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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