robotics research laboratory 1 chapter 8 polynomial approach
TRANSCRIPT
Robotics Research Labo-ratory
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Chapter 8
Polynomial Approach
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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
Robotics Research Labo-ratory
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 z
H z S z R z
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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 T
S
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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 ΓK
A z zI Φ LC
( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )c ccl c o
B z T z B z T zY z U z U z
A z A z A z
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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, o
o o cc
t B zT z t A z Y z U z
A 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 zzU z
z A z
t By k
A
t A B
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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 k
hA 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
l
hz z z r z s z s A z
h h hz r s z A zr s z s z r s
+ ( + )
( + )
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, ,
( ).
cl
cl
r s s
A 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
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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
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x y x y
x 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
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-
-
-
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
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 z
U z A z
A z z a z a z a
B z b z b z b z b
n D z
D 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
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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 0
0 0 0 0
0 0
1 0
0 1
n n
n n n n
n n n n
n
n
X Y
n n E
a b
a a b b
a a b b
a b
a a b b b b
a a
2 11
2 22
0
10 1 1 1
21 0 2 2
00 0
0
0 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 E
M
n
E D
ls and do not have
any common factors.
A B
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) ( ) .
( )
( )
( ) ( ) ( ) ( ) ( )
( ) , ( )
.
.
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 0
1 0 5 1 2
1 1 0 1
0 1 0 0
A z z z z a z a
B 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 z
α z α z α β z β z β
E
.
., ,
.
.
( ) . , ( ) . .
( ) ( ) ( ) ( )
1
0 1 0 1
3
0 1 2
0 1 0
0 0 3
1 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
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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 z
U z z
x k x k u k
y k x k
z j z j
φ z z
Regulator Design by Pole Placement – state space approach
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(
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 zG
Y z z
k
u k u k y k y k
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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 z
U z z A z
Y z α z B z
R z α z A z β z B z
z α z
α z z z β z
( )ere ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
β zA z Y z B z U z B z R z Y z
α z
Regulator Design by Pole Placement - polynomial equation approach
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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
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. . . . .
. .,
. . . . .
. . . .
.
.,
.
( )
1
1
0 1
1 0 0 02 0 0 25 0 25 0 25 0 75
2 1 0 02 0 02 0 0 0 1
1 2 0 0 02 37 5 12 5 12 5 37 5
0 1 0 0 12 5 12 5 37 5 62 5
0 0 32
0 52 1
1 2 16
1 24
E E
D M E D
α z α z α z
. , ( )
( ) ( . )
( ) ( . )
0 10 32 24 16
0 667
2 0 2
43
β z z
β z
z
z β z
z
β
α
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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 B
A 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
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( )
( )( )
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
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c
A T Su u y
B 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 BT
u z A A A
t A
A 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.
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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
BBTB B B B B B R B R
AR BS A
BBT B
AR BS A
B B
A
( )
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 T
AR BS B AR B S
B B B BB BB T BT
A A AR BS AAR B S
A BBT
A A AAR B S
A B A
A B A
AR B S A A
T A
B
B
AR BS
A S
A
T
R
B A
B
,: canceled stable process zeros observer poles , model poles o mB A A
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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 A
AR 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 B
B
B
A B
A
A
B B
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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
S
A A B A A
A B
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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 b
z z a
B B
B z b
B z
B K
B
p pH z
A A z
B
p
z
B
p z
1 2 )p p
K
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20 0 1 1 2
1 20 0 1
deg deg deg deg deg 0 2 1 2 1
deg deg deg 0
deg 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 B
S A
T A B
z z a r K s z s z p z p AR B S A A
a p p ar s s
K 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 p
K
u
z
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ζ ω rad0.7, 1 / sec
T T0.25 1.0
The process zero is canceled.
- ripple between the sampling period
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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 b
z z a
BB p pH z
A A z p z p
B
B K z
B z b
b
B
B K z b
b
p p p pB
K b
z 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
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3 21 0 1 1 2
21 2
1 2
3 21
0
0 1 2
1
1
deg deg deg deg 1
deg deg 1 1
The Diophantine equation can be written as
( -1)( - )( ) ( )( )
( )
( 1)( )
(1 )( ) 1
( )( )
( )
m oR A A A
S A
z z a z r K z b s z s z p z p z
b b p b pb
b b a
K 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) (
en
1)
o m
cu k t u k s
p pA B z t z
K
y k s k r
b
y u k
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ζ ω rad0.7, 1 / sec
T T0.25 1.0
The process zero is not canceled.
-smoother but a little slow response
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Optimal DesignB q
x k u kA q
y k x k
C qv k e k
A q
A q y k B q u k C q e k ARMA
A A A
v
C C A
k
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
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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 k
A q A q
B q C qq u k e k
A q A q
d A B pole excess
A
deg C n
- system with stable inverse
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* 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 k
A q A q
G q B qF q e k d e k u k
A q A q
C q A q F q G q q
C qF
A 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 q
A 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
Robotics Research Labo-ratory
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* *
* *
* * **
* *
222 *
* * *
* *(
Also, using ( ) ( )
(3) becomes
( ) ( ) ( ) ( )
( )
)
independent of ( ), (
(
( ) ( )
-1), , ( ), ( -1),
)
d
F e k d
A Be k y k q u k
C C
G B Fy k d F e k d y k u k
C C
E y k d E E
u
G B Fy k u k
C 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 k
B q F q F q
E y k d f f
B q
Robotics Research Labo-ratory
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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
d
F q q
G q q q
qu 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
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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