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Multivariable Control Multivariable Control SystemsSystemsSystemsSystems

Ali KarimpourpAssistant Professor

Ferdowsi University of MashhadFerdowsi University of Mashhad

Chapter 5Chapter 5Controllability, Observability and Realization

Topics to be covered include:

• Controllability and Observability of Linear Dynamical Equations

• Output Controllability and Functional Controllability• Output Controllability and Functional Controllability

• Canonical Decomposition of Dynamical Equation

• Realization of Proper Rational Transfer Function Matrices

• Irreducible RealizationsIrreducible realization of proper rational transfer functions

Irreducible Realization of Proper Rational Transfer Function Vectors

Ali Karimpour June 2012

2Irreducible Realization of Proper Rational Matrices

Chapter 5

Controllability, Observability and Realization

• Controllability and Observability of Linear Dynamical EquationsControllability and Observability of Linear Dynamical Equations

• Output Controllability and Functional Controllability





Irreducible Realization of Proper Rational Matrices


3

Chapter 5

Controllability and Observability of Linear Dynamical Equations

Definition 5-1

D fi iti 5 2Definition 5-2


4

Chapter 5


Theorem 5-1


5

Chapter 5


[ ]nn bAbAAbAbbbS 11 −−= [ ]mmm bAbAAbAbbbS 111 ............=


6

Chapter 5


Theorem 5-2


7

Chapter 5










8

Chapter 5

Output Controllability

Definition 5-3 Output Controllability


9

Chapter 5

Functional Controllability

Definition 5-4 Functional controllability.

An m-input l-output system G(s) is functionally controllable if thenormal rank of G(s), denoted r, is equal to the number of outputs; that is, if G(s) has full row rank. A plant is functionally uncontrollable if r < l.if G(s) has full row rank. A plant is functionally uncontrollable if r l.

Remark 1: The minimal requirement for functional controllability is

that we have at least many inputs as outputs i e m ≥ lthat we have at least many inputs as outputs, i.e. m ≥ l

Remark 2: A plant is functionally uncontrollable if and only if

ωωσ ∀= 0))(( jG ωωσ ∀= ,0))(( jGlRemark 3: For SISO plants just G(s)=0 is functionally uncontrollable.


10

Remark 4: A MIMO plant is functionally uncontrollable if the gain is identically zero in some output direction at all frequencies.

Chapter 5


⎤⎡ 21

Consider following transfer function:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡++=

11

11

32

11

)( sssG It is Functionally controllable

⎦⎣ ++ 11 ss

Its state space representation is:

)(2123

)(00410030

)( tutxtx⎥⎥⎥⎤

⎢⎢⎢⎡

+⎥⎥⎥⎤

⎢⎢⎢⎡

−−

=&

0010

)(

1111

)(

21001000

)( tutxtx

⎤⎡

⎥⎥

⎦⎢⎢

⎣

+

⎥⎥

⎦⎢⎢

⎣ −−

It is not state controllable


11)(

10000010

)( txty ⎥⎦

⎤⎢⎣

⎡=

Chapter 5


10000 ⎤⎡⎤⎡

Consider following state space form:

It is state controllable and output controllable

)(000110

)(010001000

)( tutxtx⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=&

p

)(100010

)(

0000

txty ⎥⎦

⎤⎢⎣

⎡=

⎥⎦⎢⎣⎥⎦⎢⎣

Its transfer function is:

⎤⎡ 11It is not functionally controllable.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

32

2

11

11

)( sssG


12

⎦⎣ 32 ss

Chapter 5


� An Example: dc-dc boost converter

6

2 51

2498 3.049 10ˆ 609 3 207 10 ˆ

sx s s

⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥

+ ×+ + × 1x1

15 82

2 5

609 3.207 10 ˆ ,ˆ 2.5 10 1.217 10

609 3.207 10

s sx s

s s

ω⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦

⎢ ⎥⎣ ⎦

+ + ×− × + ×

+ + ×

12x

1ω

functionally uncontrollable

6

:

2498 3 049 10 12 5 s + 7 44 0

or new system design

s + ×⎡ ⎤

functionally uncontrollable

2 51

2

2498 3.049 10ˆ 609 3.207 1

12.5 s + 7

ˆ

40

4 0sx s s sx⎡ ⎤

=

+ ×+ + ×

⎢ ⎥⎣ ⎦

2 51

5 8 5

2 5 2 5

ˆ609 3.207 10ˆ2.5 10 1.217 10 10

609 3 207 10 66.2

09 3 207 15

0in

svs

s s s s

ω⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥

+ + ×− × + × ×

+ + × + + ×⎣ ⎦

⎢ ⎥⎣ ⎦


1313

609 3.207 10 609 3.207 10s s s s+ + × + + ×⎣ ⎦

functionally controllable

Chapter 5


BAsICsG 1)()( −−= is functionally uncontrollable ifAn m-input l-output system

lBrank

Chapter 5


Example 5-1

⎥⎥⎥⎤

⎢⎢⎢⎡

++= 421

21

1

)( sssG⎥⎦

⎢⎣ ++ 2

42

2ss

Thi i il i l 2 f G( ) i i l 1This is easily seen since column 2 of G(s) is two times column 1.

The uncontrollable output directions at low and high frequencies are, i lrespectively,

⎥⎤

⎢⎡

∞⎥⎤

⎢⎡ 21)(

11)0( yy


15

⎥⎦

⎢⎣−

=∞⎥⎦

⎢⎣−

=15

)(,12

)0( 00 yy

Chapter 5










16

Chapter 5

Canonical Decomposition of a Linear Time-invariant Dynamical Equation

ECBuAxx

++=& uBxAPBuxPAPx +=+= −1&Pxx =EuCxy += uExCEuxCPy +=+= −1

Theorem 5-3 The controllability and observability of a linear time-invariantd i l i i i d i l f idynamical equation are invariant under any equivalence transformation.

Proof: Let we first consider controllability

[ ] [ ][ ] PSBABAABBP

PBPPAPBPPAPBPAPPBBABABABSn

nn

==

==−

−−−−−

12

1112112 ..........[ ] PSBABAABBP == .....

Similarly we can consider observability


17

Chapter 5


ECBuAxx +=&Theorem 5-4

Consider the n-dimensional linear time –invariant dynamical equation EuCxy +=

y q

If the controllability matrix of the dynamical equation has rank n1 (where n1

Chapter 5


Theorem 5-4 (Continue)

1Furthermore P=[q1 q2 … qn1 … qn]-1 where q1, q2, …, qn1 be any n1 linearly

independent column of S (controllability matrix) and the last n-n1 column of P

are entirely arbitrary so long as the matrix [q1 q2 … qn1 … qn] is nonsingular.

Proof: See “Linear system theory and design” Chi-Tsong Chen

ldimensiona+=+=

nEuCxyBuAxx&

ldimensiona≤+=

+=

EuxCyuBxAx

cc

cccc&Pxx =

ldimensiona−n ldimensiona1 −≤ nn

G(s)


19Hence, we derive the reduced order controllable equation.

Chapter 5


ECBuAxx +=&Theorem 5-5

Consider the n-dimensional linear time –invariant dynamical equation EuCxy +=

y q

If the observability matrix of the dynamical equation has rank n2 (where n2

Chapter 5


Theorem 5-5 (Continue)

Furthermore the first n row of P are any n linearly independent rows of VFurthermore the first n2 row of P are any n2 linearly independent rows of V

(observability matrix) and the last and the last n-n2 row of P is entirely arbitrary

so long as the matrix P is nonsingularso long as the matrix P is nonsingular.


ldimensiona−+=+=

nEuCxyBuAxx& Pxx =

+=

+=

EuxCyuBxAx

oo

oooo&

ldimensiona−n

G(s)

ldimensiona2 −≤ nn


21Hence, we derive the reduced order observable equation.

Chapter 5


ECBuAxx

++=&Theorem 5-6 (Canonical decomposition theorem)

Consider the n-dimensional linear time –invariant dynamical equation EuCxy +=Consider the n-dimensional linear time invariant dynamical equation There exists an equivalence transformation

Pxx = Pxxwhich transform the dynamical equation to

[ ] Ex

CCBBx

AAAAAx ocococococ

⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡⎥⎤

⎢⎡

⎥⎤

⎢⎡

001312

&

&

d th d d di i l b ti

[ ] EuxxCCyuB

xx

AAA

xx

c

coccoco

c

co

c

co

c

co +⎥⎥⎥

⎦⎢⎢⎢

⎣

=⎥⎥⎥

⎦⎢⎢⎢

⎣

+⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣

=⎥⎥⎥

⎦⎢⎢⎢

⎣

0000

0 23&

and the reduced dimensional sub-equation

EuxCyuBxAx cocococo

+=

+=&


22is observable and controllable and has the same transfer function matrix

as the first system.

EuxCy coco +

Chapter 5


Definition 5-5

A linear time-invariant dynamical equation is said to be reducible if and only if thereA linear time invariant dynamical equation is said to be reducible if and only if there

exist a linear time-invariant dynamical equation of lesser dimension that has the same

transfer function matrix. Otherwise, the equation is irreducible.q

Theorem 5-7

A linear time invariant dynamical equation is irreducible if and only if it is controllable

and observable.

Theorem 5-8


23

Chapter 5










24

Chapter 5

Realization of Proper Rational Transfer Function Matrices

Dynamical equation (state-space) description This transformation

The input-output description (transfer function matrix)

EuCxyBuAxx

+=+=& is unique

EBAsICsG +−= −1)()(

( )

The input-output description (transfer function matrix) Realization

BA&

Dynamical equation (state-space) description

This transformationEBAsICsG +−= −1)()(

EuCxyBuAxx

+=+=This transformation

is not unique

1 Is it possible at all to obtain the state-space description from the transfer function1. Is it possible at all to obtain the state-space description from the transfer function matrix of a system?

2. If yes, how do we obtain the state space description from the transfer function matrix?


25

y , p p

Chapter 5

Realization of Proper Rational Transfer Function Matrices

Theorem 5-9

A transfer function matrix G(s) is realizable by a finite dimensional

linear time invariant dynamical equation if and only if G(s) is a propery q y ( ) p p

rational matrix.



26

Chapter 5

Irreducible realizationsDefinition 5-6

Theorem 5-10


27

Chapter 5

Irreducible realizations

Before considering the general case

(irreducible realization of proper rational matrices)

we start the following parts:

1. Irreducible realization of Proper Rational Transfer Functions

2 I d ibl R li ti f P R ti l T f F ti V t2. Irreducible Realization of Proper Rational Transfer Function Vectors

3. Irreducible Realization of Proper Rational Matrices


28

Chapter 5

Irreducible realization of proper rational transfer functions

0,ˆ......ˆˆ

ˆ......ˆˆ)( 01

10

110 ≠

++++++

= −−

ααααβββ

nnn

nn

ssss

sg0

01

1

22

11

ˆ

ˆ

............

)(αβ

ααβββ

+++++++

= −−−

nnn

nn

ssss

sg

)()()(ˆ)(ˆˆ

)(ˆ)(ˆˆ

)(......

......)(0

0

0

01

1

22

11 seususgsusysusu

sssssy

nnn

nnn

+=+=+++++++

= −−−

αβ

αβ

ααβββ

......10 +++ ααα nss 01 ...... ααα +++ nss

)()(ˆ)(ˆ

susysg =

00n

)(ˆ sg

)(su

)()(ˆ)(ˆ

susysg =

ucxybuAxx0ˆ +=+=&

)()( sysg =buAxx +=

β̂

&


29

)()(

susg =

eucxuyy +=+=0

0

ˆˆ

αβ

Chapter 5


nnn

nnn

ssss

susysg

ααβββ

++++++

== −−−

............

)()(ˆ)(ˆ 1

1

22

11

0

01

1

22

11

ˆ

ˆ

............

)(αβ

ααβββ

+++++++

= −−−

nnn

nn

ssss

sgn)( 1

There are different forms of realization

01 ...... ααα +++ nss

Ob bl i l f li ti

There are different forms of realization

• Observable canonical form realization

• Controllable canonical form realization• Controllable canonical form realization

• Realization from the Hankel matrix


30

• Realization from the Hankel matrix

Chapter 5

Observable canonical form realization of proper rational transfer functions

nnn

nn sssg βββ +++= −−− ......)(ˆ 1

22

110

1

22

11

ˆ

ˆ......)(

ββββ+

++++++

= −−−

nnn

nn sssg

nnn ss αα +++ ......11

uuuyyy nnn

nnn βββαα +++=+++ −−− ............ )2(2

)1(1

)1(1

)( )))

)()( )

01

1 ...... ααα +++ nnn ss

)()( tytxn)=

)()()()()()()( 11)1(

11)1(

1 tutxtxtutytytx nnn βαβα −+=−+=−))

)()()()()()(.................................

)2()2()1( tttttt nnn −−− +++ ββ )))

)()()()()()()()()( 22)1(

122)1(

1)1(

1)2(

2 tutxtxtutytutytytx nnn βαβαβα −+=−+−+= −−)))

)()()()()()( )1()1()1()1()()1( nnn ββ )))

)()()(

)()(...)()()()(

11)1(

2

11)2(

1)2(

1)1(

1

tutxtx

tutytutytytx

nnn

nnnnn

−−

−−

−+=

−++−+=

βα

βαβα


31)()()()()()(...)()()()( )1(1

)1(1

)1(1

)1(1

)()1(1

tutxtutytutytutytytx

nnnnn

nnnnn

βαβαβαβα

+−=+−=−++−+= −−

−−

)

)))

Chapter 5

)()( tytx = )Observable canonical form realization of proper rational transfer functions

)()()()()()()()()(

)()()()()()()(

)()(

22)1(

122)1(

1)1(

1)2(

2

11)1(

11)1(

1

tutxtxtutytutytytx

tutxtxtutytytx

tytx

nnn

nnn

n

−−

−

−+=−+−+=

−+=−+=

βαβαβα

βαβα)))

))

)()()(

)()(...)()()()(.................................

)1(11

)2(1

)2(1

)1(1

tutxtx

tutytutytytx nnnnn

−−−−−

+=

−++−+=

βα

βαβα )))

)()()( 112 tutxtx nnn −− −+= βα

)()()()()()(...)()()()( )1(1

)1(1

)1(1

)1(1

)()1(1

tutxtutytutytutytytx

nnnnn

nnnnn

βαβαβαβα

+−=+−=−++−+= −−

−−

)

)))

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

−−

⎥⎥⎤

⎢⎢⎡

−− n

n

n

n

xx

xx

xx

0...010...00

2

1

12

1

12

1

ββ

αα

&

&

)()()()(y nnnnn ββ

[ ]

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

−=

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

−− n

n

n

n

xyuxx.

10...00ˆ..

100.......

0...10.

3

2

2

1

3

2

2

1

3

2

β

βα

&

&


32

⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ −⎥⎦⎢⎣ nnn xxx 1...00 11 βα

Chapter 5


nnn

nnn

sssssg

ααβββ

++++++

= −−−

............)(ˆ 1

1

22

11

⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡

[ ]⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

−

−

−

n

n

n

n

n

n

xxx

yuxxx

xxx

10...00ˆ0...100...010...00

3

2

1

2

1

3

2

1

2

1

3

2

1

βββ

ααα

&

&

&

21 β̂βββ −− nn

[ ]

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −⎥⎥⎥

⎦⎢⎢⎢

⎣ n

n

n

n

n x

y

xx...

1...00........

3

1

23

1

23

β

β

α&

0

01

1

22

11

ˆ............)(

αβ

ααβββ

+++++++

= −−−

nnn

nnn

sssssg

xxx 000 βα ⎤⎡⎤⎡⎤⎡⎤⎡ −⎤⎡ &

[ ] uxxx

yuxxx

xxx

n

n

n

n

n

n

0

03

2

1

2

1

3

2

1

2

1

3

2

1

ˆˆ

10...00ˆ0...100...010...00

αββ

ββ

ααα

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−

−

−

−

&

&


33xxx nnn

0

11

...1...00

........α

βα ⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −⎥⎥⎥

⎦⎢⎢⎢

⎣ &

Chapter 5


02

21

1ˆ......)( ββββ ++++=

−−n

nn sssg nnn sssg βββ +++=−− ......)(ˆ

22

11

01

1 ˆ......)(

ααα+

+++= −

nnn ss

sgn

nn sssg

αα +++= − ......

)( 11

xxx nn 111 0...00 βα⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡⎥⎤

⎢⎡ −

⎥⎤

⎢⎡ &

[ ] uxx

yuxx

xx

n

n

n

n

0

03

2

2

1

3

2

2

1

3

2

ˆˆ

.10...00ˆ

.........0...100...01

.αββ

βαα

+

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

=

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

+

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−−

=

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−

−

−

−

&

&

xxx nnn 11

...1...00

........βα ⎥

⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥⎥⎦⎢

⎢⎣ −⎥

⎥⎦⎢

⎢⎣ &

Th d i d d i l ti i b bl E i 1 Wh ?

The derived dynamical equation controllable as well if numerator and

The derived dynamical equation is observable. Exersise 1: Why?


34

denominator of g(s) are coprime. Exersise 2: Why?

Chapter 5

Controllable canonical form realization of proper rational transfer functions

nnn sssNsg

βββ +++==

−− ......)()( 12

21

1)01

22

11

ˆ

ˆ......)(

ββββ+

+++= −

−−

nnn

nn sssg

Let us introduce a new variable)()()()()()(

svsNsysusvsD

==

)

nnn sssD

gαα +++ − ......)(

)( 110

11 ...... ααα +++ n

nn ss

)()()( svsNsy =

We may define the state variable as: ⎥⎥⎥⎤

⎢⎢⎢⎡

=⎥⎥⎥⎤

⎢⎢⎢⎡

=)(

)()()(

)(

)1(2

1

tvtv

txtx

txWe may define the state variable as:

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

− )(....

)(....)(

)1( tvtx

tx

nn

nn xxxxxx === −13221 ,.....,, &&&Clearly

)()()()()( )1()1()( tvtvtvtutvx nn ααα −−−−== −&


35)(...)()()(

)(...)()()()(

1211

11

txtxtxtutvtvtvtutvx

nnn

nnn

αααααα

−−−−===

−

−

Chapter 5


02

21

1 ......)(ββββ)

+++ −− nnn ss n

nn sssN βββ +++ −− ......)()(2

21

1)

0

01

1

21

............

)(αβ

ααβββ

)++++

= −n

nnn

sssg

nnn

n

sssDsNsg

ααβββ

+++== − ......)(

)()( 11

21)

nn xxxxxx === −13221 ,.....,, &&&

)(...)()()()(...)()()()(

1211

)1(1

)1(1

)(

txtxtxtutvtvtvtutvx

nnn

nnn

nn

αααααα

−−−−=−−−−==

−

−−&

[ ] ⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

xx

xx

xx

00

0...1000...010

2

1

2

1

2

1

ββββ&

&

[ ]

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−−−

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

−−

−− n

nnn

nnnnn x

xyu

x

x

x

x.

...

1.0

....

1...000.......

.31213

121

3 ββββ

αααα&

&


36

Chapter 5


nnn

nnn

ssss

sDsNsg

ααβββ

++++++

== −−−

............

)()()( 1

1

22

11)

⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡

[ ]⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−− nnn xxx

yuxxx

xxx

...000

.......0...1000...010

3

2

1

1213

2

1

3

2

1

ββββ&&

&

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−⎥⎥⎥

⎦⎢⎢⎢

⎣ −− nnnnnn xxx.

1..

...1...000.

121 αααα&

0

01

1

22

11

ˆˆ

............)(

αβ

ααβββ

+++++++

= −−−

nnn

nnn

sssssg

xxx 00010 ⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡ &

[ ] uxxx

yuxxx

xxx

nnn0

03

2

1

1213

2

1

3

2

1

ˆˆ

...000

1000.......0...1000...010

αβββββ +

⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎡

=

⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎡

+

⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎡

⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎡

=

⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎡

−−&

&


37xxx nnnnnn

0

121

.1..

...1...000.αααα ⎥

⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−⎥⎥⎥

⎦⎢⎢⎢

⎣ −−&

Chapter 5


0

01

1

22

11

ˆˆ

............)(

αβ

ααβββ

+++++++

= −−−

nnn

nnn

sssssg nn

nnn

ssss

sDsNsg

ααβββ

++++++

== −−−

............

)()()( 1

1

22

11)

01 ...... ααα +++ nss nsss αα +++ ......)( 1

xx

xx

xx 111

00

01000...010

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡&

&

[ ] uxx

yuxx

xx

nnn0

03

2

1213

2

3

2

ˆˆ

....

1.00

.1...000.......0...100

.αβββββ +

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

−−

&

&

Th d i d d i l ti i t ll bl E i 3 Wh ?

xxx nnnnnn 121 1... αααα ⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ −−−−⎥⎦⎢⎣ −−

The derived dynamical equation observable as well if numerator and

The derived dynamical equation is controllable . Exersise 3: Why?


38

denominator of g(s) are coprime. Exersise 4: Why?

Chapter 5

Controllable and observable canonical form realization of proper rational transfer functions

Example 5-2 Derive controllable and observable canonical realization for following system.

3248182 23 +++ sss

2202663248182)(223

++++++ ssssssg

61163248182)( 23 +++

+++=

sssssssg

261166116

)( 2323 ++++=

+++=

sssssssg

Observable canonical form realization is:

20600 ⎤⎡⎤⎡⎤⎡⎤⎡ &Controllable canonical form realization is:

0010 ⎤⎡⎤⎡⎤⎡⎤⎡ &u

xxx

xxx

62620

6101101600

3

2

1

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

&

&

&

uxxx

xxx

100

6116100010

3

2

1

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

&

&

[ ] uxxx

y 2100 21

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= [ ] u

xxx

y 262620 21

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=


39

x3 ⎥⎦⎢⎣ x3 ⎥⎦⎢⎣It is not controllable. It is not observable.Why? Why?

Chapter 5


Example 5-3 Derive irreducible realization for following transfer function.

3248182 23 +++ sss

2202663248182)(223

++++++ ssssssg

61163248182)( 23 +++

+++=

sssssssg

2206 ++= s261166116

)( 2323 ++++=

+++=

sssssssg

Observable canonical form realization is: Controllable canonical form realization is:

2652+

++=

ss

uxx

xx

620

5160

2

1

2

1

⎤⎡

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡&

&u

xx

xx

10

5610

2

1

2

1

⎤⎡

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡&

&

[ ] uxx

y 2102

1 +⎥⎦

⎤⎢⎣

⎡= [ ] u

xx

y 26202

1 +⎥⎦

⎤⎢⎣

⎡=


40

It is controllable too. It is observable too.Why? Why?

Chapter 5


Realization from the Hankel matrix

nn βββ −1

nnn

nnn

sssssg

ααβββ

++++++

= − ............)( 1

1

110 ......)3()2()1()0()( 321 ++++= −−− shshshhsg

The coefficients h(i) will be called Markov parameters.

⎥⎥⎤

⎢⎢⎡

+ )1(...)4()3()2()(...)3()2()1(

ββ

hhhhhhhh

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢

+=

.......

.......)2(...)5()4()3(

),(β

βαhhhh

H

⎥⎥⎦⎢

⎢⎣ −+++ )1(...)2()1()(

.......βαααα hhhh


41

Chapter 5


Realization from the Hankel matrix

Theorem 5-11 Consider the proper transfer function g(s) as

nnn

nnnn

sssss

sgαα

ββββ+++

++++= −

−−

............

)( 11

22

110

( ) ( ) ....,3,2,1,everyfor),(),( =++= lklmkmHmmH ρρ

then g(s) has degree m if and only if


42

Chapter 5

Irreducible realization of proper rational transfer functions Realization from the Hankel matrixRealization from the Hankel matrix

buAxx +=&Now consider the dynamical equation

ebAsIcsebAsIcsg +−=+−= −−−− 1111 )()()(eucxy +=

g )()()(

.....3221 ++++= −−− bscAcAbscbse......,3,2,1)( 1 == − ibcAih i)(

⎥⎥⎥⎤

⎢⎢⎢⎡

+ )1(.....)2()(.....)2()1(

nhhnhhh

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−+

=+

)12(.....)1()(................

),1(

nhnhnh

nnH

⎥⎥⎦⎢

⎢⎣ ++ )2(.....)2()1( nhnhnh

Let the first σ rows be linearly independent and the (σ+1) th row of H(n+1,n) be linearly dependent on its previous rows. So


43

y p p

0),1(]0.....01.....[ 21 =+ nnHaaa σ

Chapter 5

Irreducible realization of proper rational transfer functions Realization from the Hankel matrixRealization from the Hankel matrix

0),1(]0.....01.....[ 21 =+ nnHaaa σWe claim that the σ-dimensional dynamical equationWe claim that the σ dimensional dynamical equation

hhh

)3()2()1(

0000000.....10000.....010

⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

uh

xx..

)3(

..........

..........00.....000

.

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

+

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

=& (I)

[ ] uhxyh

haaaaa

)0(00.....001)(

)1(.....

10.....000

1321

+=

⎥⎥⎥

⎦⎢⎢⎢

⎣

−⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−− − σσ

σσ

is a controllable and observable (irreducible realization).

[ ]

Exercise 5: Show that (I) is a controllable and observable (irreducible realization) of


44

Exercise 5: Show that (I) is a controllable and observable (irreducible realization) of

eucxybuAxx

+=+=&

Chapter 5


Example 5-4 Derive irreducible realization for following transfer function.3248182)(

23 +++ ssssg6116

)( 23 +++=

ssssg

.....2303410141062)( 654321 ++−−+−+= −−−−−− sssssssg

⎤⎡

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−−

−

=341014101410

14106

)3,4(H

⎥⎦

⎢⎣ −− 2303410

We can show that the rank of H(4,3) is 2. So [ ] 0)3,4(0156 =H

Hence an irreducible realization of g(s) is610⎥⎤

⎢⎡

+⎥⎤

⎢⎡

&


45[ ] uxy

uxx

2011056

+=

⎥⎦

⎢⎣−

+⎥⎦

⎢⎣ −−

=

Chapter 5

Realization of Proper Rational Transfer Function Vectors

⎤⎡ )(

Consider the rational function vector

⎤⎡⎤⎡ )()

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

= .)()(

)(2

1

sgsg

sG⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

= .)()(

.)(2

1

2

1

sgsg

ee

sG

)

)

⎥⎥⎥

⎦⎢⎢⎢

⎣ )(.sgq ⎥

⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ )(..sge qq

)

⎥⎥⎤

⎢⎢⎡

++++++

⎥⎥⎤

⎢⎢⎡

−−

−−

nnn

nnn

ssss

ee

ββββββ

....

....

1 22

221

21

12

121

11

2

1

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

++++

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

= −

n

nnn ss

sGαα

.

......

1

.

.)(

21

11


46

⎥⎦

⎢⎣ +++

⎥⎦⎢⎣−−

qnn

qn

qq sse βββ ....2

21

1

Chapter 5


⎥⎥⎥⎤

⎢⎢⎢⎡

++++++

⎥⎥⎥⎤

⎢⎢⎢⎡

−−

−−

nnn

nnn

ssss

ee

ββββββ

....

....

1 22

221

21

12

121

11

2

1

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ +++

++++

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

−−

−

qnn

qn

q

nnn

q ss

ss

e

sG

βββ

αα

......

.....1

.

.)(

22

11

11

ee

yy

nnn

nnn

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

−−

−−

...

...00

0...1000...010

2

1

21)2(2)1(22

11)2(1)1(11

2

1

ββββββββ

u

e

x

y

yuxx

nnn

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

+

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

=..

.................

.

.

0..

1...000.............. 2

1)2()1(

21)2(2)1(222

ββββ

ββββ&

ey qqnqnqqnqnnn

⎦⎣⎦⎣⎦⎣⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣ −−−−

−−−−

...1... 1)2()1(121

ββββαααα

This is a controllable form realization of G(s).


47We see that the transfer function from

u to yi is equal to nnn

inn

in

ii ss

sse

ααβββ

++++++

+ −−−

.........

11

22

11

Chapter 5


Example 5-5 Derive a realization for following transfer function vector.

⎤⎡ + 3s

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

++=

34

)2)(1(3

)(

ss

sss

sG

⎥⎦

⎤⎢⎣

⎡

+++

++++⎥

⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

+++

=)2)(1(

)3()3)(2)(1(

110

34

)2)(1(3

)(2

sss

ssssss

s

sG

⎥⎦

⎤⎢⎣

⎡

++++

++++⎥

⎦

⎤⎢⎣

⎡=

⎥⎦⎢⎣ +

2396

61161

10

3

2

2

23 ssss

sss

s

H i i l di i l li i f G( ) i i bHence a minimal dimensional realization of G(s) is given by

⎥⎤

⎢⎡

+⎥⎤

⎢⎡⎥

⎤⎢⎡

+⎥⎤

⎢⎡

016900

100010

&


48

uxyuxx ⎥⎦

⎢⎣

+⎥⎦

⎢⎣

=⎥⎥⎥

⎦⎢⎢⎢

⎣

+⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−=

113210

6116100

Chapter 5

Realization of Proper Rational Matrices

There are many approaches to find irreducible realizations for proper i l irational matrices.

1. One approach is to first find a reducible realization and then applypp pp ythe reduction procedure to reduce it to an irreducible one.

Method I, Method II, Method III, Method IV and Method V

2. In the second approach irreducible realization will yield directly.pp y y


49

Chapter 5


Method I: Given a proper rational matrix G(s), if we first find

an irreducible realization for every element gij(s) of G(s) as jijjiijij

jijijijij

uexcy

ubxAx

+=

+=&

an irreducible realization for every element gij(s) of G(s) as jijjiijijy

⎥⎤

⎢⎡⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

112

11

12

11

12

11

12

11

00

000000

ubb

xx

AA

xx&

&

⎤⎡

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦⎢⎢⎢

⎣

+

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣

=

⎥⎥⎥

⎦⎢⎢⎢

⎣2

1

22

21

12

22

21

12

22

21

12

22

21

12

00

000000 u

bb

xx

AA

xx&

&

yyy +

12111 yyy +=

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

1

2221

1211

21

12

11

2221

1211

2

1

0000

uu

eeee

xxx

cccc

yy

22222 yyy +=

⎥⎦

⎢⎣ 22x

Clearly this equation is generally not controllable and not observable.


50

To reduce this realization to irreducible one requires the application of the reductionprocedure twice (theorems 5-4 and 5-5).

Chapter 5


⎥⎤

⎢⎡⎥⎥⎤

⎢⎢⎡

+⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

=⎥⎥⎥⎤

⎢⎢⎢⎡

112

11

12

11

12

11

12

11

00

000000

ubb

xx

AA

xx&

&

⎤⎡

⎥⎦

⎢⎣⎥⎥⎥

⎦⎢⎢⎢

⎣

+

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣

=

⎥⎥⎥

⎦⎢⎢⎢

⎣

11

2

22

21

22

21

22

21

22

21

00

000000

x

ub

bxx

AA

xx&

&

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

1

2221

1211

21

12

11

2221

1211

2

1

0000

uu

eeee

xxx

cccc

yy

⎥⎦

⎢⎣ 22x

00000)(

0011

1

111

bbAsI

⎤⎡⎥⎤

⎢⎡⎥⎤

⎢⎡ −⎤⎡

−

Proof:

00

0

)(0000)(0000)(0

0000

2221

1211

22

21

12

122

121

112

2221

1211

eeee

bb

b

AsIAsI

AsIcc

cc⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣ −−

−⎥⎦

⎤⎢⎣

⎡

−

−

−


51)(

)()()()(

)()()()(

2221

1211

22221

222221211

2121

12121

121211111

1111 sGsgsgsgsg

ebAsIcebAsIcebAsIcebAsIc

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

+−+−+−+−

=−−

−−

Chapter 5


Method II: Given a proper rational matrix G(s), if we find the controllable canonical-form realization for the ith column, Gi(s), of G(s) say,

iiiii ubxAx +=&

iiiii

iiiii

uexCy +=

uu

bb

xx

AA

xx

0...00...0

0....00....0

2

1

2

1

2

1

2

1

2

1

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡&

&

This realization is always

[ ] [ ]ueeexCCCyubxAx ppppp

....00

............00

...........22222

+=⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ &

This realization is alwayscontrollable. It is however generally not observable.

[ ] [ ]ueeexCCCy pp ........ 2121 +=Proof:

[ ] [ ]....0...00...0

0...)(00...0)(

.... 212

11

2

11

21 eeeb

bAsI

AsI

CCC +⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

−−

−

−

[ ] [ ]

)()()()(...)()(

....

...00......

)(...00......

....

11211

21

1

21

sgsgsg

eee

bAsI

CCC

p

p

pp

p

⎥⎤

⎢⎡

+

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −−


52

[ ] )()(...)()(

......)(...)()(

)(.....)(

21

22221111

111 sG

sgsgsg

sgsgsgebAsICebAsIC

qpqq

ppppp =

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣

=+−+−= −−

Chapter 5


Method III: Let a proper rational matrix G(s), where consider )()()( ∞+= GsGsG)

mmm)( 21the monic least common denominator of G(s) as mmmm ssss αααψ ++++= −− ...)( 2211

Then we can write G(s) as [ ] )(...)(

1)( 221

1 ∞++++=−− GRsRsR

ssG m

mm

ψ )(sψ

Then the following dynamic equation is a realization of G(s).

I 0000 ⎤⎡⎤⎡

uxI

I

xp

p

pppp

pppp

..00

.......0...000...00

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=&

[ ] uGxRRRRyIIIII

I

p

p

ppmpmpm

pppp

)(

0......000

121

∞+=

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−− −− αααα


53

[ ] uGxRRRRy mmm )(... 121 ∞+= −−Exercise 6: Show that the above dynamical equation is a controllable realization of G(s)

Chapter 5


Method IV: Gilbert diagonal representation.

S di ti t l i lSuppose distinct real eigenvalues.

Reminders

qisBpisCBCG kkkkkkk ××= ρρ

{ }IIdiA λλ{ }r

IIdiagA r ρρ λλ ,...,11=

[ ] ⎥⎤

⎢⎡B1


54

[ ]rCCC ...1=⎥⎥⎥

⎦⎢⎢⎢

⎣

=

rBB M

Chapter 5


Gilbert diagonal representation.

⎥⎤

⎢⎡ 121

⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢

⎣ ++++

+

++

+++

=

0)3)(2)(1(

)3(21

14

12

10411

)(

sss

ssssG

[ ]02101

⎥⎥⎤

⎢⎢⎡

[ ]01010

⎥⎥⎤

⎢⎢⎡

[ ]32100⎥⎤

⎢⎡

[ ]10011

⎥⎥⎤

⎢⎢⎡

⎥⎦

⎢⎣ ++++ )3)(2)(1(1 ssss

⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡

100100

1000000

1010000

1000021

1)(G

[ ]02110⎥⎥⎦⎢

⎢⎣

[ ]0102

1⎥⎥⎦⎢

⎢⎣−

[ ]32100⎥⎥⎥

⎦⎢⎢⎢

⎣

[ ]10001⎥⎥⎦⎢

⎢⎣

⎥⎥⎥

⎦⎢⎢⎢

⎣+

+⎥⎥⎥

⎦⎢⎢⎢

⎣+

+⎥⎥⎥

⎦⎢⎢⎢

⎣ −+

+⎥⎥⎥

⎦⎢⎢⎢

⎣+

=000100

4000000

3020010

2021000

1)(

sssssG

⎥⎤

⎢⎡

⎥⎤

⎢⎡− 0210001 ⎤⎡⎤⎡− 021001

uxx

⎤⎡

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−

−=

100321010

400003000020

& uxx

⎥⎤

⎢⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

101

100010021

400020001

&

⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒nrealizatio minimal


55xy⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

002110101001 xy

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

021110

Chapter 5



( ) ⎤⎡ + 421 s( )( )( )

( )( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+++−

+++

+=

21

211

4142

11

)(

sss

sss

ssG

⎥⎤

⎢⎡⎥⎤

⎢⎡ 0121 [ ]110⎥

⎤⎢⎡ [ ]110⎥

⎤⎢⎡

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

001001211)(sG

⎥⎦

⎢⎣⎥⎦

⎢⎣− 1001

[ ]111⎥⎦⎢⎣

[ ]110⎥⎦⎢⎣

⎥⎦

⎢⎣+

⎥⎦

⎢⎣+

⎥⎦

⎢⎣−+ 004112011

)(sss

⎥⎤

⎢⎡

⎥⎤

⎢⎡− 010001

⎤⎡⎤⎡uxx

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−

−=

111110

400002000010

& uxx

⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

021

111001

200010001

&

⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒nrealizatio minimal


56xy ⎥⎦

⎤⎢⎣

⎡−

=01010021 xy ⎥

⎦

⎤⎢⎣

⎡−

=101021

Chapter 5



⎥⎤

⎢⎡

⎥⎤

⎢⎡− 0000012

( ) ( ) ( ) ⎥⎤

⎢⎡ 1111

kd

uxx

⎤⎡

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

+

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−

−=

321000000

200012000120

& ( ) ( ) ( )

( ) ( )

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢

+++

++++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

000000000

1100

21

21

210

2222

)(32

432

sss

ssss

iflhebkgda

sG

xmifclhebkgda

y⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

( ) ⎥⎦⎢⎣

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ +

++⎥⎦⎢⎣ 321

21000

2200 2

s

ssmifc

⎤⎡⎤⎡⎤⎡⎤⎡ kgda

( )[ ]

( )[ ]

( )[ ] [ ]321

21321

21321

21321

21)( 234

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+=

mlk

sihg

sfed

scba

ssG


57

Chapter 5


Method IV: Gilbert diagonal representation.

R titi l i lRepetitive real eigenvalues.

r1 Jordan block of order 3r1 Jordan block of order 3

r2 - r1 Jordan block of order 2

r3 – r2 Jordan block of order 1


58

Chapter 5


Gilbert diagonal representation. Repetitive real eigenvalues.

2 Jordan block of order 2

0 J d bl k f d 10 Jordan block of order 1


59

Chapter 5




60

Chapter 5



:),4(B)1(:,C

:),4(B

:),4(B)2(:,C )5(:,C


61

),(

:),7(B

Chapter 5



⎤⎡⎤⎡ 211100:),4(B)3(:,C )5(:,C

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

101101211

351000100

3M :),7(B

:),9(B)8(:,C ),()8(:,C

:)4(B)4(:,C )7(:,C

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

101101211

010000100

4M

:),4(B)(

:),7(B

)(

)9(B)9(C


62

:),9(B)9(:,C

Chapter 5




63

Chapter 5


Method V: It is possible to obtain observable realization of a proper G(s). Let

)2()1()0()( 21 HHHG ...........)2()1()0()( 21 +++= −− sHsHHsG

Consider the monic least common denominator of G(s) as

mmmm ssss αααψ ++++= −− ...)( 22

11

Then after deriving H(i) one can simply show

Exercise 7: Proof equation (I)

)(1)(...)2()1()( 21 IiiHimHimHimH m ≥−−−+−−+−=+ ααα

)()( 32211 BCACABCBEBAICEG

Let {A, B, C and E} be a realization of G(s) then we have


64

...........)()( 32211 ++++=−+= −−−− BsCACABsCBsEBAsICEsG

Chapter 5


Then {A, B, C and D} be a realization of G(s) if and only if

210)1()0( iBCAiHdHE i ,....2,1,0)1()0( ==+= iBCAiHandHE i

Now we claim that the following dynamical equation is a realization of G(s).

HI )1(000 ⎤⎡⎤⎡

uHH

xI

I

xqqqq

qqqq

..)2()1(

.......0...000...00

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=&

[ ] uHxIymH

mHIIII

I

q

qqmqmqm

qqqq

)0(0...00

)()1(

...

...000

121

+=

⎥⎥⎥

⎦⎢⎢⎢

⎣

−⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−− −− αααα

[ ]y q )(We can readily verify that

⎥⎤

⎢⎡

++

⎥⎤

⎢⎡

⎥⎤

⎢⎡

)2()1(

)4()3(

)3()2(

iHiH

HH

HH

)1(iHBCAi


65⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ +

+=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ +

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ +

=

)(...

)2(,...,

)2(...

)4(,

)1(...

)3( 2

imH

iHBA

mH

HBA

mH

HAB i )1( += iHBCA

i

)0(HE =

Chapter 5


Now we shall discuss in the following a method which will yield directly irreducible

realizations. This method is based on the Hankel matrices.

...........)2()1()0()( 21 +++= −− sHsHHsGLet G(s) be pq ×

Consider the monic least common denominator of G(s) as

mmmm ssss αααψ ++++= −− ...)( 22

11Define

⎥⎥⎥⎤

⎢⎢⎢⎡

qqqq

qqqq

II

0...000...00

⎥⎥⎥⎤

⎢⎢⎢⎡

−−

− pmppp

pmppp

IIII

I

1

000...00...00

αα

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−−−

=

−− qqmqmqm

qqqq

IIIII

M

121 ......000

.......

αααα⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −

−= −

pppp

pmppp

II

IIN

1

2

...00.........

.0.....0

α

α


66

Chapter 5


mmmm ssss αααψ ++++= −− ...)( 22

11

⎥⎤

⎢⎡ qqqq I 0...00

⎥⎤

⎢⎡ − pmppp I0...00 α

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

=

qqqq

qqqq

I

IM

...000.......

0...00

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−−

= −−

pmppp

pmppp

pppp

IIII

N 21

..........0.....0

0...0αα

⎥⎥⎦⎢

⎢⎣ −−−− −− qqmqmqm

qqqq

IIII 121 ... αααα⎥⎥⎦⎢

⎢⎣ − pppp II 1...00 α

We also define the two following Hankel matrices

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+=

...)1()3()2(

)()2()1(mHHH

mHHH

T

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

++

=...

)2()4()3()1()3()2(

~ mHHHmHHH

T

⎥⎦

⎢⎣ −+ )12()1()( mHmHmH

⎥⎦

⎢⎣ ++ )2()2()1( mHmHmH

It can be readily verified that


67TNMTT ==~ ,.....2,1,0== iTNTM ii

Chapter 5



TNMTT ==~ ,.....2,1,0== iTNTM iiTNMTT == ,.....2,1,0iTNTM

[ ]0II =lkI , )( kllk >×Let be as the form

[ ]0, klk II =Note that the left-upper-corner of M iT = TN i is H(i+1) so:

TT ....,2,1,0)1( ,,,, ===+ iITNITIMIiHT

pmpi

qmqT

pmpi

qmq



68But we want Irreducible Realization of Proper Rational Matrices

Chapter 5



69

Chapter 5


Example 5-6 Derive an irreducible realization for the following proper rational function.

⎤⎡ −−− 1232 2 ss

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−

+++=

153

154

1)1(

232

)(2

ss

ss

ssss

sG

⎦⎣ ++ 11 ss

2)1()( += sssψLeast common denominator of G(s), is

06050403021102 ⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡.....

2106

2105

.2104

2103

2102

2111

3402

)( 654321 −−−−−− ⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡−

−= sssssssG

⎤⎡ 030211 Non-zero singular values of T

⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎡

−−−−−

−−−−

=⎥⎥⎥⎤

⎢⎢⎢⎡

=212121040302212121

030211

)4()3()2()3()2()1(

HHHHHH

T

gare 10.23, 5.79, 0.90 and 0.23.

So, r = 4.


70⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−−−

⎥⎦⎢⎣

212121050403212121

)5()4()3( HHH,

Chapter 5


⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

0 10260 09780 10570 54960.5306-0.6022- 0.58720.10490.6574- 0.4905 0.0196-0.4003-

0 82640 10540 20780 51270.0071-0.0581-0.5238-0.2357-0.1621-0.8902-0.25450.3413-

⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢

⎣

−=

⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢

⎣ −−−

=

0.18880.38750.5432 0.13820.4522 0.2949-0.23110.6989- 0.1888-3875.00.5432-0.1382-

0.10260.0978-0.1057-0.5496

0071.00581.00.5238-0.2357-5392.04316.00.26270.6738-

0071.00581.0052380.23570.8264-0.1054-0.2078-0.5127

rr UY

⎥⎦⎢⎣⎥⎦⎢⎣

⎥⎥⎤

⎢⎢⎡

0.0034- 0.0551- 1.2598- 0.7539- 0.0770- 0.8443- 0.6121 1.0915-

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢

==

0.2560- 0.4093 0.6317 2.1553- 0.0034 0.0551 1.2598 0.7539

0.3923- 0.1000- 0.4999- 1.6398 ˆ 2/1r SYY

⎥⎥⎥⎤

⎢⎢⎢⎡

==

⎥⎥⎦⎢

⎢⎣

1.3066 0.5557 1.3066- 0.2543- 1.4124 0.0471- 0.4421 2.2356- 0.4421- 1.7579 0.3355 1.2803-

ˆ

0.0034- 0.0551- 1.2598- 0.7539-

2/1 HUSU


71⎥⎥

⎦⎢⎢

⎣ 0.0896 0.2147 0.0896- 0.0487 0.2519- 0.3121- 0.3675 0.2797- 0.3675- 0.0927- 0.5711- 0.4652 r

USU

Chapter 5


⎤⎡ 0 07370 21070 07370 16030 07370 1067

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

== −

0.0149- 1.1356- 0.0149 1.7406- 0.0149- 0.3415- 0.0613- 0.4551 0.0613 0.1112- 0.0613- 0.9386- 0.2178- 0.1092 0.2178 0.0864- 0.2178- 0.1058 0.0737- 0.2107- 0.0737 0.1603 0.0737- 0.1067-

ˆ 2/1† HrYSY

⎦⎣

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

== −0.2161 0.1031- 0.0440- 0.1718 1.1176- 0.6349- 0.2441 0.0328 1.3847- 0.5171 0.0081- 0.1251-

ˆ 2/1† SUU

⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣ 0.3976 0.4086 0.2259 0.0432 0.9525 0.3109- 0.0961 0.2185-

0.3976- 0.4086- 0.2259- 0.0432- SUU r

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

==0.8076 0.2888- 0.1800- 0.2227-

0.0772 0.1604- 1.0139- 0.1588 0.1904- 0.2155 0.0369 1.2497-

ˆ~ˆ †† UTYA

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

==0.5711- 0.4652 1.4124 0.0471- 0.3355 1.2803-

ˆ,

TpmpIUB

⎥⎦

⎢⎣ 0.4476- 0.1354 0.1181- 0.1246

⎥⎦

⎢⎣ 0.2519- 0.3121-

⎥⎤

⎢⎡

==0.0770- 0.8443- 0.6121 1.0915-

ŶIC ⎥⎤

⎢⎡−

==02

)0(HE


72

⎥⎦

⎢⎣

==0.0034- 0.0551- 1.2598- 0.7539- ,

YIC qmq ⎥⎦

⎢⎣ −

==34

)0(HE

Chapter 5

Exercises


73

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