Post on 03-Jan-2016
On the computation of the defining polynomial of the algebraic Riccati
equation
Yamaguchi Univ. Takuya Kitamoto
Cybernet Systems, Co. LTD Tetsu Yamaguchi
Outline of the presentation
• What is ARE (Algebraic Riccati Equation)?
• Properties of ARE
• Problem formulation
• Algorithm description
• Numerical experiments
• Conclusion
What is ARE (Algebraic Riccati Equation)?
0
such that matrix Find
symmetric) :, ( ,, matrices Given
:matrix ofEquation
QPWPPAPA
Pnn
QWQWAnn
T
4908.03646.0
3646.08554.0
10
01 ,
20
02 ,
11
10
:Example
P
QWA
Properties of ARE
• Important equation for control theory (H2 optimal control, etc)
• Symmetric solutions (solution matrices are symmetric) are important.
• There are 2^n symmetric solutions.• When matrices A, W, Q are numerical matrices,
a numerical algorithm to compute the solutions is already known.
• The numerical algorithm can not be applied when matrices A, W, Q contain a parameter.
Problem formulation
0 such that Find
10
01 ,
20
02 ,
11
1Given
QPWPPAPAP
QWk
A
T
012222
022
012222
22,22,2
22,12,1
2,22,12,22,11,12,12,11,1
22,12,1
21,11,1
2,22,1
2,11,1
pppp
pppppkppp
pppkp
pp
ppP
Example:
133)(
,16484)( ,484)(
,402488)( ,201244)(
0)()()()()(
) ordering (term basisGroebner Computing
230
231
22
233
234
02,212
2,223
2,234
2,24
2,22,11,1
kkkkf
kkkkfkkkf
kkkkfkkkkf
kfpkfpkfpkfpkf
kppp
We can compute the defining polynomial of entries of P, not P itself.
The method with Groebner Basis:
Effective for only small degree n (n=2),
because of its heavy numerical complexities
0
satisfying matrix
of entries of polynomial defining thefind
,parameter ain entries polynomialwith
symmetric) :, ( ,, matrices Given
QPWPPAPA
P
k
QWQWAnn
T
Algorithm description
.for ARE of solutions 2 thecomputecan we,Given .2
))()(, )(( )()()(such that )( have We1.
:sAssumption
00
22
kkk
kkfkfkfkfkkn
nn
Z
Z
jil
jirl
lr
rjirjirr
rjirjirr
rjirjir
ppkfkf
kfpkfpkfkf
kpkpkfkf
kpkpk
n
n
n
nn
n
,,
2
0
0,12,2
2,1,2
2,1,
)()(
)()()()(
)()()()(
)()()(
Z
ionsinterpolat polynomialby computed becan )()( kkfkf l Z
ARE of polynomial defining the
)()()()()()( 0,12,2
2
0, kfpkfpkfkfpkfkf jiji
l
ljil
n
n
n
ionsinterpolat polynomialby computed becan )()( kkfkf l Z
Algorithm
it. factorize and ionsinterpolat polynomial
by , )()( Compute 3.
)()(
)()()(
computeThen
y.numericall when ARE of )(,),(
solution thecompute andinteger an be Let .2
.)( Compute .1
,,
2
0
,,
2
0
2,1,
21
kppkfkf
ppkfkf
kpkpk
kkkk
k
kk
jilji
ll
jiljir
llr
rjirjir
r
r
n
n
n
n
Z
Z
Z
06421.03256.0
3256.02614.0 ,
0642.13256.1
3256.12614.0
,4908.16354.0
6354.08554.0 ,
4908.03646.0
3646.08554.0
.0 when ARE of
solution symmetric 42 compute and 0Let .2
1
21
kk
k
Example
10
01 ,
20
02 ,
11
1Given QW
kA
)52()1(256)( .1 26 kkkk
64102425625601280
0.640.10240.2560.25600.1280
)06421.0)(0642.1)(4908.1)(4908.0)(0(
22222
322
422
22222
322
422
22222222
pppp
pppp
pppp
64)0()0( ,1024)0()0(
256)0()0( ,2560)0()0( ,1280)0()0(
31
234
ffff
ffffff
8192)1()1( ,32768)1()1(
32768)1()1( ,131072)1()1( ,65536)1()1(
31
234
ffff
ffffff
1For 2 kk
8192327683276813107265536 22222
322
422 pppp
)14()1(64)()(
)43()1(256)()(
)1(256)()(
)52()1(512)()(
)52()1(256)()(
5,5,4,4,3,3,2,2,1,0 .3
260
261
72
263
264
kkkkfkf
kkkkfkf
kkfkf
kkkkfkf
kkkkfkf
k
1443414
528524)1(64)()(
222
2222
322
2422
264
0
kkpkkpk
pkkpkkkkfkfl
l
Conversion from floating point numbers to integers
64102425625601280
0.640.10240.2560.25600.1280
)06421.0)(0642.1)(4908.1)(4908.0)(0(
22222
322
422
22222
322
422
22222222
pppp
pppp
pppp
• Arbitrary precision arithmetic can be used.
• Precision required is unknown.
.integer to
9,,1,0, .
number real Conversion
21
2121
e
jice
ddd
hdhhhdddr
r
0 part) (decimalFraction 2.
) of (magnitude arithmetic theofprecision The .1
:Conditions
21
chhh
re
result. same the
obtain and arithmetic theofprecision theIncrease .3
:Conditions Additional
Conversion from integers to polynomials
• Polynomial interpolation can be used.
• The degree of the polynomial is unknown.
kkfkflkfkf lrlr ZZ )()(,2,1 )()(
01)(
polynomial to
,,2,1 )(
integers from Conversion
apakakg
qrkg
pp
r
Z
),,2,1( )()( .1
:Conditions
miiqq
)2()1( have Then we
.,,2,1 )( integers with edinterpolat
polynomial theof degree thebe )(Let
ppp
qrkgq
q
r
.polynomial defining theof candidate thefrom
obtained ones ith thesolution w thecompareThen
. when ARE ofsolution the
compute and integers generatedrandomly be Let .2
:Conditions Additional
0
0
kk
k
polynomial defining theof Candidate
)()()()(
)()(
0,12,2
,
2
0
kfpkfpkfkf
pkfkf
jiji
lji
ll
n
n
n
Computation of )(k
)()()(
)()()(
)()()(
),,(
matrixcertain a of
seigenvalue are ,,
),,()(
theorycontrolin AREfor algorithm numerical theFrom
21
22212
12111
11
1111
1
nnnn
n
n
n
n
jjiiji
nn
s
yvyvyv
yvyvyv
yvyvyv
yy
ss
ssk
l
).( compute toused be
canion interpolat polynomial and arithmeticpoint Floating
k
Numerical experiments (1)
5. and 5-between integer generatedrandomly :
, , ,
BEQBBW
k
A T
1
0 ,
10
01 ,
10
00 ,
21
332
2for exampleAn
BQWk
A
n
Numerical experiments (2)
paper. in this method The (M2)
basis.Groebner using method The (M1)
Environments:
Maple 10 on the machine with
Pentium M 2.0GHz, 1.5Gbyte memory
n 2 3 4 5
M10.88
4× × ×
M22.04
416.7
1766.
6×
Computation time (in seconds)
Conclusion
• An algorithm to compute the defining polynomial of ARE with a parameter is given.
• The algorithm uses polynomial interpolations and arbitrary precision arithmetic.
• Numerical experiments suggest that the algorithm is practical for the system with size n<5.
• The algorithm is suitable for multi-CPU environments.