MODERN CONTROL THEORYLecturer：鲍其莲 Bao Qilian

1

Chapter 6 Controllability and

Observability

Objectives:

• Definition of controllability

• Definition of observability

• Detective methods for controllability

• Detective methods for observability

• Decomposition

2Chapter 6

6.1 controllability

• Definition:

• The state equation or the pair (A,B) is said to

be controllable if for any initial state x(0)=x0 and any final

state x1, there exists an input that transfer x0 to x1 in a

finite time. Otherwise is said to be uncontrollable.

3

BuAxx

Chapter 6

Example: uncontrollable network.

4

xCxy

uxbuAxx

u, xux cc

10

1

1

21

12

Let 2211

tttt

tttt

t

33

33

eeee

eeee

2

1eA

0

0)0(xIf then ττutx

tτt d)(e

1

1)(

0

)(

21 xx

No matter what the input is, the states are always equivalent.

So the states of the system are uncontrollable.

Chapter 6

• Theorem: The following statements are equivalent.

1. The n-dimensional pair (A,B) is controllable.

2. The following nn matrix is non-singular for any t.

3. The following nnp controllability matrix (Kalman matrix)

has rank n (full row rank).

5

dτeBBedτeBBe(t)W τ)(tAT)A(t-τt

τATAτt

C

TT

00

]1BAB AB C

n-[

Chapter 6

A system is controllable if the following nnp controllability matrix

has rank n (full row rank).

][ BABAABBQ1n2

C

nCQrank

Proof: Caley-Hamilton theorem

1

0

1

110 )()()()(n

i

i

i

n-

n

τ τaτaτaτae AAAIA

τττaxt

i

n

i

i d)()()0(1

0

1

0

uBA

i

ir

i

i

t

i

β

β

β

τττa

2

1

0d)()(

1

u )1,,1,0( ni

Chapter 6

7

Therefore

1

1

0

1 ][)0(

n

n-

β

β

β

BAABBx

n ]rank[rank BABAABBQ1n2

C

The solution of exists according to initial state x(0) only

if following matrix has rank n.

Chapter 6

8

Example: Determine the controllability of following systems

u

9

0

2

10

5

07

xx

u

57

04

10

10

5

07

xx

（1）

（2）

Solution:

(1) Uncontrollable.

(2) Controllable.

Chapter 6

• Example: Determine the controllability of following

systems

• (1)

• (2)

9

u

3

4

0

200

040

014

xx

u

03

00

24

200

040

014

xx

• Solution:

• (1) Controllable.

• (2) Uncontrollable.

Chapter 6

10

Theorem: If all the eigenvalues of a system are different, then the

system can be transformed to the following form using similarity

transformation. The system is controllable if there is no all zero

row in .

uBxx

nλ

λ

λ

0

0

2

1

B

Chapter 6

Other conditions of controllability

11

Theorem: If a system can be transformed to the following Jordan

matrix form using similarity transformation. It is controllable if

each row in matrix corresponding to the last rows of each

Jordan block is not all zero row.B

Chapter 6

uBx

J

J

J

x

k0

0

2

1

i

i

i

i

λ

λ

λ

0

1

01

J

12

Example: Determine the controllability of following systems

using alternative method.

u

9

0

2

10

5

07

xx

u

57

04

10

10

5

07

xx

（1）

（2）

Solution:

(1) Uncontrollable.

(2) Controllable.

Chapter 6

13

Example:Determine the controllability of following systems.

u

3

4

0

200

040

014

xx

u

03

00

24

200

040

014

xx

（1）

（2）

Solution:

（1）Controllable

（2）Uncontrollable

Chapter 6

6.2 Observability

• Definition:

• A linear state system or pair (A,C) is said to be observable

if for any unknown initial state x(0), there exist a finite t1>0

such that the acknowledge of input u and output y over

[0, t1] suffices to determine uniquely the initial state x(0).

Otherwise the system is said to be unobservable.

14Chapter 6

Example: Unobservable network

15

xCxy

ux-

-BuAxx

11

0

1

21

12

tttt

tttt

t

33

33

eeee

eeee

2

1eA

ττtut τtt

t d)(e)0(e)( )(

0

bxx AA

0)( tu

• If then

)0(e)( xxAtt

tt xxty 3

21 e)]0()0([)0(e)( xCA

• The value of output only determined by the difference of

initial states. Therefore, the states are unobservable.

Chapter 6

• Theorem: The following statements are equivalent.

1. The n-dimensional pair (A,C) is observable.

2. The following nn matrix is nonsingular for any t>0.

3. The nqn observability matrix has rank n ( full column

rank)

16

tAτTτA

C dτCeCe(t)WT

0

nnm

n

1CA

CA

C

QO

Chapter 6

A system is observable if the nqn observability matrix has rank n

( full column rank)

nnm

n

1CA

CA

C

QO

nOQrank

Proof: if then0)( tu

)0(e)()( xCCxyAttt

)0(e)( xxAtt

1

0

)(en

i

i

i

τ τa AA )0()()(

1

0

xACy

n

i

i

i τat

)0()()()()(

1

110 x

CA

CA

C

y

n

n tatatat

known

Condition: rank is n

unknown

Chapter 6

• Example: Is the following state equation observable?

18

u

2

1

50

02xx

x10y

Solution:

nrankrank

21

50

10

CA

C

Unobservable.

Chapter 6

Example: Determine the observability of following systems

（1） xx

10

5

07

x540y

（2） xx

10

5

07

x

130

023y

Solution:

（1）Unobservable.

（2）Observable.

Chapter 6

Other conditions of observability

20

Theorem: If all the eigenvalues of a system are different, then the

system can be transformed to the following form using similarity

transformation. The system is observable if there is no all zero

column in .

uBxx

nλ

λ

λ

0

0

2

1

Chapter 6

C

xCy

Other conditions of controllability

21

Theorem: If a system can be transformed to the following Jordan

matrix form using similarity transformation. It is controllable if

each column in matrix corresponding to the first column of

each Jordan block is not all zero column.

Chapter 6

uBx

J

J

J

x

k0

0

2

1

i

i

i

i

λ

λ

λ

0

1

01

J

xCy

C

22

Example: Are the following systems observable?

（1） xx

10

5

07

x540y

（2） xx

10

5

07

x

130

023y

Solution:

(1)unobservable.

(2)Observable

Chapter 6

23

Example: is the state equation observable?

xx

20000

12000

00300

00130

00013

-

-

xy

00110

01111

Using two different ways to determine the observability.

(Observable)

Chapter 6

24Chapter 6

Example : Is the following state equation controllable?

duβββy n x110

u

aaa n

1

0

0

0

100

10

010

110

xx

25Chapter 6

duβββy n x110

Theorem: If a system (A, B) is controllable, it can be

tranformed into canonical controllable form as follows:

u

aaa n

1

0

0

0

100

10

010

110

xx

01

1

1

1

1

2

121

1

n

n

n

a

a

aaa

BAABBP

01

1

1]det[ aλaλaλAλI n

n

n

26

Example: Find the controllable form for the following

system

u

1

1

0

001

010

101

xx x011y

（1）

Solution:

101

111

110

][ 2bAAbbQC

3rank CQ Controllable

（2） 12]det[ 23 λλλ AI

Chapter 6

27

（3）Compute matrix P

121

111

011

001

01

1

][][ 2

21

2

321 a

aa

bAAbbppp

213

112

111

121

111

011

][

1

1

321 pppP

（4） 102

121

111

011

0111

CPC

（5）0 1 0 0

x 0 0 1 x 0

-1 0 2 1

u

2 0 1y x

Chapter 6

28Chapter 6

Example : Is the following state equation observable?

u

β

β

β

a

a

a

nn

1

1

0

1

1

0

100

0

10

01

0

xx

x100 y

29Chapter 6

Theorem: If a system(A,C) is observable, it can be tranformed

into canonical observable form as follows:

u

β

β

β

a

a

a

nn

1

1

0

1

1

0

100

0

10

01

0

xx

x100 y

11

2

121

01

1

1

1

nn

n

a

a

aaa

CA

CA

C

P

01

1

1]det[ aλaλaλAλI n

n

n

30

u

1

1

0

001

010

101

xx

x011y

Chapter 6

Example: Find the observable form for the following

system

6.3 Canonical decomposition

System decomposition

• (1) controllable subsystem and uncontrollable subsystem

• (2) observable subsystem and unobservable subsystem

• (3) Transfer function

31Chapter 6

If not all the states of a linear system are controllable, then

the system can be decomposed into two subsystem, one is

controllable, and another one is un-controllable.

Assume:

32

Cxy

BuAxx

12

0 0

[ ]

CC C C

CC C

C

C C

C

xx A A Bu

xx A

xy C C

x

12

1

C C C CC

C C

x A x A x B u

y C x

Transform

Controllable states

Transfer function

xPx C

nnBAABBQ 1

2

C ]rank[rank

Then

Transfer function of the system is the same as controllable subsystem

Chapter 6

11 1 1 1 1

1 1

C

=

C C

G(s) C sI A B CP sI A P B C sP IP P AP B

C sI A B C sI A B

33

11 1 1 1 1

1

-1

12

12

C

=

0 0

1

( )( )

010

0

C C

C C

C

C C C C

C C

C

C C CCC

G(s) C sI A B CP sI A P B C sP IP P AP B

C sI A B

sI A A BC C

sI A

A

sI A sI A sI A BC C

sI A

B

C C C sI AsI A

1

CB

Transfer function

Transfer function of the system is the same as controllable subsystem

34

C

C

CC

C

C

C

C

C

C

C

x

x]CC[y

uB

x

x

A

AA

x

x

1

12

00

controllable

uncontrollable

Chapter 6

• Theorem: ( Chosen of transformation matrix)

• Consider the n-dimensional state equation with

• We form the nn matrix

• Where the first n1 columns are any n1 independent

columns of QC and remaining columns can arbitrarily be

chosen as long as P is nonsingular.

• Then the system will be decomposed by the following

equivalence transformation .

35

nnBAABBQ 1

2

C ]rank[rank

][: nn qqqP 11

1

Pxx

Chapter 6

36

Example: decompose the following system:

u

1

1

21

12xx x10y

Solution

(1)

(2) Chose independent column vector

2111

11rankrankrank

nAbbQC

1

11p

1

02p

11

01

11

011

1

21 ppPC

1 CC APPA BPB C1 CCPC

C

C

C

C

C

C

x

xy

ux

x

x

x

11

0

1

30

11

Chapter 6

37

uB

B

x

x

AA

A

x

x

O

O

O

O

O

O

O

O

21

0

O

O

Ox

xCy ]0[

uBxAx OOOO

OO xCy

Transfer functionOOO BAICBAICG

11 )()()( sss

If not all the states of a linear system are observable, then the

system can be decomposed into two subsystem by equivalence

transformation, one is observable, and another one is

unobservable. The transfer function will be the same as the

controllable subsystem.

Assume:2

O

C

rankQ rank CA n

CA

Cxy

BuAxx Transform

xPx o

Observable subsystem

Chapter 6

• Theorem: ( Chosen of transformation matrix)

• Consider the n-dimensional

• state equation with

• We form the nn matrix

• Where the first n2 rows are any n2 independent rows of QO

and remaining rows can arbitrarily be chosen as long as P

is nonsingular.

• Then the system will be decomposed by the following

equivalence transformation .

38

n

n

p

p

p

P

2

1

1 :

Pxx

nn

n

2

1

rankrank

CA

CA

C

QO

39

O

O

Ox

xCy ]0[

uB

B

x

x

AA

A

x

x

O

O

O

O

O

O

O

O

21

0

observable

unobservable

Chapter 6

40

Example:

u

1

0

0

342

100

010

xx x011y

Solution32

242

110

011

CA

CA

C

Q2

O

nrankrankrank

100

110

011

OP

100

110

0111-

OP

1 OO APPA

BPB O1 OCPC

u

1

1

0

122

022

010

O

O

O

O

x

x

x

x

OC

O

O

x

xy ]001[

}

100

110

011

OP

An independent row

Chapter 6

• For both uncontrollable and unobservable system, it can

be decomposed into

• Transfer function

• The same as the controllable and observable subsystem.

41

uB

B

x

x

x

x

AA

A

AAAA

AA

x

x

x

x

OC

CO

OC

OC

OC

CO

OC

OC

OC

CO

OC

OC

OC

CO

0

0

00

000

00

43

242321

13

OC

OC

OC

CO

OCCO

x

x

x

x

CC 00y

COCOCO BAICBAICG11 )()()( sss

leunobservab and ableuncontroll :

observablebut abeluncontroll:

leunobservabbut lecontrollab:

observable and elcontrollab:

OC

OC

OC

CO

x

x

x

x

Chapter 6

现代控制理论 42Chapter 6

43

Kalman decomposition

Chapter 6

6.4 Discrete-time state equation

• The state model of discrete-time systems

• or

44

)k(Cx)k(y

)k(Bu)k(Ax)k(x

1

)k(Du)k(Cx)k(y

)k(Bu)k(Ax)k(x

1

Chapter 6

1.Controllability of discrete-time system

• Definition:

• The state equation or the pair (A,B)

is said to be controllable if for any initial state x(0)=x0 and

any final state x1, there exists an input sequence of finite

length that transfer x0 to x1. Otherwise is said to be

uncontrollable.

45

)k(Bu)k(Ax)k(x 1

Chapter 6

• Theorem: The following statements are equivalent.

1. The n-dimensional pair (A,B) is controllable.

2. The following nn matrix is non-singular.

3. The following nnp controllability matrix has rank n (full

row rank).

46

1

0

1n

m

mTTm

dc )(ABBA)n(W

]1BAB AB C

n-[

Chapter 6

47

现代控制理论 47

Example: Is the state equation controllable?

)(

1

0

1

)(

011

220

001

)1( kukk

xx

Sol.

nrankBAABBrankrank

3

111

620

111

Q 2

C

Controllable

Chapter 6

2.Observability of discrete-time system

• Definition:

• A discrete time state equation is said to be observable if

for any unknown initial state x(0), there exist a finite integer

k1>0 such that the acknowledge of input sequence u(k)

and output y(k) from k=0 to k1 suffices to determine

uniquely the initial state x(0). Otherwise the system is said

to be unobservable.

48Chapter 6

• Theorem: The following statements are equivalent.

1. The n-dimensional pair (A,C) is observable.

2. The following nn matrix is nonsingular.

3. The nqn observability matrix has rank n ( full column

rank)

49

1

0

1n

m

mTmT

do ACC)A()(nW

nnm

n

1CA

CA

C

QO

Chapter 6

现代控制理论

Example : Is the following state equation observable?

)(

1

0

1

)(

011

220

001

)1( kukk

xx )(111)( kky x

nrank

CA

CA

C

rankrankQO

3

642

230

111

2

Solution:

Observable

Chapter 6

MATLAB

Homework:

• Learn and practice functions related to :

1. Controllability matrix

2. Observability matrix

51

Summary

• Concepts of controllability and observability

• Rules to determine the controllability and observability of linear

system

• Decomposition of LTI system

• Controllability and observability of discrete-time linear system

52Chapter 6