Modern Control Systems 1

Lecture 08 State Feedback Controller Design

8.1 State Feedback and Stabilization

8.2 Full-Order Observer Design

8.3 Separation Principle

8.4 Reduced-Order Observer

8.5 State Feedback Control Design with Integrator

Modern Control Systems 2

00)( , xtxBuAxx

nBFAI 21det

Fxu

xBFA

BFxAxBuAxx

Stabilization by State Feedback: Regulator Case

Given ,n,ii 1 ,

Controllable )(A,B

There exists a state feedback matrix, F, such that

Plant:

State Feedback Law:

Closed-Loop System:

Theorem

State Feedback and Stabilization

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State Feedback System (Regulator Case)

CC

AA

BB

DD

FF

xu x y

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nn

cc

fafafa

FBA

2211

000

010

1

0

0

cB

xfffFxu n 21

State Feedback Design in Controllable Form

111det fasfasFBAsI n

nnn

cc

n

nc

aaa

IA

21

1

0

0

(8.1)

Modern Control Systems 5

11

1

cn

cnn

n

ii asass

111

111

aaf

aaf

aaf

c

ncnn

ncnn

ncncc aaaaaaF 2211

Comparing (8.1) and (8.2), we have

Suppose the desired characteristic polynomial

(8.2)

(8.3)

Modern Control Systems 6

CC

AA

BB

DD

FF

State Feedback: General Case (Non-Zero Input Case)

rFxu nfffF 21

State Feedback Control System

r xu x y

Modern Control Systems 7

Cxy

BuAxx

n

ii

n

s1

1

: Poly.Char. Desired

,, :poles Desired

zCy

uBzAz

c

cc

Tc

n

nc

B

aaa

IA

100

0

0

21

1

zFu c Tncnccc aaaaaaF 2211

xTFzFu cc-1

121

1

det ccn

cnn

n

ii asasassBFAsI

CTC

BTB

ATTA

c

c

c

1

1

Tzx

State Feedback Design with Transformation to Controllable Form

Controllable From:

Modern Control Systems 8

n

nnc aaa

IA

21

110

1

0

0

cB

ncccc CCCC

21

Tc

Tc

Tc BCCBAA ooo

Cxy

RxBuAxx n

nvvvT 21

TzxxTz 1

11

1

1 where

V

l

l

L

Al

Al

l

TT

n

T

nTn

Tn

Tn

Transform to Controllable Form

xCy

RxuBxAx

c

ncc

CTC

BTBATTA

c

cc

1

11

lecontrollab is ),( ,)(

1-

BAnUrank

BAABBU n

Coordinate Transform Matrix

Controllable Form:

Modern Control Systems 9

uxx

1

0

22

10

18633 :l PolynomiaChar. Desired

33 :poles Desired222

sss

j

Example

41626218 F

(A, B) is in controllable from, we can derive the state feedback gain from eq. (8.3)

xFxu 416

1 1)(sU

2

-1s -1s 1x2x

2

164

y

Modern Control Systems 10

BuAxx 1 nRx RuCxy Ry

)()()( trtKxtu

BrxBKAx )(

0 BKAsI

nBABAABB n ] [rank 12

nRK 1

Plant:

State Feedback:

Closed Loop System:

Char. Equation:

Suppose that the system is controllable, i.e.

Obtain the State Feedback Matrix by Comparing Coefficients

Modern Control Systems 11

n ,,1

)()( 1 nss

BKAsI )()( 1 nss

Then, for any desired pole locations:

(8.4)

We can obtain the desired char. polynomial

By controllability, there exists a state feedback matrix K, such that

From (8.4), we can solve for the state feedback gain K.

Modern Control Systems 12

)10)(1(

8

)(

)(

ssssU

sY

122

33

1

8)810()811(

8

)(

)(

ksksks

k

sR

sY

Example

Plant:

r

10

1 y1x 2x

1s 1s8u1 1s3x

1

2k

1k

3k

1

Fig. State Feedback Design Example

)()( tKxtu State Feedback: 321 kkkK

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ploes)(dominant 882,1 js 403 s

Desired pole locations:

707.0,8nPercent Overshoot 5%, Settling Rise time 5 sec.

)40)(88)(88( 8)810()811( 12

23

3

sjsjsksksks

-6401 k 94.752 k 5.6253 k

By comparing coefficients on the both sides of 8.5), we obtain

From (8.4), we get

(8.5)

5.62594.75640-K

Spec. for Step Response:

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0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

Simulation Results

Fig. Step response of above example

)(ty

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IaAaAaAAA ccn

cnn

n

iic 12

1

1

)(

100 K 112 BABAABB n )(Ac

n ,, :poles Desired 1

The Matrix Polynomial

BuAxx 1 nRx RuCxy Ry

)()()( trtKxtu nRK 1

Plant:

State Feedback:

Then the state feedback gain matrix is

121

1

: Poly.Char. Desired ccn

cnn

n

ii asasass

Ackermann Formula for SISO Systems

nR 1

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Steady State Error

)()()()()(

tCxtytBrtAxtx

From (3.6)

)()()( sYsRsE

])(-)[1(lim

)( lim)(lim1-

0

0

BsI-ACssR

ssEtee

s

stss

][1 )()( 1 BAsICsRsE

By Final Value Theorem

Lapalce Transform of the Error Variable

Error Variable )(-)()( tytrte

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CxyxtxBuAxx

00)( ,

LyBuxLCA

xCyLBuxAx

ˆ

ˆˆ

Full-Order Observer

Suppose x is the observer state

xxe ˆ

CxyeLCA

LyBuxLCABuAxxxe

ˆˆ

Estimation error:

Error Dynamics Equation:

Plant:

L: Observer gain

Full-Order Observer Design

Modern Control Systems 18

Hence if all the eigenvalues of (A-LC) lie in LHP, then the error system

te as ,0

eLCAe

is asy. stable and

Fig. Full-Order Observer

)(ˆ tx

)(tu)(ty

)(tx)(txC

A

B s

1

C

A

B s

1

+L

)(ˆ ty

)(ˆ tx

)(tey

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By duality between controllable from and obeservable form

Tc

Tc

Tc BCCBAA ooo

nLCAI 21det

Given ,n,ii 1 , Observable )(A,C

There exists a observer matrix, L, such that

Theorem

we have the following theorem.

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The eigenvalues of can be assigned arbitrarily by proper choice of K. Since

)( KCA TT

)( and ),( CKAKCA TTT

have same eigenvalues, if we choose

TKL

then the eigenvalues of (A-LC) can be arbitrarily assigned.

Modern Control Systems 21

Cxy

BuAxx

LyxLCBFA

xxLCxBFxAxCyLBuxAx

ˆ

ˆˆˆˆˆ

eLCAxxe

xxe

ˆ

ˆ

BFexBFA

BFeBFxAxxBFAxBuAxx

ˆ

Separation Principle

xFu ˆ

Plant:

State Feedback Law using estimated state:

Observer:

Error Dynamics:

State Equation:

(8.6)

(8.7)

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e

xA

e

x

LCA

BFBFA

e

x ~0

xI

C

y

yy

2

1

LCAIBFAI

LCAI

BFBFAIAI

0

~

Equation (8.9) tells us that the eigenvalues of the observer-based state feedback system is consisted of eigenvalues of (A-BF) and (A-LC).Hence, the design of state feedback and observer gain can be done independently.

From (8.6) and (8.7), we obtain the overall state equation

Eigenvalues of the overall state equation (7.17)

(8.8)

(8.9)

Separation Principle (Cont.)

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Observer-Based Control SystemObserver-Based Control System

)()(

)()()(

tCxty

tButAxtx

Plant:

BuLCxxLCABuxCyLxAx

ˆ)()ˆ(Observer ：

State Feedback Law: )()(ˆ)( trtxKtu

Modern Control Systems 24

Fig. Observer-based control system

)(ˆ tx

)(tu)(ty

)(tx)(txC

A

B s

1

C

A

B s

1

L

)(ˆ ty

)(ˆ tx

)(te

)(tr

K

Modern Control Systems 25

)(ˆ tx

)(tu)(ty

)(tx)(txC

A

B s

1

C

A

B s

1

L

)(ˆ ty

)(ˆ tx

)(te

)(tr

K

ck

Fig. Observer-based control system with compensating gain

Modern Control Systems 26

)()(

)()()(

tCxty

tButAxtx

Consider the n-dimensional dynamical equation

nqpnnn RCRBRA ,,

(8.10a)

(8.10b)

Here we assume that C has full rank, that is, rank C =q. Then, there exists a coordinate transformation Pxx

xIxCQxCPy

PBuxPAPx

q 01

1

which can be partitioned as

1

2

1

2

1

2221

1211

2

1

0 xxIy

uB

B

x

x

AA

AA

x

x

q

Reduced-Order Observer Design

(8.11)

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Since 1xy , we have

uBxAxAx

uBxAyAy

22221212

121211

which become

uByAyw

uByAu

111

221

212

2222

xAw

uxAx

Observer

(8.12a)

(8.13a)

(8.12b)

(8.13b)

)()(

)()()(

tCxty

tButAxtx

BuLyxLCA

BuxCyLxAx

ˆ)(

)ˆ(Observer:

Plant:

uwLxALAx 212222 ˆ)(

where

Modern Control Systems 28

Note that and w are function of known signals u and y. Now if the dynamical equation above is observable, an estimator of can be constructed.

u2x

Theorem:The pair {A, C} in (8.10) or, equivalently, the pair in (8.12) is observable if and only if the pair in (8.13) is observable.

} ,{ CA} ,{ 1222 AA

Modern Control Systems 29

uwLxALAx 212222 ˆ)(

Such that the eigenvalues of can be arbitrarily assigned by a proper choices of . The substitution of w and into (8.143)

yields L u

1222 ALA

)()(

ˆ)(ˆ

221111

212222

uByAuByAyL

xALAx

To eliminate the term of the derivative of y, by defining

yLxz 2ˆ

(8.14)

(8.15)

(8.16)

Let the estimate of be2x

Modern Control Systems 30

Using (8.15), then the derivative of (8.16) becomes

yALALALA

uBLBzALA

uBLByALA

yLzALAz

)()(

)()(

)()(

)()(

11211222

121222

121121

1222

From (8.15), we see that

yLzx 2ˆ

is an estimate of .2xDefine the following matrices

yDzCL

y

zx

xx ˆˆ

y

0

ˆˆ

2

1

)()(ˆ

,)(ˆ ,)(ˆ

11211222

121222

ALALALAJ

BLBBALAA

Modern Control Systems 31

yJuBzAz ˆˆˆ

yDzCx ˆˆˆ

Reduced-Order Observer:

where

)()(ˆ

,)(ˆ ,)(ˆ

11211222

121222

ALALALAJ

BLBBALAA

yLzx 2ˆ,ˆˆy

0

ˆˆ

2

1 yDzCL

y

zx

xx

Modern Control Systems 32

Fig. Reduced-Order Observer

)(tz

)(tu)(ty

)(tx)(txC

A

B s

1

s

1 + )(ˆ tx)(tz

A

B C

DJ

+

Modern Control Systems 33

)(

ˆ

2

22

yLzx

xxe

then we have

eALA

xLzxALA

uBLxALxALuBLBxALA

xLzALAuBxAxA

xLzxyLzxe

)(

))((

)()(

))((

)()(

1222

121222

12121111211121

112222222121

122

Define Error Variable

Modern Control Systems 34

Since the eigenvalues of can be arbitrarily assigned, the r

ate of e(t) approaching zero or, equivalently, the rate of )( 1222 ALA

yLz

approaching can be determined by the designer. Now we combine with to form

2x1x yLzx 2ˆ

yLz

y

x

xx

2

1

ˆ

ˆˆ

xQPxx 1

We get

z

y

IL

IQQ

yLz

yQQxQx

qn

q 0

ˆ

21

21

Then from

Modern Control Systems 35

)()(

)()()(

tCxty

tButAxtx

Consider the n-dimensional dynamical equation

nqpnnn RCRBRA ,,

Here we assume that C has full rank, that is, rank C =q. Define

R

CP

where R is any (n-q)n real constant matrix so that P is nonsingular.

(8.17a)

(8.17b)

How to transform state equation to the form of (8.11)

Modern Control Systems 36

Compute the inverse of P as

211 QQPQ

where Q1 and Q2 are nq and n(n-q) matrices. Hence, we have

qn

q

n

I

I

RQRQ

CQCQ

QQR

CPQI

0

0

21

21

21

Modern Control Systems 37

Now we transform (8.17) into (8.11),

by the equivalence transformation Pxx

xIxCQxCPy

PBuxPAPx

q 01

1

which can be partitioned as

1

2

1

2

1

2221

1211

2

1

0 xxIy

uB

B

x

x

AA

AA

x

x

q

Modern Control Systems 38

BuAxx

z

xCy

ruB

z

x

C

A

z

x n

0

1

0

00

0 1

Cxy

Cxryrz

rCxz

nnnn RCRBRA 11 ,,

CxyBuAxx

RyRx n

Ru

SISO State Space System

Rz

Integral Control:

Augmented Plant:

Modern Control Systems 39

zKKxu e

z

xKK e

z

xCy

rz

x

C

BKBKA

z

x ne

0

1

0

01

Closed-Loop System:

State Feedback Control Design with Integrator

Modern Control Systems 40

)(tu )(ty)(tx)(txC

A

B s

1)(te)(tr

K

s

1

)(tz

Block diagram of the integral control system

Fig.Block diagram of the integral control system

eK

Modern Control Systems 41

xy

uxx

011

0

53

10

Example

0.598,nPercent Overshoot 10%, Settling time 0.5 sec.Spec. for Step Response:

State Feedback Design: )()()( trtKxtu 21 kkK

3.11816 :l PolynomiaChar. Desired

10.918 :poles Desired2

ss

j

Modern Control Systems 42

BrxBKAx rx

1

0

1683.11

10

xCxy 01

BBKAC 11

0.995

1

0

16183.1

10011

1

])(-)[1(lim

)( lim)(lim1-

0

0

BsI-ACssR

ssEtee

s

stss

)1

)( (s

sR

From the steady state analysis in Sec. 3.4

Modern Control Systems 43

State Feedback Design with Error Integrator:

z

x

x

y

r

z

x

x

Kkk

r

z

x

xKkk

z

x

x

e

e

2

1

2

1

21

2

1

212

1

001

1

0

0

001

)(5)(3

010

1

0

0

0011

0

1

0

53

10

(8.18)

Closed-Loop System:

100)3.1)(1816( :l PolynomiaChar. Desired100 10.91,8 :poles Desired

2

sssj

Modern Control Systems 44

1780.11 k 1112 k 18310eK

By comparing coefficients on left hand sides of (8.19) and (8.20), we obtain

From (8.18), we get the char. eq. of the closed-loop system is

(8.19)

111 1780.1K

0)3()5( 12

23 eKsksks

0)183.116)(100( 2 sss (8.20)The desired char. eq. of the closed-loop system is

18310eK

Modern Control Systems 45

xy

r

z

x

x

z

x

x

00011

0

0

001

831011161783.1

010

2

1

2

1

Closed-Loop System:

Final Value Theorem

)(

)(

183101783.1116

18310)(

23 sR

sY

ssssT

1183101783.1116

18310lim

1)(lim)(lim

2300

ssssssTty

sst

0sse

Steady State Error