MODERN CONTROL THEORYLecturer：Qilian Bao鲍其莲

1

Chapter 7 State Feedback and State

Estimator

Objectives:

• State feedback and design of feedback matrix

• Design of state estimator

• Feedback control with estimated states

2Chapter 7

7.1 State Feedback

• Output feedback control system

3Chapter 7

Output feedback system

4

let HyVu

DuCxy

BuAxx

VDDHIBHBxCDHIBHAHyVBAxx ])([])([)( 11

DVDHICxDHIy11 )()(

Chapter 7

State feedback system• A linear system:

• Then

5

DuCxy

BuAxx

Let: KxVu

DVxDKCy )(

BVxBKAKxVBAxx )()(

Kx - State feedback

V – input of system

u – output of controller/input of plant

Chapter 7

7.2 Controllability of Feedback system

• Theorem: The system (A-BK, B) is controllable IFF system

(A,B) is controllable.

• Note: The observability may be changed.

6

Linear system:

Cxy

BuAxx

State feedback

n

i

iin xkrxkkr-Kxru1

1

Cxy

BrBK)x(AxFeedback system

Chapter 7

• Example:

• Let

• Feedback system

• Controllability matrix controllable

• Observability matrix un-observable

•

7

xy

uxx

21

1

0

13

21

xru 13

xy

uxx

21

1

0

00

21

01

20fC

21

21fO

Chapter 7

7.3 Pole placement and design of

feedback matrix

• State feedback can be used to place eigenvalues(pole)of

A in any position if the system is controllable.

• Pole placement: choosing state feedback gains to place

poles in desired positions.

• Theorem: If the n-dimensional linear system is

controllable, then the eigenvalues of A-BK can arbitrarily

assigned by state feedback u=r-Kx, where k is a 1n real

constant vector.

8Chapter 7

9Chapter 7

10Chapter 7

Example:

• Consider a linear system

11

uxx

0

1

13

31

2

1 2( ) ( 1) 9 ( 4)( 2), 4, 2s s s s s s

rxkk

uxkk

x

0

1

13

31

0

1

0013

31 2121

)kk(s)k(s)s( 832 211

2

State feedback

n

i

iin xkrxkkV-Kxru1

1

New s1 s2 is determined by values of k1 and k2.

Chapter 7

• If linear system is controllable, then it can be transfer to

canonical controllable form by linear transformation:

• Transfer function:

12

u

aaa n

1

0

0

1000

0100

0010

110

xx

x110 nβββy

xPx1

α(s)

β(s)

asasas

βsβsβsβ

b]A[sICbA]C[sIg(s)

n

n-

n

n

n-

n

n-

01

1

1

01

2

2

1

1

11

Chapter 7

• Let state feedback

• K is unknown gains.

• Then

13

xKrxKPrKxru 1

110

1

nkkk KPK

)k(a)k(a)k(a

kkk

aaa

KbA

nn

n

n

111100

110

110

10

010

1

0

0

10

010

)k(a)sk(a)sk(as)]KbA([sIdet(s)Δn

nn

n

K 0011

1

11

Chapter 7

• For feedback system

• Assume that desired eigenvalues are s1, s2, …, sn

Feedback gains after transformation

Feedback gains or feedback matrix

14

)k(a)sk(a)sk(as)]KbA([sIdet(s)Δn

nn

n

K 0011

1

11

*

0

*

1

1*

1

1

* )()(Δ asasassss n

n

nn

i

iK

1

*

11

*

10

*

0 nn aaaaaa K

110 nkkk PKK

Chapter 7

Example: Consider a linear system

15

uxx

0

1

13

31

242491 21

2 s,s),s)(s()s()s(

rxkk

uxkk

x

0

1

13

31

0

1

0013

31 2121

)kk(s)k(s)s( 832 211

2

State feedback

n

i

iin vkrxkkV-Kxru1

1

New s1 s2 are determined by values of k1 and k2.

Suppose desired eigenvalue is -1, -2. Then

2321 2 ss)s)(s()s(*

283

32

21

1

kk

k

5

5

2

1

k

k

Chapter 7

16

A cart with inverted pendulum:

;0

1

000

1000

000

0010

1

1

4

3

2

1

)(

4

3

2

1

u

x

x

x

x

x

x

x

x

Ml

M

Ml

gmM

M

mg

4

3

2

1

0001

x

x

x

x

y

Chapter 7

• Exercise: for a cart with inverted pendulum system

• Assume the desired eigenvalues

• Try to find the feedback gain matrix K.

17

;u

x

x

x

x

x

x

x

x

2

0

1

1

0500

1000

0100

0010

4

3

2

1

4

3

2

1

4

3

2

1

0001

x

x

x

x

y

1 2 3 411, 12, 13, 14s s s s

Chapter 7

• Exercise:

18

;u

x

x

x

x

x

x

x

x

2

0

1

1

0500

1000

0100

0010

4

3

2

1

4

3

2

1

4

3

2

1

0001

x

x

x

x

y

00505 23422 ssss)s(s)s(

)s)(s)(s)(s()s(* 14131211

Chapter 7

• Note: State feedback can shift the poles of a plant but has

no effect on the zeros. This can be used to explain why a

state feedback may alter the observability property of a

state equation.

19Chapter 7

Control system design using state feedback

A rough guide for system design

• Place all eigenvalues inside the

region denoted by G in the right

figure

• Better to place all eigenvalues

evenly around a circle with radius r

inside the sector as shown

• A final selection may involve

compromises among many

conflicting requirements

20Chapter 7

7.4 State estimator

• Why do we need state estimator or state observer?

– State feedback requires real-time values of state variables

– State variables may not be accessible for direct connection

in practice

– Sensing devices or transducers may be unavailable or very

expensive

21Chapter 7

• Why we can estimate state variable?

State vector x(t) may be estimated from u and y over any

time interval [t, t+t1].

22Chapter 7

23

Open-loop estimator

Closed-loop estimator

Chapter 7

How to estimate state variables?

• If the states of the linear system can not be measured

directly, alternative approach is to construct an equivalent

system and measure the states of equivalent system

instead.

24Chapter 7

25

closed-loop estimator

Chapter 7

• The original system

• Design a system with same parameters

• Then

• Let

• Then

26

)x()x(t,Cxy

BuAxx0

0

xCy

BuxAx

ˆˆ

ˆ

)ˆ(ˆ

)ˆ(ˆ

xxCyy

xxAxx

lyBuxlC)(A

)xlC(xBuxA)yl(yBuxAx

)xlC)(x-(Aly]BuxlC)[(ABuAxx-xe

Chapter 7

27

State

estimator

State

estimator

Chapter 7

)yl(y

a) Can be

rewrote as b)

• If (A-lC) is stable, i.e. all the eigenvalues of (A-lC) have

negative real parts, then

• So the states of equivalent system can be used for state

feedback of original system.

• Theorem: All eigenvalues of (A-lC) can be assigned

arbitrary by selecting a real constant vector l if and only if

(A, C) is observable.

28Chapter 7

0)ˆ(lim

xxt

lC)e(A

)xlC)(x-(Aly]BuxlC)[(ABuAxx-xe

• Example: Design a state estimator with desired

eigenvalues as -3,-4,-5 for the following system.

Solution:

(1) observability of the system

It is observable, then the state estimator can be assigned

eigenvalues.

29

u

1

0

1

200

120

001

xx

x011y

441

121

011

OQ 3OQrank

Chapter 7

(2) Design of state estimator

Let

Then for desired eigenvalues

The state estimator

The vector is

30

2

1

0

l

l

l

L

604712)5)(4)(3()(Δ 23* sssssssG

)()()(

detΔ

4248345 210210

2

10

3

lllslllslls

LCAsIG

210

103

120

2

1

0

l

l

l

L

60424

47834

125

210

210

10

lll

lll

ll

Chapter 7

7.5 Reduced-dimensional state estimator

(optional)• Theorem: If (A, C) is observable, and rank(C)=m<n, then

the minimal dimension of state estimator is (n-m).

• m states can be observed from outputs, so we only

estimate the rest (n-m) states.

31

21 CCC

mCrank

Chapter 7

• If the following system is observable

• Let transformation matrix

• After transform

• Only need to be estimated. The estimator is called

reduced-dimensional state estimator.

32

Pxx

1 PAPA

PBB 1CPC

Cxy

BuAxx

21

0

CC

IP

2

2

1

2

1

2

1

2221

1211

2

1

0 xx

xIy

uB

B

x

x

AA

AA

x

x

1x

Chapter 7

• Rewrite the above system as

• Let

• Subsystem

• Design the state estimator as

33

2

2

1

2

1

2

1

2221

1211

2

1

0 xx

xIy

uB

B

x

x

AA

AA

x

x

2221212

11211112121111

uByAxAyx

uByAxAuBxAxAx

xAy

u)ByA(xAx

121

1121111

yGyAGAuBGBxAGA

yGuByAxAGAx

122112211121111

11121211111

)()(ˆ)(

)(ˆ)(ˆ

121222 xAuByAyy

Chapter 7

• Let

• Then

• So the estimate of is .

• and

• Where

34

yGxz 11

ˆ

yGxz 11ˆ

yGzx 11

ˆ

yGzx 11ˆ

yAGAGAGAuBGBzAGAz ])[()()( 2211212111121121111

2

1

x

xx 0

0

ˆlimlim 111

xx

y

yGzx

tt

y

yGz 1x

y

yGzQQxQxPx

1

21

1

])[(~

221121211111 AGAGAGAG

Chapter 7

35Chapter 7

7.6 State feedback from estimated states

36

State

estimator

Chapter 7

• Linear system

• State estimator

• Choose state feedback control

• Then the system equations are

37

Cxy

buAxx

xK ˆVu

bVxbK)GC(AlCxx VbxbKAxx ˆ

Cxy

Vb

b

x

x

bKlCAlC

bKA

x

x

x

xC

ˆ0y

Chapter 7

State

estimator

GybuxlC)(Ax

• Equivalent transformation

• Then

• The poles of system are

• Eigenvalues of (A-bK) and (A-lC) can be assigned

independently.

38

II

IP

0

II

IP

01

Vb

x~x

lCA

bKbKA

Vb

b

II

I

x~x

II

I

bKlCAlC

bKA

II

I

x~x

00

000

x

xC

x

x

II

IC ~0

ˆ

00y

lC)A(sIdetbK)A(sIdetlCAsI

bKbKAsIdet

0

Chapter 7

Transfer function

Note:

• The transfer function of the above system equals the transfer

function of the original state feedback system without using a

state estimator.

39

bbKAsICb

lCAsI

bKbKAsIC(s)gK

1

1

000

Chapter 7

Remark:

1. The eigenvalues are the union of those of A-bk and A-lc.

2. Inserting the state estimator does not affect the

eigenvalues of the original state feedback; nor are the

eigenvalues of the state estimator affected by the

connection.

3. The design of state feedback and the design of state

estimator can be carried out independently. This is

called the separation property.

4. Better to choose bigger-magnitude eigenvalues of (A-

lC) than those of (A-bK) to make the convergences of

state estimator faster than the state feedback control

system.

40Chapter 7

MATLA B

Homework:

• Learn and practice functions related to :

1. Pole placement

2. Computation of state feedback matrix

3. State estimator

41Chapter 7

Summary

• State feedback

• Determination of feedback gains

• State estimator

• Design of state estimators

• State feedback with state estimator

42Chapter 7