections
Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
M t i F ti D i ti (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
2
Bandwidth and Crossover Frequency
Chapter 2
Introduction to Multivariable Control
Topics to be covered include: Multivariable Connections
M lti i bl P l d Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
Matrix Fraction Description (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
3
Bandwidth and Crossover Frequency
Chapter 2
Multivariable Connections
• Cascade (series) interconnection of transfer matrices( )
)(su )(sy )()()()()()( 12 susGsusGsGsy ==)(1 sG )(2 sG
)()()( 12 sGsGsG = )()( 21 sGsG≠
• Parallel interconnection of transfer matricesGenerally
)(1 sG)(su+
)(sy )()()())()(()( 21 susGsusGsGsy =+=
)(2 sG
++
)()()())()(()( 21y
)()()( 21 sGsGsG +=
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21
Chapter 2
Multivariable Connections
• Feedback interconnection of transfer matrices
)(1 sG )(2 sG)(sy)(su
+
− ( ))()()()()( 12 sysusGsGsy −= ( ))()()()()( 12 yy
)()()()()())()(()( 121
12 susGsusGsGsGsGIsy =+= −
)()())()(()( 121
12 sGsGsGsGIsG −+=
A useful relation in multivariable is push-through rule
12122
112 ))()()(()())()(( −− +=+ sGsGIsGsGsGsGI
Ali Karimpour Feb 2012
5Exercise 1: Proof the push-through rule
Chapter 2
Multivariable Connections
MIMO rule: To derive the output of a system, start from the output and write down the blocks as you meet them whenmoving backward (against the signal flow) towards the inputmoving backward (against the signal flow) towards the input.If you exit from a feedback loop then include a term ( ) 1−− LI
or according to the feedback sign where L is the transfer function around that loop (evaluated against the
( ) 1−+ LI
transfer function around that loop (evaluated against the signal flow starting at the point of exit from the loop). Parallel branches should be treated independently and their contributions added together.
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6
g
Chapter 2
Multivariable Connections
Example 2-1: Derive the transfer function of the system shown in figurp y g
( )ω211
221211 )( PKPIKPPz −−+=
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7
Chapter 2
Introduction to Multivariable Control
Topics to be covered include:
Multivariable Connections
l i i bl l d Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
Matri Fraction Description (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
8
Bandwidth and Crossover Frequency
Chapter 2
Multivariable Poles ( State Space Description )
Definition 2 1: The poles of a system with state space descriptionipDefinition 2-1: The poles of a system with state-space description are eigenvalues of the matrix A.
i i i i i fi
niAi ,...,2,1,)( =λ),,,( DCBA
ip
The pole polynomial or characteristic polynomial is defined as
)det()( AsIs −=φ
Thus the system’s poles are the roots of the characteristic polynomial
0)det()( =−= AsIsφ
Note that if A does not correspond to a minimal realization then the poles by this definition will include the poles (eigenvalues)
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p y p ( g )corresponding to uncontrollable and/or unobservable states.
Chapter 2
Multivariable Poles ( Transfer Function Description )
Theorem 2-1The pole polynomial corresponding to a minimal realization of a system with transfer function G(s) is the least common denominator
)(sφ
of all non-identically-zero minors of all orders of G(s).A minor of a matrix is the determinant of the square matrix obtained byq ydeleting certain rows and/or columns of the matrix.
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10
Chapter 2
Multivariable Poles ( Transfer Function Description )
Example 2-4
⎥⎦
⎤⎢⎣
⎡
+−+−++−−+−
−++=
)1)(1()1)(1()2)(1()1(0)2)(1(
)1)(2)(1(1)(
2
sssssssss
ssssG
⎦⎣ ))(())(())(())()((
The non-identically-zero minors of order 1 are11111
21,
21,
11,
)2)(1(1,
11
++−−
++−
+ ssssss
s
Th id ti ll i f d 2The non-identically-zero minor of order 2 are
2)2)(1()1(,
)2)(1(1,
)2)(1(2
++−−
++++ sss
ssss )2)(1()2)(1()2)(1( ++++++ ssssss
By considering all minors we find their least common denominator
Ali Karimpour Feb 2012
11)1()2)(1()( 2 −++= ssssϕ
Chapter 2
Multivariable Poles
Exercise 2: Consider the state space realization BuAxx +=&
DuCxy +=
⎤⎡⎤⎡⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
−−
0001021010321
002.24.0004.08.2
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
⎥⎥⎥
⎦⎢⎢⎢
⎣
=
⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−−
=000000
00111021
100111010
0.108.06.700.36.18.0002.24.0
DCBA
a- Find the poles of the system directly through state space form.
b- Find the transfer function of system ( note that the numerator and the denominator of each element must be prime ).
c- Find the poles of the system through its transfer function.
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12
Chapter 2
Multivariable Zeros
Importance of multivariable zeros
• Dynamic response. • Stability of inverse system.
• Closed loop poles are on the open loop zeros at high gain.
• Blocking the inputs. • Not affected by feedback.
p p p p g g
Different definition of multivariable zeros
• Element zeros. • Decoupling zeros (input and output).
• Transmission zero or blocking zero. • System zeros.
Ali Karimpour Feb 2012
13• Invariant zeros.
Chapter 2
Multivariable Zeros
Element zeros
Decoupling zeros (input and output)
They are not very important in MIMO design.
Transmission ero or blocking eroThey are clearly subset of poles.
Transmission zero or blocking zero
They are the zeros that block the output in special condition.
System zero
zerosi.o.d.zeroso.d.zerosi.d.zerosonTransmissizerosSystem −++=
Invariant zero
zerosi.o.d.zeroso.d.zerosi.d.zeroson TransmissizerosSystem ++
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14Invariant zeros=System zeros (In the case of square plant)
Chapter 2
Multivariable Zeros ( State Space Description )
⎥⎤
⎢⎡ −
⎥⎤
⎢⎡
⎥⎤
⎢⎡ BAsI
Px
P )(0
)(BuAxx +=& Laplace transform⎥⎦
⎢⎣ −
=⎥⎦
⎢⎣−
=⎥⎦
⎢⎣− DC
sPyu
sP )(,)(DuCxyBuAxx
+=+
Rosenbrock system matrix
The system zeros are then the values of s=z for which the polynomial system matrix P(s) (if P(s) is square) loses rank resulting in zero outputf ifor some nonzero input.
⎥⎤
⎢⎡
=⎥⎤
⎢⎡ −
=0I
IBA
MLet
⎥⎦
⎢⎣
=⎥⎦
⎢⎣ −
=00
, IDC
M g
Then the zeros are found as non trivial solutions of
0)( =⎥⎦
⎤⎢⎣
⎡− z
g ux
MzI
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15
⎦⎣ zu
This is solved as a generalized eigenvalue problem.
Chapter 2
Multivariable Zeros ( State Space Description )
⎥⎤
⎢⎡ −
=BAsI
sP )(System zero⎥⎦
⎢⎣ − DC
sP )(
Rosenbrock system matrix
I t d li
) P(s) squarefor (just rank. losses )( that of valueThe sPs
Input decoupling zeros
[ ] rank. losses that of valueThe A BsIs −
Output decoupling zeros⎤⎡ − AsI
rank. losses that of valueThe ⎥⎦
⎤⎢⎣
⎡−C
AsIs
In ariant erosInvariant zeros
( ) ( ))()(min)(result that s of values thoseare zerosInvariant
CnBnsP ρρρ ++<
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16
( ) ( ))(),(min)( CnBnsP ρρρ ++<
)( of minorsgreatest all of gcdOr sP
Chapter 2
Multivariable Zeros
Element zeros
Decoupling zeros (input and output)10001⎥⎤
⎢⎡
⎥⎤
⎢⎡− 655)4)(3)(2()( ++++ ssss
uxx
001
400003000020
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−
−=&
15
)4)(3)(2)(1())()(()(
+=+
++++=
ssssssg
s= -2 is output decoupling zero (o.d.z)
[ ] uxy 5010104000
+=⎦⎣⎦⎣ − s= -3 is input decoupling zero (i.d.z)
s= -4 is input-output decoupling zero (i.o.d.z)Transmission zero or blocking zeroTransmission zero or blocking zero
s= -6/5 is transmission or blocking zeroSystem zeros and invariant zeros
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System zeros and invariant zeross= -6/5, s=-2, s=-3 and s=-4 are system zero(invariant zeros)
Chapter 2
Multivariable Zeros
Let00005 ⎤⎡⎤⎡− ⎥
⎤⎢⎡ + 00005s
[ ] [ ]
uxx
1210112
1000
300020005
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=&
⎥⎦⎤
⎢⎣⎡
+−−
++
=34
382)(
ss
sssG
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
−++
=
1210112300
10020)(
ss
sP
[ ] [ ]uxy 12101 −+=
Element zeross= -4, -4
⎦⎣ −12101
Decoupling zeros (input and output)s= -2 is output decoupling zero (o.d.z)s= -5 is input decoupling zero (i.d.z)
Transmission zero or blocking zero
s= -4 is transmission or blocking zeroInvariant zeros
s= -4 s=-5 are invariant zeros
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s 4, s 5 are invariant zeros.System zeros
s= -4, s=-5, s=-2 are system zeros.
Chapter 2
Multivariable Zeros
Generally we have:
zeros i.o.d.zeros o.d.zeros i.d.zeroson Transmissizeros System −++=
S tI i tT i i zerosSystemzerosInvariant zeroson Transmissi ⊂⊂
System zerosInvariant zeros Decoupling zeros
Transmission zero
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Chapter 2
Multivariable Zeros
For square systems with m=p inputs and outputs and n states limitsFor square systems with m p inputs and outputs and n states, limits on the number of transmission zeros are:
zerosCBrankmnmostAtDzerosDrankmnmostAtD
+−=+−≠
)(2:0)(:0
Ali Karimpour Feb 2012
20zerosmnExactlymCBrankandD −== :)(0)(
Chapter 2
Multivariable Zeros ( State Space Description )
Example 2-5: Consider the state space realization BuAxx +=&
DuCxy +=
[ ] 001400
100010
==⎥⎥⎤
⎢⎢⎡
=⎥⎥⎤
⎢⎢⎡
= DCBA [ ] 001410
560100 ==
⎥⎥⎦⎢
⎢⎣
=⎥⎥⎦⎢
⎢⎣ −−
= DCBA
⎤⎡⎤⎡ 00010010
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=⎥⎦
⎤⎢⎣
⎡=
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−−−=⎥
⎦
⎤⎢⎣
⎡−−
=010000100001
000
156001000010
II
DCBA
M g
),( gIMeig
4
zerosDrankmnmostAtD +−≠ )(:0
⎥⎦
⎢⎣
⎥⎦
⎢⎣ 00000014 4−=z
zerosmnExactlymCBrankandDzerosCBrankmnmostAtD−==
+−=:)(0
)(2:0
Ali Karimpour Feb 2012
2110123 =−⋅−≤zerosofNumber
Chapter 2
Multivariable Zeros ( Transfer Function Description )
Definition 2-2
zi is a zero (transmission zero) of G(s) if the rank of G(zi) is less than
the normal rank of G(s). The zero polynomial is defined as
)()( n zssz z −Π= )()( 1 ii zssz −Π= =
Where nz is the number of finite zeros (transmission zero) of G(s).z ( ) ( )
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Chapter 2
Multivariable Zeros ( Transfer Function Description )
Theorem 2-2
The zero polynomial z(s) corresponding to a minimal realization of the
Theorem 2-2
The zero polynomial z(s) corresponding to a minimal realization of the
system is the greatest common divisor of all the numerators of allsystem is the greatest common divisor of all the numerators of all
order-r minors of G(s) where r is the normal rank of provided that
these minors have been adjusted in such a way as to have the pole
polynomial as their denominators.)(sφ
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Chapter 2
Multivariable Zeros ( Transfer Function Description )
Example 2-9
⎥⎦
⎤⎢⎣
⎡
+−+−++−−+−
−++=
)1)(1()1)(1()2)(1()1(0)2)(1(
)1)(2)(1(1)(
2
sssssssss
ssssG
⎦⎣ ))(())(())(())()((
according to example 2-4 the pole polynomial is:
)1()2)(1()( 2 −++= ssssϕ
)(φThe minors of order 2 with as their denominators are)(sφ
)1()2)(1()2)(1(2 2−−+−+− sssss)1()2)(1(
,)1()2)(1(
,)1()2)(1( 222 −++−++−++ sssssssss
The greatest common divisor of all the numerators of all order-2 minors is
Ali Karimpour Feb 2012
241)( −= ssz
Chapter 2
Multivariable Poles
Exercise 4: Consider the state space realization BuAxx +=&
DuCxy +=
⎤⎡⎤⎡⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
−−
0001021010321
002.24.0004.08.2
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
⎥⎥⎥
⎦⎢⎢⎢
⎣
=
⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−−
=000000
00111021
100111010
0.108.06.700.36.18.0002.24.0
DCBA
a- Find the transmission zeros of the system directly through state space form.
b- Find the transfer function of system ( note that the numerator and the denominator of each element must be prime ).
c- Find the transmission zeros of the system through its transfer function.
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25
Chapter 2
Directions of Poles and Zeros
Zero directions:
Let G(s) have a zero at s = z, Then G(s) losses rank at s = z and
there will exist nonzero vectors and such that zu zy
0)(0)( == zGyuzG Hzz
is input zero direction and is output zero directionzu zy
We usually normalize the direction vectors to have unit length
1 1
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26
12=zu 1
2=zy
Chapter 2
Directions of Poles and Zeros
Pole directions:
Let G(s) have a pole at s = p. Then G(p) is infinite and we may
somewhat crudely write
)()( GG H ∞=∞= )()( pGyupG Hpp
is input pole direction and is output pole directionpu py
HH Hp
Hppp pqAqptAt ==
Ali Karimpour Feb 2012
27pH
ppp qBuCty ==
Chapter 2
Directions of Poles and Zeros
Example:⎥⎤
⎢⎡ − 411)(s
G ⎥⎦
⎢⎣ −+
=)1(25.42
)(ss
sG
It has a zero at z = 4 and a pole at p = -2 . p pH
GsG ⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −=⎥
⎦
⎤⎢⎣
⎡==
653.0757.0757.0653.0
271.000475.1
588.0809.0809.0588.0
45.442
51)3()( 0
⎦⎣⎦⎣⎦⎣⎦⎣ 653.0757.0271.00588.0809.045.45H
GzG ⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −=⎥
⎦
⎤⎢⎣
⎡==
60808.06.0
00050.1
55083083.055.0
65443
61)4()( ⎥
⎦⎢⎣⎥⎦
⎢⎣⎥⎦
⎢⎣
⎥⎦
⎢⎣ 6.08.00055.083.065.46
)()(
zuzyNo let the inp t as: Input zero directionOutput zero directionNow let the input as:
)()()( susGsy =tetu 4
6.08.0
)( ⎥⎦
⎤⎢⎣
⎡−= ⇒
41
608.0
)1(25441
21
−⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡−
−+
=ss
ss ⎥
⎦
⎤⎢⎣
⎡−+
=218.0
21
s
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28
6.0 ⎦⎣ 46.0)1(25.42 ⎦⎣⎦⎣+ sss ⎦⎣+ 2.12s
⎥⎦
⎤⎢⎣
⎡−=
−
−
t
t
ee
ty 2
2
2.18.0
)( which does not contain any component of the input signal e4t
Chapter 2
Directions of Poles and Zeros
Example:⎤⎡⎥⎦
⎤⎢⎣
⎡−
−+
=)1(25.4
412
1)(s
ss
sG
⎤⎡It has a zero at z = 4 and a pole at p = -2 . ⎥⎦
⎤⎢⎣
⎡∞∞∞∞
=−= )2()( GpG
??!!
⎥⎦
⎤⎢⎣
⎡ +−=+−=+
)3(254431)2()(
εεε GpG
??!!
001.0 examplefor Let =ε⎥⎦
⎢⎣ +− )3(25.4 εε
H⎤⎡⎤⎡⎤⎡− 8.06.00901083.055.0
p
uy
G ⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=+−
6.08.08.06.0
00.0009010
55.083.083.055.0
)001.02(
Ali Karimpour Feb 2012
29
pu
Input pole direction
py
Output pole direction
Chapter 2
Minimality
A state space system (A,B,C,D) is minimal if it is a system with the least number of states giving its transfer function.
If there is another system (A1,B1,C1,D1) with fewer states having the y ( 1 1 1 1) gsame transfer function, the given system is not minimal.
For SISO systems, a system is minimal iff the transfer functionFor SISO systems, a system is minimal iff the transfer function numerator polynomial and denominator polynomial have no common roots.
For MV systems one must use the definition of MV zeros.
Theorem. A system (A,B,C,D) is minimal iff it has no input decoupling zeros and no output decoupling zeros.
Ali Karimpour Feb 2012
30
p g p p g
Chapter 2
Introduction to Multivariable Control
Topics to be covered include:
Multivariable Connections
Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
M t i F ti D i ti (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
31
Bandwidth and Crossover Frequency
Chapter 2
Transmission zero assignment
Zeros position depends on the location of sensors and actuators.
Theorem: Transmission zeros cannot be assigned by state feedback.
Proof ⎤⎡ − BAsIProof ⎥⎦
⎤⎢⎣
⎡−
=DC
sP )(by definedisSystem
⎤⎡ + BBKAsI⎥⎦
⎤⎢⎣
⎡+−+−
=DDKCBBKAsI
sPf )( :isfeedback with System
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−−
=⎥⎦
⎤⎢⎣
⎡+−+−
=IK
IsP
IKI
DCBAsI
DDKCBBKAsI
sPf
0)(
0)(
( ) ( ))()( sPsPf ρρ =
Ali Karimpour Feb 2012
32Theorem: Transmission zeros cannot be assigned by output feedback.
Chapter 2
Transmission zero assignment
What about following structures?
⎩⎨⎧
=−+=
CxyurBAxx
sG)(
)(&
⎩⎨⎧
+=+=
DyHwuGyFww
sK&
)(⎩ Cxy
Theorem: Transmission zeros cannot be assigned by dynamic feedback.
⎩ ywu
⎥⎤
⎢⎡ −− 0GCFsI
⎥⎤
⎢⎡
+−− 0
)( BBDCAIBHGCFsI
P
⎥⎥⎥
⎦⎢⎢⎢
⎣ −+−=
00)(
CBBDCAsIBHsPf
)()( sPFsIsP −=⎥⎥⎥
⎦⎢⎢⎢
⎣ −+−=
00)(
CBBDCAsIBHsPf
)()( sPFsIsPf =
⎥⎥⎤
⎢⎢⎡
−⎥⎥⎤
⎢⎢⎡ −
= 0000BAsI
IBDIBHGFsI
Ali Karimpour Feb 2012
33⎥⎥⎦⎢
⎢⎣ −⎥⎥⎦⎢
⎢⎣ 0000 CI
Chapter 2
Transmission zero assignment
What about following structures?
⎩⎨⎧
=+=
CxyBuAxx
sG&
)(⎩⎨⎧
+=+=
uDHwuuGFww
sKˆˆ
)(&
⎩ Cxy
Theorem: Transmission zeros cannot be assigned by dynamic feedback.
⎩
⎥⎥⎤
⎢⎢⎡
−−−
=0
)( BDAsIBHGFsI
sP ⎥⎥⎤
⎢⎢⎡
−−−
= BDAsIBHGFsI
sP0
)(⎥⎥⎦⎢
⎢⎣ −
=00
)(C
BDAsIBHsPs
)()()( sPsPsP ′=⎤⎡⎤⎡
⎥⎥⎦⎢
⎢⎣ −
GFII
CBDAsIBHsPs
000
00)(
)()()( sPsPsPs =
⎥⎥⎥⎤
⎢⎢⎢⎡ −
⎥⎥⎥⎤
⎢⎢⎢⎡
−= IGFsI
BAsII
000
000
Ali Karimpour Feb 2012
34⎥⎦⎢⎣ −⎥⎦⎢⎣ − DHC 000
Chapter 2
Transmission zero assignment
Ali Karimpour Feb 2012
35
Chapter 2
Introduction to Multivariable Control
Topics to be covered include:
Multivariable Connections
Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
M t i F ti D i ti (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
36
Bandwidth and Crossover Frequency
Chapter 2
Smith Form of a Polynomial Matrix
)(sΠSuppose that is a polynomial matrix.)(pp p y
Smith form of is denoted by , and it is a pseudo diagonali th f ll i f
)(sΠ )(ssΠ
in the following form
⎥⎤
⎢⎡Π
=Π0)(
)(s
s ds⎥⎦
⎢⎣
=Π00
)(ss
)(,,........)(,)()( Where 21 sssdiags rds εεε=Π
)(siε )(1 si+εis a factor of and1
)(−
=i
ii s
χχ
ε
Ali Karimpour Feb 2012
37
Chapter 2
Smith Form of a Polynomial Matrix
)(,,........)(,)()( Where 21 sssdiags rds εεε=Π
)(sε )(sεis a factor of and )( = isχ
ε)(siε )(1 si+εis a factor of and1
)(−
=i
i sχ
ε
10 =χ
=1χ gcd all monic minors of degree 1
=χ gcd all monic minors of degree 2=2χ gcd all monic minors of degree 2
………………………………………………………..
=rχ gcd all monic minors of degree r
………………………………………………………..
Ali Karimpour Feb 2012
38
rχ g g
Chapter 2
Smith Form of a Polynomial Matrix
The three elementary operations for a polynomial matrix are usedThe three elementary operations for a polynomial matrix are used to find Smith form.
• Multiplying a row or column by a constant;
• Interchanging two rows or two columns; andInterchanging two rows or two columns; and
• Adding a polynomial multiple of a row or column to another lrow or column.
)().....()()()()().......()( 121122 sRsRsRssLsLsLs nns Π=Π
)()()()( sRssLss Π=Π
Ali Karimpour Feb 2012
39
Chapter 2
Smith Form of a Polynomial Matrix
Example 2-11⎤⎡ +− )2(4 s
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−+
+=Π
21)2(2
)2(4)(
s
ss
⎦⎣
10 =χ 11,2,2,1gcd1 =++= ssχ
)3)(1(34gcd 22 ++=++= ssssχ
χ χ1)(
0
11 ==
χχ
ε s )3)(1()(1
22 ++== sss
χχ
ε
⎥⎦
⎤⎢⎣
⎡++
=Π)3)(1(0
01)(
ssss
Ali Karimpour Feb 2012
40
⎦⎣Exercise 5: Derive R(s) and L(s) that convert to)(sΠ )(ssΠ
Chapter 2
Smith-McMillan Forms
Theorem 2-3 (Smith-McMillan form)
Let be an m × p matrix transfer function, where
are rational scalar transfer functions G(s) can be represented by:
)]([)( sgsG ij= )(sgij
)()(
1)( sd
sG Π=
are rational scalar transfer functions, G(s) can be represented by:
)(sd
Where Π(s) is an m× p polynomial matrix of rank r and d(s) is the least
common multiple of the denominators of all elements of G(s) .
)(~GTh i S ith M Mill f f G( ) d b d i d di tl b)(sGThen, is Smith McMillan form of G(s) and can be derived directly by
⎥⎤
⎢⎡
⎥⎤
⎢⎡Π
Π0)(0)(1)(1)(~ sMs
ssG ds
Ali Karimpour Feb 2012
41⎥⎦
⎢⎣
=⎥⎦
⎢⎣
=Π=0000)(
)()(
)(sd
ssd
sG s
Chapter 2
Smith-McMillan Forms
Theorem 2-3 (Smith-McMillan form)
)()(
1)( ssd
sG Π= ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡Π=Π=
000)(
000)(
)(1)(
)(1)(~ sMs
sds
sdsG ds
s)( ⎦⎣⎦⎣)()(
⎭⎬⎫
⎩⎨⎧
=)()(,,........
)()(,
)()()( 21 sssdiagsM r
δε
δε
δε
⎭⎬
⎩⎨ )(
,,)(
,)(
)(21 sss
grδδδ
~ )()()()(~ sRsGsLsG = )(~)(~)(~)( sRsGsLsG =
11 )()(~)()(~ RRLL
and unimodular are )( and )(~,)(,)(~ matrices The sRsRsLsL
Ali Karimpour Feb 2012
42
11 )()(,)()( −− == sRsRsLsL
Chapter 2
Smith-McMillan Forms
Example 2-13⎤⎡ 14
⎥⎥⎥⎤
⎢⎢⎢⎡
−+−
++= 12)1(
1)2)(1(
4
)( ssssG
⎥⎥⎦⎢
⎢⎣ +++ )2)(1(2)1( sss
))(()()2(4
)()(1)(⎤⎡ +−
ds
)2)(1()(,5.0)2(2
)()(,)(
)(1)( ++=⎥
⎦
⎤⎢⎣
⎡−+
=ΠΠ= sssds
sssd
sG
⎤⎡ 01⎥⎦
⎤⎢⎣
⎡++
=Π)3)(1(0
01)(
ssss
⎥⎥⎤
⎢⎢⎡
++=Π=
0)2)(1(
1
)(1)(~ ssssG
Ali Karimpour Feb 2012
43⎥⎥⎥
⎦⎢⎢⎢
⎣ ++
Π
230
)()(
)(
ss
ssd
sG s
Chapter 2
Smith-McMillan Forms
Example 2-14⎥⎤
⎢⎡ −
Π 82411
1)(1)( 22G⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−−−+
++=Π=
824824
)2)(1()(
)()(
22
22
ssssss
sss
sdsG
⎤⎡ ⎤⎡ ⎤⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−−+
−=Π
824824
11)(
22
22
sssssss
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=Π
)4(30)4(30
11)(
2
21
sss
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=Π
00)4(30
11)( 2
2 ss
⎦⎣ ⎦⎣ ⎦⎣
⎥⎤
⎢⎡
Π )4(3001
)( 2 ⎥⎤
⎢⎡
ΠΠ )4(001
)()( 2
⎥⎥⎥
⎦⎢⎢⎢
⎣
−=Π00
)4(30)( 23 ss
⎥⎥⎥
⎦⎢⎢⎢
⎣
−=Π=Π00
)4(0)()( 24 sss s )()()()()()( 4312 sRsRssLsLss Π=Π
⎤⎡ 01
)(~)(~)(~)(~)()(~)(
1)()(
1)( sRsGsLsRssLsd
ssd
sG s =Π=Π=⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
+−
++
=120
0)2)(1(
)(~ss
ss
sG
Ali Karimpour Feb 2012
44
)()( sdsd
⎥⎥
⎦⎢⎢
⎣ 00
Chapter 2
Introduction to Multivariable Control
Topics to be covered include: Multivariable Connections
M lti i bl P l d Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
Matrix Fraction Description (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
45
Bandwidth and Crossover Frequency
Chapter 2
Matrix Fraction Description (MFD)
function transferSISOan is 1)(Let 2
+=
ssg65
)( 2 ++ ssg
( )( ) 12 651)(can writeWe −+++= ssssg ( )( )651)( can write We +++= ssssg
polynomial polynomialp y p y
This is a Right Matrix Fraction Description (RMFD)
( ) ( )165)( writealsocan We 12 +++=− ssssg
polynomial polynomial
Ali Karimpour Feb 2012
46This is a Left Matrix Fraction Description (LMFD)
Chapter 2
Matrix Fraction Description (MFD)
A model structure that is related to the Smith-McMillan form isA model structure that is related to the Smith-McMillan form is matrix fraction description (MFD).
• Right matrix fraction description (RMFD)
• Left matrix fraction description (LMFD)Left matrix fraction description (LMFD)
Let is a matrix and its the Smith McMillan is )(sG mm× )(~ sG)( )(
( )0,...,0,)(,...,)()( 1 ssdiagsN rεε∆
= ( )1,...,1,)(,...,)()( 1 ssdiagsD rδδ∆
=Let define:
or)()()(~ 1−= sDsNsG )()()(~ 1 sNsDsG −=
Ali Karimpour Feb 2012
47
Chapter 2
Matrix Fraction Description (MFD)
)()()(~ 1−DNG )()()(~ 1 NDG −or)()()( 1= sDsNsG )()()( 1 sNsDsG =
We know thatWe know that
)(~)(~)(~)( sRsGsLsG = )(~)()()(~ 1 sRsDsNsL −= 1)()( −= sGsG DN( ) 1)()()()(~ −= sDsRsNsL
This is known as a right matrix fraction description (RMFD) whereg p ( )
~ )()()(,)()(~)( sDsRsGsNsLsG DN ==
Ali Karimpour Feb 2012
48
Chapter 2
Matrix Fraction Description (MFD)
)()()(~ 1−DNG )()()(~ 1 NDG −or)()()( 1= sDsNsG )()()( 1 sNsDsG =
We know thatWe know that
)(~)(~)(~)( sRsGsLsG = )(~)()()(~ 1 sRsNsDsL −= )()( 1 sGsG ND−=( ) )(~)()()( 1 sRsNsLsD −=
This is known as a left matrix fraction description (LMFD) wherep ( )
)(~)()(,)()()( sRsNsGsLsDsG ND ==
Ali Karimpour Feb 2012
49
Chapter 2
Matrix Fraction Description (MFD)
We can show that the MFD is not unique, because, for any nonsingularq y g
mm× )(sΩmatrix we can write G(s) as:
( ) ( )( ) 111( ) ( )( ) 111 )()()()()()()()()( −−− ΩΩ=ΩΩ= ssGssGsGsssGsG DNDN
Where is said to be a right common factor. When the only right )(sΩ g y gcommon factors of and is unimodular matrix, then, weSay that the RMFD is irreducible
)(
)(sGN )(sGD
( ))()( sGsGSay that the RMFD is irreducible. ( ))(),( sGsG DN
It is easy to see that when a RMFD is irreducible, then
zssGsGzs N ==∗ at rank losses )( ifonly and if )( of zero a is
zssGsGps atsingularis)(ifonlyandif)(ofpoleais ==∗
Ali Karimpour Feb 2012
50( )(s)Gφ(s)zssGsGps
D
D
det is G(s) of polynomial pole that themeans This at singular is )( ifonly and if )(ofpolea is
===∗
Chapter 2
Matrix Fraction Description (MFD)
Example 2-16 ⎥⎤
⎢⎡ −
)2)(1(1
)2)(1(1
⎥⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢⎢
++−−
++−+
++++
=)2)(1(
82)2)(1(
4)2)(1()2)(1(
)(22
ssss
ssss
ssss
sG
⎥⎥
⎦⎢⎢
⎣ +−
+−
)1(42
)1(2
ss
ss
⎥⎤
⎢⎡ 01
=⎥⎦
⎤⎢⎣
⎡ −
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
+−
++
⎥⎥⎥⎤
⎢⎢⎢⎡
−+==3011
120
0)2)(1(
114014001
)(~)(~)(~)(2
2
ss
ss
sssRsGsLsG⎦⎣
⎥⎥⎥
⎦⎢⎢⎢
⎣
⎥⎦⎢⎣ −
00
1142s
11
2
21
2
2
10
31)1)(2(
2424
01
3011
100)1)(2(
0020
01
114014001
−
−
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+
+++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−+=⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡+
++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+s
sss
sssss
sss
ss
ss
Ali Karimpour Feb 2012
513
02400114 ⎥⎦⎢⎣⎥⎦⎢⎣ −−⎥⎦⎢⎣⎥⎦⎢⎣ − sss
)(sGN )(sGDRMFD:
Chapter 2
Matrix Fraction Description (MFD)
Example 2-16 ⎥⎤
⎢⎡ −
)2)(1(1
)2)(1(1
⎥⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢⎢
++−−
++−+
++++
=)2)(1(
82)2)(1(
4)2)(1()2)(1(
)(22
ssss
ssss
ssss
sG
⎥⎥
⎦⎢⎢
⎣ +−
+−
)1(42
)1(2
ss
ss
⎥⎤
⎢⎡ 01
=⎥⎦
⎤⎢⎣
⎡ −
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
+−
++
⎥⎥⎥⎤
⎢⎢⎢⎡
−+==3011
120
0)2)(1(
114014001
)(~)(~)(~)(2
2
ss
ss
sssRsGsLsG⎦⎣
⎥⎥⎥
⎦⎢⎢⎢
⎣
⎥⎦⎢⎣ −
00
1142s
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−−+
++
=⎥⎦
⎤⎢⎣
⎡ −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−+
−−
00)2(30
11
1101)4)(1(00)1)(2(
3011
0020
01
10001000)1)(2(
114014001 1
2
1
2
2 ss
ssssss
ssss
sss
Ali Karimpour Feb 2012
52)(sGN)( sG DLMFD:
⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣ 114s
Chapter 2
Matrix Fraction Description (MFD)
Example 2-17 Consider⎥⎤
⎢⎡ +
⎥⎤
⎢⎡ +−
−
− )()()(212
1 sssssssGsGsG ⎥
⎦⎢⎣ ++⎥
⎦⎢⎣ +−
==2222
)()()(ssss
sGsGsG ND
? eirreducibl )( and )( Are sGsG ND
⎤⎡⎤⎡⎥⎦
⎤⎢⎣
⎡
+++
=⎥⎦
⎤⎢⎣
⎡+
+⎥⎦
⎤⎢⎣
⎡=
221111
110
)(2
sssss
sss
sGN⎥⎦
⎤⎢⎣
⎡
+−+−
=⎥⎦
⎤⎢⎣
⎡+−
−⎥⎦
⎤⎢⎣
⎡=
221111
110
)(2
sssss
sss
sGD
Now let:
⎥⎦
⎤⎢⎣
⎡+−
−=
1111
)(~s
ssGD ⎥
⎦
⎤⎢⎣
⎡+
+=
1111
)(~s
ssGN
Now let:
? eirreducibl )(~ and )(~ Are sGsG ND
⎥⎤
⎢⎡ −
=⎥⎤
⎢⎡ −⎥⎤
⎢⎡
=11111
)(~ sssGD ⎥
⎤⎢⎡ +
=⎥⎤
⎢⎡ −⎥⎤
⎢⎡
=11111
)(~ sssGN⎥
⎦⎢⎣ +−⎥
⎦⎢⎣ +−⎥⎦
⎢⎣ 111110
)(ss
sGD ⎥⎦
⎢⎣ +⎥
⎦⎢⎣ +⎥⎦
⎢⎣ 111110
)(ss
sGN
Now let:⎥⎤
⎢⎡ −11
)(G ⎥⎤
⎢⎡ −11
)(ˆ sG
Ali Karimpour Feb 2012
53? eirreducibl )(ˆ and )(ˆ Are sGsG ND
but why? Yes
⎥⎦
⎢⎣ +−
=11
)(s
sGD ⎥⎦
⎢⎣ +
=11
)(s
sGN
Chapter 2
Matrix Fraction Description (MFD)
Checking irreducibility:
Form:[ ] ⎥
⎦
⎤⎢⎣
⎡+−+−−
=1111
1111)(ˆ)(ˆ
sssGsG DN
Do preliminary transformation to make right part zero:
1) Add C on C ⎤⎡ − 0111 2) Add -C1 on C3 ⎤⎡ − 00111) Add -C2 on C4 ⎥⎦
⎤⎢⎣
⎡−+ 0111
0111s
2) Add C1 on C3
3) Add C1 on C2
⎥⎦
⎤⎢⎣
⎡−+ 0211 s
⎥⎤
⎢⎡ 00013) Add C1 on C2
4) Add 0.5(s+2)C3 on C2
⎥⎦
⎢⎣ −+ 0221 s
⎥⎤
⎢⎡ 0001) ( ) 3 2
5) Change C3 and C2 ⎥⎤
⎢⎡
=Ω01
)(:isfactorCommon s
⎥⎦
⎢⎣ − 0201
⎥⎤
⎢⎡ 0001
Ali Karimpour Feb 2012
54
⎥⎦
⎢⎣ −
Ω21
)( :isfactor Common s
e.irreducibl are )(ˆ and )(ˆ so unimodular is (s) Since sGsG NDΩ
⎥⎦
⎢⎣ − 0021
Chapter 2
Matrix Fraction Description (MFD)
Deriving common factor:
Form:[ ] ⎥
⎦
⎤⎢⎣
⎡
+−+++−+
=2222
)()(22
ssssssssss
sGsG DN
Do preliminary transformation to make right part zero:
1) Add C on C ⎤⎡ −+ 02 ssss 2) Add C1 on C3 ⎤⎡ + 002 sss1) Add -C2 on C4 ⎥⎦
⎤⎢⎣
⎡
−+++
02220
sssssss 2) Add C1 on C3
⎥⎦
⎤⎢⎣
⎡
+++
022200
ssssss
3) Add –(s+1)C1 on C2 ⎥⎤
⎢⎡ 000s3) Add (s+1)C1 on C2 ⎥
⎦⎢⎣ +−+ 02)2(2 ssss
4) Add 0.5(s+2)C3 on C2 ⎥⎤
⎢⎡ 000s) ( ) 3 2 ⎥
⎦⎢⎣ + 0202 ss
5) Change C3 and C2 ⎥⎤
⎢⎡ 000s
⎥⎤
⎢⎡
=Ωs
s0
)(:isfactorCommon
Ali Karimpour Feb 2012
55
⎥⎦
⎢⎣ + 0022 ss ⎥
⎦⎢⎣ +
=Ωss
s22
)( :isfactor Common
reducible. are )( and )( so unimodularnot is (s) Since sGsG NDΩ
Chapter 2
Introduction to Multivariable Control
Topics to be covered include: Multivariable Connections
M lti i bl P l d Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
Matrix Fraction Description (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
56
Bandwidth and Crossover Frequency
Chapter 2
Scaling
ryedGuGy d ˆˆˆ;ˆˆˆˆˆ −=+= ryedGuGy d ;+
rDreDeyDyuDudDd eeeud ˆ,ˆ,ˆ,ˆ,ˆ 11111 −−−−− =====
rDyDeDdDGuDGyD eeeddee −=+= ;))
yy eeeddee
ddedue DGDGDGDG ˆ,ˆ 11 −− == ryedGGuy d −=+= ;ddedue , yy d ;
Ali Karimpour Feb 2012
57
Chapter 2
Introduction to Multivariable Control
Topics to be covered include: Multivariable Connections
M lti i bl P l d Multivariable Poles and zeros
Transmission zero assignment
Smith-McMillan Forms
Matrix Fraction Description (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
Ali Karimpour Feb 2012
58
Bandwidth and Crossover Frequency
Chapter 2
Performance Specification
• Nominal stability NS: The system is stable with no model• Nominal stability NS: The system is stable with no model uncertainty.
• Nominal Performance NP: The system satisfies the performance specifications with no model uncertainty.
• Robust stability RS: The system is stable for all perturbed plants about the nominal model up to the worst case model uncertaintyabout the nominal model up to the worst case model uncertainty.
• Robust performance RP: The system satisfies the performance p y pspecifications for all perturbed plants about the nominal model up to the worst case model uncertainty.
Ali Karimpour Feb 2012
59
Chapter 2
Performance Specification
• Time Domain PerformanceTime Domain Performance
• Frequency Domain Performance
Ali Karimpour Feb 2012
60
Chapter 2
Time Domain Performance
Although closed loop stability is an important issue,Although closed loop stability is an important issue,
the real objective of control is to improve performance, that is, to make the output y(t) behave in a more desirable manner.
Actually, the possibility of inducing instability is one of the disadvantages of feedback control which has to be traded off against performance improvement.
The objective of this section is to discuss ways of evaluatingclosed loop performance.
Ali Karimpour Feb 2012
61
closed loop performance.
Chapter 2
Time Domain Performance
Step response of a system
• Rise time, tr , the time it takes for the output to first reach 90% of i fi l l hi h i ll i d b llits final value, which is usually required to be small.
• Settling time, ts , the time after which the output remains withing , s , p5% ( or 2%) of its final value, which is usually required to be small.
O ershoot P O th k l di id d b th fi l l hi h
Ali Karimpour Feb 2012
62
• Overshoot, P.O, the peak value divided by the final value, whichshould typically be less than 20% or less.
Chapter 2
Time Domain Performance
Step response of a system
• Decay ratio, the ratio of the second and first peaks, which should typically be 0.3 or less.
• Steady state offset, ess, the difference between the final value and the desired final value, which is usually required to be small.
Ali Karimpour Feb 2012
63
y q
Chapter 2
Time Domain Performance
Step response of a system
E i ti th t t l i ti (TV) di id d b th ll• Excess variation, the total variation (TV) divided by the overall change at steady state, which should be as close to 1 as possible.
The total variation is the total movement of the output as shown.
/i tiE TVTV ∑Ali Karimpour Feb 2012
640/variationExcess vTVvTV
ii ==∑
Chapter 2
Time Domain Performance
• ISE : Integral squared error ττ deISE 2
0)(∫
∞=
• IAE : Integral absolute error ττ deIAE ∫∞
=0
)(
• ITSE : Integral time weighted squared error τττ deITSE 2)(∫∞
=• ITSE : Integral time weighted squared error τττ deITSE0
)(∫=
• ITAE : Integral time weighted absolute error τττ deITSE )(0∫∞
=
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Chapter 2
Frequency Domain Performance
Let L(s) denote the loop transfer function of a system which isLet L(s) denote the loop transfer function of a system which is closed-loop stable under negative feedback.
Bode plot of )( ωjL
1G
180)( 180 =∠ ωjL
)(1
180ωjLGM =
1)( =cjL ω
180)( +∠= cjLPM ω
Ali Karimpour Feb 2012
66cPM ωθ /max =
Chapter 2
Frequency Domain Performance
Let L(s) denote the loop transfer function of a system which isLet L(s) denote the loop transfer function of a system which is closed-loop stable under negative feedback.
1G
180)( 180 =∠ ωjLNyquist plot of )( ωjL
)(1
180ωjLGM =
1)( =cjL ω
180)( +∠= cjLPM ω
Ali Karimpour Feb 2012
67cPM ωθ /max =
Chapter 2
Frequency Domain Performance
Stability margins are measures of how close a stable closed-loop system is to instability.
From the above arguments we see that the GM and PM provide stabilityFrom the above arguments we see that the GM and PM provide stabilitymargins for gain and delay uncertainty.
More generally, to maintain closed-loop stability, the Nyquist stability condition tells us that the number of encirclements of the critical point-1 by must not change. )( ωjL
Thus the actual closest distance to -1 is a measure of stability
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Chapter 2
Frequency Domain Performance
One degree-of-freedomconfiguration
nGKIGKdGGKIrGKIGKsy d111 )()()()( −−− +−+++=
Complementary sensitivity function )(sT
Sensitivity function )(sS
The maximum peaks of the sensitivity and complementary sensitivity functions are defined as
Ali Karimpour Feb 2012
69)(max)(max ωωωω
jTMjSM Ts ==
Chapter 2
Frequency Domain Performance
( ) ( ) 11 )()()()( −− +=+= ωωωω jLIjKjGIjS ( ) ( )
sM1
)(max ωω
jSM s =
Thus one may also view Ms as a robustness measure.
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y s
Chapter 2
Frequency Domain Performance
There is a close relationship between MS and MT and the GM and PM.
][12
1sin2;1
1 radMM
PMM
MGM s ≥⎟
⎠
⎞⎜⎜⎝
⎛≥≥ −
21 MMM sss ⎠⎜⎝−
For example, with MS = 2 we are guaranteed GM > 2 and PM >29˚.
⎞⎛
For example, with MS 2 we are guaranteed GM 2 and PM 29 .
][12
1sin2;11 1 radMM
PMM
GMTTT
≥⎟⎟⎠
⎞⎜⎜⎝
⎛≥+≥ −
For example, with MT = 2 we are guaranteed GM > 1.5 and PM >29˚.
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Chapter 2
Trade-offs in Frequency Domain
One degree-of-freedomconfiguration
nGKIGKdGGKIrGKIGKsy d111 )()()()( −−− +−+++=
Complementary sensitivity function )(sT
Sensitivity function )(sS
)(sT
IsSsT =+ )()(
TdSGS + KSdKSGKSAli Karimpour Feb 2012
72
TndSGSrrye d −+−=−= KSndKSGKSru d −−=
Chapter 2
Trade-offs in Frequency DomainTndSGSrrye + KSndKSGKSru =
IsSsT =+ )()(TndSGSrrye d −+−=−= KSndKSGKSru d −−=
( ) 1)()( −+= sLIsS
• Performance, good disturbance rejection ∞→→→ LITS or or 0
• Performance, good command following ∞→→→ LITS or or 0
• Mitigation of measurement noise on output 0oror0 →→→ LIST• Mitigation of measurement noise on output 0or or 0 →→→ LIST
• Small magnitude of input signals 0Tor0or0 →→→ LKSmall magnitude of input signals 0Tor 0or 0 →→→ LK
• Physical controller must be strictly proper 0Tor 0or 0 →→→ LK
• Nominal stability (stable plant) small be L
Ali Karimpour Feb 2012
73• Stabilization of unstable plant ITL →or large be
Chapter 2
Introduction to Multivariable Control
Topics to be covered include:
Multivariable Connections
Multivariable Poles and zeros
Transmission zero assignment g
Smith-McMillan Forms
M i F i D i i (MFD) Matrix Fraction Description (MFD)
Scaling
Performance Specification
Bandwidth and Crossover Frequency
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Bandwidth and Crossover Frequency
Chapter 2
Bandwidth and Crossover Frequency
Definition 2-3
below. from db 3- crossesfirst )S(j wherefrequency theis , ,bandwidth The B
ωω
40
60g
L
Definition 2-4 0
20
L
TDefinition 2 4
bfdb3)T(jt hi hfrequency highest theis , ,bandwidth The BTω
-40
-20S
above. from db3-crosses )T(jat which ω
Definition 2-510
010
110
210
3-60
Frequency (rad/sec)
bfdb0)L(jh frequency theis , ,frequency crossover gain The cω
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75above. from db0crosses )L(j where ω
Chapter 2
Bandwidth and Crossover Frequency
Specifically, for systems with PM < 90˚ (most practical systems) we have
BTcB ωωω ≤≤60
g
20
40
60
L
-20
0T
S
100
101
102
103
-60
-40
S
In conclusion ωB ( which is defined in terms of S ) and also ωc
10 10 10 10Frequency (rad/sec)
Ali Karimpour Feb 2012
76( in terms of L ) are good indicators of closed-loop performance, while ωBT ( in terms of T ) may be misleading in some cases.
Chapter 2
Bandwidth and Crossover Frequency
Example 2-18: Comparison of and as indicators of performance.BTωBω
1,1.0;1
1;)2(
Let ==++
+−=
+++−
= ττττ
zszs
zsTzss
zsL1)2( ++++ τττ szszss
036.0=Bω054.0=cω 1=BTω 1.0 zero RHPan is There =zc BT
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Chapter 2
Bandwidth and Crossover Frequency
Example 2-17: Comparison of and as indicators of performance.BTωBω
1,1.0;1
1;)2(
Let ==++
+−=
+++−
= ττττ
zszs
zsTzss
zsL1)2( ++++ τττ szszss
036.0=Bω054.0=cω 1=BTω 1.0 zero RHPan is There =zc BT
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Chapter 2
Exercises
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Chapter 2
Exercises(Continue)( )
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