x60 y345 w7 h9">
X
, V and
f
V
,
,.
T
TT
f
, we
change to operations with their square roots (denoted by
generic symbol S):
ˆˆ
ˆ,,
TT
VV ff
X
SSX SS
VSSVS


 XSS
S

 
 (25)
Modified procedure srRicsiup operates with the
square roots of (25) like it was first introduced in [63]:
Stage I:

ˆ
,,, ,r
SsSK
CC
CLC
LC
D
T
C
LY
:1
k
:
CGsrRicsiup
Name: square-root Riccati scalarized instant update
(instead of (16) (19)):
begin
T
C
DL obtain ,
T
G obtain Y
begin
while do cycle ks
begin continue
Tk
f
Sy a column
:Tf
k
df a scalar
1
:1 k
d

 a scalar
:TT
k
KfS
a row
ˆ:TT
k
SS Kf

ˆ
S
ˆ
:SS
matrix
S
:1kk
update
increment
end finish
1
:
TTT
s
K
KK
  collect T
K
T
Cr
LK K
obtain r
K
end
Correspondingly, we change from procedure Rictup
to srRictup using orthogonal transformations:
Stage II: ˆ
,,,,,
Vrr
SASKsSG
srRictup
ˆ,
0
TT
rr
T
V
SSAGKA
S




 
 
Name: square-root Riccati temporal update
orthogonalized (instead of (20) (18)):
(26)
where is one of the orthogonal transformations (Hau-
sholder or Givens or Gram-Schmidt) reducing matrix in
the right-hand side of (26) to the upper triangular form.
Theorem 4. Algorithm srRicsiup is equivalent to
algorithm Ricsiup and algorithm srRictup is equi-
valent to algorithm Rictup.
Proof. Selecting from (25) the proper substitution for
matrix
X
in algorithm Ricsiup and then factoring
X
difference kk
Xy K
 ˆ
(denoted
X
for each k in
Ricsiup) into with SS yields
ˆˆT
SS

ˆT
I ff


2
:.
TT
fI ffIff

 
This results in the quadratic equation with respect to
 
11
21
20.
kk
dd
 


From its two solutions one selects
1
11k
d


as being numerically stable, and introduces the interme-
diate notation

. The first equation in (26) can be
proved by premultiplying it by itself transposed. The
product coincides with (20).
Remark 12. Some comparative insight into numerical
stability of the above algorithm as well as of the two al-
gorithmic modifications that follow in Sections 9 and 10,
can be gained from [64] , pp. 163-167 and pp. 198-201.
Copyright © 2012 SciRes. IJCNS
I. V. SEMUSHIN
618
9. Bierman Style Modification
The algorithm to be presented here is conceptually the
Bierman’s algorithm originally developed for the U-D
matrix decomposition used for the KF covariance fac-
torizations. It was motivated by the work of Agee and
Turner about the one-rank modification of the UDUT
Cholesky factorization [40] which we convert into the
LDLT formulation as follows.
Theorem 5 (Agee-Turner PD Factorization Update).
Let
TTT
LDL caa
1,,
n
Ddd
diag 


aadimnP
PLDL
where L is unit lower triangular, ,
c is a scalar, , and .

12
,,, T
n
a a
If P is PD (positive definite), then factors L (unit
lower triangular) and 0D (diagonal) can be calcu-
lated in the following algorithm:
1) Initialization: .
1
2) Computation: for to cycle:
cc1i1n
a) 2
ii
a
ii
b) for to n
cycle:
dd
c
1ki
:
kkiki
aaal
;
i) ;
ii) kikiiiki ; In matrices L, nontrivial
entries exist only below their unit diagonal.
llcaad
c) 1iiii
3) Concluding assignment:
ccdd
. 2
ddca
nn
nn
Proof. The algorithm is validated by representing
.
T
x
Px as a sum of complete squares with substitution of
the equation =,
T
PPcaaT
PLDL
in this quadratic
form. Details can be found in [40] or [60].
We apply Bierman’s algorithm to LQR design in the
context of Remark 11, thereby presenting another modi-
fication of procedure Ricsiup named here ldRis-
ciup that avoids potentially unstable numerical differ-
encing. What is required for that is conversion of the
aforesaid UD Bierman’s algorithm into its LD analogue
and writing it in terms of LQR. In doing this, we obtain
the following result.
Theorem 6 (Bierman style ldRisciup algorithm).
Let :kk
X
XXyK
 written as ˆ:kk
X
XXyK

ˆ
: fol-
lowed by
X
X
in procedure Ricsiup be using
factorizations ˆˆˆˆ
T
X
LDLT
and
X
LDL

 where any
and all L are unit lower triangular and any and all D posi-
tive diagonal. Then Ricsiup is equivalent to the fol-
lowing procedure.
Stage I:

ˆˆ
,, ,,r
DsLDK
T
CCC
CLDLC
LC
D
TT
C
LY G
:1k
,,
CGL

1dRicsiup
Name: L-D Riccati scalarized instant update
(instead of (16) (19)):
begin
obtain ,
obtain Y
begin
kswhile
do cycle

1,,:
TT
nk
begin continue
ff Ly
f
1,, :
T
n
vv vDf
:k
d
K
:0 0
kn
v

in
for
down to 1 do
begin
:ii
vf
:1
ˆ:
ii
dd
:i
f

1ji
to n do for
begin
,
ˆ:T
j
ijijk
ll K

,,
:
TT
K
j
kjkjii
Klv
:
end
end
ˆ
:
LL
ˆ
:
DD;
L
D
:1kk update and
increment
end finish
1
:
TTT
s
K
KK
  collect T
K
T
Cr
LK K
obtain r
K
end
Remark 13. Recall that k is the k-th column of
matrix Y and k is the k -th diagonal element of matrix
D, both introduced in Section 7;
y
d
,
T
j
k
K
is the j-th element
of column T
k
K
that exists within each repetition of cy-
cle while.
Proof. Given in [60] similarly to the UD-version of
[40].
Forming the matrices
ˆˆ
,
LD
,LD

ˆ
L L
from
is illus-
trated schematically by Figure 4. It shows that: 1) this
computation is columnwise starting from the last column
and moving backwards; 2) the diagonal positions are
used to store elements of D because the predetermined
unit diagonals of both and need no storing; (3)
output data
ˆˆ
,
LD
,LD

can supersede
in the same
array; and (4) the upper triangular part of the array is
zero and so may not be stored thus saving memory.
We now turn to the LQ implementation of Stage II in
the form of a new procedure ldRictup which is to be
equivalent to Rictup.
At entry to ldRictup, we have two pairs of factors:
Copyright © 2012 SciRes. IJCNS
I. V. SEMUSHIN 619
1
ˆ
d
2
ˆ
d
3
ˆ
d
4
ˆ
d
1
d
1
ˆ
l
2
ˆ
l
3
ˆ
l
1
l
1dRicsiup
2
d
3
d
4
d
2
l
3
l
Figure 4. Forming of matrices
ˆˆ
,LD
ˆ,
LD ˆˆˆˆ
T
.
1) instead of
ˆ
X
LDL
,
V
LD T
VVV
VLDL
, and
2) instead of in (20).
V
Using them, re-write (20) as
ˆˆ
0T
T
VV
DT
W
DLA
DL





ˆ0
TV
W
XA
LL


 
(27)
The problem of Stage II sounds as follows: Given are
factors W and D for which (27) holds, find factors L
and D, L unit lower triangular and D positive di-
agonal, such that for matrix
X
T to be represented in the
factored form
X
LD


L

the following factorization
holds: T
X
LDL
. In other words, we seek to have an
algorithm yielding the pair

,LD so as to immediately
get results: LL
and DD
. So, in equation
T
LDL
,,
n
ww
T
WDW , the left hand side is given and the right
hand side is what we wish to find. This is exactly what is
known as Weighted Gram-Schmidt Orthogonalization
(WG-SO). It is presented in [40] (pp. 125-126) in the
UD-version. For our needs, we convert it into the LD-
version as follows.
Lemma 3. Let
1 be a linear independent
set of (column) M-vectors,
M
n, and let 1,,
M
DD
be positive scalars in a diagonal matrix
1M. If 1 are defined by the
following algorithm, then none of the v’s are zero and
for :
,,DD
diag
0
kj
v
D
j
T
vD

,,
n
vv
k
MG-SO:

-,,WDLD
1k
:
kk
vw
1k
1dMG SOrt
Name: L-D Modified Weig hted Gram-Schmidt
Orthogonalization:
begin
for to n do
for to n do
begin
:T
kkk
DvDv
1jk
for to n do
begin

:T
j
kjkk
LvDvD
:
j
jjkk
vvLv
end
end
end

0
i
Vt 
Proof. Can be obtained by a straightforward calcula-
tion.
Remark 14. The above procedure is called modified
because it works columnwise (Figure 5).
Finally for the case of , we obtain
ˆˆ
,,,,,, ,,
VV rr
LD ALDKsLDG


1dRictup
Stage II:
Name: L-D Riccati temporal update orthogonalized
(instead of (20) (18)):
Compute 1
ˆT
TnT
V
LA
WwwL

 


2
M
n(with
).
Compute

1
ˆ0
,, .
0
M
V
D
DDD
diag D




Call ldMG-SOrt
,,WDL D


rr
GKA
.
Compute .
10. Kailath Style Modification
There exists another a comparatively new class of algo-
rithms in Kalman filtering (LQG estimation) area [65],
the so-called array algorithms. They alleviate some
computational problems associated with Riccati itera-
tions by using the well-known QR-decomposition in nu-
merical linear algebra with an appropriate orthogonal
matrix Q where R is upper triangular (R indicates here
the right corner of a matrix). Below, we show how to
adapt such algorithms for LQR implementations, and we
refer to them as Kailath style paying thus a tribute to
works by Kailath and co-authors [43]. Starting out again
from Remark 11, we choose now (from several alterna-
tives recently serveyed in [60]) a square-root array
modification.
Theorem 7 (Kailath style asrRisciup algorithm).
Let ˆ:
X
:kk
XXyK
 written as
X
kk
XXyK

ˆ
: fol-
lowed by
X
X
ˆˆ
ˆT
in procedure Ricsiup be using fac-
torizations like in (25), that is SS T
and
X
X
SS
Figure 5. Modified WG-SO procedure, LD-version.
Copyright © 2012 SciRes. IJCNS
I. V. SEMUSHIN
620
with both S lower triangular. Then Ricsiup is equiva-
lent to the following procedure.
Stage I:

ˆ
,,, ,r
SsSK
CC
CLC
LC
D
T
C
LY
:1k
CGasrRicsiup
Name: array square-root Riccati scalarized instant
update (instead of (16) (19)):
begin
T
C
DL obtain ,
T
G obtain Y
begin
while do cycle ks
begin continue
:k
d

:
TT
nk
1,,
f
ff Sy
0
T
k
QfS




:
S
S
1
:
0
k
T
K
S





(T)
ˆ
S update
:kk increment
end finish
1
TT
:T
s
K
KK
collect T
K
T
Cr
LK r
K obtain
K
end
Proof. Assignment operator (T) in the above algorithm
contains two arrays: pre-array B (on the right) and
post-array A (on the left). At each cycle, let the latter be
obtained from the first in the upper triangular form by
means of an appropriate orthogomal transformation k
(it may be Hausholder reflections or Givens rotations):
k
Q
A
QB. Since kk
QQ , we have and
via straightforward calculations, we are done.
TTT
AAIBB
Stage II in this modification coincides with Stage II in
the Potter style modification, that is expression (26).
11. Applications Challenges
Possible applications of adaptation capability of stochas-
tic systems are numerous and can be found in almost
every field of modern engineering. While considering
applicability of the above results, one should select the
cases that seem to fit perfectly in the pattern of Figure 2.
In this pattern, the very necessity for adaptation is con-
sidered as a factual constraint the occurrence of which in
time is comparatively rare resulting from an abrupt fault
against the long lasting nominal mode of system opera-
tion. This can be exemplified by the development and
implementation of a high integrity navigation system
based on the combined use of an inertial measurement
unit aided by different outer sources of data [20,66] some
of which are fault-susceptible or working in an accident-
prone situation.
Famous industrial/technological study cases to be
brought forward as applications to theoretical/computa-
tional work are those on advanced MPC collected in
[59].
Overall, we have to admit that practical problems are
much more challenging than theoretical ones. One barrier
to overcome is the nonlinearity of the original system
(Data Source) models. The traditional remedy for this is
to invoke a linearized perturbation model or equation of
the first variation about a nominal (or reference) solution
to the nonlinear model on the assumption that such a
solution is known (as in [67 ]) or deliver ed by an External
(more precise) Data Source. In the strict sense, equation
(1) has been written yet in the form of perturbation
model, as it can be seen from criterion (3). The latter case,
combining the use of the Global Positioning System
(GPS acting as an external data source) and an inertial
measurement unit for vehicle applications, can be viewed
as a modeling technique for online estimation of the error
between the reference model and the real dynamics [68].
Another challenge to be considered is a set of con-
straints representing the physical limitations of the proc-
ess variables as is the case in MPC and optimization for
paperma king machines [69].
12. Concluding Remarks
The emphasis in this paper has been on the robust linear
quadratic regulator computations where the single Ric-
cati iteration algorithm is an integral part and where
seeking a steady-state Riccati solution (Algebraic Riccari
Equation) does not apply.
Main novelty of the results is technical: we have shown
that linear algebra methods of input scalarization, matrix
factorization and array orthogonalization earlier known
for the robustified linear quadratic estimators due to [40,
63,65] and many other works, now are successfully ex-
tended to the robust LQR computation problems includ-
ing LQ Regulator Modification phase in the adaptive
control systems. The new algorithmic LQ regulator for-
mulations based on these methods enhance LQR numeric
robustness and generate a productive perspective for fur-
ther investigations into the regulator modification (re-
design) methods within the structure of adaptive control.
Further research is encouraged into the advancement
of new insights about the numerics of LQR/ARE/DARE
procedures, thus leading to Adaptive Control System
CAD that is expected to include all three ACS phases—
Modifier/Identifier/Classifier.
13. Acknowledgements
The author would like to express his gratitude to an
C
opyright © 2012 SciRes. IJCNS
I. V. SEMUSHIN 621
anonymous reviewer for a number of suggestions that
helped to improve the style of this paper.
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