Chapter 8 Polynomial Approach

Robotics Research Labo-ratory

1

Chapter 8

Polynomial Approach


2

I/O Model

where and are polynomials in forward-shift operator q.

Basic assumptionsi) deg B(q) < deg A(q)ii) A(q) and B(q) do not have any common factors.

(coprime)iii) The polynomial of A(q) is monic.

(normalized for uniqueness)

Note: Pulse transfer functionB(z)/A(z)

A q

A q y k B q u k open-loop system 1

B q


where R(q), T(q) and S(q) are polynomials in forward-shift opera-tor. R(q) can be chosen so that the coefficient of the term of the highest power in q is unity.

Notes:

deg R(z) deg T(z)deg R(z) deg S(z) causal controller

cu yu ( ) ( )R q u k T q u k S q y kc ( )

( )

B q

A q

3

Controller ( ) ( )cR q u k T q u k S q y k 2

( ) ( ) / ( )( ) ( ) / ( )

ff

fb

H z T z R zH z S z R z


4

The characteristic polynomial of the closed-loop system

(( ( ) ( ) ( ) ( )) ( )) ( ) () 3cy kA q R q B B qq T q uS q k

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

4

5

cl

cl c

A z A z R z B z S z

A q y k B q T q u k

if there is a time delay in the control law of one sampling period

deg deg 1 deg 1

R TS


5

Pole Placement Design

Algebraic problem of finding polynomials R(z) and S(z) that satisfy (4) for given A(z), B(z) and Acl(z)

( ) ( ) ( )cl c oA z A z A z

where ( ) det( ) ( ) det( )

c

o

A z zI Φ ΓKA z zI Φ LC

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )c c

cl c o

B z T z B z T zY z U z U zA z A z A z


6

It is natural to choose the polynomial T(z) so that it cancels the observer polynomial Ao(z).

where to is the desired static gain of the system.

( )( ) ( )( )

Then, oo o c

c

t B zT z t A z Y z U zA z

1

1

0

1

0

0

lim ( ) lim(1 ) ( )

( )1 lim ( )( )

If unit step is applied,(1)

lim ( ) 1(1)

1 / (1)

k z

czc

kc

c

y k z Y z

t B zz U zz A z

t By k

At A B


7

ex) Control of a double integrator

22

22

0

22 0

1 0 1

( ) ( ) ( ) ( )

( ) 1 ( ) 1 2

2 1 12

For ( ) 1, ( ) (proportional controller)

2 1 12

For ( ) , ( )

cl

cl

A q y k B q u khA z z B z z

hz z R z z S z A z

R z S z s

s hz z z

R z z r S z

A

s z s

z

22

1 0 1

2 2 23 2

1 0 1 0 1 1 1

2 1 12

( 2) 1 22 2 2

c

cl

lhz z z r z s z s A z

h h hz r s z A zr s z s z r s

＋　（＋）

（＋）


8

, ,( ).

cl

cl

r s sA z

z p z p z p

hr s

A z

1 0 1

3 21 2 3

2

1 0

It is possible to select the controller coefficients from the desired characteristic polynimial

(known poles by pole assignment)

2

( )

, ,

p

hr s s p

hr s p

p p p p p p p p pr s s

h h

1

2

1 0 1 2

2

1 1 3

1 2 3 1 2 3 1 2 31 0 12 2

2

2 12

2

This equation has the solution.3 5 3 3 3

4 2 2


9

Let A, B, and C be polynomials with real coefficients and X and Y unknown polynomials.

Then the above equation has a solution iff the greatest common factor of A and B divides C.

Notes:i) The Diophantine equation has many other names in literature,

the Bezout identity or the Aryabhatta’s identity.

ii)

iii) The extended Euclidean algorithm is a straightforward method to solve the Diophantine equation.

( )AX BY C

( ) ( ) ( ) ( ) ( )clA z R z B z S z A z

Diophantine Equation


10

x y x yx y

x x n y y n x y0 0 0 0

ex) 3 2 5 where and are integers. Then 1 and 1 is a solution.

2 and 3 where 3 2 5 are another solutions. - in

x y

finite number of solutions

ex) 4 6 1 (Diophantine equation without a solution)

Note: Diophantus of Alexandreia ( A.D. 246? ~ 330?) - one of the original invento

rs of algebra


11

-

-

-

( ) ( )( ) ( )

( )

( )

( ) ( )

( )

11 1

10 1 1

2 1 2 20 1

where

Let define a stable 2 1 th-degree polynimial as follows.

1 n nn n

n nn n

n n

Y z B zU z A z

A z z a z a z aB z b z b z b z b

n D zD z d z d z

-

-

( - )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

2 2 2 1

1 20 1 2

Then there exist unique 1 th-degree polynomials (z) and (z) such that i.e.,

where

c

n n

n n

l

n

d z d

n α β

α z A z β z B z D z A z B z

α z α z α z α

R z S A

z α

z z

-( )1

1 20 1 2 1

n

n nn nβ z β z β z β z β


12

1 1

1 1

1 1

1 2 0 1 2

1

In MATLAB, the Diophantine equation can be solved for and by using of the 2 2 Sylvester matrix .

0 0 0 0 0 00 0 0 0

0 0

1 00 1

n n

n n n n

n n n n

n

n

X Yn n E

a ba a b b

a a b ba b

a a b b b ba a

2 11

2 22

0

10 1 1 1

21 0 2 2

00 0

00 0 1 0 0

0 0 0 1 0 0 0

nn

nn

nn

n nn

n n n

dα

dα

αd

βb b b d

βa b b d

βb d

2 2 Sylverster matrix Note: It is nonsingular i

f

f the polynomia

n EM

nE D

ls and do not have any common factors.

A B


13

) ( ) .

( )

( )

( ) ( ) ( ) ( ) ( )( ) , ( )

..

2 21 2

20 1 2

3 3 20 1 2 3

0 1 0 1

ex 0 5

2

where

0 5 0 2 01 0 5 1 2

1 1 0 10 1 0 0

A z z z z a z aB z z b z b z b

D z z d z d z d z d

A z α z B z β z D zα z α z α β z β z β

E

..

, ,..

( ) . , ( ) . .

( ) ( ) ( ) ( )

1

0 1 0 1

3

0 1 20 1 0

0 0 31 0 2

1 2 0 2 0 3

Notice that

D M E D

α z α z α z β z β z β z

A z α z B z β z z


14

2

1 2

( ) 0.02( 1)ex) ( ) ( 1)

1 0.2 0.02 or ( 1) ( ) ( )

0 1 0.2

( ) 1 0 ( )

Desired poles: 0.6 0.4, 0.6 0.4 Reduced observer : ( ) 0

Y z zU z z

x k x k u k

y k x k

z j z jφ z z

Regulator Design by Pole Placement – state space approach


15

(

1

)

2

( )( ) ( ) 8 3.2

( )

( 1) 8 ( 1) 3.2 5 ( 1) ( 1)

24 ( 1) 16 ( ) .032 ( )( 1) 0.32 ( ) 24 (

( ) ( 0.6667)24

( ) ( 0.32)

1) 16 ( )

D z

x ku k Kx k

x k

u k y k y k η k

y k y k u

U z zGY z z

ku k u k y k y k


16

2

2

( ) 0.02( 1) ( ) ex) open-loop system

( ) ( 1) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )0.02( 1) ( )

closed-loop system( )( 1) 0.02( 1) ( )

wh

Y z z B zU z z A z

Y z α z B zR z α z A z β z B z

z α zα z z z β z

( )ere ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )β zA z Y z B z U z B z R z Y zα z

Regulator Design by Pole Placement - polynomial equation approach


17

2

Desired closed-loop characteristic polynomial

( ) ( 0.6 0.4)( 0.6 0.4) 1.2 0.52

Desired minimum-drder observer polynomial ( )

To determine ( ) and ( ) systematically, we solve the fol

clA z z j z j z z

F z z

α z β z

3 2 3 20 1 2 3

2 3 2

lowing Diophantine equation.

( ) ( ) ( ) ( ) ( ) ( ) ( )

where ( ) 1.2 0.52

( )( 1) 0.02 ( )( 1) 1.2 0.52

clα z A z β z B z F z A z D z

D z d z d z d z d z z z

α z z β z z z z z


18

. . . . .

. .,

. . . . .. . . .

..

,.

( )

1

1

0 1

1 0 0 02 0 0 25 0 25 0 25 0 752 1 0 02 0 02 0 0 0 1

1 2 0 0 02 37 5 12 5 12 5 37 50 1 0 0 12 5 12 5 37 5 62 5

0 0 320 52 1

1 2 161 24

E E

D M E D

α z α z α z

. , ( )

( ) ( . )( ) ( . )

0 10 32 24 16

0 6672 0 2

43

β z z

β z

z

z β z

z

β

α


Pole Placement Design- More Realistic Assumptions

.

19

Poles and zeros between the plant and controller can be cancelled.

,

where and are the factors (stable modes) that can be canceled.

i) For unique factorization, and a

A A A B B BA B

A B

re chosen to be monic.

ii) and must have all their roots inside the unit disc.

( )( ) ( )( )

where , and

process pole controller zero

process zero con

A B

AR BS A A B R B B A S

R B R S A S T A T

troller pole


20

( )

( )( )

minimum-degree causal controller

deg deg for uniqueness

cl

cl

cl c o C o

cl c o

c

c

A AR BS A A B R B B A S

A B A R B S A B A

A A A B A A A

A R B S A A A

S A

Ru Tu Sy

B Ru A Tu A S

y


21

cA T Su u yB R R

Pulse transfer function (uc to y)

o o

c cl c o

o o

c o c c

B B t B B t By z BT BTu z A A A

t AA A B A A

( )( )

Remarks:

i) Causalitydeg R deg Tdeg R deg Sdeg A deg B

ii) Uniquenessdeg A > deg S, deg B > deg R

iii) The cancelled factors must correspond to stable modes.


22

For ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ), consider the open-loop zeros. (a zero in processor is a pole in controller)

where , ,

c

mm

m

m

m

m

m

m

A q y k B q u k R q u k T q u k S q y k

BBT B B B B B B R B RAR BS A

BBT BAR BS A

B BA

( )

after cancelation

before cancelation

Note

m mm

m m m

o mm

m o m

o o

o o

o m

o m

o m

T B B TAR BS B AR B S

B B B BB BB T BTA A AR BS AAR B S

A BBTA A AAR B S

A B AA B A

AR B S A A

T A

B

B

AR BS

A S

A

TR

B A

B

,: canceled stable process zeros observer poles , model poles o mB A A


23

Causality Solution

Since deg deg and deg deg

deg deg( ) deg

(1)

(2)

Since deg deg

deg deg deg de

deg deg deg

g deg

deg deg

eg e

eg

d

d

d

o

o

m

o m

o

o

m

m

m

R B A

AR BS B A A

S R B AAR A

A

R BS B A A

A

T A B

R T

A A B A

g deg

deg deg deg deg

deg deg deg deg deg deg deg de

deg deg (3g )m

o m

m m

m m

m

A B

A B A B

A BB

BA B

AAB B


24

Since deg deg

(4)

deg deg

deg deg deg deg deg

deg deg 1

deg 2deg deg deg 1

1

(5)o

o m

m

S A

R

S A

A A

SA A B A A

A B


25

ex) DC motor with cancellation of process zero

1 22

1 2

( )( ) , 1( 1)( )

The desired closed-loop system is assumed that

Then

(1 )( ) simple struct

(1

uremm

m

m

m

mm

K z bH z z bz z a

B B

B z b

B z

B KB

p pH z

A A z

B

p

zB

p z

1 2 )p pK


26

20 0 1 1 2

1 20 0 1

deg deg deg deg deg 0 2 1 2 1

deg deg deg 0deg deg 1 1

deg 0

deg deg deg 1

( 1)( ) ( ) , i.e., 11, ,

( ) ( ) ( ) 1

( ) 1

o m

o m

o m

o m

o o

R A A B A

R R BS AT A B

z z a r K s z s z p z p AR B S A Aa p p ar s sK K

zT z A z B

A A z

z

1

0 0 1

2

( ) (

(1

) ( ) ( 1 (

)

) 1)c

o

u k t u k s y k s y k b

t

k

p pK

u

z


27

ζ ω rad0.7, 1 / sec

T T0.25 1.0

The process zero is canceled.- ripple between the sampling period


28

ex) DC motor with no cancellation of process zero

1 22

1

1

2

2

1

1 21

1( )

( )( ) , 1( 1)( )

The desired closed-loop system is ass

(1 )( )(1 )

um

1( )

ed that

1

1

( )(1 )

Hence (1 )

mm

m m

m

m

m

K z bH z z bz z a

BB p pH zA A z p z p

BB K z

B z bb

BB K z b

bp p p pB

K bz b

b

deg

The degree of the observer polynomia

2deg deg deg 1 1 )

is

(

l

o m oA A A B A z z


29

3 21 0 1 1 2

21 2

1 2

3 21

0

0 1 2

1

1

deg deg deg deg 1deg deg 1 1

The Diophantine equation can be written as

( -1)( - )( ) ( )( )

( )( 1)( )

(1 )( ) 1

( )( )

( )

m oR A A AS A

z z a z r K z b s z s z p z p z

b b p b pb

b b aK b p p

K a b a a p a p

r

s

a

s

s

z

s

T

1

0 0 1 1

20

1(1 )

The control law is th ( ) ( ) ( ) ( 1) (

en1)

o m

cu k t u k s

p pA B z t z

K

y k s k r

b

y u k


30

ζ ω rad0.7, 1 / sec

T T0.25 1.0

The process zero is not canceled.-smoother but a little slow response


31

Optimal DesignB q

x k u kA q

y k x kC q

v k e kA q

A q y k B q u k C q e k ARMAA A A

v

C C Ak

k

B B A

1

1

1

2

1 2 1 2 1 1

( )Process: ( ) ( )

( )output: ( ) ( )

( )noise: ( ) (

( ) (c

)( )

( ) ( ) ( ) ( ) ( ) ( ) ( )where , , e(

olored

)

noi

i

se)

s a sequence of inde

σ

J E y k

J E y k ρu k

2

2

2 2

with zero mean and variance of (white noise).Criteria

( ) minimum variance c

pendent or

ontrol

( ) ( ) linear quadra

uncorrelated random variable

t

ic control


32

Minimum Variance Control

* 1 * 1

* 1 * 1

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) forward( ) ( )

( ) ( )( ) ( ) backward (1)( ) ( )

where deg deg ( ) deg

d

A q y k B q u k C q e k

B q C qy k u k e kA q A qB q C qq u k e kA q A q

d A B pole excessA

deg C n

- system with stable inverse


33

* 1 * 1

* 1 * 1

* 1 * 1* 1

* 1 * 1

* 1 * 1 * 1 * 1

* 1*

* 1

( ) ( )( ) ( ) ( ) (2)( ) ( )

( ) ( )( ) ( ) ( ) ( ) (3)( ) ( )

where ( ) ( ) ( ) ( )

( ) or ( )

d

C q B qy k d e k d u kA q A q

G q B qF q e k d e k u kA q A q

C q A q F q G q qC q FA q

* 11

* 1

1 21 1

1 1 11 1

1 20 1 1

1 1 10 1 1

( )( )( )

Note: ( )

( ) 1

( )

( )

d

d dd

dd

n nn

nn

G qq qA q

F q q f q f

F q f q f q

G q g q g q g

G q g g q g q


34

* *

* *

* * **

* *

222 *

* * *

* *(

Also, using ( ) ( )

(3) becomes

( ) ( ) ( ) ( )

( )

)

independent of ( ), (

(

( ) ( )

-1), , ( ), ( -1),

)

d

F e k d

A Be k y k q u kC C

G B Fy k d F e k d y k u kC C

E y k d E E

u

G B Fy k u kC C

y k y k u k u k

k

* 1

* 1 * 1

2 2 2 21 1

( ) ( )( ) ( )

( ) ( ) ( )

Notes: 1) ( ) 1

( )

2) minimum phase system plant zero must be stable. d

G q G qy k y kB q F q F q

E y k d f f

B q


35

3 2

3 2

2

ex) Minimum variance control

( ) 1.7 0.7

( ) 0.9

pole excess 2 ( ) 0.8

(

( ) 0.5

( 0

) 0.66 0.56(0.66 0.56) ( ) ( )

( 0.8))

.

5

A q q q q

C q q q

dF q q

G q q qqu k y k

q

B q q

q

2 2 2 2

( ) 1.3 ( 1) 0.4 ( 2) 0.6

Note: (1 (0.8)

6 ( ) 0.56

) 64

1)

1.

(u k u k u k y k y

E y σ

k

σ


36

References:

1. Discrete-time control systems, 2nd Ed., K. Ogata, Prentice-Hall,1995

2. Computer-controlled systems, 3rd Ed., K.J. Astrom, B. Wittenmark, Prentice-Hall, 1997

Documents

Chapter 8 Polynomial Approach