Stabilization of linear time invariant systems, Factorization Approach

Stabilization of Linear Time-Invariant Systems

Factorization Approach

SOLO HERMELIN

Updated: 17.12.10

1

Table of Content

SOLO Stabilization of Linear Time-Invariant Systems


Introduction

Well-PosednessInternal 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. SystemState Space Realization of the System

Stability of the SystemTransmission Zeros of the L.T.I. System

Controllability, Observability Stabilizability, Detectability

Transfer Function of a L.T.I. System:

Table of Content (continue)



3

State-Space Realization of All Coprime Matrices

Operations on Linear SystemsChange of VariablesCascade of Two Linear Systems G1(s)G2(s)

Para-HermitianPseodoinverse of G (s) for rank Dpxm=min (p,m) is G (s)

†

The Equivalence Between Any Stabilizing Compensator and the Observer Based CompensatorThe Eigenvalues of Closed-Loop SystemThe 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

• Assume a Linear Time-Invariant Plant, not necessary stable with m inputs and p outputs.

SOLO Stabilization of Linear Time-Invariant SystemsFactorization 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:

sGslim ( G(s) is analytic as s →∞ )

Therefore G (s) is in the Ring of Real-rational proper (Rp) matrices:

pxmpRsG

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 parameterizationof all compensators K (s)mxp that stabilize the given plant (and conversely all plants that can be stabilized by a given compensator). This parameterization willallow to define methods of optimizing the desired performances of the design overall possible stabilizing compensators. 4


sK sG 2e

1u2u

1e0r

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

usH

u

u

IsK

sGI

e

e

mp

mp

eummxp

pxmp

5


sK sG 2e

1u2u

1e0rWell-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

usH

u

u

IsK

sGI

e

eeu

mmxp

pxmp

Well-Posed

1

mmxp

pxmp

IsK

sGIexists and is proper

1

m

p

IK

GIis invertible KGI p is invertible

GKIm is invertible

6


Internal Stability

Internally Stable mpmpeu HsH

2

1

11

11

2

1

2

1

1

2

1

u

u

GKIKGIK

GKIGKGI

u

usH

u

u

IsK

sGI

e

e

mp

mp

eummxp

pxmp

Definition: The System in Figure internally stable if it is well-posed and Heu (s) is analytic forall Real (s) ≥ 0 Cs

If in addition Heu (s) is real-rational and proper we will write: mpmppeu RsH

Internally Stable mpmpeu RHsH

and

Internally Stability of a Real-Rational and Proper System

RH

GKIKGIK

GKIGKGIRH

IsK

sGI

mp

mp

mmxp

pxmp

11

111

7

sK sG 2e

1u2u

1e0r


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

mmg

mpg RHsDRHsN

& ppg

mpg RHsDRHsN

~&

~

pmg

mmg RHsYRHsX

~&

~ mpg

ppg RHsYRHsX

&

1 1

2 2

3 3

s.t.: s.t.:

m

g

g

gg IsN

sDsYsX

~~ p

g

g

gg IsY

sXsNsD

~~Bézout-DiophantineIdentities

8


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


Right and Left Coprime Factorization of a Transfer Matrix (continue – 1)

m

g

g

gg IsN

sDsYsX

~~ p

g

g

gg IsY

sXsNsD

~~Bézout-DiophantineIdentities

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)

mppRsG

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:

GeneralizedBézoutIdentity

10



p

m

gg

gg

gg

gg

I

I

sXsN

sYsD

sDsN

sYsX

0

0~~

~~

0

0

0

0we obtained:

GeneralizedBézoutIdentity

We can see that:

mpmp

gg

gg

gg

ggRH

sXsN

sYsD

sDsN

sYsX

0

0

1

0

0

~~

~~

mpmp

gg

gg

gg

ggRH

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

qqRHsU

1

11



Proof Proof

1

111

UDUN

DUUNDNsG

gg

gggg

Post-multiply by U: mg

g

gg IN

DYX

~~

UUN

UDYX

g

g

gg

~~

Pre-multiply this by U-1:

mg

g

gg IUN

UDYUXU

~~ 11

1

111

~~~~

~~~~~~

gg

gggg

DUNU

NUUDNDsG

UUN

UDYX

g

g

gg

~~

p

g

g

gg IUY

UXNUDU

1

1

~

~~~~~

Pre-multiply by : pg

g

gg IY

XND

~~ U~

Post-multiply this by :1~ U


12



Theorem 1a:If is a r.c.f. ofthen is also a r.c.f. of G (s)for every U(s) mxm unimodular.

gg ND , UNUD gg ,

mppRsG

Theorem 1b:If is a l.c.f. ofthen is also a l.c.f. of G (s)for every unimodular.

gg ND~

,~

gg NUDU~~

,~~

mppRsG

Proof (continue – 1) Proof (continue – 1)

mg

g

gg IUN

UDYUXU

~~ 11 p

g

g

gg IUY

UXNUDU

1

1

~

~~~~~

ppsU ~

Because RHU 1

RHYUY

RHXUX

gg

gg

~:

~

~:

~

11

11

and

Because RHU 1~

RHUYY

RHUXX

gg

gg

11

11

~:

~:

and mg

g

gg IUN

UDYX

11

~~ pg

g

gg IY

XNUDU

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.


13 gg NUDU~~

,~~

14



We can use the coprime-factorization to find necessary and sufficient conditionss.t. the Linear Time-Invariant System G(s), K(s) is internally stable. The definition ofan internal stable system is:

RH

GKIKGIK

GKIGKGIsH

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

sK sG 2e

1u2u

1e0r



RH

GKIKGIK

GKIGKGIsH

mp

mp

eu 11

11

:

mxpkmxmkpxpkmxpkmxp

pxmgpxpgmxmgpxmgpxm

sNsDsDsNsK

sNsDsDsNsG~~

~~

11

11

Use Matrix Inverse Lemma:

gkgkgkgkkgkkg

gkkggkkggggkkggp

kgkgkgpkkggpp

DNNDDDDDNNDDD

DDNNDDNNDDDDNNDI

NNNDDNINDDNIKGI

~~~~~~

~~~~~~~~~

~~~~~

1111

111111111

11111

mmmnnmmmmnnnnmmmmmmnnnnmmm CBDCBADCCBADC

1111

kgkgkgkgkgkgkk

kgkgkgkmkkgkgkgpkk

gkgkgkgkgkgkkkp

NNNDDDNNNDDDDD

NNNDDNNIDNNNDDNIND

DNNDDNDNNDDDDNKGIK

~~~~~~~~

~~~~~~~~~~

~~~~~~

111

1111

1111

15

sK sG 2e

1u2u

1e0r



RH

GKIKGIK

GKIGKGIsH

mp

mp

eu 11

11

:

mxpkmxmkpxpkmxpkmxp

pxmgpxpgmxmgpxmgpxm

sNsDsDsNsK

sNsDsDsNsG~~

~~

11

11

Use Matrix Inverse Lemma:


1111

kgkgkgkggkggk

kggkkggkkkkggkkm

gkgkgkmggkkmp

DNNDDDDDNNDDD

DDNNDDNNDDDDNNDI

NNNDDNINDDNIGKI

~~~~~~

~~~~~~~~~

~~~~~

1111

111111111

11111

gkgkgkgkgkgkgg

gkgkgkgmggkgkgkmgg

kgkgkgkgkgkgggm

NNNDDDNNNDDDDD

NNNDDNNIDNNNDDNIND

DNNDDNDNNDDDDNGKIG

~~~~~~~~

~~~~~~~~~~

~~~~~~

111

1111

1111

16

sK sG 2e

1u2u

1e0r


Stabilization of Linear Time-Invariant Systems We found for:

gkkgpp DDNNIKGI~~ 1

21

11

gkkgp DNNDKGIK~~ 1

21

11

gkmkgp NNIDDGKI~~ 1

11

11

gkkgm NDDNGKIG~~ 1

21

11

RHN

DNDNNDD

RHN

DNDNNDD

g

k

kgkgkg

g

g

kkgkgk

~~~~:

~~~~:

2

1

p

m

kg

kg

kg

kg

I

I

DN

ND

DN

ND

0

0~~

~~

0

01

2

11

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

sK sG 2e

1u2u

1e0r



gkkgpp DDNNIKGI~~ 1

21

11

gkkgp DNNDKGIK~~ 1

21

11

gkmkgp NNIDDGKI~~ 1

11

11

gkkgm NDDNGKIG~~ 1

21

11

RHN

DNDNNDD

RHN

DNDNNDD

g

k

kgkgkg

g

g

kkgkgk

~~~~:

~~~~:

2

1

From those Equations we can write:

ggk

k

mgkmgk

gkgk

kkg

gp

kgkg

kgkgp

mp

mp

eu

NDN

D

INNIDN

NDDD

DND

NI

DDND

DNNNI

GKIKGIK

GKIGKGIsH

~~0

00~~

~~

~~

00

0~~

~~

:

121

21

2

12

12

111

11

1

11

11

11

11

18

sK sG 2e

1u2u

1e0r



RHN

DNDNNDD

RHN

DNDNNDD

g

k

kgkgkg

g

g

kkgkgk

~~~~:

~~~~:

2

1

ggk

k

mkk

g

gpeu ND

N

D

IDN

D

NIsH

~~0

00~~

00

01

21

1

Theorem 2The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are RHsH eu

RHRH 12

112121 ,&,UnimodularUnimodular

Proof Theorem 2

(1) If Δ1(s) (or Δ2(s) ) is Unimodular; i.e. Δ1(s)ϵRH∞ and Δ1(s)-1ϵRH∞ , then from the Equation above we can se that Heu(s) ϵRH∞

19

sK sG 2e

1u2u

1e0r



Theorem 2The Necessary and Sufficient Conditions that Heu(s) is Stable, i.e. are

ggk

k

mkk

g

gpeu ND

N

D

IDN

D

NIsH

~~0

00~~

00

01

21

1

RHsH eu

RHRH 12

112121 ,&,UnimodularUnimodular

Proof Theorem 2 (continue)

(2) Use and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by

m

g

g

gg IsD

sNsXsY

00

~~ m

k

kkk I

sX

sYsDsN

0

0~~

sXsY gg 00

~~

sX

sY

k

k

0

0

RHsHRH

Y

X

IsHYX eu

g

g

meukk

0

0

001

2 0

00~~

RHsHRH

X

YIsHXY eu

k

kpeugg

0

000

11

00

0~~

Use and pre-multiply Heu(s) ϵRH∞ by and post-multiply it by

pk

kkk I

N

DYX

00

~~ pg

g

gg IY

XND

0

0~~

00

~~kk YX

0

0

g

g

Y

X

q.e.d.20

sK sG 2e

1u2u

1e0r



Corollary

Unimodular~~

~~

Unimodular

kg

kg

kg

kg

euDN

ND

DN

NDRHsH

Proof

p

m

kg

kg

kg

kg

I

I

DN

ND

DN

ND

0

0~~

~~

0

01

2

11

kg

kg

kg

kg

DN

ND

DN

ND~~

~~

0

01

2

11

1

12

11

1

0

0~~

~~

kg

kg

kg

kg

DN

ND

DN

ND

We found :

Therefore:

RHDN

NDRHsRHs

RHDN

NDRHsRHs

kg

kg

kg

kg

1

11

11

1

11

11

~~

~~

&

&

q.e.d.21

sK sG 2e

1u2u

1e0r



Theorem 3 The set of all Proper Linear Compensators K (s)mxp achieving Internal Stability is given by:

1

01

01

01

00

0

1

01

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

sK sG 2e

1u2u

1e0r



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

sK sG 2e

1u2u

1e0r




(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

NDRH

DN

NDeu

gg

kk

kg

kg

1

0

1

~~

~~

&

i.e., K (s) stabilizes the System. 24

sK sG 2e

1u2u

1e0r




(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

01

01

01

00

0

1

01

00

~

~~~~

gggpggg

gggggggg

XQNXIQXXY

DQYNQXQNXQDYsK

If K (s) stabilizes the System then Δ2 is Unimodular

RHandRHNNDD kgkg1

22

~~:

Let calculate

1

21

211

221

12

11200

1

120

10

120

100

~~~~

~~~~~~

~~

kkggkgg

kgmgkgggggg

kggggkgggggg

DDDDNND

NNIDNNYNXDD

NYNDXNYDNXQNX

100

112

12

QNXQDYDNsK ggggkkTherefore

Let define Q (s) using the equation 120 kgg NQDY 1

201: kgg NYDQ

25

sK sG 2e

1u2u

1e0r




(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: mg

g

gg IN

DYX

00 00 gg YX

RH

DX

NYYXQ

kg

kg

gg 120

120

00

120

120

kgg

kgg

DQNX

NQDYThe chosenQ (s) satisfies

120

120

kg

kg

g

g

DX

NYQ

N

D

In the same way we can define Q1 (s) using the equation:

gkgk

kgg

NNDD

RHNDQY~~

:

~~~

1

1110

11101

~~~: gkg DNYQ

26

sK sG 2e

1u2u

1e0r




(2) We defined Q1 (s) using the equation:

gkgk

kgg

NNDD

RHNDQY~~

:

~~~

1

1110

11

101

~~~: gkg DNYQ

RHDDDDDNN

DNNIDNNNYDX

DNNYXNDNYXNQX

kggkggk

ggkmggkgggg

ggkggggkgggg

~~~

~~~~

~~~~~~~~~~

11

111

11

11

111

11100

11100

1110010

Therefore for the chosen Q1 (s) we obtain:

(2)

kkkk NDNDsK~~~~ 1

1

111

1 gggg DQYNQX~~~~

10

1

10

We still have to prove that RHsQsQ1

RHsQsQ1

kgggggkgggg NDYDDDNYDDQD~~~~~~

:~ 1

1011

101

Use the fact that gggg DYYD~~

00 Bezout Id.

gggkggkgggg DQDDNYDNDYDQD~~~~~ 1

201

201

gkkg DNND~~ 1

21

1 and

27

sK sG 2e

1u2u

1e0r




(3) The last part of the proof is to show that K (s) has the form

10

110

10

100

~ gggpggg XQNXIQXXYsK

Use the Inversion Matrix Lemma

10

110

10

100

100

gggmggggggggg XQNXQINXXQDYQNXQDYsK

10

110

10

10

110

100

10

100

110

110

110

g

QNXII

QNXIpQ

ggmgggg

QNXIQ

ggmggggggg XQNXQINXQDXQNXQINXYXQDXY

ggpp

ggpggp

10

110

10

110

100

10

100

gggmpggggpggggggg XQNXIIQDXQNXIQNXYXQDXY

10

110

10

110

100

10

100

gggppggggmggggggg XQNXIIQDXQNXIQNXYXQDXY

10

110

100

100

gggpgggggg XQNXIQNXYDXYsK 28

sK sG 2e

1u2u

1e0r


1111




(3) We want to prove is to show that K (s) has the form

10

110

10

100


10

110

100

100

gggpgggggg XQNXIQNXYDXYsK

We found

1000

10

1000

~~~~~ ggggggggmgggg XYDNYXDXINYDX

Therefore:

10

110

10

100


Since 01

01

00

~~gggg YXXY

10

110

100

10

~~~ gggpggg XQNXIQXYXsK

Uze Bezout Identity: 01

01

00

~~gggg YXXY

q.e.d.29

sK sG 2e

1u2u

1e0r




(3) We have shown that

10

110

100

10

10

110

10

100

~~~~ gggpggggggpggg XQNXIQXYXXQNXIQXXYsK

This is equivalent to

ggg

ggg

NXX

XXY

sJ1

01

0

10

100

~

fy

sQ

1e 1y

fe sK

1e 1y

fff

ggg

ggg

f

yQee

e

NXX

XXY

y

y

1

10

10

10

100

1

~

To prove this, let develop

11

0

110

101

10 eXQNXIyyQNXeXy gggffgggf

111

0

110

10

100

101

1001

~~esKeXQNXIQXXYeXeXYy ggggggfggg

11

0

110 eXQNXIQyQe gggff

q.e.d.30

sK sG 2e

1u2u

1e0r



Theorem 4 (Dual of Theorem 3) The set of all Proper Linear Time-Invariant Systems G (s) pxm stabilized by theController is given by:

1

0

110

10

100

0

1

01

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

sK sG 2e

1u2u

1e0r

sNsDsDsNsK kkkk

~~ 11



32

sK sG 2e

1u2u

1e0r

From

we obtain

gggggggg DQYNQXQNXQDYsK~~~~

0

1

01

00

KNDKXYQDQYKNQKX

XKYNKDQQDYQNKXK

gggggggg

gggggggg

~~~~~~~0000

001

00

KNDKXYXKYNKDQ gggggggg ~~~0000

1

From

we obtain

kkkkkkkk DSYNSXSNXSDYsG~~~~

0

1

01

00

GNDGXYSDSYGNSGX

XGYNGDSSDYSNGXG

kkkkkkkk

kkkkkkkk

~~~~~~~0000

001

00

GNDGXYXGYNGDS kkkkkkkk ~~~0000

1

Go back to Q (s)Realization



33

sK sG 2e

1u2u

1e0r

Let calculate Δ1 (s) and Δ2 (s) for

kkkk N

gg

D

gg

D

gg

N

gg DQYNQXQNXQDYsK~

0

1

~

01

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 2Heu(s) is Stable for this choise of K (s).



34

sK sG 2e

1u2u

1e0r

Let calculate Heu(s) for

kkkk N

gg

D

gg

D

gg

N

gg DQYNQXQNXQDYsK~

0

1

~

01

00

~~~~

ggk

k

mkk

g

gpeu ND

N

D

IDN

D

NIsH

~~0

00~~

00

01

21

1

we obtained

gg

gg

gg

mgggg

g

gpND

QDY

QNX

INQXDQY

D

NI ~~0

00~~~~

00

0

0

0

00

RHNDQD

NND

Y

X

I

NDQD

NXY

D

NI

ggg

g

ggg

g

m

ggg

g

ggg

gp

~~~~0

00

~~~~

00

0

0

0

00


Stabilization of Linear Time-Invariant Systems - State-Space Approach

Cs

1 yu B

A

D

xTransfer Function of a L.T.I. System:

pxmnxmnxnnpxnpxm DBAIsCsG 1

State Space Realization of the System:

00 xxu

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 numbersλ such that:

nAIsrank nxnn

Stability of the System:

The System is stable if ii Real

35



Cs

1 yu B

A

D

x

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.:

pmnDC

BAIzrank

pxmpxn

nxmnxnn,min

36


Stabilization of Linear Time-Invariant Systems - State-Space Approach Cs

1 yu B

A

D

x

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-1B] has independent rows

1nAC

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(3) The matrix has independent columns (This is so called P.B.H. test)

C

IA

C

(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



Cs

1 yu B

A

D

x

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 C

C

IA The modes λ of A for which loses rank are called uncontrollable modes. All other modes are controllable

(3) The matrix has independent columns C

C

IA

Stabilizability: Detectability:

The Poles of the System are all the modes that are both Controllable and Observable.

38



Cs

1 yu B

A

D

xTransfer 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



Cs

1 yu B

A

D

xOperations 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



Operations on Linear Systems

Usefull Notation:

†

Suppose we have a partition:

INIR

I

TIT

I

TIT

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

rn x

xx

DCTCC

BTAAA

BTBTATTAATAATA

DTC

BTTAT

DC

BA

sG pxm

|

|

|

|

|

|

|

2111

21212221

21112111112211221111

1

1

221

1211

1

1,,0 columncolumnTcolumn

rowrowTrow

rn

rINIR

I

TIT rnrx

41



sG2

1yy 2uu 12 uy

sG1


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

sGsGto 1sDsN gg

42

43




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

TTTn

Tmxp

Tmxp

H

DB

CA

DCAIsBsGsG

|

|

:1

ppxppxp

ppxppxp

pxp

IsGsG

IDD

DDmpif

1-

1-

1†

&

1sDgto




Proof for p ≥ m

††

††

†

|

|

DCD

BDCBDA

sG mxpPseodoinverse 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

INIRI

IIT

I

IIT

n

nn

n

nn

,,

001

mmnn

m

pxmmxp IIBAsICBDAsICD

ICD

BA

CBDA

sGsG

0

1

0

1††

†

†

† 00

|0

|0

0|0Then:

q.e.d.44




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.

mxnF1 nxp

H145



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. Zmxm and Wpxp are any Nonsingular Matrices

RHsYsXsNsDsYsXsNsD gggggggg

~,

~,

~,

~,,,, 46



State-Space Realization of All Coprime Matrices † sNsDsDsNsG ggggpxm

~~ 11


1

1

0

0

|

0|

|

|

WDZDFC

ZF

HWBZBFA

sXsN

sYsD

gg

gg

WWDWC

ZFZ

HHDBHCA

sDsN

sYsX

gg

gg

|

0|

|

|

~~

~~

11

0

0

Proof of (the same way for all others)

1 sDsNsG ggpxm

1111

111

1

|

|

|

|

|

|

|

|

|

ZFZ

BA

ZFZ

ZBZFZBZBFA

ZF

BZBFA

sDgTo Pseudoinverse

RH

ZF

BZBFA

sDg

|

|

|

47




~~ 11

Proof of (continue - 1) 1 sDsNsG ggpxm

DDFDFC

BA

BBFBFA

ZFZ

BA

DZDFC

BZBFA

sDsN gg

|

|

|0

|

|

|

|

|

|

|

11

1

q.e.d.

to sGsG 21

sG

DC

BA

DFC

BFA

DC

BA

DCDFC

BA

BFA

VariablesChange

columncolumncolumn

rowrowrow

|

|

|

0|

|

0|

|

|

|

|

|

|0

0|0

0

221

121

48




~~ 11


1

1

0

0

|

0|

|

|

WDZDFC

ZF

HWBZBFA

sXsN

sYsD

gg

gg

WWDWC

ZFZ

HHDBHCA

sDsN

sYsX

gg

gg

|

0|

|

|

~~

~~

11

0

0

1

1

11

111

1

|

|

|

|

|

|

|

|

|~

WC

WHA

WCWW

WHCWWHHCA

WWC

HHCA

sDg

RH

WWC

HHCA

sDg

|

|

|~

Proof of (the same way for all others)

sNsDsG ggpxm

~~ 1

49




~~ 11

DCC

HDBHCA

DHHCA

WDWC

HDBHCA

WC

WHA

sNsDsee

GGgg

|

|

|0

|

|

|

|

|

|

|~~

211

1

1

Proof of (continue - 1) sNsDsG ggpxm

~~ 1

sG

DC

BAHDBHCA

DC

BA

DC

HDBHCA

BA

VariablesChange

columncolumncolumn

rowrowrow

|

|

|

0|0

|

|

|

|

|

|0

|

|0

|0

0

221

121

q.e.d.

50



Coprime Factorization Example † 0,0.,.

121

212

aaeiRHasas

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

WZh

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



Coprime Factorization Example (continue) †

0,0.,.1

2121

2

aaeiRH

asassG

By developing the previous Matrix representations we obtain

fasas

hfsY

fasas

hfasassX

fasassN

fasas

asassD

g

g

g

g

2120

212

212

0

212

212

212

1

hasas

hfsY

hasas

hfasassX

hasassN

hasas

asassD

g

g

g

g

2120

212

212

0

212

212

212

~

~

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



The Equivalence Between Any Stabilizing Compensatorand the Observer Based CompensatorWe have shown that

10

110

100

10

10

110

10

100

~~~~

00

gggpg

K

gggggpg

K

gg XQNXIQXYXXQNXIQXXYsK

This is equivalent to

ggg

ggg

NXX

XXY

sJ1

01

0

10

100

~

fy

sQ

1e 1y

fe sK

1e 1y

fff

sJ

ggg

ggg

f

yQee

e

NXX

XXY

y

y

1

10

10

10

100

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

1000



The Equivalence Between Any Stabilizing Compensatorand 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

1000

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

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:

Cs

1 ey

u

B

A

D

ex

H

F

1

Filter Gain

Control Gain

sK0e




56

We want to find the State-Space Realization of

pp

gg

IFDC

HFBA

IFDC

HFDHCHFBA

NX

|

|

|

|1

0

ggg

ggg

NXX

XXY

sJ1

01

0

10

100

~

VariablesChange

columncolumncolumn

rowrowrow

pIFDCFDC

BFBA

DHFDCHFDCHFBA

221

121

|

|0

|

DFDC

DHBFDCHFBA

|

|




57


ggg

ggg

NXX

XXY

sJ1

01

0

10

100

~

mm

g

IF

DHBFDHBCHA

IF

DHBCHA

X

|

|

|

|~

1

10

Finally




58


ggg

ggg

NXX

XXY

sJ1

01

0

10

100

~

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|

|~

10

10

10

100




59

We found

DIFDC

IF

DHBHFDHCHFBA

NXX

XXY

sJ

p

m

ggg

ggg

|

0|

|~

10

10

10

100

Cs

1 ey

u

B

A

D

ex

H

F

1

Filter Gain

Control Gain

sK

sQ1u 1y

e

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

Cs

1 yu

B

A

D

ex

H

F

1

Filter Gain

Control Gain

sK

sQ

1u 1y

Cs

1 B

A

D

qCs

1 qB

qA

qD

Observer

System

e

Compensator

r

eyx

qx

The Eigenvalues of Closed-Loop System



61


The State-Space Equation of the System are

sQuDxCy

uBxAx

EstimatoruDxCy

uHuBxAx

SystemuDxCy

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


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

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

C

s

1 u

B

A

D

ex

H

F

1

Filter Gain

Control Gain

sK

sQ

1u 1yqCs

1 qB

qA

qD

Observere

Compensator

ey

qx

From the last two equations

qpkkqqkeqk

qmkqqqeqk

DDIReRxCRDxCDFRDCu

DDIReDxCxCDFRu

:

:

2211

11

1111

1

111

111

111

211

212

211

|

|

|

kqqkqk

kqqkqqkq

kqqkqk

RDCRCDFR

RBCRDBAFDCRB

RDBHCRDHBCDFRDHBCHA

sK

Therefore we obtain:



64

The Transfer Function of the Compensator K (s) with O.B.C. Realization

Cs

1 u

B

A

D

ex

H

F

1

Filter Gain

Control Gain

sK

sQ

1u 1yqCs

1 qB

qA

qD

Observere

Compensator

ey

qx

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

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

11

11

11

11

11

12

11

1

001

gggg XKYNKDQ



66



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

001

gggg XKYNKDQ



67



kk

kk

DC

BA

sK

|

|

|

We found that

kkk

kkk

kk

kk

kkkkk

gggg

DFDCDFC

HFBA

BFDCBA

RCDRFCR

BCDRBACRB

DBCRBCRDBA

XKYNKDQ

|

|

|0

|

|

|

|

|

11

11

11

11

11

12

11

001

001

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

11

11

11

11

11

11

12

12

11

001

|

|

|

|



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

121

12

212

aaeiRHaa

asassG

k

kasas

hkfk

kfk

hkaka

DRCDRFCR

HDRBCDRBACRB

RBCRBCRDBA

Q

kkk

kkk

kkkkk

212

12

11

11

11

11

11

11

12

12

11

|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

Realization of Heu (s) Given G (s) and K (s)

Suppose we have a Minimal State-Realization of a Compensatorand of the System

kk

kk

DC

BA

sK

|

|

|

DC

BA

DC

BA

sGmp

n

INIR

IT

|

|

|

|

|

|

We want to find a Realization of 1

m

p

euIK

GIsH

11

12

11

11

11

12

12

12

11

11

11

11

12

12

12

11

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

Realization of Heu (s) Given G (s) and K (s)

11

12

11

11

11

12

12

12

11

11

11

11

12

12

12

11

11

11

|

|

|

|

|

RRDCDRCR

RDRCRCDR

RBHDRBCDRBACRB

DRBRBCRBCRDBA

GKIKGIK

GKIGKGIsH

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:

11

12

11

12

1

limRRD

RDR

IK

GIsH

km

p

eus

1

11

11

11

11

11

11

12

12

11

|

|

|

|

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

sHkp

km

m

p

eu :

:

2

1

References



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

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

Jhon C. DoyleCalifornia Institute

of Technology

Kemin ZhouLouisiana State

University

Bruce A. FrancisUniversity of

Toronto

Mathukumalli Vidyasagar

University ofTexas

Keith GloverUniversity of

Cambridge

Pramod KhargonekarUniversity of

Florida

73

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