Deterministic Kalman Filtering on Semi Infinite Interval

(1)

Volume 2012, Article ID 490139,11pages doi:10.1155/2012/490139

Research Article

Deterministic Kalman Filtering on

Semi-Infinite Interval

L. Faybusovich

1

_{and T. Mouktonglang}

2, 3

1_{Mathematics Department, University of Notre Dame, Notre Dame, IN 46556, USA}

2_{Mathematics Department, Faculty of Science, Chiang Mai University, Chiang Mai 50220, Thailand}

3_{The Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand}

Correspondence should be addressed to T. Mouktonglang,[email protected]

Received 28 March 2012; Accepted 9 May 2012

Academic Editor: Aloys Krieg

Copyrightq2012 L. Faybusovich and T. Mouktonglang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We relate a deterministic Kalman filter on semi-infinite interval to linear-quadratic tracking control model with unfixed initial condition.

1. Introduction

In1, Sontag considered the deterministic analogue of Kalman filtering problem on finite

interval. The deterministic model allows a natural extension to semi-infinite interval. It is of a special interest because for the standard linear-quadratic stochastic control problem extension to semi-infinite interval leads to complications with the standard quadratic objective function

see, e.g.,2. According to1, the model which we are going to consider has the following

form:

Jx, u, x0

_∞

0

uTRuCx−yTQCx−y dt, 1.1

˙

xAxBu, 1.2

x0 x0. 1.3

Here we assume that the pairx, u∈ax0 Z, whereZis a vector subspace of the Hilbert

spaceLn

(2)

functionsis defined as follows:

Zx, u∈Ln

20,∞×L2m0,∞:xis absolutely continuous,

˙

x∈Ln₂0,∞,x˙ AxBu, x0 0. 1.4

HereAis annbynmatrix;Bis annbymmatrix;RRT_{is an}_n_by_n_{and positive definite;}_Q

QT_{is an}_r_by_r_{and positive definite;}_C_{is an}_r_by_n_matrix;_y_∈_Lr

20,∞. Notice that in1.1–

1.3x0is not fixed and we minimize over all triplex, u, x0 ∈Ln

20,∞×L2m0,∞×Rn

satisfying our assumption.

Notice also that we interpret1.1–1.3as an estimation problem of the form

˙

xAxBu,

yCxv, 1.5

where we try to estimate xwith the help of observationyby minimizing perturbationsu,

vand choosing an appropriate initial conditionx0.

2. Solution of the Deterministic Problem

Consider the algebraic Riccati equation

KAATKKLK−CTQC0, 2.1

whereLBR−1BT. Assuming that the pairA, Bis stabilizable and the pairC, Ais

detec-table, there exists a negative definite symmetric solution Kst to 2.1 such that the matrix

ALKstis stablesee, e.g., Theorem 12.3 in3. According to4, we have described a

com-plete solution of the linear-quadratic control problem on a semi-infinite interval with the

linear term in the objective function. The major motivation for this extension comes from5

where we consider applications of primal-dual interior-point algorithms to the computational analysis of multicriteria linear-quadratic control problems in mini-max form. To compute a primal-dual direction it is required to solve linear-quadratic control problems with the same

quadratic and diﬀerent linear parts on each iteration. Using the results in5, we can describe

the optimal solution to1.1–1.3with fixedx0as follows.

There exists a unique solutionρ0∈Ln

20,∞satisfying the diﬀerential equation

˙

ρ−ALKstTρ−CTQy. 2.2

Moreover,ρ0can be explicitly described as follows:

ρ0t

_∞

0

expALKstTτ

(3)

The optimal solutionx, uto1.1–1.3has the form

˙

x ALKstxLρ0, x0 x0, 2.4

uR−1BTKstxρ0

. 2.5

For details see5.

Notice thatρ0does not depend onx0. To solve the original problem1.1–1.3we need

to express the minimal value of the functional1.1in term ofx0.

Theorem 2.1. Letx, ube an optimal solution of 1.1–1.3with fixedx0 given by2.2–2.5.

Then

Jx, u, x0 &−xT₀Kstx0−2ρ00Tx0

_∞

0

yTQy−ρ₀TLρ0 dt. 2.6

Remark 2.2. Notice thatJx, u, x0is a strictly convex function ofx0and hence minimum ofJ

as a function ofx0is attained at

xopt₀ −Kst−1ρ00. 2.7

Hence2.2–2.5gives a complete solution of the original problem1.1–1.3.

Proof. Lety, w∈ax0 Zbe feasible solution to1.1–1.3, wherex0is fixed. Consider

Δy, ww−R−1BTKstyρ0

T

·R·w−R−1BTKstyρ0

, 2.8

where we suppressed an explicit dependence on time. Notice that by2.5

Δx, u≡0,

Δy, w≡0, 2.9

for any feasible solutiony, wimplies thaty, w x, u. Furthermore, letΔy, w Δ1

Δ2 Δ3, where

Δ1wTRw,

Δ2 −2

Kstyρ0 T

Bw,

Δ3

Kstyρ0

T

LKstyρ0

.

(4)

NowBwy˙ −Ay, and consequently

Δ2 −2

yT_K

stρ₀T

˙

y−AyyT_K

stAATKst

y−2yT_K

sty˙−2ρT₀y˙2ρTAy,

Δ3yTKstLKstyρT0Lρ02ρT0LKsty.

2.11

Consequently,

Δy, wwTRwρT0Lρ0yT KstLKstKstAATKst

y− d dt y

T_K

sty

−2d

dt ρ

T

0y

2 ˙ρT

0y2ρ0TLKsty2ρT0Ay.

2.12

Using2.1and2.2, we obtain

Δy, wwTRwρT0Lρ0yTCTQCy−2 d dt ρ T 0y − d dt y T_K sty

−2 CTQyTy

wTRwρT0Lρ0−2 d dt ρ T 0y − d dt y T_K sty

y−CyTQy−Cy−yTQy.

2.13

Hence, taking into account thatρ0t → 0, yt → 0, t → ∞see, for details5, we

obtain

_∞

0

Δy, wdt

_∞

0

wTRwy−CyTQy−Cydt

_∞

0

ρT

0Lρ0−yTQy

dt2ρ00Tx0x0Kstx0

Jy, w, x02ρ00Tx0x0Kstx0c,

2.14

wherec₀∞ρT

0Lρ0−y

T

Qydt.

Notice, thatΔy, w≥0 andΔx, u≡0. This shows that, indeed,x, uis an optimal

solution to1.1–1.3 with fixedx0and proves2.6.

Remark 2.3. By 2.14 and Δx, u ≡ 0, we have Jy, w, x0 ≥ Jx, u, x0 and the equality

occurs if and only ify, w ≡ x, u see also2.9. Hencex, uis a unique solution to the

(5)

3. Steady-State Deterministic Kalman Filtering

In light of2.7, it is natural to consider the process

zt −K_st−1ρ0t, t∈0,∞ 3.1

as a natural estimate for the optimal solution to problem1.1–1.3. Let us find the diﬀ

eren-tial equation forz.

Proposition 3.1. One has

˙

zAzKst−1CTQ

y−Cz. 3.2

Remark 3.2. Notice thatK−1

st is a solution to the algebraic equation

L−P CTQCPAPP AT 0. 3.3

In other words, the diﬀerential equation 3.2 is a precise deterministic analogue for the

stochastic diﬀerential equation describing the optimalsteady-state estimation in Kalman

filtering problem. See, for example,2.

Proof. Using2.2and3.1, we obtain

˙

zK−_st1ALKstTρ0Kst−1CTQy

− K−st1ATL

Kstz K−st1CTQy.

3.4

SinceKstis a solution to2.1, we have

−K_st−1ATKst−LKstA−Kst−1CTQC. 3.5

Hence,

˙

zAz−K−1

stCTQCzK−st1CTQy. 3.6

Hence, we obtain3.2.

Remark 3.3. Notice that due to3.1 Δz,0≡0 and consequentlyz,0would be an optimal

(6)

4. The Solution of the Discrete Deterministic Problem

It is natural to consider the discrete version for the problem1.1–1.3. In this case, the

pro-blem can be reformulated as follows:

Jx, u, x0 1

2 ∞

k1

uT_kRuk

Cxk−yk T

QCxk−yk

−→min, 4.1

xk1AxkBuk, 4.2

x0xo. 4.3

Here we letxdenote a sequence{xk} ⊂ Rn _for_k ₀_{, . . . ,}_{∞. We say that}_x _∈ _ln

2N

if∞_i₁xi2 < ∞, where · is a norm induced by an inner product ,inRn_{. Let}_{x, u} _∈

ln₂N×lm₂N.

Like in the continuous case, we assume that the pairx, u∈ ax0 Z, whereZis a

vector subspace of the Hilbert spaceln

2N×lm2N.

Observe now the inner product inHhas the following form:

x, y,u, v_H

∞

k0

xk, uk

yk, vk

. 4.4

The vector subspaceZnow takes the following form:

Z{x, u∈H: xk1AxkBuk, k0,1, . . . , x00}. 4.5

HereAis annbynmatrix.Bis annbymmatrix.RRT _{is an}_n_by_n_{and positive definite.}

QQT_{is an}_r_by_r_{and positive definite.}_C_{is an}_r_by_n_{matrix and}_y_∈_lr

2N.

As in the continuous case, we interpret4.1–4.3 as an estimation problem of the

form

xk1AxkBuk,

y_kCxkvk,

4.6

where we try to estimate xwith the help of observationyby minimizing perturbationsu,

vand choosing an appropriate initial conditionx0.

According to4, a general cost function for a discrete linear-quadratic control problem

with linear term on the cost function has the following form:

Jx, u, x0

∞

k1

1 2

xT

kQxkuTkRuk

xT

(7)

whereψ ∈ln₂Nandφ ∈l₂mN. The solution to the particular class of problems can be com-pletely described by solving several system of recurrence relations and the following discrete

algebraic Riccati equationDARE:

KATKA− ATKB RBTKB−1 ATKBTQ. 4.8

We assume that this equation has a positive definite stabilizing solutionKst. For suﬃcient

conditions, see6.

In our situation, we have

Jx, u, x0 1

2 ∞

k1

uT_kRuk

Cxk−y_k

T

QCxk−y_k

−→min. 4.9

It is easy to see thatψk −CTQy_k andφk 0, k 0,1, . . .. By4, there is a unique solution

ρ{ρ_k} ∈ln

2Nof the following recurrence relations

ρk

AT− ATKstB RBTKstB

−1

BT

ρk1CTQy_k. 4.10

For details on an explicit solution of the above recurrence relation, see4. For simplicity, we

let

R RBT_K

stB

, 4.11

and we also let

LBR−1BT. 4.12

So our recurrence relation forρnow takes the form

ρk

AT−ATKstL

ρk1CTQy_k 4.13

with the corresponding DARE

K ATKA−ATKBR−1ATKBTCTQC,

KATKA−ATKLKACTQC.

4.14

The optimal solution to4.1–4.3has the following form:

xk1 AT−ATKstL T

xkLρ_k1, 4.15

uk−R

−1

BTKstAxkR

−1

(8)

For details, see4. To solve the original problem4.1–4.3we need to express the minimal

value of the functional4.1in terms ofx0.

Theorem 4.1. Letx, ube an optimal solution of 4.1–4.3with fixedx0given by4.15-4.16.

Then

Jx, u, x0 1

2x

T

0Kstx0−ρT0x0

1 2

∞

k0

2yT_kQy_k−ρ_kT ₁Lρ_k₁. 4.17

Proof. For simplicity of notation, we useKforKst. Let

Δyk, wk

wkR

−1

BTKAyk−ρ_k₁

T

·R·wkR

−1

BTKAyk−ρ_k₁

Δ1 Δ2 Δ3,

4.18

where

Δ1 wT_kRwk,

Δ2 2

KAyk−ρ_k1

T

Bwk

2KAyk−ρ_k₁

T

yk1−Ayk

2yT

kATKyk1−2ykTATKAyk−2ρ

T

k1yk12ρT_k1Ayk,

Δ3

KAyk−ρ_k1

T

LKAyk−ρ_k1

y_kTATKLKAykρT_k₁Lρ_k₁−2y_kTATKLρ_k₁.

4.19

We assume that y, w ∈ ax0 Z. Since AT_KA ₋_AT_KLKA _K ₋_CT_QC _and _AT ₋

AT_KL_ρ

k1 ρk−CTQyk, we have

Δyk, wk

w_kTRwk−2yTk

ATKA−ATKLKAyk−ykTATKLKAyk

−2ρT_k₁yk12ρT_k1AykρT_k1Lρk1−2ykTATKLρk12yTkATKyk1

w_kTRwk−2yT_k

K−CTQCyk−yT_kATKLKAyk−2ρT_k₁yk1

2yT_kATρ_k₁ρT_k₁Lρ_k₁−2yT_kATKLρ_k₁2y_kTATKyk1

w_kTRwk−2yTk

K−CTQCyk−yTkATKLKAyk−2ρT_k1yk1

2yT k

AT₋_AT_KL_ρ

(9)

wT

kRwk−2yTk

K−CT_QC_y

k−yT_kATKLKAyk−2ρ_kT ₁yk1

2yT_kρ_k−CTQy_kρT_k₁Lρ_k₁2y_kTATKyk1

w_kTRwk−2yT_k

K−CTQCyk−yT_kATKLKAyk−2ρT_k₁yk1

2yT_kρ_k−2y_kTCTQy_kρT_k₁Lρ_k₁2y_kTATKyk1.

4.20

By recalling now the definition ofR, we have

wT_kRwkwT_k RBTKB

wk

wT_kRwkwT_kBTKBwk

wT_kRwk

yk1−Ayk

T

Kyk1−Ayk

wT_kRwkyT_k1Kyk1yTkATKAyk−2yTkATKyk1.

4.21

Therefore,

Δyk, wk

wT

kRwk−yTkKyk−yTkCTQCyk−2ρT_k1yk12yTkρk

−2yT_kCTQy_kρ_kT ₁Lρ_k₁yT_k₁Kyk1.

4.22

We then rearrange the terms and complete the square to obtain a useful expression forΔ:

Δyk,wk

wT_kRwk−ykTKyky_kT 1Kyk1−2ρT_k1yk12ρTkyk

ρT_k₁Lρ_k₁yT_kCTQCyk−2yT_kCTQy_k

wT_kRwk−ykTKyky_kT 1Kyk1−2ρT_k1yk12ρTkyk

ρT_k₁Lρ_k₁Cyk−y_k

T

QCyk−y_k

−2yT_kQy_k.

4.23

Notice, since we fixedx0, we lety0x0and take summation of both sides:

∞

k0

Δyk, wk

−x₀TKx02ρT₀x0

∞

k0

w_kTRwk

Cyk−yk T

QCyk−yk

∞

k0

ρT_k₁Lρ_k₁−2yT_kQy_k.

(10)

By the definition ofΔyk, wk,Δxk, uk 0. Therefore,

0−xT₀Kx02ρT₀x02Jx, u, x0

∞

k0

ρT_k₁Lρ_k₁−2yT_kQy_k. 4.25

As a result,

Jx, u, x0 1

2x

T

0Kx0−ρT0x0

1 2

∞

k0

2yT_kQy_k−ρ_kT ₁Lρ_k₁. 4.26

Then the proof is completed.

As in continuous case, for the discrete case,Jx, u, x0is a strictly convex function of

x0and hence minimum ofJas a function ofx0is attained at

xopt₀ K_st−1ρ₀, 4.27

whereρis the uniquel2solution to4.13.

Since we have4.27, it is natural to consider the process

zkK−st1ρk, 4.28

as an estimate for the optimal solution to problem4.1−4.3. Let us find the recurrence

rela-tion forzk.

Proposition 4.2. Assuming that the closed loop matrixA-LKAis invertible, one has

zk1 Azk−K−st1

AT−ATKstL

₋1

y_k−Czk

. 4.29

Proof. We can rewrite4.13in the form

Kstzk

AT−ATKstL

Kstzk1CTQy_k. 4.30

Using the algebraic Riccati equation

KstATKstA−ATKstLKstACTQC, 4.31

we can rewrite4.30in the form

CT_QCz

k ATKst−ATKstLKst

Azk AT−ATKstL

(11)

which is equivalent to

AT₋_AT_K

stL

Kstzk1 AT−ATKstL

KstAzk−CTQ

y_k−Czk

. 4.33

The result follows.

Remark 4.3. Notice that4.29is the analogue of the “limiting” discrete Kalman filter6, Page

384,17.6.1.

5. Concluding Remarks

In this paper, we relate a deterministic Kalman filter on semi-infinite interval to linear-quad-ratic tracking control model with unfixed initial condition. Solutions of the deterministic pro-blems both continuous and discrete cases are described. This extends the result of Sontag to semi-infinite interval.

Acknowledgments

The research of L. Faybusovich was partially supported by National science foundation. Grant DMS07-12809. The research of T. Mouktonglang was partially supported by the

Cen-tre of Excellence in Mathematics and the Commission for Higher EducationCHE, Sri

Ayut-thaya Road, Bangkok, Thailand.

References

1 E. D. Sontag, Mathematical Control Theory, Springer, 1990.

2 M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall, London, UK, 1977.

3 W. M. Wonham, Linear Multivariable Control, a Geometric Approach, Springer, 1974.

4 L. Faybusovich, T. Mouktonglang, and T. Tsuchiya, “Implementation of infinite-dimensional interior-point method for solving multi-criteria linear-quadratic control problem,” Optimization Methods &

Soft-ware, vol. 21, no. 2, pp. 315–341, 2006.

5 L. Faybusovich and T. Mouktonglang, “Linear-quadratic control problem with a linear term on semi-infinite interval: theory and applications,” Tech. Rep., University of Notre Dame, 2003.

(12)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of

Deterministic Kalman Filtering on Semi Infinite Interval

Research Article