Post on 24-Sep-2020
Multivariable Control Multivariable Control SystemsSystemsSystemsSystems
Ali KarimpourpAssistant Professor
Ferdowsi University of MashhadFerdowsi University of Mashhad
Chapter 4Chapter 4Stability of Multivariable Feedback Control SystemsS b y o u v b e eedb c Co o Sys e s
Topics to be covered include:
• Well - Posedness of Feedback Loop
• Internal Stability• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
C i F t i ti St bl T f F ti• Coprime Factorizations over Stable Transfer Functions
• Stabilizing Controllers
Ali Karimpour Feb 2011
2• Strong and Simultaneous Stabilization
Chapter 4
Stability of Multivariable Feedback Control Systems
W ll P d f F db k L• Well - Posedness of Feedback Loop
• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
• Coprime Factorizations over Stable Transfer Functionsp
• Stabilizing Controllers
St d Si lt St bili ti
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3
• Strong and Simultaneous Stabilization
Chapter 4
Well - Posedness of Feedback Loop
Assume that the plant P and the controller K are fixed real rational t f t iproper transfer matrices.
The first question one would ask is whether the feedback interconnectionmakes sense or is physically realizablemakes sense or is physically realizable.
1,21Let =
+−
−= KssP
idsdnrsu3
1)(3
2 −+−−
+=
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4
2+s 33
Hence, the feedback system is not physically realizable!
Chapter 4
Well - Posedness of Feedback Loop
Definition 4-1 A feedback system is said to be well-posed if all closed-loop transfer matrices are well-defined and propertransfer matrices are well-defined and proper.
Now suppose that all transfer matrices from the signals r n d and d toNow suppose that all transfer matrices from the signals r, n, d and di tou are respectively well-defined and proper.
Thus y and all other signals are also well-defined and the related transfery gmatrices are proper.
So the system is well-posed if and only if the transfer matrix from di
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5
y p y iand d to u exists and is proper.
Chapter 4
Well - Posedness of Feedback Loop
So the system is well-posed if andonly if the transfer matrix from dionly if the transfer matrix from diand d to u exists and is proper.
Theorem 4-1 The feedback system in Figure is well-posed if and only if
invertible is )()( ∞∞+ PKI
( ) [ ] ⎥⎦
⎤⎢⎣
⎡+−= −
idd
KPKKPIu 1Proof
Thus well - posedness is equivalent to the condition that exist( ) 1−+ KPIThus well - posedness is equivalent to the condition that existand is proper.
( )+ KPI
And this is equivalent to the condition that the constant term of the
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6
And this is equivalent to the condition that the constant term of the transfer matrix is invertible.)()( ∞∞+ PKI
Chapter 4
Well - Posedness of Feedback Loop
Transfer matrix is invertible.)()( ∞∞+ PKI
i i l i h f h f ll i di iis equivalent to either one of the following two conditions:
invertible is )(⎥⎤
⎢⎡ ∞KI
invertibleis)()( ∞∞+ KPIve b es)( ⎥
⎦⎢⎣ ∞− IP
invertible is )()(+ KPI
The well- posedness condition is simple to state in terms of state-space realizations. Introduce realizations of P and K:
⎥⎦
⎤⎢⎣
⎡≅
DCBA
P⎥⎥⎦
⎤
⎢⎢⎣
⎡≅
DC
BAK ))
ˆˆ
⎦⎣ DC ⎥⎦⎢⎣ DC
SO well- posedness is equivalent to the condition that
Ali Karimpour Feb 2011
7invertible is ⎥⎦
⎤⎢⎣
⎡
− IDDI)
Chapter 4
Stability of Multivariable Feedback Control Systems
W ll P d f F db k L• Well - Posedness of Feedback Loop
• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
• Coprime Factorizations over Stable Transfer Functionsp
• Stabilizing Controllers
St d Si lt St bili ti
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8
• Strong and Simultaneous Stabilization
Chapter 4
Internal Stability
Assume that the realizations for P(s) and K(s) are stabilizable and detectable. Let x and denote the state vectors for P and K, respectively.x̂
ˆˆ&
yDxCuDuCxy
yBxAxBuAxxˆˆˆ
ˆˆ
−=+=
−=+=&
Definition 4-2
ht ti iti lllft)ˆ(t tthit blllt tii (0,0))ˆ,(origin theif stable internally be tosaid is Figure of system The =xx
Ali Karimpour Feb 2011
90 and0,0,0whenstatesinitialallfromzero togo),(states thei.e. stableally asymptotic is
==== nddrxx
i
Chapter 4
Internal Stability
⎤⎡⎤⎡⎤⎡⎤⎡− ˆˆ 1
e
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−=⎥
⎦
⎤⎢⎣
⎡xx
CC
IDDI
yu
ˆ00
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−⎥⎦
⎤⎢⎣
⎡
−+⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
xx
Axx
CC
IDDI
B
B
A
A
x
xˆ
~ˆ0
ˆ0ˆˆ0
0ˆ0
0
ˆ
1
&
&
⎠⎝
Theorem 4-2
The system of above Figure with given stabilizable and detectable
A~
The system of above Figure with given stabilizable and detectable
realizations for P and K is internally stable if and only if is a A~
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Hurwitz matrix (All eigenvalues are in open left half plane).
Chapter 4
Internal Stability
⎥⎤
⎢⎡
⎥⎤
⎢⎡⎥⎤
⎢⎡ duKI ip
e
⎥⎦
⎢⎣−
=⎥⎦
⎢⎣−⎥⎦
⎢⎣− reIP
ip
Theorem 4-3The system in Figure is internally stable if and only if the transfer matrixThe system in Figure is internally stable if and only if the transfer matrix
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−+−=⎥
⎦
⎤⎢⎣
⎡ −−−
11
111
)()()()( PKIKPPKIKI
IPKI
⎥⎥⎦⎢
⎢⎣ ++⎥
⎦⎢⎣− −− 11 )()( PKIPPKIIP
from (di , -r) to (up , - e) be a proper and stable transfer matrix.
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11
( i , ) ( p , ) p p
Chapter 4
Internal Stability
⎤⎡⎤⎡ −−− 111 )()( PKIKPPKIKIKI
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
+−+−=⎥
⎦
⎤⎢⎣
⎡− −− 11
11
)()()()(
PKIPPKIPKIKPPKIKI
IPKI
Note that to check internal stability it is necessary (and sufficient) to testNote that to check internal stability, it is necessary (and sufficient) to test whether each of the four transfer matrices is stable.
⎤⎡ ++ ss 11
11,
11Let
−=
+−
=s
KssP ⎥
⎦
⎤⎢⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++
+−
+−−
+=⎥
⎦
⎤⎢⎣
⎡
− rd
sssss
e
u ip
21
21
)2)(1(2
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⎥⎦⎢⎣ ++ ss 22Stability cannot be concluded even if three of the four transfer matrices are stable.
Chapter 4
Internal Stability
Theorem 4-4insystemThen thestable.isSuppose K
Stable
stable. is )( ifonly and if stable internally is figure the
in system Then the stable. is Suppose
1 PPKI
K
−+⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
+−+−−−
−−
11
11
)()()()(
Remember PKIPPKI
PKIKPPKIKI
)(
Proof: The necessity is obvious. To prove the sufficiency let
PPKIQ 1)( −+=
KQIPPKIKI −=+− −1)( Stable
)()( 1 QKIKPKIK −−=+− −
QPPKI =+ −1)(
Stable
Stable
Ali Karimpour Feb 2011
13QKIPKPKIIIPKIIPKI −=+−=−++=+ −−− 111 )()()(
QPPKI + )( Stable
Stable
Chapter 4
Internal Stability
Theorem 4-5insystemThen thestableisSuppose P
Stable
stableis)(ifonly and if stable internally is figure the
in system Then the stable. is Suppose
1−+ PKIK
P
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
+−+−−−
−−
11
11
)()()()(
Remember PKIPPKI
PKIKPPKIKI
stable. is )( + PKIK
Proof: The necessity is obvious. To prove the sufficiency let
1)( −+= PKIKQ
QPIPPKIKI −=+− −1)( Stable
QPKIK −=+− −1)(
PPQIPPKI )()( 1 −=+ −
Stable
Stable
Ali Karimpour Feb 2011
14PQIPKIPKIIPKIIPKI −=+−=−++=+ −−− 111 )()()(
PPQIPPKI )()( + Stable
Stable
Chapter 4
Internal Stability
Theorem 4-6Thenstable.bothareandSuppose KP
Stable Stable
stable.is )( ifonly and if stable internally is figure in the system the
Then stable.both are and Suppose
1−+ PKI
KP
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
+−+−−−
−−
11
11
)()()()(
Remember PKIPPKI
PKIKPPKIKI
)(y
Proof: The necessity is obvious. To prove the sufficiency let
1)( −+= PKIQ
KQPIPPKIKI −=+− −1)( Stable
KQPKIK −=+− −1)(
QPPPKI =+ −1)(
Stable
Stable
Ali Karimpour Feb 2011
15QIPKIIPKI =−++=+ −− 11 )()(
QPPPKI + )( Stable
Stable
Chapter 4
Internal StabilityN l ll ti
Theorem
No pole zero cancellation
)()(ofpolesRHPopenofnumberthei) and posed- wellisit ifonly and if
stableinternallyisfigure in the system he
+= nnsKsP
T
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
+−+−−−
−−
11
11
)()()()(
Remember PKIPPKI
PKIKPPKIKI
stable. is )( ii)
)()(ofpolesRHPopen ofnumber thei)1−+
+=
PKI
nnsKsP PK
Proof:
of polse RHP ofnumber theis of polse RHP ofnumber theis PnKn PK
See: “Essentials of Robust control” written by Kemin Zhou
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Chapter 4
Stability of Multivariable Feedback Control Systems
W ll P d f F db k L• Well - Posedness of Feedback Loop
• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
• Coprime Factorizations over Stable Transfer Functionsp
• Stabilizing Controllers
St d Si lt St bili ti
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• Strong and Simultaneous Stabilization
Chapter 4
The Nyquist Stability Criterion
f ibhfbilih k hiN
lRHPi thdlh)](d t[L t PPkGI
. of valuesreal sfor variousystemabove theofstability check the togoing are weNow
k
plane.RHPin the zerosandpoleshave )](det[Let co PPskGI +
tii ithi l))(det()( ofplot Nyquist thesystems, SISO asjust Then
PPskGIs +=φ
times. origin, theencircles oc PP −
However, we would have to draw the Nyquist locus of for each value of k in which we
Ali Karimpour Feb 2011
18
were interested, whereas in the classical Nyquist criterion we draw a locus only once,and then infer stability properties for all values of k.
Chapter 4
The Nyquist Stability Criterion
∏ +=+i
i skskGI )](1[)](det[ λ
)(of eigenvaluean is)(Where sGsiλ )(g)(i
{ } ( )∑ +=+ i skskGI )(1)](det[ λϑϑ{ } ( )∑i
i )()]([
1- 0 →
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Chapter 4
The Nyquist Stability Criterion
Theorem 4-7 (Generalized Nyquist theorem)Theorem 4 7 (Generalized Nyquist theorem)
If G(s) with no hidden unstable modes, has Po unstable (Smith-McMillan)
poles, then the closed-loop system with return ratio –kG(s) is stable if
and only if the characteristic loci of kG(s) taken together encircle theand only if the characteristic loci of kG(s), taken together, encircle the
point -1, Po times anticlockwise.
Ali Karimpour Feb 2011
201)(1)(1 −++ =− NPZ skGskG 1)()(1 −+ =− NPZ sGskG⇒
Chapter 4
The Nyquist Stability Criterion
Example 4-1: Let
⎤⎡ 2
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+−−
+++
=01
)1()1(5.0
)1(1)1(5.0
)(3
2
ss
ss
sG
⎥⎦⎢⎣ +−0
)1)(1( 3ss
Suppose that G(s) has no hidden modes, check the stability of system for different values of k.
0)( =− sGIλ
)1(5.0
1 +=
sλ 32 )1(
1+
=s
λ
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)( )(s
Chapter 4
The Nyquist Stability CriterionStability Criterion
)1(5.0
1 +=
sλ
)1( +s
32 )1(1+
=s
λ125.0−
Since G(s) has one unstable poles, we will have closed l bili if h l iloop stability if these loci give one net encirclementsof -1/k (ccw) when a negative feedback kI is appliedfeedback kI is applied.
pole. loop closed RHP one is theresont encircleme no 125.0-/1-- <<∞ kpoleloopclosedRHPthreeistheresontencirclemetwo0 5k/11250- <−<
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pole.loopclosedRHP threeis theresont encircleme two0.5k/1125.0 <<pole. loop closed RHP twois theresont encircleme one 1k/10.5 <−<pole. loop closed RHP threeis theresont encircleme two 11/k- >
Chapter 4
The Nyquist Stability Criterion
Example 4-2: Let
⎥⎤
⎢⎡ −11)(
ssG ⎥
⎦
⎤⎢⎣
⎡−−++
=26)2)(1(25.1
)(sss
sG
Suppose that G(s) has no hidden modes, check the stability of system for different values of k.
0)( =− sGIλ
24132 −−− ssλ 24132 −+− ss)2)(1(5.2
241321 ++=
ssssλ
)2)(1(5.224132
2 ++−+−
=ss
ssλ
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Chapter 4
The Nyquist Stability Criterion
)2)(1(5.224132
1 ++−−−
=ss
ssλ
)2)(1(5.224132
2 ++−+−
=ss
ssλ
Since G(s) has no unstable poles, we will have closed l bili if h l iloop stability if these loci give zero net encirclementsof -1/k when a negative feedback kI is appliedfeedback kI is applied.
nt.encircleme no /153.0 and 0/14.0 , 8.0/1 ∞<−<<−<−−<−<∞− kkk
Ali Karimpour Feb 2011
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system. loop closedin pole RHPonesont encirclemeone 4.0/18.0 −<−<− ksystem. loop closedin pole RHP twoso ntsencircleme two53.0/10 <−< k
Chapter 4
The Inverse Nyquist Stability Criterion
Theorem 4-7 (Generalized Nyquist theorem)
f ( ) i h hidd bl d h bl ( i h ill ) l h hIf G(s) with no hidden unstable modes, has Po unstable (Smith-McMillan) poles, then the closed-loop system with return ratio –kG(s) is stable if and only if the characteristic loci of kG(s), taken together, encircle the point -1, Po times anticlockwise.
NPZ 1)()(1 −+ =− NPZ sGskG
Theorem 4-7’ (Generalized Inverse Nyquist theorem)
1)(1)(1 −++ =− NPZ skGskG ⇒
If G(s) with no hidden unstable modes, has Zo unstable (Smith-McMillan) zeros, then the closed-loop system with return ratio –kG(s) is stable if and
l if th h t i ti l i f i G( )/k t k t th i l th i t
NPZ
only if the characteristic loci of invG(s)/k, taken together, encircle the point -1, Zo times anticlockwise.
= NZZ⇒
Ali Karimpour Feb 2011
25
1/)(inv1/)(inv1 −++ =− NPZ ksGksG 1)()(1 −+ =− NZZ sGskG⇒
Chapter 4
The Nyquist Stability Criterion
Example 4-3: Let1)( =sG
)10)(5()(
++=
ssssG
Suppose that G(s) has no hidden modes, check the stability of system for different values of k.1
)10)(5(1
++=
sssλ0)( =− sGIλ )10)(5(1 ++=− sssλ
polesunstableTwo)750(750 >−<− kk poles.unstableTwo)750(750 >−<− kk
Stable.)7500(0750 <<<−<− kk
-750 pole. RHP One)0(0 <−< kk
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Chapter 4
Nyquist arrays and Gershgorin bands
The Nyquist array of G(s) is an array of graphs (not necessarily closed) th ijth h b i th N i t l f ( )curve), the ijth graph being the Nyquist locus of gij(s) .
Theorem 4-8 (Gershgorin’s theorem)
:follows as elements diagonal thearound circles of unions in two containedare of seigenvalue The . dimensions ofmatrix complex a be Let ZmmZ iλ×
mizzm
ijj
ijiii ,....,2,1,1
=<− ∑≠=
λ mizzm
ijj
jiiii ,....,2,1,1
=<− ∑≠=
λ
The ‘bands’ obtained in this way are called Gershgorin bands, each is composed of Gershgorin circles.
Ali Karimpour Feb 2011
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composed of Gershgorin circles.
Chapter 4
Nyquist arrays and Gershgorin bandsNyquist array, with Gershgorin bands for a sample systemNyquist array, with Gershgorin bands for a sample system
If all the Gershgorin bands l d th i t 1 thexclude the point -1, then
we can assess closed-loop stability by counting the
encirclements of -1 by theencirclements of -1 by the Gershgorin bands, since this
tells us the number of encirclements made by the y
characteristic loci.
f h h i b d f ( ) l d h i i h h ( ) iIf the Gershgorin bands of G(s) exclude the origin, then we say that G(s) is diagonally dominant (row dominant or column dominant).
The greater the degree of dominant ( of G(s) or I+G(s) ) – that is, the narrower
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the Gershgorin bands- the more closely does G(s) resembles m non-interacting SISO transfer function.
Chapter 4
Nyquist arrays and Gershgorin bands
And in general:
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29
Chapter 4
Nyquist arrays and Gershgorin bands
Diagonal dominance of a matrix G(s): mm×
Row Diagonal dominance: If for all s on the Nyquist contour,
misgsgm
ijj
ijii ...,,2,1)()(1
=>∑≠=
Column Diagonal dominance: If for all s on the Nyquist contour,
ij≠
misgsgm
jiii ...,,2,1)()( =>∑Ali Karimpour Feb 2011
30ijj
jiii1∑≠=
Chapter 4
Stability of Multivariable Feedback Control Systems
W ll P d f F db k L• Well - Posedness of Feedback Loop
• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
• Coprime Factorizations over Stable Transfer Functionsp
• Stabilizing Controllers
St d Si lt St bili ti
Ali Karimpour Feb 2011
31
• Strong and Simultaneous Stabilization
Chapter 4
Coprime Factorizations over Stable Transfer Functions
Two polynomials m(s) and n(s), with real coefficients, are said to be coprime ifp• their greatest common divisor is 1 or
• they have no common zeros or• there exist polynomials x(s) and y(s) such that 1)()()()( =+ snsysmsx
Exercise1 : Let n(s)=s2+5s+6 and m(s)=s find x(s) and y(s) if n and m are coprime
Si il l t t f f ti ( ) d ( ) i th t f t bl t f
Exercise1 : Let n(s) s +5s+6 and m(s) s find x(s) and y(s) if n and m are coprime. Exercise 2 : Let n(s)=s2+5s+6 and m(s)=s+2 show that one cannot find x(s) and y(s) in
x(s)m(s)+y(s)n(s)=1
Similarly, two transfer functions m(s) and n(s) in the set of stable transfer functions are said to be coprime over stable transfer functions if there exists x(s) and y(s) in the set of stable transfer functions such that
Ali Karimpour Feb 2011
321)()()()( =+ snsysmsx Bezout identities
Chapter 4
Coprime Factorizations over Stable Transfer Functions
Definition 4-3 Two matrices M and N in the set of stable transfer matrices are right coprime over the set of stable transfer matricesmatrices are right coprime over the set of stable transfer matrices if they have the same number of columns and if there exist matrices X and Y in the set of stable transfer matrices s tXr and Yr in the set of stable transfer matrices s.t.
[ ] INYMXNM
YX rrrr =+=⎥⎦
⎤⎢⎣
⎡N ⎦⎣
Similarly, two matrices and in the set of stable transfer matrices are left coprime over the set of stable transfer matrices if they have the
N~M~
are left coprime over the set of stable transfer matrices if they have the same number of rows and if there exist two matrices Xl and Yl in the setof stable transfer matrices such that
Ali Karimpour Feb 2011
33[ ] IYNXMYX
NM lll
l =+=⎥⎦
⎤⎢⎣
⎡ ~~~~
Chapter 4
Coprime Factorizations over Stable Transfer Functions
Now let P be a proper real-rational matrix. A right-coprime factorization(rcf) of P is a factorization of the form
1−= NMP
where N and M are right-coprime in the set of stable transfer matrices.
Similarly, a left-coprime factorization (lcf) of P has the form
NMP ~~ 1−=
stableofsetin thecoprimeareand~~lcfandrcfIn NMNM
Ali Karimpour Feb 2011
34matricesfunction ansfer tr stable ofset in the coprimeare and,, lcf and rcfIn NMNM
Chapter 4
Coprime Factorizations over Stable Transfer Functions
Remember: 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 −=
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35
In Matrix Fraction Description D(s) and N(s) are polynomial matrices
Chapter 4
Coprime Factorizations over Stable Transfer Functions
Theorem 4-9 Suppose P(s) is a proper real-rational matrix and
⎤⎡⎥⎦
⎤⎢⎣
⎡≅
DCBA
sP )(
is a stabilizable and detectable realization. Let F and L be such that A+BF and A+LCare both stable, and define
⎤⎡ −+⎤⎡
LBBFAYM ⎥
⎤⎢⎡ +−+
⎤⎡LLDBLCA
YX)(
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+≅⎥
⎦
⎤⎢⎣
⎡ −
IDDFCIF
XNYM
l
l 0⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−≅⎥
⎦
⎤⎢⎣
⎡
− IDCIF
MN
YX rr 0)(
~~
Then rcf and lcf of P are:
NMP ~~ 1−=1−= NMP
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36
NMP= NMP
Exercise: Let P(s)=(s+1)/(s+2) find two different rcf for P.
Chapter 4
Coprime Factorizations over Stable Transfer Functions
The right coprime factorization of a transfer matrix can be given a feedback control interpretation.
BA&
P
DuCxyBuAxx
+=++&
Fxvu +=BvxBFAx ++= )(& the transfer matrix from v to u is
⎤⎡ BBFAand that from v to y is
⎤⎡ BBFADvxDFCy
vFxu++=
+=)( ⎥
⎦
⎤⎢⎣
⎡ +≅
IFBBFA
sM )( ⎥⎦
⎤⎢⎣
⎡++
≅DDFCBBFA
sN )(
)()()( N )()()( 1MNP)()()( 1MN )()(P
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37
)()()( svsNsy = )()()( 1 sMsNsP −=⇒)()()( 1 susMsN −= )()( susP=
Exercise3 : Derive a similar interpretation for left coprime factorization.
Chapter 4
Stability of Multivariable Feedback Control Systems
W ll P d f F db k L• Well - Posedness of Feedback Loop
• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
• Coprime Factorizations over Stable Transfer Functionsp
• Stabilizing Controllers
St d Si lt St bili ti
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• Strong and Simultaneous Stabilization
Chapter 4
Stabilizing Controllers
Theorem 4-10 Suppose P is stable.Then the set of all stabilizing controllersThen the set of all stabilizing controllersin Figure can be described as
1)( −= PQIQK )( −= PQIQK
for any Q in the set of stable transfer matrices and non singular. )()( ∞∞− QPI
Proof:11 )()()( −− +=⇒=−⇒−= PKIKQQPQIKPQIQK System is stable.
Now suppose the system is stable, so define then stable, is )( 1−+ PKIK
KKPIPKIKQ 11 )()( −− +=+= KKPQQ =+⇒
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39
KKPIPKIKQ )()( +=+= KKPQQ +⇒
1)( −−= PQIQKsor nonsingula is )()( ∞∞− QPI
Chapter 4
Stabilizing Controllers
Example 4-4 For the plant
)2)(1(1)(
++=
sssP
Suppose that it is desired to find an internally stabilizing controller so that ySuppose that it is desired to find an internally stabilizing controller so that yasymptotically tracks a ramp input.
Solution: Since the plant is stable the set of all stabilizing controller is derived from
let so r,nonsingula is )()( such that Q stableany for )( 1 ∞∞−−= − QPIPQIQK
+ bas3++
=s
basQ
)3)(2)(1()()3)(2)(1(
)3)(2)(1(11)(11 1
++++−+++
=+++
+−=−=+−=−= −
sssbassss
sssbasPQPKIPKTS
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40
)3)(2)(1()3)(2)(1( ++++++ ssssss
3611
++
=ssQ
Chapter 4
Stabilizing Controllers
NMNMP
P11 .~~ andmatrix rational
-realproper a be Let :11-4 Theorem−− ==
rrll YXXYK 11
controller gstabilizin a exists Then there−− ==
IXNYM
MN
YX
l
lrr =⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡
− ~~ Where
Proof. See “Multivariable Feedback Design By Maciejowski”
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Chapter 4
Remember
Theorem 4-9 Suppose P(s) is a proper real-rational matrix and
⎤⎡⎥⎦
⎤⎢⎣
⎡≅
DCBA
sP )(
is a stabilizable and detectable realization. Let F and L be such that A+BF
and A+LC are both stable, and defineand A LC are both stable, and define
⎤⎡ −+⎤⎡
LBBFA ⎤⎡ +−+⎤⎡
LLDBLCAYX
)(
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+≅⎥
⎦
⎤⎢⎣
⎡ −
IDDFCIF
XNYM
l
l 0⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−≅⎥
⎦
⎤⎢⎣
⎡
− IDCIF
MN
YX rr 0)(
~~
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Chapter 4
Stabilizing Controllers
Ali Karimpour Feb 2011
43Proof. See “Multivariable Feedback Design By Maciejowski”
Chapter 4
Stabilizing Controllers
Example 4-5 For the plant
1)2)(1(
1)(−−
=ss
sP
The problem is to find a controller that
1. The feedback system is internally stable.2. The final value of y equals 1 when r is a unit step and d=0.3. The final value of y equals zero when d is a sinusoid of 10 rad/s and r=0.
0]01[10
3210
Clearly ==⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
= DCBA
To derive coprime factorization let F = [1 -5] and L = [-7 -23]T
clearly A+BF and A+LC are stable.
⎤⎡ −+⎤⎡
LBBFA ⎤⎡ +−+⎤⎡
LLDBLCA )(
Ali Karimpour Feb 2011
44⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
+≅⎥
⎦
⎤⎢⎣
⎡ −
IDDFCIF
LBBFA
XNYM
l
l 0⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
++≅⎥
⎦
⎤⎢⎣
⎡
− IDCIF
LLDBLCA
MN
YX rr 0)(
~~
Chapter 4
Stabilizing Controllers
Solution: The set of all stabilizing controller is:Solution: The set of all stabilizing controller is:
1))(( −−+= llll NQXMQYK
⎥⎥⎥⎤
⎢⎢⎢⎡ −+
≅⎥⎦
⎤⎢⎣
⎡ −IF
LBBFA
XNYM
l
l 0⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ +−+≅⎥
⎦
⎤⎢⎣
⎡
− IDCIF
LLDBLCA
MN
YX rr 0)(
~~⎥⎦⎢⎣ +⎦⎣ IDDFCl ⎥⎦⎢⎣ −⎦⎣ IDCN
2 38972108)1)(2(1 ++=
−=
−−==
ssXsYssMN 2222 )1(,
)1(,
)1(,
)1( +=
+=
+=
+=
sX
sY
sM
sN ll
2 38972108)1)(2(~1~ ++−−− sssss
Ali Karimpour Feb 2011
452222 )2(389,
)2(72108,
)2()1)(2(,
)2(1
+++
=+
=+
=+
=s
ssXs
sYs
ssMs
N rr
Chapter 4
Stabilizing Controllers
Solution: The set of all stabilizing controller is:
1))(( −−+= llll NQXMQYK
Clearly for any stable Q the condition 1 satisfied
To met condition 2 the transfer function from r to y must satisfyTo met condition 2 the transfer function from r to y must satisfy
To met condition 3 the transfer function from di to y must satisfy
5.36)0(1))0(~)0()0()(0( =⇒=+ rrr QMQYN→+= )()~()( srMQYNsy rr
To met condition 3 the transfer function from di to y must satisfy
jjQjNjQjXjM rrr 9062)10(0))10(~)10()10()(10( +−=⇒=−Now define 11)( ++= xxxsQ
→−= )()~()( sdNQXNsy irr
Ali Karimpour Feb 2011
46
2321 )1(1)(
++
++=
sx
sxxsQr
Exercise 6: Derive QrExercise 5: Derive transfer function from di to y.Exercise 4: Derive transfer function from r to y.
Chapter 4
Stability of Multivariable Feedback Control Systems
W ll P d f F db k L• Well - Posedness of Feedback Loop
• Internal Stability
• The Nyquist Stability CriterionThe Generalized Nyquist Stability CriterionNyquist arrays and Gershgorin bands
• Coprime Factorizations over Stable Transfer Functionsp
• Stabilizing Controllers
St d Si lt St bili ti
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47
• Strong and Simultaneous Stabilization
Chapter 4
Strong and Simultaneous
Practical control engineers are reluctant to use unstable controllers, especially when
the plant itself is stable.p
If the plant itself is unstable, the argument against using an unstable controller is
less compelling.p g
However, knowledge of when a plant is or is not stabilizable with a stable controller
is useful for another problem namely, simultaneous stabilization, meaning stabilizationis useful for another problem namely, simultaneous stabilization, meaning stabilization
of several plants by the same controller.
Simultaneous stabilization of two plants can also be viewed as an example of aSimultaneous stabilization of two plants can also be viewed as an example of a
problem involving highly structured uncertainty.
A plant is strongly stabilizable if internal stabilization can be achieved with a controller
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48
A plant is strongly stabilizable if internal stabilization can be achieved with a controller
itself is a stable transfer matrix.
Chapter 4
Strong and Simultaneous
Theorem 4-13: P is strongly stabilizable if and only if it has an even number
of real poles between every pairs of real RHP zeros( including zeros at infinity)of real poles between every pairs of real RHP zeros( including zeros at infinity).
Proof. See “Linear feedback control By Doyle”.
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Chapter 4
Exercises
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