CAREER POINT UNIVERSITY

MAJOR ASSIGNMENT ON

CONTROLABILITY AND OBSERVABILITY

SUBMITTED TOMR. SOMESH CHATURVEDY

SUBMITTED BYDeepak NagarKID-K10880B.Tech. (M.E.) 6th Semester

Motivation1

2

1

2

1

2

1

01

)(01

1012

xx

y

tuxx

xx

1s 1s 1

1 2

u y1x2x

sx )0(2

sx )0(1

1 1x2x

1controllable

uncontrollable

Motivation2

2

1

2

1

2

1

01

)(13

1002

xx

y

tuxx

xx

1s 1s 1

1 2

u y1x2x

sx )0(2

sx )0(1

1 1x2x

3 observableunobservable

Definition

A linear system is said to be completely controllable if, for all initial times and all initial states , there exists some input function (or sequence for discrete systems) that drives the state vector to any final state at some finite time .

0t )( 0tx

)( 1tx 10 tt

A linear system is said to be completely observable if, for all initial times , the state vector can be determined from the output function (or sequence) , defined over a finite time .

Definition

0t )( 0tx)( 1ty

10 tt

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

uu

u

BABBAxAx

BuABuBuABuAxAx

BuABuBuABuAxAx

BuABuxABuBuAxAx

BuAxxBuAxx

Initial condition

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

CAC

n

nDuCBuBuCABuCAxCAy

DuCBuCAxDuBuAxCyDuCxy

DuCxyBuAxx

Inputs & outputs

)()()()()()(tDutCxtytButAxtx

1

2

12

n

n

CA

CACAC

V

BABAABBU

Controllability matrix

Observability matrix

Then: (1) controllable (2) observable nVrank

nUrank

)()(

2

1

2

1

2

1

01

)(01

1012

xx

y

tuxx

xx

Example 1

2][1201

1][0021

][

VrankCAC

V

UrankABBU uncontrollable

observable

The rank of a matrix is defined by the number of linearly independent rows and/or the number of linearly independent columns

Example 2

2

1

2

1

2

1

01

)(13

1002

xx

y

tuxx

xx

1][0201

2][1163

][

VrankCAC

V

UrankABBU controllable

unobservable

Theorem III

)()()( tuBtxAtx cccc

Controllable canonical form Controllable

Theorem IV

)()()()()(

txCtytuBtxAtx

oo

oooo

Observable canonical form Observable

example

c

cc

xy

uxx

1210

3210

Controllable canonical form

1212

3110

CAC

V

ABBU

nVranknUrank

1][2][

o

oo

xy

uxx

1012

3120

Observable canonical form

3110

1122

CAC

V

ABBU

nVranknUrank

2][1][

)2)(1(2)(

ssssT

Theorem V

)()()()()()(tDutCxty

tButJxtx

Jordan form

321

3

2

1

3

2

1

CCCC

BBB

BJ

JJ

J

Jordan block

Least row has no zero row

First column has no zero column

Example

xccy

ubb

xx

3

1100020012

1211

12

11

If 012 b uncontrollable

If 011 c unobservable

xy

uxx

210

203102200

201101211

100010211100010001000

11

1

2

2

2

1

1

1

1

11b

12b13b

21b

11C 12C 13C 21C

....

....

21131211

21131211

ILCILCCCILbILbbb

controllable

observable

In the previous example

....

....

21131211

21131211

DLCILCCCILbILbbb

controllable

unobservable

001

001002113

111111122

01

00

11

001123

111112112

1111

22

212

y

uxxL.I.

L.I.

L.I. L.D.

Example