Deterministic Kalman Filtering on Semi-Infinite Interval

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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}

1Mathematics Department, University of Notre Dame, Notre Dame, IN 46556, USA

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

3The 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

u^TRu

Cx−y_T Q

Cx−y

dt, 1.1

x˙AxBu, 1.2

x0 x0. 1.3

Here we assume that the pairx, u∈ax0 Z, whereZis a vector subspace of the Hilbert spaceLⁿ₂0,∞×L^m₂0,∞ withLⁿ₂0,∞a Hilbert space ofRⁿ-value square integrable

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functionsis defined as follows:

Z

x, u∈Lⁿ₂0,∞×L^m₂0,∞:xis absolutely continuous, x˙∈Lⁿ₂0,∞,x˙ AxBu, x0 0

. 1.4

HereAis annbynmatrix;Bis annbymmatrix;RR^Tis annbynand positive definite;Q Q^Tis anrbyrand positive definite;Cis anrbynmatrix;y∈L^r₂0,∞. Notice that in1.1–

1.3x0is not fixed and we minimize over all triplex, u, x0 ∈Lⁿ₂0,∞×L^m₂0,∞×Rⁿ satisfying our assumption.

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

x˙AxBu,

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

KAA^TKKLK−C^TQC0, 2.1

whereLBR⁻¹B^T. 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 complete 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∈Lⁿ₂0,∞satisfying the diﬀerential equation

ρ˙−ALKst^Tρ−C^TQy. 2.2

Moreover,ρ0can be explicitly described as follows:

ρ0t _∞

0

exp

ALKst^Tτ

C^TQytτ dτ. 2.3

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

x˙ ALKstxLρ0, x0 x0, 2.4 uR⁻¹B^T

Kstxρ0

. 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 &−x^T₀Kstx0−2ρ00^Tx0 _∞

0

y^TQy−ρ₀^TLρ0

dt. 2.6

Remark 2.2. Notice thatJx, u, x0is a strictly convex function ofx0and hence minimum ofJ as a function ofx0is attained at

x^opt₀ −K⁻¹_stρ00. 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

Δ y, w

w−R⁻¹B^T

Kstyρ0

^T

·R·

w−R⁻¹B^T

Kstyρ0

, 2.8

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

Δx, u≡0, Δ

y, w

≡0, 2.9

for any feasible solutiony, wimplies thaty, w x, u. Furthermore, letΔy, w Δ1 Δ2 Δ3, where

Δ1w^TRw, Δ2 −2

Kstyρ0

T

Bw, Δ3

Kstyρ0

_T L

Kstyρ0

.

2.10

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NowBwy˙ −Ay, and consequently

Δ2 −2

y^TKstρ₀^T

y˙−Ay y^T

KstAA^TKst

y−2y^TKsty˙−2ρ^T₀y˙2ρ^TAy,

Δ3y^TKstLKstyρ^T₀Lρ02ρ^T₀LKsty.

2.11

Consequently,

Δ y, w

w^TRwρ^T₀Lρ0y^T KstLKstKstAA^TKst

y− d

dt y^TKsty

−2d dt ρ^T₀y 2 ˙ρ^T₀y2ρ₀^TLKsty2ρ^T₀Ay.

2.12

Using2.1and2.2, we obtain

Δ y, w

w^TRwρ^T₀Lρ0y^TC^TQCy−2d dt ρ^T₀y

− d

dt y^TKsty

−2 C^TQy_T y

w^TRwρ^T₀Lρ0−2d dt ρ^T₀y

− d

dt y^TKsty

y−Cy_T Q

y−Cy

−y^TQy.

2.13

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

_∞

0

Δ y, w

dt _∞

0

w^TRw

y−Cy_T Q

y−Cy dt

_∞

0

ρ^T₀Lρ0−y^TQy

dt2ρ00^Tx0x0Kstx0

J y, w, x0

2ρ00^Tx0x0Kstx0c,

2.14

wherec_∞

0 ρ^T₀Lρ0−y^TQydt.

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 problem1.1–1.3. Similary reasoning works in discrete time case.

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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 diﬀeren- tial equation forz.

Proposition 3.1. One has

z˙AzK_st⁻¹C^TQ

y−Cz

. 3.2

Remark 3.2. Notice thatK⁻¹_st is a solution to the algebraic equation

L−P C^TQCPAPP A^T 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

z˙ K⁻¹_stALKst^Tρ0K_st⁻¹C^TQy − K⁻¹_stA^TL

Kstz K⁻¹_stC^TQy.

3.4

SinceKstis a solution to2.1, we have

−K_st⁻¹A^TKst−LKstA−K_st⁻¹C^TQC. 3.5

Hence,

z˙Az−K_st⁻¹C^TQCzK⁻¹_stC^TQy. 3.6

Hence, we obtain3.2.

Remark 3.3. Notice that due to3.1 Δz,0≡0 and consequentlyz,0would be an optimal solution to1.1–1.3if it were feasible for this problem.

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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 problem can be reformulated as follows:

Jx, u, x0

1 2

∞ k1

u^T_kRuk

Cxk−y_kT

Q

Cxk−y_k

−→min, 4.1

xk1AxkBuk, 4.2

x0xo. 4.3

Here we letxdenote a sequence{xk} ⊂ Rⁿ fork 0, . . . ,∞. We say thatx ∈ lⁿ₂N if_∞

i1xi² < ∞, where · is a norm induced by an inner product ,inRⁿ. Letx, u ∈ lⁿ₂N×l^m₂N.

Like in the continuous case, we assume that the pairx, u∈ ax0 Z, whereZis a vector subspace of the Hilbert spacelⁿ₂N×l^m₂N.

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.RR^T is annbynand positive definite.

QQ^Tis anrbyrand positive definite.Cis anrbynmatrix andy∈l^r₂N.

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

x_k1AxkBuk,

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

x^T_kQxku^T_kRuk

x_k^Tψku^T_kφk−→min, 4.7

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whereψ ∈lⁿ₂Nandφ ∈l₂^mN. 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:

KA^TKA− A^TKB RB^TKB₋₁

A^TKB_T

Q. 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

u^T_kRuk

Cxk−y_k_T Q

Cxk−y_k

−→min. 4.9

It is easy to see thatψk −C^TQy_k andφk 0, k 0,1, . . .. By4, there is a unique solution ρ{ρ_k} ∈l₂ⁿNof the following recurrence relations

ρk

A^T− A^TKstB RB^TKstB₋₁ B^T

ρk1C^TQy_k. 4.10

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

R RB^TKstB

, 4.11

and we also let

LBR⁻¹B^T. 4.12

So our recurrence relation forρnow takes the form

ρk

A^T−A^TKstL

ρk1C^TQy_k 4.13

with the corresponding DARE

K A^TKA− A^TKB

R⁻¹

A^TKBTC^TQC, KA^TKA−A^TKLKAC^TQC.

4.14

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

x_k1 A^T−A^TKstLT

xkLρ_k1, 4.15

uk−R⁻¹B^TKstAxkR⁻¹B^Tρ_k1. 4.16

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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^T₀Kstx0−ρ^T₀x01 2

∞ k0

2y^T_kQy_k−ρ^T_k1Lρ_k1

. 4.17

Proof. For simplicity of notation, we useKforKst. Let

Δ yk, wk

wkR⁻¹B^T

KAyk−ρ_k1^T

·R·

wkR⁻¹B^T

KAyk−ρ_k1 Δ1 Δ2 Δ3,

4.18

where

Δ1 w^T_kRwk, Δ2 2

KAyk−ρ_k1T

Bwk

2

KAyk−ρ_k1_T

yk1−Ayk

2y^T_kA^TKy_k1−2y_k^TA^TKAyk−2ρ^T_k1y_k12ρ^T_k1Ayk, Δ3

KAyk−ρ_k1_T L

KAyk−ρ_k1

y_k^TA^TKLKAykρ^T_k1Lρ_k1−2y_k^TA^TKLρ_k1.

4.19

We assume that y, w ∈ ax0 Z. Since A^TKA −A^TKLKA K −C^TQC and A^T − A^TKLρ_k1 ρ_k−C^TQy_k, we have

Δ yk, wk

w_k^TRwk−2y^T_k

A^TKA−A^TKLKA

yk−y_k^TA^TKLKAyk

−2ρ^T_k1yk12ρ^T_k1Aykρ^T_k1Lρ_k1−2y_k^TA^TKLρ_k12y^T_kA^TKyk1

w_k^TRwk−2y^T_k

K−C^TQC

yk−y^T_kA^TKLKAyk−2ρ^T_k1yk1

2y^T_kA^Tρ_k1ρ^T_k1Lρ_k1−2y^T_kA^TKLρ_k12y_k^TA^TKyk1

w_k^TRwk−2y^T_k

K−C^TQC

yk−y^T_kA^TKLKAyk−2ρ^T_k1y_k1 2y^T_k

A^T−A^TKL

ρ_k1ρ^T_k1Lρ_k12y^T_kA^TKy_k1

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w_k^TRwk−2y^T_k

K−C^TQC

yk−y^T_kA^TKLKAyk−2ρ_k1^T y_k1 2y^T_k

ρ_k−C^TQy_k

ρ^T_k1Lρ_k12y_k^TA^TKyk1

w_k^TRwk−2y^T_k

K−C^TQC

yk−y^T_kA^TKLKAyk−2ρ^T_k1yk1

2y^T_kρ_k−2y_k^TC^TQy_kρ^T_k1Lρ_k12y_k^TA^TKyk1.

4.20

By recalling now the definition ofR, we have

w^T_kRwkw^T_k RB^TKB wk

w^T_kRwkw^T_kB^TKBwk

w^T_kRwk

yk1−Ayk

_T K

yk1−Ayk

w^T_kRwky^T_k1Kyk1y^T_kA^TKAyk−2y^T_kA^TKyk1.

4.21

Therefore,

Δ yk, wk

w_k^TRwk−y^T_kKyk−y^T_kC^TQCyk−2ρ^T_k1y_k12y^T_kρ_k

−2y^T_kC^TQy_kρ^T_k1Lρ_k1y^T_k1Kyk1.

4.22

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

Δ yk,w_k

w^T_kRwk−y_k^TKyky_k1^T Kyk1−2ρ^T_k1yk12ρ^T_kyk

ρ^T_k1Lρ_k1y^T_kC^TQCyk−2y^T_kC^TQy_k

w^T_kRwk−y_k^TKyky_k1^T Ky_k1−2ρ^T_k1y_k12ρ^T_kyk

ρ^T_k1Lρ_k1

Cyk−y_k_T Q

Cyk−y_k

−2y^T_kQy_k.

4.23

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

∞ k0

Δ yk, wk

−x₀^TKx02ρ^T₀x0^∞

k0

w_k^TRwk

Cyk−y_kT

Q

Cyk−y_k

^∞

k0

ρ^T_k1Lρ_k1−2y^T_kQy_k .

4.24

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

0−x^T₀Kx02ρ^T₀x02Jx, u, x0

∞ k0

ρ^T_k1Lρ_k1−2y^T_kQy_k

. 4.25

As a result,

Jx, u, x0 1

2x^T₀Kx0−ρ^T₀x01 2

∞ k0

2y^T_kQy_k−ρ^T_k1Lρ_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

x^opt₀ K_st⁻¹ρ₀, 4.27

whereρis the uniquel2solution to4.13.

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

zkK⁻¹_stρ_k, 4.28

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

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

zk1 Azk−K⁻¹_st

A^T−A^TKstL₋₁

y_k−Czk

. 4.29

Proof. We can rewrite4.13in the form

Kstzk

A^T−A^TKstL

Kstzk1C^TQy_k. 4.30

Using the algebraic Riccati equation

KstA^TKstA−A^TKstLKstAC^TQC, 4.31

we can rewrite4.30in the form

C^TQCzk A^TKst−A^TKstLKst

Azk A^T−A^TKstL

Kstz_k1C^TQy_k, 4.32

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which is equivalent to

A^T−A^TKstL

Kstz_k1 A^T−A^TKstL

KstAzk−C^TQ

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-quadratic tracking control model with unfixed initial condition. Solutions of the deterministic problems 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.

6 P. Lancaster and L. Rodman, Algebraic Riccati Equations, The Clarendon Press Oxford University Press, Oxford, UK, 1995.

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Deterministic Kalman Filtering on Semi-Infinite Interval

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