Similarity transformation

DDCPCBPBAPPA

uDxCy

DuxCPy

uBxAx

BuPxAPPx

BuxAPxP

xPxxPx

DuCxy

BuAxx

,,,

,let weIf

)(#

11

11

same

system

as(#)

(#)

(*)

x Ax Bu

y Cx Du

x Ax Bu

y Cx Du

Controllability:

n

nBBAABB

nABAABB

x

txtu

x

DuCxy

BuAxx

n

n

)BAIrank(or

)1 is (if 0]|det[or

B])|||[rank( iff c.c. :Thm

time.finitegiven any in 0

to)( bringcan which )( control

,)0(any if lecontrollab completely is

1

12

Example:

01)det(or

2rank ind.linearly

231

10rank

3

1

|

|

1

0

1

0

32

10

|

|

1

0][

2

1

0 ,

32

10

ABB

ABB

n

BA

1 2 1 0

1 2 1 0

2 2 1

1

1

1

1

1 0 0 0 0

0

0 1 0 0 0

0 0 1 0 0

[0]

1 ## ## ##

0 1 ##

0 0 1

0 0 1

0 0 0 1

n n

n n

n n

n

n

n

a a a a

x x u

y b b b b x u

B AB A B A B A B

a

a

a

det 1

ControllerCanonical Form:

CompletelyControllable

Controllability:

C.C. is B) (A, zero,non all are,,

.0,1

100

020

001

100

020

001

:Example

)BA-Irank( iff c.c. is B) (A, :Thm

321

1

3

2

1

3

2

1

bbbIf

bifrankfullhasmatrixWhen

b

b

b

BAI

u

b

b

b

xx

n

Only need to check this for eigenvalues

Controllability:

C.C. is B) (A, zero,non are,

.0,1

.0,2

1000

1100

0110

0002

1000

1100

0110

0002

:Example

)BA-Irank( iff c.c. is B) (A, :Thm

41

4

1

4

3

2

1

4

3

2

1

bbIf

bifrankfullhasmatrixWhen

bifrankfullhasmatrixWhen

b

b

b

b

AI

u

b

b

b

b

xx

n

thisofty multiplici toequalrank have should

identicalan toingcorrespond B of rows those2)or

nonzero B of roweach anddistinct s' all 1)

:if C.C. is B) (A,

00

00

00

:diagonal isA If

k

2

1

n

A

PBH test for diagonal case

1

2

k

If A is Jordan block diagonal:

0 0 1 0 0

0 0 0 0 ,

0 1

0 0 0 0 0

(A, B) is C.C. if:

1) for different Jordan block is distinct and each row

k

kk

n k

J

JA J

J

of corresponding to the last row of is nonzero

or 2) those rows of B corresponding to the last row of

's with an identical should have rank equal to

t

k

k

B J

J he number of 's with the same kJ

PBH test for block Jordan diagonal case

3

2

1

,

100

020

001

BA

Are the following (A, B) pairs C.C.?

3

0

1

,

100

020

001

BA

3

2

1

,

100

020

001

BA

3

1

0

0

2

1

,

100

020

001

BA

3

2

1

,

100

010

011

BA

Are the following (A, B) pairs C.C.?

3

0

1

,

100

010

011

BA

1

2

1

2

1

,

10000

11000

01100

00010

00011

BA

30

02

20

01

10

,

10000

11000

01100

00010

00011

BA

Observability

,)C

A-Irank(or

)1 is (if 0detor ,rank iff c.o. :Thm

0set can ,generality of lossWithout

(0). determine tous enablecan timefinite aover

)(),( of knowledge theif obserrable completely is

11

n

nC

CA

CA

C

n

CA

CA

C

u

x

tytu

DuCxy

BuAxx

nn

Example:

c.o.

01det(

10

01

32

1001

01

2

01 ,32

10

CA

C

CA

C

n

CA

Observability

C.O. is A)(C, zero,non all are,,

.0,1

321

100

020

001

321

100

020

001

:Example

)C

A-Irank( iff c.o. is A) (C, :Thm

321

1

3

2

1

cccIf

cifrankfullhasmatrixWhen

ccc

C

AI

cccy

u

b

b

b

xx

n

C.C. is B) (A, zero,non are,

.0,1

.0,2

4321

1000

1100

0110

0002

4321

1000

1100

0110

0002

:Example

)C

A-Irank( iff c.o. is A)(C, :Thm

21

2

1

4

3

2

1

ccIf

cifrankfullhasmatrixWhen

cifrankfullhasmatrixWhen

cccc

C

AI

ccccy

u

b

b

b

b

xx

n

thisofty multiplici toequalrank have should

identicalan toingcorrespond C of cols those2)or

nonzero C of col.each anddistinct s' all 1)

:if C.O. is A) (C,

00

00

00

:diagonal isA If

k

2

1

n

A

PBH test for diagonal case

same with thes'such ofnumber the

toequalrank have should identicalan with s' all

of colfirst the toingcorrespond C of cols those2)or

nonzero is each of colfirst the toingcorrespond

of coleach anddistinct isJk different for 1)

:if C.O. is A) (C,

000

10

00

001

,

00

00

00

:diagonalJordan block isA If

k

2

1

k

k

k

k

k

k

k

n

J

J

J

C

J

J

J

J

A

PBH test for block Jordan diagonal case

121

100

020

001

C

A

Are the following (C, A) pairs C.O.?

020

100

020

001

C

A

121

100

020

001

C

A

100

121

100

020

001

C

A

101

121

100

020

001

C

A

Are the following (C, A) pairs C.O.?

110

100

010

011

C

A

101

100

010

011

C

A

011

100

010

011

C

A

01010

10101

10000

11000

01100

00010

00011

C

A

00100

01001

10000

11000

01100

00010

00011

C

A

Controllability and Observability

n

nBBAABB

nABAABBn

n

)BA-Irank(or

)1 is (if 0]|det[or

B])|||[rank(

iff lecontrollab completely is B) (A,

1

12

)1 is (if 0detor ,rankor

eseigen valu,)C

A-Irank(

iff obserrable completely is A) (C,

11

nC

CA

CA

C

n

CA

CA

C

n

nn

C.C., C.O. and TF poles/zeros

C.O. is A)(C, and C.C. is B) (A, iffon cancellati

pole/zero no has )(

Consider

1BAsICDH(s)

DuCxy

BuAxx

bothor C.O.or C.C.either loses system the

on,cancellati pole/zero no has )( If 1BAsICDH(s)

State Feedback

law controlfeedback state a called is

:law the

Given

rKxu

DuCxy

BuAxx

B 1

s C

D

A

K

r u x x y+ +

+ ++

-

feedback from state x to control u

BkA

BkAA

DuCxy

BrxBkAx

BrBkxAx

rkxBAx

BuAxx

of thoseofeedback t state

by changed valuess/char.eigenvalue

tochangedMatrix only the

)(

)(

equation space state loop-closed

k

BkAnQC of choiceby any tochanged

becan of seigenvalue)(rank

i.e.

true.also is converse The

location.arbitrary

any toeigenvalueor valueschar.

thechangecan feedback statethen

lecontrollab completely is system theIf :Thm

Pole placement

)())(() det(sI

) det(sI of roots are seigenvalue loop-closed

)(

equation space state loop-closed

21 nsssBKA

BKA

DuCxy

BrxBKAx

Solve this to get k’s.

Example

1,2

22)1(1

1det

))(() det(sI

Let

1at poles loop-closedWant

001

1

0

10

10

21

212

21

21

21

1,2

kk

sskkssksk

s

ssBKA

kkK

j

uxy

rxx

Pole placementIn Matlab:

Given A,B,C,D

①Compute QC=ctrb(A,B)

②Check rank(QC)

If it is n, then

③Select any n eigenvalues(must be in complex conjugate pairs)

ev=[λ1; λ2; λ3;…; λn]

④Compute:

K=place(A,B,ev)

A+Bk will have eigenvalues at these values

Invariance under state feedback

Thm: Controllability is unchanged after state feedback.

But observability may change!