2. Table of Content
SOLO
Stabilization of Linear Time-Invariant Systems
Factorization Approach
Introduction
Well-Posedness
Internal Stability
Right and Left Coprime Factorization of a Transfer Matrix
2
Stabilization of Linear Time-Invariant Systems
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Eigenvalues (modes) of the System
Transfer Function of a L.T.I. System
State Space Realization of the System
Stability of the System
Transmission Zeros of the L.T.I. System
Controllability, Observability
Stabilizability, Detectability
Transfer Function of a L.T.I. System:
3. Table of Content (continue)
SOLO
Stabilization of Linear Time-Invariant Systems
Factorization Approach
3
State-Space Realization of All Coprime Matrices
Operations on Linear Systems
Change of Variables
Cascade of Two Linear Systems G1(s)G2(s)
Para-Hermitian
Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s)
†
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
The Eigenvalues of Closed-Loop System
The Transfer Function of the Compensator K (s)
with O.B.C. Realization
Realization of Q (s) Given G (s) and K (s)
Realization of Heu (s) Given G (s) and K (s)
References
4. • Assume a Linear Time-Invariant Plant, not necessary stable with m inputs and p
outputs.
SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
• Assume that the plant can be represented by a set of n linear ordinary differential
equations with constant coefficients (to assure time-invariance) or by the corresponding
transfer matrix G (s)pxm.
• Assume also that G (s) is proper:
( ) ∞<
∞→
sG
s
lim ( G(s) is analytic as s →∞ )
Therefore G (s) is in the Ring of Real-rational proper (Rp) matrices:
( ) pxm
pRsG ∈
Using the fact that under those assumptions all G (s) can be factorized in two
Real-rational-proper and stable (RH∞) matrices, we will obtain a parameterization
of all compensators K (s)mxp that stabilize the given plant (and conversely all
plants that can be stabilized by a given compensator). This parameterization will
allow to define methods of optimizing the desired performances of the design over
all possible stabilizing compensators. 4
5. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Given the Feedback System, we have:
( )
( )
=
− 2
1
2
1
u
u
e
e
IsK
sGI
mmxp
pxmp
from which: ( )
( )
( )
( ) ( )
( ) ( )
++
+−+
=
=
−
=
−−
−−
−
2
1
11
11
2
1
2
1
1
2
1
u
u
GKIKGIK
GKIGKGI
u
u
sH
u
u
IsK
sGI
e
e
mp
mp
eu
mmxp
pxmp
5
6. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Well-Posedness
Definition:
The System in Figure is well-posed if the transfer-matrix from u to e; i.e. Heu
exists and is proper.
( )
( )
( )
=
−
=
−
2
1
2
1
1
2
1
u
u
sH
u
u
IsK
sGI
e
e
eu
mmxp
pxmp
Well-Posed
( )
( )
1−
− mmxp
pxmp
IsK
sGI
exists and is proper
( )
( )
1−
∞−
∞
m
p
IK
GI
is invertible ( ) ( )[ ]∞∞+ KGIp
is invertible
( ) ( )[ ]∞∞+ GKIm
is invertible
6
7. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Internal Stability
Internally Stable ( ) ( ) ( )mpmp
eu HsH +×+
∞∈
( )
( )
( )
( ) ( )
( ) ( )
++
+−+
=
=
−
=
−−
−−−
2
1
11
11
2
1
2
1
1
2
1
u
u
GKIKGIK
GKIGKGI
u
u
sH
u
u
IsK
sGI
e
e
mp
mp
eu
mmxp
pxmp
Definition:
The System in Figure internally stable if it is well-posed and Heu (s)is analytic for
all Real (s) ≥ 0 ( )+∈Cs
If in addition Heu (s) is real-rational and proper we will write:( ) ( ) ( )
( )mpmp
peu RsH +×+
∈
Internally Stable ( ) ( ) ( )mpmp
eu RHsH +×+
∞∈
and
Internally Stability of a Real-Rational and Proper System
( )
( )
( ) ( )
( ) ( )
∞−−
−−
∞
−
∈
++
+−+
⇔∈
−
RH
GKIKGIK
GKIGKGI
RH
IsK
sGI
mp
mp
mmxp
pxmp
11
111
7
8. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Right and Left Coprime Factorization of a Transfer Matrix
Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.)
is a r.c.f. of G(s) pxm if( ) ( )( )mmgmpg sDsN ×× , is a l.c.f. of G(s) pxm if( ) ( )( )ppgmpg sDsN ××
~
,
~
( ) ( ) ( ) mmgmpgmp sDsNsG ×
−
×× =
1 ( ) ( ) ( ) mpgppgmp sNsDsG ××
−
× =
~~ 1
( ) ( ) mm
g
mp
g RHsDRHsN ×
∞
×
∞ ∈∈ & ( ) ( ) pp
g
mp
g RHsDRHsN ×
∞
×
∞ ∈∈
~
&
~
( ) ( ) pm
g
mm
g RHsYRHsX ×
∞
×
∞ ∈∈∃
~
&
~ ( ) ( ) mp
g
pp
g RHsYRHsX ×
∞
×
∞ ∈∈∃ &
1 1
2 2
3 3
s.t.: s.t.:
( ) ( )[ ] ( )
( ) m
g
g
gg I
sN
sD
sYsX =
~~ ( ) ( )[ ] ( )
( ) p
g
g
gg I
sY
sX
sNsD =
~~
Bézout-Diophantine
Identities
8
9. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
In number theory, Bézout's identity or Bézout's lemma is a linear diophantine equation.
It states that if a and b are nonzero integers with greatest common divisor d,
then there exist integers x and y (called Bézout numbers or Bézout coefficients)
such that
Additionally, d is the least positive integer for which there are integer
solutions x and y for the preceding equation.
In mathematics, a Diophantine equation is an indeterminated polynomial equation
that allows the variables to be integers only. Diophantine problems have fewer
equations than unknown variables and involve finding integers that work correctly
for all equations. In more technical language, they define an algebraic curve,
algebraic surface or more general object, and ask about the lattice points on it.
Diophantus of Alexandria (cca 200 – 280)
Diophantus, often known as the 'father of algebra', is
best known for his Arithmetica, a work on the solution
of algebraic equations and on the theory of numbers.
However, essentially nothing is known of his life and
there has been much debate regarding the date at
which he lived.
Title page of the 1621
edition of Diophantus'
Arithmetica, translated
into Latin by Claude
Gaspard Bachet de
Méziriac.
Étienne Bézout
1730 - 1783
dybxa =+
9
10. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Right and Left Coprime Factorization of a Transfer Matrix (continue – 1)
( ) ( )[ ] ( )
( ) m
g
g
gg I
sN
sD
sYsX =
~~ ( ) ( )[ ] ( )
( ) p
g
g
gg I
sY
sX
sNsD =
~~
Bézout-Diophantine
Identities
For a real-rational and proper transfer-matrix right and left coprime
factorizations always exist (proof in Vidyasagar M., “Control System Synthesis:
A Factorization Approach”, MIT Press, 1985)
( ) mp
pRsG ×
∈
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )( )
−
=
−
− p
ggggm
gg
gg
gg
gg
I
sYsXsXsYI
sXsN
sYsD
sDsN
sYsX
0
~~
~~
~~
or
Define: ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( )sDsYsXsXsYsYsY
sNsYsXsXsYsXsX
ggggggg
ggggggg
~~~~
:
~
~~~~
:
~
0
0
−+=
−+=
Pre-multiply by and redefine:( ) ( ) ( ) ( )( )
−−
p
ggggm
I
sYsXsXsYI
0
~~
4
4
( ) ( )
( ) ( )
( ) ( )
( ) ( )
=
−
− p
m
gg
gg
gg
gg
I
I
sXsN
sYsD
sDsN
sYsX
0
0
~~
~~
0
0
0
0
( ) ( )
( ) ( )sYsY
sXsX
gg
gg
=
=
:
:
0
0
to obtain:
Generalized
Bézout
Identity
10
11. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Right and Left Coprime Factorization of a Transfer Matrix (continue – 2)
( ) ( )
( ) ( )
( ) ( )
( ) ( )
=
−
− p
m
gg
gg
gg
gg
I
I
sXsN
sYsD
sDsN
sYsX
0
0
~~
~~
0
0
0
0
we obtained:
Generalized
Bézout
Identity
We can see that:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )mpmp
gg
gg
gg
gg
RH
sXsN
sYsD
sDsN
sYsX +×+
∞
−
∈
−
=
− 0
0
1
0
0
~~
~~
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )mpmp
gg
gg
gg
gg
RH
sDsN
sYsX
sXsN
sYsD +×+
∞
−
∈
−
=
−
0
0
1
0
0
~~
~~
Definition:
A square transfer-matrix s.t. is called unimodular,
or a unit in the ring of stable real rational and proper transfer matrices.
( ) qqqq
RHsU ×
∞
×
∈ ( ) qq
RHsU ×
∞
−
∈1
11
12. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Right and Left Coprime Factorization of a Transfer Matrix (continue – 3)
Proof Proof
( )
( ) ( ) 1
111
−
−−−
=
==
UDUN
DUUNDNsG
gg
gggg
Post-multiply by U:[ ] m
g
g
gg I
N
D
YX =
~~
[ ] U
UN
UD
YX
g
g
gg =
~~
Pre-multiply this by U-1
:
[ ] m
g
g
gg I
UN
UD
YUXU =
−− ~~ 11
( )
( )( ) 1
111
~~~~
~~~~~~
−
−−−
=
==
gg
gggg
DUNU
NUUDNDsG
[ ] U
UN
UD
YX
g
g
gg =
~~
[ ] p
g
g
gg I
UY
UX
NUDU =
−
−
1
1
~
~
~~~~
Pre-multiply by :[ ] p
g
g
gg I
Y
X
ND =
~~ U
~
Post-multiply this by :1~−
U
Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.)
12
13. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Right and Left Coprime Factorization of a Transfer Matrix (continue – 4)
Theorem 1a:
If is a r.c.f. of
then is also a r.c.f. of G (s)
for every U(s) mxm unimodular.
( )gg ND ,
( )UNUD gg ,
( ) mp
pRsG ×
∈
Theorem 1b:
If is a l.c.f. of
then is also a l.c.f. of G (s)
for every unimodular.
( )gg ND
~
,
~
( )gg NUDU
~~
,
~~
( ) mp
pRsG ×
∈
Proof (continue – 1) Proof (continue – 1)
[ ] m
g
g
gg I
UN
UD
YUXU =
−− ~~ 11
[ ] p
g
g
gg I
UY
UX
NUDU =
−
−
1
1
~
~
~~~~
( ) ppsU ×
~
Because ∞
−
∈ RHU 1
∞
−
∞
−
∈=
∈=
RHYUY
RHXUX
gg
gg
~
:
~
~
:
~
1
1
1
1
and
Because ∞
−
∈ RHU 1~
∞
−
∞
−
∈=
∈=
RHUYY
RHUXX
gg
gg
1
1
1
1
~
:
~
:
and
[ ] m
g
g
gg I
UN
UD
YX =
11
~~ [ ] p
g
g
gg I
Y
X
NUDU =
1
1~~~~
Hence is an r.c.f of G (s).( )UNUD gg , Hence is an l.c.f of G (s).
q.e.d. q.e.d.
Right Coprime Factorization (r.c.f.) Left Coprime Factorization (l.c.f.)
13
( )gg NUDU
~~
,
~~
14. 14
SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
We can use the coprime-factorization to find necessary and sufficient conditions
s.t. the Linear Time-Invariant System G(s), K(s) is internally stable. The definition of
an internal stable system is:
( )
( ) ( )
( ) ( )
∞−−
−−
∈
++
+−+
= RH
GKIKGIK
GKIGKGI
sH
mp
mp
eu 11
11
:
Suppose we have any coprime-factorization of G(s) and K(s):
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )mxpkmxmkpxpkmxpkmxp
pxmgpxpgmxmgpxmgpxm
sNsDsDsNsK
sNsDsDsNsG
~~
~~
11
11
−−
−−
==
==
The corresponding generalized Bezout identities are:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
=
−
−
=
−
− p
m
gg
gg
gg
gg
gg
gg
gg
gg
I
I
sDsN
sYsX
sXsN
sYsD
sXsN
sYsD
sDsN
sYsX
0
0
~~
~~
~~
~~
00
0
0
0
000
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
=
−
−
=
−
−
p
m
kk
kk
kk
kk
kk
kk
kk
kk
I
I
sXsN
sYsD
sDsN
sYsX
sDsN
sYsX
sXsN
sYsD
0
0
~~
~~
~~
~~
0
00000
0
0
17. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
We found for:
( ) gkkgpp DDNNIKGI
~~ 1
2
1
1
1 −−−
∆=∆−=+
( ) gkkgp DNNDKGIK
~~ 1
2
1
1
1 −−−
∆=∆=+
( ) gkmkgp NNIDDGKI
~~ 1
1
1
1
1 −−−
∆−=∆=+
( ) gkkgm NDDNGKIG
~~ 1
2
1
1
1 −−−
∆=∆=+
[ ]
[ ] ∞
∞
∈
=+=∆
∈
=+=∆
RH
N
D
NDNNDD
RH
N
D
NDNNDD
g
k
kgkgkg
g
g
kkgkgk
~~~~
:
~~~~
:
2
1
=
−
∆
∆
−
−
−
p
m
kg
kg
kg
kg
I
I
DN
ND
DN
ND
0
0
~~
~~
0
0
1
2
1
1
Rearrange those Equations in Matrix Form
∆
∆
=
−
− 2
1
0
0
~~
~~
kg
kg
kg
kg
DN
ND
DN
ND
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )sNsDsDsNsK
sNsDsDsNsG
kkkk
gggg
~~
~~
11
11
−−
−−
==
==
Also we have:
17
18. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
We found for:
( ) gkkgpp DDNNIKGI
~~ 1
2
1
1
1 −−−
∆=∆−=+
( ) gkkgp DNNDKGIK
~~ 1
2
1
1
1 −−−
∆=∆=+
( ) gkmkgp NNIDDGKI
~~ 1
1
1
1
1 −−−
∆−=∆=+
( ) gkkgm NDDNGKIG
~~ 1
2
1
1
1 −−−
∆=∆=+
[ ]
[ ] ∞
∞
∈
=+=∆
∈
=+=∆
RH
N
D
NDNNDD
RH
N
D
NDNNDD
g
k
kgkgkg
g
g
kkgkgk
~~~~
:
~~~~
:
2
1
From those Equations we can write:
( )
( ) ( )
( ) ( )
[ ]
[ ]gg
k
k
mgkmgk
gkgk
kk
g
gp
kgkg
kgkgp
mp
mp
eu
ND
N
D
INNIDN
NDDD
DN
D
NI
DDND
DNNNI
GKIKGIK
GKIGKGI
sH
~~
0
00
~~
~~
~~
00
0
~~
~~
:
1
21
2
1
2
1
2
1
2
1
11
1
1
1
1
1
1
1
11
11
−∆
+
=
∆−∆
∆−∆
=
∆
−
+
=
∆∆
∆−∆−
=
++
+−+
=
−
−−
−−
−
−−
−−
−−
−−
18
19. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
We found for:
[ ]
[ ] ∞
∞
∈
=+=∆
∈
=+=∆
RH
N
D
NDNNDD
RH
N
D
NDNNDD
g
k
kgkgkg
g
g
kkgkgk
~~~~
:
~~~~
:
2
1
( ) [ ] [ ]gg
k
k
m
kk
g
gp
eu ND
N
D
I
DN
D
NI
sH
~~
0
00~~
00
0 1
2
1
1 −∆
+
=∆
−
+
= −−
Theorem 2
The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are( ) ∞∈ RHsHeu
∞
−−
∞ ∈∆∆∈∆∆⇔∆⇔∆ RHRH 1
2
1
12121 ,&,UnimodularUnimodular
Proof Theorem 2
(1) If Δ1(s) (or Δ2(s) ) is Unimodular; i.e. Δ1(s)ϵRH∞ and Δ1(s)-1
ϵRH∞ , then from the
Equation above we can se that Heu(s) ϵRH∞
19
20. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
We found for:
Theorem 2
The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are
( ) [ ] [ ]gg
k
k
m
kk
g
gp
eu ND
N
D
I
DN
D
NI
sH
~~
0
00~~
00
0 1
2
1
1 −∆
+
=∆
−
+
= −−
( ) ∞∈ RHsHeu
∞
−−
∞ ∈∆∆∈∆∆⇔∆⇔∆ RHRH 1
2
1
12121 ,&,UnimodularUnimodular
Proof Theorem 2 (continue)
(2) Use
and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by
( ) ( )[ ] ( )
( ) m
g
g
gg I
sD
sN
sXsY =
−
− 00
~~
( ) ( )[ ] ( )
( ) m
k
k
kk I
sX
sY
sDsN =
0
0~~
( ) ( )[ ]sXsY gg 00
~~
−
( )
( )
sX
sY
k
k
0
0
[ ] ( ) ( ) ∞∞
−
∈⇔∈
−
−=∆ RHsHRH
Y
X
I
sHYX eu
g
g
m
eukk
0
0
00
1
2
0
00~~
[ ] ( ) ( ) ∞∞
−
∈⇔∈
−−=∆ RHsHRH
X
YI
sHXY eu
k
kp
eugg
0
0
00
1
1
00
0~~
Use
and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by
[ ] p
k
k
kk I
N
D
YX =
00
~~
[ ] p
g
g
gg I
Y
X
ND =
−
−
0
0~~
[ ]00
~~
kk YX
− 0
0
g
g
Y
X
q.e.d.
20
21. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Corollary
( ) Unimodular~~
~~
Unimodular
−
⇔
−
⇔∈ ∞
kg
kg
kg
kg
eu
DN
ND
DN
ND
RHsH
Proof
=
−
∆
∆
−
−
−
p
m
kg
kg
kg
kg
I
I
DN
ND
DN
ND
0
0
~~
~~
0
0
1
2
1
1
−
∆
∆
=
−
−
−−
kg
kg
kg
kg
DN
ND
DN
ND
~~
~~
0
0
1
2
1
1
1
∆
∆
−
=
− −
−
−
1
2
1
1
1
0
0
~~
~~
kg
kg
kg
kg
DN
ND
DN
ND
We found :
Therefore:
( ) ( )
( ) ( ) ∞
−
∞
−
∞
−
∞
−
∞
−
∞
−
∈
−
⇔∈∆∈∆
∈
−
⇔∈∆∈∆
RH
DN
ND
RHsRHs
RH
DN
ND
RHsRHs
kg
kg
kg
kg
1
1
1
1
1
1
1
1
1
1
~~
~~
&
&
q.e.d.
21
22. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Theorem 3
The set of all Proper Linear Compensators K (s)mxp achieving
Internal Stability is given by:
( ) ( )( ) ( ) ( )
( ) ( ) 1
0
1
0
1
0
1
00
0
1
0
1
00
~
~~~~
−−−−
−−
+−=
−+=+−=
gggpggg
gggggggg
XQNXIQXXY
DQYNQXQNXQDYsK
where Q (s) ϵ RH∞ is a free parameter and 0
~~
det&0det 00 ≠+≠+ gggg NQXQNX
Proof Theorem 3 is given in three parts
(1) Define
ggkggk
ggkggk
DQYNQDYN
NQXDQNXD
~~
:
~
:
~~
:
~
:
00
00
−=−=
+=+=
We want to prove that are coprime, and that they internally
stabilize the System.
( ) ( )kkkk NDandND
~
,
~
,
From the definition of K (s) above is clear that: ( ) ( ) ( ) ( ) ( )sNsDsDsNsK kkkk
~~ 11 −−
==
and: ( ) ( ) ( ) ( ) ∞∈ RHsNsDsNsD kkkk
~
,
~
,,
22
23. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 1)
ggkggk
ggkggg
DQYNQDYN
NQXDQNXD
~~
:
~
:
~~
:
~
:
00
00
−=−=
+=+=
(1) To prove that defined previously satisfy the Bezout identity
start with
( ) ( )kkkk NDandND
~
,
~
,
=
−
− p
m
gg
gg
gg
gg
I
I
XN
YD
DN
YX
0
0
~~
~~
0
0
0
00
Pre-multiply by and post-multiply by
−
p
m
I
QI
0
p
m
I
QI
0
to obtain:
( )
( )
=
+
−−
−
−+
p
m
ggg
ggg
gg
gggg
I
I
QNXN
QDYD
DN
DQYNQX
0
0
~~
~~~~
0
000
Which gives by above definition:
=
−
− p
m
kg
kg
gg
kk
I
I
DN
ND
DN
ND
0
0
~~
~~
0
Bezout Identity 23
24. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 2)
(1) To prove that K (s) stabilizes the System start from the Bezout Identity
=
−
− p
m
kg
kg
gg
kk
I
I
DN
ND
DN
ND
0
0
~~
~~
0
∞
−
∈
−
=
−
RH
DN
ND
DN
ND
kg
kg
gg
kk
1
0
~~
~~
∞
−
∈
−
=
−
RH
DN
ND
DN
ND
gg
kk
kg
kg
0
1
~~
~~
According to the Corllary
( ) ∞∞
−
∞
−
∈⇒∈
−
∈
−
RHsHRH
DN
ND
RH
DN
ND
eu
gg
kk
kg
kg
1
0
1
~~
~~
&
i.e., K (s) stabilizes the System. 24
25. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 3)
(2) We want to prove that if stabilizes the System then a Q (s)
in RH∞ can be found s.t. K (s) will have the form
kkkk NDDNK
~~ 11 −−
==
( ) ( )( ) ( ) ( )
( ) ( ) 1
0
1
0
1
0
1
00
0
1
0
1
00
~
~~~~
−−−−
−−
+−=
−+=+−=
gggpggg
gggggggg
XQNXIQXXY
DQYNQXQNXQDYsK
If K (s) stabilizes the System then Δ2 is Unimodular ∞
−
∞ ∈∆∈+=∆ RHandRHNNDD kgkg
1
22
~~
:
Let calculate
( ) ( )
( ) ( )
( ) 1
2
1
2
11
22
1
1
2
11
200
1
1
20
1
0
1
20
1
00
~~~~
~~~~~~
~~
−−−−−
−−−−
−−−−
∆=∆=∆−∆=
∆−=∆−+=
∆−+=∆−+=+
kkggkgg
kgmgkgggggg
kggggkgggggg
DDDDNND
NNIDNNYNXDD
NYNDXNYDNXQNX
( ) ( )( ) ( )( ) 1
00
11
2
1
2
−−−−
+−=∆∆= QNXQDYDNsK ggggkk
Therefore
Let define Q (s) using the equation ⇒∆=− −1
20 kgg NQDY ( )1
20
1
: −−
∆−= kgg NYDQ
25
26. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 4)
(2) To complete the proof we must show that Q (s) that has been chosen is in RH∞.
Pre-multiply by and use the fact that to obtain:[ ] m
g
g
gg I
N
D
YX =
00
[ ]00 gg YX
[ ] ∞−
−
∈
∆+−
∆−
= RH
DX
NY
YXQ
kg
kg
gg 1
20
1
20
00
∆=+
∆=−
−
−
1
20
1
20
kgg
kgg
DQNX
NQDY
The chosen
Q (s) satisfies
∆+−
∆−
=
−
−
1
20
1
20
kg
kg
g
g
DX
NY
Q
N
D
In the same way we can define Q1 (s) using the equation:
gkgk
kgg
NNDD
RHNDQY
~~
:
~~~
1
1
110
+=∆
∈∆=− ∞
−
⇒ ( ) 11
101
~~~
:
−−
∆−= gkg DNYQ
26
⇒
27. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 5)
(2) We defined Q1 (s) using the equation:
gkgk
kgg
NNDD
RHNDQY
~~
:
~~~
1
1
110
+=∆
∈∆=− ∞
−
( ) 11
101
~~~
:
−−
∆−= gkg DNYQ
( ) ( )
( ) ( )
( ) ∞
−−−−−
−−−−
−−−−
∈∆=∆=−∆∆=
∆−=∆−+=
∆−+=∆−+=+
RHDDDDDNN
DNNIDNNNYDX
DNNYXNDNYXNQX
kggkggk
ggkmggkgggg
ggkggggkgggg
~~~
~~~~
~~~~~~~~~~
1
1
11
1
1
1
1
1
11
1
11
100
11
100
11
10010
Therefore for the chosen Q1 (s) we obtain:
(2)
( ) ( ) ( )kkkk NDNDsK
~~~~ 1
1
11
1
1 −−−−
∆∆== ( ) ( )gggg DQYNQX
~~~~
10
1
10 −+=
−
We still have to prove that ( ) ( ) ∞∈= RHsQsQ1
( ) ( ) ∞∈= RHsQsQ1
( ) kgggggkgggg NDYDDDNYDDQD
~~~~~~
:
~ 1
10
11
101
−−−
∆−=∆−=
Use the fact that gggg DYYD
~~
00 =
Bezout Id.
( ) gggkggkgggg DQDDNYDNDYDQD
~~~~~ 1
20
1
201 =∆−=∆−= −−
gkkg DNND
~~ 1
2
1
1
−−
∆=∆and
27
28. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 6)
(3) The last part of the proof is to show that K (s) has the form
( ) ( ) ( ) 1
0
11
0
1
0
1
00
~ −−−−−
+−= gggpggg XQNXIQXXYsK
Use the Inversion Matrix Lemma
( ) ( )( ) ( ) ( )[ ]1
0
11
0
1
0
1
00
1
00
−−−−−−
+−−=+−= gggmggggggggg XQNXQINXXQDYQNXQDYsK
( )
( )
( )
( )
( )
1
0
11
0
1
0
1
0
11
0
1
00
1
0
1
00
11
0
11
0
11
0
−
+−
+
−−−−
+
−−−−−
−−
−−−−
+++−−= g
QNXII
QNXIpQ
ggmgggg
QNXIQ
ggmggggggg XQNXQINXQDXQNXQINXYXQDXY
ggpp
ggpggp
( ) ( )[ ] 1
0
11
0
1
0
11
0
1
00
1
0
1
00
−−−−−−−−−
+−++−−= gggmpggggpggggggg XQNXIIQDXQNXIQNXYXQDXY
( ) ( )[ ] 1
0
11
0
1
0
11
0
1
00
1
0
1
00
−−−−−−−−−
+−++−−= gggppggggmggggggg XQNXIIQDXQNXIQNXYXQDXY
( ) ( ) ( ) 1
0
11
0
1
00
1
00
−−−−−
++−= gggpgggggg XQNXIQNXYDXYsK 28
( ) ( ) mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC ××
−
×××××××
−
×
−
××
−
× +−=+
1111
29. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 7)
(3) We want to prove is to show that K (s) has the form
( ) ( ) ( ) 1
0
11
0
1
0
1
00
~ −−−−−
+−= gggpggg XQNXIQXXYsK
( ) ( ) ( ) 1
0
11
0
1
00
1
00
−−−−−
++−= gggpgggggg XQNXIQNXYDXYsK
We found
1
000
1
0
1
000
~~~~~ −−−
+=+=⇒=+ ggggggggmgggg XYDNYXDXINYDX
Therefore:
( ) ( ) ( ) 1
0
11
0
1
0
1
00
~ −−−−−
+−= gggpggg XQNXIQXXYsK
Since 0
1
0
1
00
~~
gggg YXXY
−−
=
( ) ( ) ( ) 1
0
11
0
1
00
1
0
~~~ −−−−−
+−= gggpggg XQNXIQXYXsK
Uze Bezout Identity: 0
1
0
1
00
~~
gggg YXXY
−−
=
q.e.d.
29
30. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Proof Theorem 3 (continue - 8)
(3) We have shown that
( ) ( ) ( ) ( ) ( ) 1
0
11
0
1
00
1
0
1
0
11
0
1
0
1
00
~~~~ −−−−−−−−−−
+−=+−= gggpggggggpggg XQNXIQXYXXQNXIQXXYsK
This is equivalent to
ff
f
ggg
ggg
f
yQe
e
e
NXX
XXY
y
y
=
−−
=
−−
−−
1
1
0
1
0
1
0
1
00
1
~
To prove this, let develop
( ) 1
1
0
11
0
1
01
1
0 eXQNXIyyQNXeXy gggffgggf
−−−−−
+−=⇒−−=
( )[ ] ( ) 11
1
0
11
0
1
0
1
00
1
01
1
001
~~
esKeXQNXIQXXYeXeXYy ggggggfggg =+−=+=
−−−−−−−
( ) 1
1
0
11
0 eXQNXIQyQe gggff
−−−
+−==⇒
q.e.d.30
31. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
Theorem 4 (Dual of Theorem 3)
The set of all Proper Linear Time-Invariant Systems G (s) pxm stabilized by the
Controller is given by:
( ) ( )( ) ( ) ( )
( ) ( ) 1
0
11
0
1
0
1
00
0
1
0
1
00
~
~~~~
−−−−−
−−
+−=
−+=+−=
kkkpkkk
kkkkkkkk
XSNXISXXY
DSYNSXSNXSDYsG
where S (s) ϵ RH∞ is a free parameter and 0
~~
det&0det 00 ≠+≠+ kkkk NSXSNX
Proof Theorem 4
The duality of Theorem 4 to Theorem 3 is evident because by replacing in
Theorem 3 g to k and Q (s) to S (s), K (s) to G (s) we obtain Theorem 4. Therefore
the proof is similar, by performing the above mentioned replacement and
interchanging between G (s) and K (s).
31
( ) ( ) ( ) ( ) ( )sNsDsDsNsK kkkk
~~ 11 −−
==
32. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
32
From
we obtain
( ) ( )( ) ( ) ( )gggggggg DQYNQXQNXQDYsK
~~~~
0
1
0
1
00 −+=+−=
−−
( ) ( )
( ) ( )KNDKXYQDQYKNQKX
XKYNKDQQDYQNKXK
gggggggg
gggggggg
+−=⇒−=+
−+=⇒−=+
−
~~~~~~~
0000
00
1
00
( ) ( ) ( ) ( )KNDKXYXKYNKDQ gggggggg +−=−+=
− ~~~
0000
1
From
we obtain
( ) ( )( ) ( ) ( )kkkkkkkk DSYNSXSNXSDYsG
~~~~
0
1
0
1
00 −+=+−=
−−
( ) ( )
( ) ( )GNDGXYSDSYGNSGX
XGYNGDSSDYSNGXG
kkkkkkkk
kkkkkkkk
+−=⇒−=+
−+=⇒−=+
−
~~~~~~~
0000
00
1
00
( ) ( ) ( ) ( )GNDGXYXGYNGDS kkkkkkkk +−=−+=
− ~~~
0000
1
Go back to Q (s)
Realization
33. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
33
Let calculate Δ1 (s) and Δ2 (s) for
( ) ( )( ) ( ) ( )
kkkk N
gg
D
gg
D
gg
N
gg DQYNQXQNXQDYsK
~
0
1
~
0
1
00
~~~~
−+=+−=
−−
( ) ( ) ( )
( ) ( ) mgggg
I
gggg
gggggggkgk
INDDNQNYDX
NDQYDNQXNNDDs
m
=−++=
−++=+=∆
0
00
001
~~~~
~~~~~~
( ) ( ) ( )
( ) ( ) pgggg
I
gggg
ggggggkgkg
IQDNNDYNXD
QDYNQNXDNNDDs
p
=−++=
−++=+=∆
0
00
002
~~~~
~~~~
We can see that Δ1 (s) and Δ2 (s) are Unimodular and according to Theorem 2
Heu(s) is Stable for this choise of K (s).
34. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
34
Let calculate Heu(s) for
( ) ( )( ) ( ) ( )
kkkk N
gg
D
gg
D
gg
N
gg DQYNQXQNXQDYsK
~
0
1
~
0
1
00
~~~~
−+=+−=
−−
( ) [ ] [ ]gg
k
k
m
kk
g
gp
eu ND
N
D
I
DN
D
NI
sH
~~
0
00~~
00
0 1
2
1
1 −∆
+
=∆
−
+
= −−
we obtained
( ) ( )[ ]
( )
( ) [ ]gg
gg
gg
m
gggg
g
gp
ND
QDY
QNX
I
NQXDQY
D
NI ~~
0
00~~~~
00
0
0
0
00 −
−
+
+
=+−
−
+
=
[ ] [ ]
[ ] [ ] ∞∈−
−
−−
+
=
−
−
−
−
+
=
RHNDQ
D
N
ND
Y
X
I
NDQ
D
N
XY
D
NI
gg
g
g
gg
g
g
m
gg
g
g
gg
g
gp
~~~~
0
00
~~~~
00
0
0
0
00
35. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Transfer Function of a L.T.I. System:
( ) ( ) pxmnxmnxnnpxnpxm DBAIsCsG +−=
−1
State Space Realization of the System:
( ) 00 xx
u
x
DC
BA
y
x
pxmpxn
nxmnxn
=
=
Eigenvalues (modes) of the System:
The Eigenvalues (modes) of the System are the collection of the n complex numbers
λ such that:
( ) nAIsrank nxnn <−
Stability of the System:
The System is stable if ( ) ii ∀λReal
35
36. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Transmission Zeros of the L.T.I. System:
( ) ( ) ( ) 0,000exp 0 >∀=≠≠= ttyhavewegandxsomeforandtzgtuFor
The following definitions are equivalent:
(1) The Transmission Zeros of the System are the collection of the complex numbers z s.t.:
(2) The Transmission Zeros of the System are the collection of the complex numbers z s.t.:
( )pmn
DC
BAIz
rank
pxmpxn
nxmnxnn
,min+<
−−
36
37. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Controllability:
The System, or the pair (A,B) is
Controllable if, for each time tf > 0 and
given state xf, there exists a continuous
input u (t) for t ϵ [t0,tf] s.t. x(tf) = xf.
Observability:
The System, or the pair (A,C) is
Observable if, for each time tf > 0 the
function y(t), t ϵ [t0,tf] uniquely
determines the initial state x0.
(1) (A,B) is Controllable (1) (C,A) is Observable
(2) The matrix [B AB … An-1
B] has
independent rows
−1n
AC
AC
C
(2) The matrix
has independent
columns
(3) The matrix [A – λI B ] has
independent rows
(This is so called P.B.H. test
- Popov-Belevitch-Hautus test)
C∈∀λ
(3) The matrix
has independent columns
(This is so called P.B.H. test)
−
C
IA λ
C∈∀λ
(4) The eigenvalues of (A+BF) can be
freely assigned by suitable choice
of F (state feedback)
(4) The eigenvalues of (A+HC) can be
freely assigned by suitable choice
of H (output injection feedback)
(5) [BT
,AT
] is Observable (5) [AT
,CT
] is Controllable
37
38. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Un-Controllability:
The modes λ of A for which [A – λI B ]
loses rank are called uncontrollable
modes. All other modes are
controllable
Un-Observability:
(1) (A,B) is Stabilizable (1) (C,A) is Detectable
(2) Exists F s.t. (A+BF) is stable (2) Exists H s.t. (A+HC) is stable
(3) The matrix [A - λI B] has
independent rows +∈∀ Cλ
−
C
IA λ
The modes λ of A for which
loses rank are called uncontrollable
modes. All other modes are
controllable
(3) The matrix
has independent columns +∈∀ Cλ
−
C
IA λ
Stabilizability: Detectability:
The Poles of the System are all the modes that are both Controllable and Observable.
38
39. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Transfer Function of a L.T.I. System:
( ) ( ) pxmnxmnxnnpxnpxm DBAIsCsG +−=
−1
(A,B,C,D) is called a Realization of the L.T.I. System. The Realization is
Minimal if n is the minimal possible degree. That happens if and only if
(A,B) is Controllable and (C,A) is Observable.
Doyle and Chu introduced the following notation:
( ) ( )
−−−−−−−=+−=
−
pxmpxn
nxmnxn
pxmnxmnxnnpxnpxm
DC
BA
DBAIsCsG
|
|
1
39
40. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Operations on Linear Systems
rnonsingula~
rnonsingula~
rnonsingula~
NzNz
RyRy
TxTx
mxm
pxp
nxn
=
=
=
Suppose we make the following change of variables:
( ) N
NDRTCR
NBTTAT
R
DC
BA
sG pxm
−−−−−−−−−=
−−−=
−−
−−
−
11
11
1
|
|
|
|
Change of Variables:
Then:
=
=
=
=
−−
−−
−
−
u
x
NDTTCR
NBTTAT
u
x
N
T
DC
BA
R
T
u
x
DC
BA
R
T
y
x
R
T
y
x
~
~
~
~
0
0
0
0
0
0
0
0
~
~
11
11
1
1
40
41. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Operations on Linear Systems
Usefull Notation:
†
Suppose we have a partition:
( )
( ) ( )
INIR
I
TI
T
I
TI
T
rn
r
rn
r rnrx
==
−
=⇒
=
−
−
−
−
,,
00
111
( )
( )
( ) ( ) ( ) ( )
( )
−−−−−−−−−−−−−−
=
−
−−−−
−
pxmDCC
BAA
BAA
sG
rnpxpxdr
xmrnrnxrnxrrn
rxmrnrxrxr
|
|
|
21
22221
11211
Then:
Suppose we want to Change Variables according to:
=
−rn
r
n
x
x
x
( )
+−
−−−−−−−−−−−−−−−−−−−−−−−−−−−
−
+−−++
=
−−−−−−−=
−−−=
−
−
DCTCC
BTAAA
BTBTATTAATAATA
DTC
BTTAT
DC
BA
sG pxm
|
|
|
|
|
|
|
2111
21212221
21112111112211221111
1
1
( )
( )
221
1211
1
1
,,
0 columncolumnTcolumn
rowrowTrow
rn
r
INIR
I
TI
T
rnrx
→+•−
→•+
−
→⇔
==
=
−
41
42. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Operations on Linear Systems
We have:
Cascade of Two Linear Systems G1(s)G2(s):
Then:
uuyyyu
u
x
DC
BA
y
x
u
x
DC
BA
y
x
===
=
=
2121
2
2
22
22
2
2
1
1
11
11
1
1
=
1
2
1
21211
22
21211
1
2
1
0
u
x
x
DDCDC
BA
DBCBA
y
x
x
=
1
1
2
21121
21121
22
1
1
2 0
u
x
x
DDCCD
DBACB
BA
y
x
x
or:
From which:
( ) ( )
−−−−−−−−−
=
−−−−−−−−−
=
−−−−
−−−−=
21121
21121
22
21211
22
21211
22
22
11
11
21
|
|
|0
|
|0
|
|
|
|
|
DDCCD
DBACB
BA
DDCDC
BA
DBCBA
DC
BA
DC
BA
sGsG
to ( ) ( ) 1−
sDsN gg
42
43. 43
SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Operations on Linear Systems
Para-Hermitian:
†
( )
−−−−−−−−−−
−−
=
††
††
†
|
|
|
DCD
BDCBDA
sG mxp
where is the pseudoinverse (not uniquely defined if p≠m) of D.
†
D
( ) ( ) mpxmmxp
mpxmmxp
IsGsG
IDDmpif
=
=⇒>
†
†
& ( ) ( ) ppxmmxp
pmxppxm
IsGsG
IDDmpif
=
=⇒<
†
†
&
Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) is:
†
( ) ( ) ( )
−
−−−−−−−
−
=+−−=−=
−
TT
TT
TTT
n
T
mxp
T
mxp
H
DB
CA
DCAIsBsGsG
|
|
:
1
( ) ( ) ppxppxp
ppxppxp
pxp
IsGsG
IDD
DDmpif
=
=⇒
== −
1-
1-
1†
&
( ) 1−
sDgto
44. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Operations on Linear Systems
Proof for p ≥ m
( )
−−−−−−−−−−
−−
=
††
††
†
|
|
DCD
BDCBDA
sG mxp
Pseodoinverse of G (s) for rank Dpxm=min (p,m) is G(s) is:
†
( ) ( )
−−−−−−−−−−−−−
−−−
=
−−−
−−−−−−−−−−
−−
=
m
pxmmxp
ICDCD
BA
BCBDCBDA
DC
BA
DCD
BDCBDA
sGsG
|
|0
|
|
|
|
|
††
††
††
††
†
Change of Variables
INIR
I
II
T
I
II
T
n
nn
n
nn
==
−
=⇒
= −
,,
00
1
( ) ( ) ( ) ( ) mmnn
m
pxmmxp IIBAsICBDAsICD
ICD
BA
CBDA
sGsG =+−++−=
−−−−−−−−−
−
=
−−
0
1
0
1††
†
†
†
00
|0
|0
0|0
Then:
q.e.d.
44
45. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Operations on Linear Systems
since is not uniquely defined if p≠m, is not uniquely defined. Moreover it
is easy to check that
( )
( )
( )
−−
−−−−−−−−−−−−−−−−−−−
−−+−
=
†
1
††
†
1
††
†
|
|
|
DFDDICD
BDFDDIBCBDA
sG
m
m
mxp
†
D
Pseodoinverse of G(s) for rank Dpxm=min (p,m) is G(s) is:
†
( )sG†
( )
( ) ( )
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−+−−+−
=
††
†
1
††
1
†
†
|
|
|
DCD
DDIHBDCDDIHCBDA
sG
pp
mxp
and
Are pseudoinverses of G (s), where and are any matrices of the
given dimensions.
mxn
F1 nxp
H1
45
46. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
State-Space Realization of All Coprime Matrices
†
We have any coprime-factorization of G(s): ( ) ( ) ( ) ( ) ( )sNsDsDsNsG gggg
~~ 11 −−
==
The corresponding generalized Bezout identities are:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
=
−
−
=
−
− p
m
gg
gg
gg
gg
gg
gg
gg
gg
I
I
sDsN
sYsX
sXsN
sYsD
sXsN
sYsD
sDsN
sYsX
0
0
~~
~~
~~
~~
0
0
0
0
0
0
0
0
Nett, Jacobson and Balas gave the following State-Space Realizations of those Matrices:
( ) ( )
( ) ( )
( )
( )
+
−−−−−−−−−−−−−
−+
=
−
−
−
1
1
0
0
|
0|
|
|
WDZDFC
ZF
HWBZBFA
sXsN
sYsD
gg
gg
( ) ( )
( ) ( )
( ) ( )
−
−−−−−−−−−−−−−−
+−+
=
−
−−
WWDWC
ZFZ
HHDBHCA
sDsN
sYsX
gg
gg
|
0|
|
|
~~
~~
11
0
0
( ) ( ) ( ) ( ) ( )sNsDsDsNsG gggg
~~ 11 −−
==
where:
Fmxn is any Matrix s.t. (A+BF) is stable for (A,B) Stabilizable
Hnxp is any Matrix s.t. (A+HC) is stable for (C,A) Detectable
s.t.
Z and W are any Nonsingular Matrices
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∞∈RHsYsXsNsDsYsXsNsD gggggggg
~
,
~
,
~
,
~
,,,, 46
51. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Coprime Factorization Example †
( ) ( )0,0.,.
1
21
21
2
>>∈
++
= ∞ aaeiRH
asas
sG
The Observability Canonical State-Space Realization of G (s) (see Kailath, Linear
Systems, pg. 41) is:
−−
=
u
x
x
aa
y
x
x
2
1
122
1
001
1
010
We choose [ ] 1,
0
,0 ==
−
=−= WZ
h
HfF
( ) ( )
( ) ( )
−
−−−−−−−−−−−−
−−−
=
−
10|01
01|0
|
1|
00|10
12
0
0
f
hafa
sXsN
sYsD
gg
gg ( ) ( )
( ) ( )
−
−−−−−−−−−−−−
−−−−−
=
−
10|01
01|0
|
1|
00|10
~~
~~ 12
0
0
f
haha
sDsN
sYsX
gg
gg
51
52. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems - State-Space Approach
Coprime Factorization Example (continue) †
( ) ( )0,0.,.
1
21
21
2
>>∈
++
= ∞ aaeiRH
asas
sG
By developing the previous Matrix representations we obtain
( )
( )
( )
( )
fasas
hf
sY
fasas
hfasas
sX
fasas
sN
fasas
asas
sD
g
g
g
g
+++
=
+++
++++
=
+++
=
+++
++
=
21
20
21
2
21
2
0
21
2
21
2
21
2
1
( )
( )
( )
( )
hasas
hf
sY
hasas
hfasas
sX
hasas
sN
hasas
asas
sD
g
g
g
g
+++
=
+++
++++
=
+++
=
+++
++
=
21
20
21
2
21
2
0
21
2
21
2
21
2
~
~
1~
~
We can see that are in RH∞ for every f >-a2, and
that are in RH∞ for every h >-a2.
( ) ( ) ( ) ( ),,,, sYsXsNsD gggg
( ) ( ) ( ) ( )sYsXsNsD gggg
~
,
~
,
~
,
~ 52
53. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
We have shown that
( ) ( ) ( ) ( ) ( ) 1
0
11
0
1
00
1
0
1
0
11
0
1
0
1
00
~~~~
00
−−−−−−−−−−
+−=+−= gggpg
K
gggggpg
K
gg XQNXIQXYXXQNXIQXXYsK
This is equivalent to
( )
ff
f
sJ
ggg
ggg
f
yQe
e
e
NXX
XXY
y
y
=
−−
=
−−
−−
1
1
0
1
0
1
0
1
00
1
~
53
We want to find the State-Space Realization of K0 (s) and J (s)
( )
+
−−−−−−−−−−−−−−−−−
−+++
−−−−−−−
+
=
+
−−−−−−−−−
−+
−−−−−−−
+
==
−
−
pp
gg
IFDC
HFDHCHFBA
F
HFBA
IFDC
HFBA
F
HFBA
XYsK
|
|
0|
|
|
|
0|
|
1
1
000
54. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
54
We want to find the State-Space Realization of K0 (s)
( )
+
−−−−−−−−−−−−−−−−−
+++
−−−−−−−
+
=
+
−−−−−−−−−
−+
−−−−−−−
+
==
−
−
pp
gg
IFDC
HFDHCHFBA
F
HFBA
IFDC
HFBA
F
HFBA
XYsK
|
|
0|
|
|
|
0|
|
1
1
000
( )
−−−−−−−−−−−−−−−−
+++
+
−−−−−−−−
+
=
−−−−−−−−−−−−−−−−−−−−−−
+++
+
=
→
−−−−−−−−−−−−−−−−−−−−−−
+++
++
=
→+
→−
0|
|
0|
0|
0|
|0
0|0
0|0
|0
|
0
221
121
F
HFDHCHFBA
F
FBA
FF
HFDHCHFBA
FBA
F
HFDHCHFBA
HFDCHFBA
VariablesChange
columncolumncolumn
rowrowrow
55. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
55
We want to find the State-Space Realization of K0 (s)
( )
−−−−−−−−−−−−−−−−
+++
==
−
0|
|
1
000
F
HFDHCHFBA
XYsK gg
K0 (s) given above is the transfer Matrix of the Estimator-Regulator Compensator
(E.R.C.) , well known from the LQG Design Method, given in Figure bellow:
57. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
57
We want to find the State-Space Realization of ( )
−−
=
−−
−−
ggg
ggg
NXX
XXY
sJ
1
0
1
0
1
0
1
00
~
( ) ( ) ( )
−−−−−−−−−−−−−−−−−−
++++
=
−−−−−−−−−−−−−
+−+
=
−
−
mm
g
IF
DHBFDHBCHA
IF
DHBCHA
X
|
|
|
|
~
1
1
0
Finally
58. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
58
We want to find the State-Space Realization of ( )
−−
=
−−
−−
ggg
ggg
NXX
XXY
sJ
1
0
1
0
1
0
1
00
~
( ) ( )
−−−−−−−−−−−−−−−−−−
++++
=
−
m
g
IF
DHBFDHBCHA
X
|
|
~ 1
0
We found
( )
( )
−+−
−−−−−−−−−−−−−−−−−−
++++
=−
−
DFDC
DHBFDCHFBA
NX gg
|
|
1
0
−+
−−−−−−−−−−−−−−−−−
−+++
=−
−
p
g
IFDC
HFDHCHFBA
X
|
|
1
0
−−−−−−−−−−−−−−−−
+++
=
−
0|
|
1
00
F
HFDHCHFBA
XY gg
Therefore
( )
( )
−−+−
−−−−−−−−−−−−−−−−−−−−−−−
++++
=
−−
=
−−
−−
DIFDC
IF
DHBHFDHCHFBA
NXX
XXY
sJ
p
m
ggg
ggg
|
0|
|~
1
0
1
0
1
0
1
00
59. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
The Equivalence Between Any Stabilizing Compensator
and the Observer Based Compensator
59
We found
( )
( )
−−+−
−−−−−−−−−−−−−−−−−−−−−−−
++++
=
−−
=
−−
−−
DIFDC
IF
DHBHFDHCHFBA
NXX
XXY
sJ
p
m
ggg
ggg
|
0|
|~
1
0
1
0
1
0
1
00
( )
−−+−
++++
=
11
0
y
e
x
DIFDC
IF
DHBHFDHCHFBA
u
u
x e
p
m
e
We can see that
the realization of
K (s) consists of
Estimator
Regulator
Compensator
plus the feedback
Q (s).
60. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
60
The Eigenvalues of Closed-Loop System
61. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
61
The Eigenvalues of Closed-Loop System
The State-Space Equation of the System are
( )sQ
uDxCy
uBxAx
Estimator
uDxCy
uHuBxAx
System
uDxCy
uBxAx
q
qq
ee
ee
+=
+=
+=
−+=
+=
+=
11
1
1
( ) ( ) ( )
( ) rDxCxCDFxCDyxFu
rxCxC
ruDxCuDxCyyru
qeeeqqe
e
ee
−+−+=+=
−−=
−+−+=−−−=
1
1
( ) ( )
−−+
−−
−−+++−
−−+
=
r
x
x
x
DDCDCDFDCDDI
BACBCB
DBHCBCDBCHFBACDBCH
DBCBCDBFBCDBA
y
x
x
x
q
e
qqqqp
qqqq
qqqq
qqqq
q
e
with
62. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
62
The Eigenvalues of Closed-Loop System
( ) ( )
VariablesChange
columncolumncolumn
rowrowrow
qqqqp
qqqq
qqqq
qqqq
DDCDCDFDCDDI
BACBCB
DBHCBCDBCHFBACDBCH
DBCBCDBFBCDBA
221
121
|
|
|
|
→+
→−
→
−−+
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−−
−−+++−
−−+
( )
( )
( ) ( )
( ) ( )
−++
−−−−−−−−−−−−−−−−−−−−−−−−−
−
−++−
−+
=
qqqp
qqq
qqq
cl
DDCDFDCCDDI
BACB
DBHCBFBACDBH
HCHA
sG
|
|0
|
|
|00
We can see that the Eigenvalues of the Closed Loop System are the Eigenvalues of
(A+B F), (A+H C) and Aq. We can see the Separation between the Eigenvalues of the
Regulator, Estimator and Q (s). The separation of Eigenvalues of Q (s) is due to the fact
that the Input to Q (s) is from the Estimator Error, also Input to Estimator.
63. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
63
The Transfer Function of the Compensator K (s) with O.B.C. Realization
We want to compute the Transfer Matrix
of K (s) using the State-Space Representation
euDxCu
uDxCxFu
uBxAx
uHuBxAx
e
qqqe
qq
ee
−−−=
++=
+=
−+=
1
1
1
1
From the last two equations
( )[ ]
( )[ ] qpkkqqkeqk
qmkqqqeqk
DDIReRxCRDxCDFRDCu
DDIReDxCxCDFRu
+=−−−+−=
+=−+−=
−−−
−
:
:
2211
11
111
1
1
( )
( ) ( ) ( ) ( ) ( )
( )
( )
−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−−+−
−+−+++
=
−−−
−−−
−−−
111
111
111
211
212
211
|
|
|
kqqkqk
kqqkqqkq
kqqkqk
RDCRCDFR
RBCRDBAFDCRB
RDBHCRDHBCDFRDHBCHA
sK
Therefore we obtain:
64. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
64
The Transfer Function of the Compensator K (s) with O.B.C. Realization
( )
( ) ( ) ( ) ( ) ( )
( )
( )
−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
−−+−
−+−+++
=
−−−
−−−
−−−
111
111
111
211
212
211
|
|
|
kqqkqk
kqqkqqkq
kqqkqk
RDCRCDFR
RBCRDBAFDCRB
RDBHCRDHBCDFRDHBCHA
sK
We obtained:
It is easy to prove that:
( ) ( ) ( )
( ) ( ) ( )FDCRDBHFBA
CDFRDHBCHA
kq
qk
+−++=
−+++
−
−
1
1
2
1
Therefore a certain duality exists between
(A+B F) and (A+H C), (B+H D) and (C+D F),
(F-DqC) and (H-B Dq), respectively.
The Realization of K (s) has n + nq States, where nq is number of states of Q (s).
But this realization may not be minimal.
65. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
65
Realization of Q (s) Given G (s) and K (s)
Suppose we have a Minimal State-Realization of a Compensator ( )
−−−−−−−=
kk
kk
DC
BA
sK
|
|
|
We found that
[ ]
( ) ( )
( )
( )
( )
( )
( )
( )
+++
−−−−−−−−−−−−−−−−−−−
+
+
=
+
−−−−−−−−−−−−−−
+
+
+
−−−−−−−−
+
=
+
−−−−−−−
+
−−−−−−−+
−−−−−−−−
+
=+
DDIFDCDFC
BBFA
DBFDCBA
DDFDCDC
BBFA
DBFDCBA
IF
BZBFA
DDFC
BBFA
DC
BA
IF
BZBFA
NKD
kmkk
kkk
kkk
kkk
m
kk
kk
m
gg
|
|
|0
|
|
|
|0
|
|
|
|
|
|
|
|
|
|
|
|
|
[ ]
+=
+=
+
−−−−−−−−−−−−−−−−−−−−−−−
−−
−
=+
−−−
−−
−−
−
kp
km
kk
kk
kkkkk
gg
DDIR
DDIR
RCDRFCR
BCDRBACRB
DBCRBCRDBA
NKD
:
:
|
|
|
|
2
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
( ) ( )00
1
gggg XKYNKDQ −+=
−
66. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
66
Realization of Q (s) Given G (s) and K (s)
Suppose we have a Minimal State-Realization of a Compensator ( )
−−−−−−−=
kk
kk
DC
BA
sK
|
|
|
We found that
( )
( ) ( )
( )
( )
( )
( )
( )
( )
−++
−−−−−−−−−−−−−−−−
+
−+
=
−+
−−−−−−−−−−−−−
+
−+
−
−−−−−−−
+
=
+
−−−−−−−
+
−−−−−−−−
−−−−−−−
+
=−
kkk
kkk
kkk
kkk
pkk
kk
gg
DFDCDFC
HFBA
BFDCBA
DFDCDC
HFBA
BFDCBA
F
HBFA
IDFC
HBFA
DC
BA
F
HBFA
XKY
|
|
|0
|
|
|
|0
|
0|
|
|
|
|
|
|
|
|
0|
|
|
00
( ) ( )00
1
gggg XKYNKDQ −+=
−
67. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
67
Realization of Q (s) Given G (s) and K (s)
Suppose we have a Minimal State-Realization of a Compensator ( )
−−−−−−−=
kk
kk
DC
BA
sK
|
|
|
We found that
( ) ( )
( )
( )
−++
−−−−−−−−−−−−−−−−
+
−+
+
−−−−−−−−−−−−−−−−−−−−−−−
−−
−
=−+=
−−−
−−
−−
−
kkk
kkk
kk
kk
kkkkk
gggg
DFDCDFC
HFBA
BFDCBA
RCDRFCR
BCDRBACRB
DBCRBCRDBA
XKYNKDQ
|
|
|0
|
|
|
|
|
1
1
1
1
1
1
1
1
1
1
1
2
1
1
00
1
( ) ( )00
1
gggg XKYNKDQ −+=
−
- Perform the Multiplication
- Change Variables according to
VariablesChange
columncolumncolumn
rowrowrow
331
131
→+−
→+
→
VariablesChange
columncolumncolumn
rowrowrow
442
242
→+−
→+
→- Change Variables according to
- Delete Unobservable Modes, to obtain
( ) ( )
+
−−−−−−−−−−−−−−−−−−−−−−−−−−−
+−−
−−
=−+=
−−−
−−−
−−−
−
kkk
kkk
kkkkk
gggg
DRCDRFCR
HDRBCDRBACRB
RBCRBCRDBA
XKYNKDQ
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
2
1
1
00
1
|
|
|
|
68. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
68
Example
Suppose we have the System G (s) and the Compensator K (s)
( )
−−−−−−==
k
ksK
|0
|
0|0
( ) ( )0,0.,.
0|01
1|
0|10
1
21
12
21
2
>>∈
−−−−−−−−−
−−
=
++
= ∞ aaeiRH
aa
asas
sG
( ) ( ) k
kasas
hkfk
kfk
hkaka
DRCDRFCR
HDRBCDRBACRB
RBCRBCRDBA
Q
kkk
kkk
kkkkk
−
+++
−−
=
−−
−−−−−−−−−−−−−
−−−−
=
+
−−−−−−−−−−−−−−−−−−−−−−−−−−−
+−−
−−
=
−−−
−−−
−−−
21
2
12
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
2
1
1
|0
|
0|10
|
|
|
|
Note 1: The Poles of Q (s) are the Poles of the Closed-Loop System
Note 2: The order of the System is n = 2 and the order of Q is nq = 2. A O.B.E.
State-Space Realization of the Compensator K will have nk = n + nq = 4. But the
minimum realization of K (s) is nk = 0. This shows that the realization of K (s) by
O.B.E. is not minimal.
Using the previous result we obtain:
69. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
69
Realization of Heu (s) Given G (s) and K (s)
Suppose we have a Minimal State-Realization of a Compensator
and of the System
( )
−−−−−−−=
kk
kk
DC
BA
sK
|
|
|
( )
−
−−−−−−
−
→
−−−−−−=
=−=
=
DC
BA
DC
BA
sG
mp
n
INIR
IT
|
|
|
|
|
|
We want to find a Realization of ( )
1−
−
=
m
p
eu
IK
GI
sH
( )
−−
−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
+−−
−−
=
−−−−−−−−−−
−
=
−−−−
−−−−
−−−−
−−−−−
1
1
1
2
1
1
1
1
1
1
1
2
1
2
1
2
1
1
1
1
1
1
1
1
1
2
1
2
1
2
1
1
1
|
|
|
|
|
|0
0|0
|
0|0
0|0
RRDCDRCR
RDRCRCDR
RBHDRBCDRBACRB
DRBRBCRBCRDBA
IDC
IC
BA
BA
sH
kkk
k
kkk
kkkkkk
mkk
p
kk
eu
Note 1: We can see that Heu (s) and Q (s) have the same eigenvalues, therefore
{ } ( ){ } ( ){ }StablesQStablesHStableInternallySystem eu ⇔⇔
+=
+=
kp
km
DDIR
DDIR
:
:
2
1
70. SOLO Stabilization of Linear Time-Invariant Systems
Factorization Approach
Stabilization of Linear Time-Invariant Systems
70
Realization of Heu (s) Given G (s) and K (s)
( )
( ) ( )
( ) ( )
−−
−−
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
+−−
−−
=
++
+−+
=
−−−−
−−−−
−−−−
−−−−
−−
−−
1
1
1
2
1
1
1
1
1
1
1
2
1
2
1
2
1
1
1
1
1
1
1
1
1
2
1
2
1
2
1
1
11
11
|
|
|
|
|
RRDCDRCR
RDRCRCDR
RBHDRBCDRBACRB
DRBRBCRBCRDBA
GKIKGIK
GKIGKGI
sH
kkk
k
kkk
kkkkkk
mp
mp
eu
Note 2: If G (s) is stable, and therefore, if we choose for F and H the particular
values F = 0 and H = 0, we obtain:
( )
( )
( )
−
=
∞−
∞
= −−
−−−
∞→ 1
1
1
2
1
1
1
2
1
lim
RRD
RDR
IK
GI
sH
km
p
eu
s
( ) 1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
2
1
1
|
|
|
|
−
−−−
−−−
−−−
+−=
−−−−−−−−−−−−−−−−−−−−−−−−−
−−
−−
= KGIK
DRCDRCR
DRBCDRBACRB
RBCRBCRDBA
Q p
kkk
kkk
kkkkk
That means that for a stable G (s) is enough to check the stability of K (Ip+G K)-1
or
any other entry of Heu (s).
Note 3:
+=
+=
kp
km
DDIR
DDIR
:
:
2
1
( )
( )
( )
+=
+=
⇔
∞−
∞
⇔
−
Invertible
DDIR
DDIR
Invertible
IK
GI
PosedWell
sH
kp
km
m
p
eu
:
:
2
1
71. References
SOLO
Stabilization of Linear Time-Invariant Systems
Factorization Approach
S. Hermelin, “Robustness and Sensitivity Design of Linear Time-Invariant Systems”,
PhD Thesis, Stanford University, 1986
M. Vidyasagar, “Control System Synthesis: A Factorization Approach MIT Press, 1985”,
71
K. Zhou, J.C. Doyle, K. Glover, “Robust and Optimal Control”, Lecture Notes , 1993
B.A. Francis, “A Course in H∞ Control Theory”, Lecture Notes in Control and
Information Sciences, vol .88, Springer-Verlag, 1987
K. Zhou, “Essential of Robust Control”, Pdf Slides on Homepage at Louisiana
University, 2000
J.C. Doyle, B.A. Francis, A.R. Tannenbaum, “Feedback Control Theory”,
Macmillan Publishing Company, 1992
A. Weinmann, “Uncertain Models and Robust Control”, Springer-Verlag, 1991
D.C.McFarlane, K. Glover, “Robust Controller Design Using Normalized Coprime
Factor Plant Descriptions”, Lecture Notes in Control and
Information Sciences, vol .138, Springer-Verlag, 1990
72. 72
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
73. Jhon C. Doyle
California Institute
of Technology
Kemin Zhou
Louisiana State
University
Bruce A. Francis
University of
Toronto
Mathukumalli
Vidyasagar
University of
Texas
Keith Glover
University of
Cambridge
Pramod
Khargonekar
University of
Florida
73