modern contral systems1 lecture 06 analysis (ii) controllability and observability 6.1...

Modern Contral Systems 1

Lecture 06 Analysis (II) Controllability and Observability

6.1 Controllability and Observability

6.2 Kalman Canonical Decomposition

6.3 Pole-zero Cancellation in Transfer Function

6.4 Minimum Realization


Motivation1

2

1

2

1

2

1

01

)(0

1

10

12

x

xy

tux

x

x

x

1s 1s 1

1 2

u y1x2x

s

x )0(2

s

x )0(1

1 1x2x

1controllable

uncontrollable


DuCxy

RxBuAxx n

,

Controllability and Observability

Plant:

Definition of Controllability

A system is said to be (state) controllable at time , if there exists a finite such for any and any , there exist an input that will transfer the state to the state at time , otherwise the system is said to be uncontrollable at time .

0t01 tt )( 0tx 1x

][ 1,0 ttu )( 0tx

1x 1t

0t


1

1

)(sU

)(sY1

-1s

3 -1s 2

Example: An Uncontrollable System

xy

uxx

21

0

1

30

01

1x

2x

※ State is uncontrollable.2x

BABAABBU

nUrankBAn 12

,)(leControllab ,

0)det( U

Controllability Matrix

BABAABBU n 12Matrix ility Controllab

Ru if


Proof of controllability matrix

)1(

)2(

1

)1()2(121

)1()2(121

12

12

112

1

)(

nk

nk

k

nk

nnk

nknkkn

kn

kn

nk

nknkkn

kn

kn

nk

kkkkkkk

kkk

kkk

u

u

u

BABBAxAx

BuABuBuABuAxAx

BuABuBuABuAxAx

BuABuxABuBuAxAx

BuAxx

BuAxx

Initial condition


Motivation2

2

1

2

1

2

1

01

)(1

3

10

02

x

xy

tux

x

x

x

1s 1s 1

1 2

u y1x2x

s

x )0(2

s

x )0(1

1 1x2x

3 observableunobservable


Definition of Observability

A system is said to be (completely state) observable at time , if there exists a finite such that for any at time , the knowledge of the input and the output over the time interval suffices to determine the state , otherwise the system is said to be unobservable at .

0t 01 tt )( 0tx][ 1,0 ttu

],[ 10 tt

0x

0t

0t][ 1,0 tty


Example: An Unobservable System

xy

uxx

40

1

0

20

10

※ State is unobservable.1x

1)(sU -1s -1s 1x2x

2

4

)(sY

nVrankCA )(Observable , 0)det( V

1

2Matrix ity Observabil

nCA

CA

CA

C

V

Observability Matrix

Ry if


Proof of observability matrix

)1()2()3(11

1

)1()2(1321

1

111

111

1

)(),2(),1(

)(

)2()(

)1(

nknknkkkkkk

k

n

nknkkn

kn

kn

nk

kkkkkkk

kkk

kkk

kkk

DuCBuCABuDuCBuyDuy

x

CA

CA

C

n

nDuCBuBuCABuCAxCAy

DuCBuCAxDuBuAxCy

DuCxy

DuCxy

BuAxx

Inputs & outputs


10,1

0,

01

10

CBA

2)()( NrankVrank

01

10 ty Matrix Obervabili

01

10ix ility MatrControllab

CA

CN

ABBV

Example

DuCxy

RxBuAxx n

,

Plant:

Hence the system is both controllable and observable.


Theorem I

)()()( tuBtxAtx cccc

Controllable canonical form Controllable

Theorem II

)()(

)()()(

txCty

tuBtxAtx

oo

oooo

Observable canonical form Observable


example

c

cc

xy

uxx

12

1

0

32

10

Controllable canonical form

12

12

31

10

CA

CV

ABBU

nVrank

nUrank

1][

2][

o

oo

xy

uxx

10

1

2

31

20

Observable canonical form

31

10

11

22

CA

CV

ABBU

nVrank

nUrank

2][

1][

)2)(1(

2)(

ss

ssT


Theorem III

)()()(

)()()(

tDutCxty

tButJxtx

Jordan form

321

3

2

1

3

2

1

CCCC

B

B

B

B

J

J

J

J

Jordan block

Least row has no zero row

First column has no zero column

Linear system1. Analysis


Example

xccy

ub

b

xx

3

1100

020

012

1211

12

11

If 012 b uncontrollable

If 011 c unobservable


xy

uxx

2

1

0

203

102

200

201

101

211

100010

211

100010

001

000

1

1

1

2

2

2

1

1

1

1

11b

12b13b

21b

11C 12C 13C 21C


....

....

21131211

21131211

ILCILCCC

ILbILbbb

controllable

observable

In the previous example

....

....

21131211

21131211

DLCILCCC

ILbILbbb

controllable

unobservable


0

0

1

001

002

113

111

111

122

0

1

0

0

1

1001

123111

112

112

11

112

22

12

y

uxxL.I.

L.I.

L.I. L.D.

Example


zCy

uBzAz

m

mm

)0()0()( 111

nt

nt zevzevtz n

BuAxx DuCxy

Txz

observable and lecontrollab is mode 0, and 0 If i mimi cb

Kalman Canonical Decomposition

Diagonalization:

There exists a coordinate transform (See Sec. 4.4)

n 321All the Eigenvalues of A are distinct, i.e.

such that

mnmm

mn

m

m

ccCTC

b

b

BTB

1

11

.

0

0

where1

1

n

mm AATTA

System in z-coordinate becomes

Homogeneous solution of the above state equation is


][ 21 n,v, , vvT

The coordinate matrix for diagonalization

t.independen are , rs,Eigenvecto 21 n,v, , vv

n 321All the Eigenvalues of A are distinct, i.e.

How to construct coordinate transformation matrix for diagonalization

ubzλz

ubzz

ubzz

mnnnn

m

m

2222

1111

nmnmm zczczcy 2211

Consider diagonalized system


n

mnmnmmn

i i

mimi

s

bc

s

bc

s

bcsH

1

11

1

)(

Transfer function is

le,unobservabor ableuncontroll is mode 0,or 0 If i mimi cb

H(s) has pole-zero cancellation.

1

1mc1mb

2

2mc2mb

n

mncmnb

∑

)(tu

)(ty


)(ty

OCS

COS

OCSOCS

)(tu

Kalman Canonical Decomposition

Subsystem ble Unobservale,Controllab:OCS

Subsystem Observable able,Uncontroll:OCS

Subsystem Observable le,Controllab :COS

Subsystem ble Unobservaable,Uncontroll:OCS


xCCy

uB

B

x

x

x

x

A

A

A

A

x

x

x

x

OCCO

OC

CO

OC

OC

OC

CO

OC

OC

OC

CO

OC

OC

OC

CO

00

0

0

000

000

000

000

Kalman Canonical Decomposition: State Space Equation

(5.X)


Example

xcccy

u

b

b

b

xx

311211

31

12

11

300

020

001

3 mode 0, If 13 b

3 mode 0, If 13 c

The same reasoning may be applied to mode 1 and 2.

Plant:

is uncontrollable.

is unobservable.


n

mnmnmmn

i i

mimi

s

bC

s

bC

s

bCsH

1

11

1

)(

From Sec. 5.2, state equation

Hence, the T.F. represents the controllable and observableparts of the state variable equation.

may be transformed to

BuAxx

DuCxy

T.F.. in vanishesand ableuncontroll is mode , 0 If imib

T.F.. in vanishesand leunobservab is mode 0, If imic

Pole-zero Cancellation in Transfer Function


BAsICsHsU

sY 1)()(

)(

4

1

2

2210

42

1

1

2

41

0210

42

1

s

s

s

ss

s

s

ss

4,2 21

Example

xy

uxx

10

1

2

21

04

Transfer Function

T.F.. in vanishes"-2" Mode

Plant:


uxx

1

2

11

60

xy 10

3

1)(

)(

)( 1

sBAsICsT

sU

sY

Example 5.6

Plant:

Transfer Function

T.F.. in vanishes"2" Mode

-3,2 21


Minimum Realization

Realization: Realize a transfer function via a state space equation.

3

1)(

)(

)(

ssT

sU

sY

2

2

3

1)(

)(

)(

s

s

ssT

sU

sY

1 1)(sU )(sY

3

1 1)(sU )(sY-1s

3

2

-1s

-1s1

ExampleRealization of the T.F.

3

1)(

ssT

Method 1:

Method 2:

※There is infinity number of realizations for a given T.F. .


Minimum realization: Realize a transfer function via a state space equation with elimination of its uncontrollable and unobservable parts.

3

5)(

)(

)(

ssT

sU

sY1 5)(sU )(sY

3

-1s

Example 5.8

Realization of the T.F.3

5)(

ssT

modern contral systems1 lecture 06 analysis (ii) controllability and observability 6.1...

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modern contral systemsfrom

state variable

state equationhence

state space equation5

transfer function6

given t

unobservable parts

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