Research Article Least-Mean-Square Receding Horizon Estimation

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1 Mathematical Problem in Engineering Volume 212, Article ID , 19 page doi:1.1155/212/ Reearch Article Leat-Mean-Square Receding Horizon Etimation Bokyu Kwon 1 and Soohee Han 2 1 Department of Control and Intrumentation Engineering, Kangwon ational Univerity, Samcheock , Republic of Korea 2 Department of Electrical Engineering, Konkuk Univerity, Seoul , Republic of Korea Correpondence hould be addreed to Soohee Han, hhan@konkuk.ac.kr Received 31 October 211; Accepted 2 December 211 Academic Editor: M. D. S. Aliyu Copyright q 212 B. Kwon and S. Han. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. We propoe a leat-mean-quare LMS receding horizon RH etimator for tate etimation. The propoed LMS RH etimator i obtained from the conditional expectation of the etimated tate given a finite number of input and output over the recent finite horizon. Any aprioritate information i not required, and exiting artificial contraint for eay derivation are not impoed. For a general tochatic dicrete-time tate pace model with both ytem and meaurement noie, the LMS RH etimator i explicitly repreented in a cloed form. For numerical reliability, the iterative form i preented with forward and backward computation. It i hown through a numerical example that the propoed LMS RH etimator ha better robut performance than conventional Kalman etimator when uncertaintie exit. 1. Introduction Several criteria have been often employed for the deign of optimal etimator. In particular, the mean-quare-error criterion i the mot popular and ha many application ince it offer a imple cloed-form olution a well a important geometric and phyical interpretation. It i well known that the optimal etimator baed on the mean-quare-error criterion i obtained from the conditional expectation of the etimated variable given the known meaurement. For tate etimation, many trial have been conducted to obtain a receding horizon RH etimator baed on the mean-quare-error criterion. At the current time k, theleatmean-quare LMS RH etimator i to etimate the tate x k h at the time k h from the input u k and the output y k over the recent finite horizon k, k 1, which can be written a x k h k E x k h u k u k 1,y k y k 1, 1.1

2 2 Mathematical Problem in Engineering Sytem Input u k Unkown tate: x k Output y k u k 1 u k +1 u k D D D D Etimated tate ꉱx k h/k Ex k h /u k,,y k 1,u k,,u k 1 D D D D y k y k +1 y k 1 Receding horizon etimator Figure 1: The tructure of the receding horizon etimator D i a unit delay component. where h and, that i, the lag ize and the memory ize are deign parameter to be determined, repectively. Available input and output are regarded a given condition and the correponding conditional expectation of the tate at time k h i obtained a it optimal etimate. Practically, input and output are known variable and hence can be conidered a condition. The tructure of the LMS RH etimator 1.1 i depicted in Figure 1. A mentioned before, the LMS RH etimator 1.1 minimize the mean-quare-error criterion, E x k h k x k h T x k h k x k h u k u k 1,y k y k 1. For h 2, h 1, and h, the etimator 1.1 are often called the moother, the filter, and the predictor, repectively. Receding horizon i traced from the fact that the finite-time horizon, where input and output neceary for etimating unknown tate are available, recede with time. In deigning the control, receding horizon cheme have already been popular 1, 2. If approache, the etimator 1.1 reduce to the well-known tationary Kalman etimator with infinite-memory. So, the LMS RH etimator 1.1 can be alo called a finite memory etimator for the finite and fixed memory ize. It ha been illutrated through numerical imulation and analyi that the LMS RH etimator with finite memory have better robut performance than conventional Kalman etimator with infinite memory 3, 4. Aloin input/output model ariing in ignal proceing area, it i acknowledged that the finitememory or finite-impule-repone FIR filter have been preferable for practical reaon 5. In pite of the good performance and the uefulne of the LMS RH etimator, no one ha propoed a general reult on the conditional expectation 1.1. Since it wa difficult to obtain the conditional expectation 1.1 for a general tate pace model, ome aumption have been made to implify the problem and then obtain a olution eaily. At firt, ytem or meaurement noie were et to zero, which offer a cloed-form olution eaily 6 1. In 11, 12, aprioriinformation on the initial tate on the horizon wa aumed to be known for obtaining a olution eaily. Intead of directly obtaining a cloed-form olution, the duality of a control and the complicated cattering theory from a phyical phenomenon were ued to how the feaibility of implementation for a general tate pace model 13, 14. For eay derivation, the Kalman filter wa alo employed with infinite covariance 4. However, thi approach i o heuritic that the optimality i not guaranteed in the ene of the meanquare-error criterion. Beide, the ytem matrix i required to be noningular. A in other conventional etimator, there were alo ome trial that unbiaed and linear contraint are impoed to obtain the optimal etimator 3, However, external control input are not conidered 3 and the ytem matrix i required to be noningular 15. Furthermore, it i not

3 Mathematical Problem in Engineering 3 guaranteed that even though uch contraint are removed, the optimality i till preerved. Though computing the conditional expectation 1.1 look like a very imple problem, there i no reult on a cloed-form olution correponding to 1.1 for a general tate pace model without any artificial aumption and requirement. In thi paper, exiting artificial aumption for obtaining a olution eaily are not made and any condition are not required. Unlike the exiting reult, the ytem matrix i not required to be noningular. Both ytem and meaurement noie are conidered together with external control input. The LMS RH etimator will be directly obtained from the conditional expectation 1.1, which automatically guarantee it optimality. It turn out that the propoed LMS RH etimator ha the deadbeat property and the linear tructure with input and output over the recent finite horizon. The ret of thi paper i organized a follow. In Section 2, the LMS RH etimator i obtained from the conditional expectation and it iterative computation i introduced in Section 3. A numerical imulation i carried out in Section 4 to illutrate the performance of the propoed LMS RH etimator. Finally, concluion are preented in Section LMS RH Etimator Conider a linear dicrete-time tate pace model with an external control input: x i1 Ax i Bu i Gw i, y i Cx i v i, 2.1 where x i R n, u i R l,andy i R q are the tate, the input, and the output, repectively, and the ytem noie w i R p and the meaurement noie v i R q are aumed to be zero-mean white Gauian with wi w T E v i i v T i Q S, Q, R >, 2.2 R S T where nonzero S often happen when I/O model are converted to tate pace model. It i alo aumed that A, C of the ytem i obervable. Through thi paper, the current time i denoted by k. For mathematical tractability, the tate pace model 2.1 i equivalently changed to x i1 A x i Bu i Gw,i GSR 1 y i, y i Cx i v i, 2.3 where A A GSR 1 C and w,i w i SR 1 v i. It can be eaily hown that w,i and v i in 2.3 are not correlated. In other word, we have w,i E w T v,i i v T i Q SR 1 S T O, 2.4 O R where Q SR 1 S T. It i noted that off-diagonal block are filled with zero.

4 4 Mathematical Problem in Engineering For imple repreentation, everal variable are defined a C B C CA CA 2. CA 1, Y k 1 y k y k 1 y k 2. y k 1, 2.5 CB CA B CB..., B CA 3 B CB CA 2 Q {}}{ Q SR 1 S T Q SR 1 S T, R {}}{ R R R, where denote the direct um of the matrice. Additionally, W k 1, V k 1,and U k 1 are defined by replacing y in 2.5 with w, v, andu, repectively. G and S are alo defined by replacing B in B with G and GSR 1, repectively. Firt, we conider the cae of h 1in1.1. A prediction problem for h 1 will be dicued later on. By uing the defined variable, the tate x k h to be etimated i repreented in term of the initial tate x k on the horizon, input, and ytem noie on the recent finite horizon k, k 1 a x k h A h x k L g, W k 1 L b, U k 1 L, Y k 1, 2.9 where L g, i given by L g, A h 1 G A h 2 G G h {}}{ O O, 2.1 and L b, and L, are obtained by replacing G in L g, with B and GSR 1, repectively. From 2.9, the conditional expectation Ex k h U k 1,Y k 1 in 1.1 can be repreented a Ex k h U k 1,Y k 1 A h Ex k U k 1,Y k 1 L g, EW k 1 U k 1,Y k 1 L b, U k 1 L, Y k ote that if Ex k U k 1,Y k 1 and EW k 1 U k 1,Y k 1 are known, Ex k h U k 1,Y k 1 can be obtained. In order to compute Ex k U k 1,Y k 1 and EW k 1 U k 1,Y k 1, we firt try

5 Mathematical Problem in Engineering 5 to find the relation among x k, W k 1, U k 1,andY k 1. For thi purpoe, the ytem 2.3 i repreented in a batch form on the recent finite horizon k, k 1 a follow: Y k 1 C x k B U k 1 G W k 1 S Y k 1 V k 1, 2.12 from which we obtain Wk 1 Wk 1 I V k 1 G C x k Y k 1 S Y k 1 B U k We can ee from 2.13 that W k 1 and V k 1 are linearly, more correctly affinely, dependent on W k 1 and x k. The oint probability denity function of W k 1 and x k, that i, f wx W k 1,x k, can be expreed a kf wv W k 1,V k 1 U k 1,Y k 1 kf wv W k 1,Y k 1 C x k B U k 1 G W k 1 S Y k 1 for an appropriate caling factor k. How to chooe k will be dicued later on. By uing the probability denity function of W k 1 and V k 1 given a 1 f wv W k 1,V k 1 2π pq detq detr ( exp 1 T 1 Wk 1 Q Wk 1, 2 V k 1 R V k Ex k U k 1,Y k 1 can be computed from Ex k U k 1,Y k 1 x k f wx W k 1,x k U k 1,Y k 1 dw k 1 dx k x k kf wv (W k 1,Y k 1 C x k B U k 1 G W k 1 S Y k 1 dw k 1 dx k, 2.15 where a contant k i choen to atify the following normalization condition: f wx W k 1,x k U k 1,Y k 1 dw k 1 dx k ( k f wv W k 1,Y k 1 C x k B U k 1 G W k 1 S Y k 1 dw k 1 dx k

6 6 Mathematical Problem in Engineering In the ame way, EW k 1 U k 1,Y k 1 can be obtained. It i noted that f wv W k 1,Y k 1 C x k B U k 1 G W k 1 S Y k 1 i an exponential function with the following exponent: Y T k 1 T (I S R 1 U T R 1 B R 1 G R 1 k 1 B x T T R 1 B B T R 1 G B T R 1 W 1,1 W 1,2 k W T W 2,2 k 1 C C Y T k 1 T T (I S U T k 1 x T, 2.17 k W T k 1 where denote ymmetric part and W 1,1, W 1,2,andW 2,2 are given by W 1,1 C T R 1 C, W 1,2 C T R 1 G, W 2,2 G T R 1 Q By completing the quare of the term of the exponent and recalling the integration of Gauian function from to, we can compute Ex k U k 1,Y k 1 and EW k 1 U k 1,Y k 1. To begin with, we introduce the following completion of quare: a b T α β a β γ T b ( Tγ ( b γ 1 β T a b γ 1 β T a a T( α βγ 1 β T a 2.19 to obtain b exp ( a b T α β a β T γ b db exp ( a b T α β a β T γ b db γ 1 β T a, 2.2 for ome vector a and b, and ome matrice α, β, andγ of appropriate dimenion. The relation 2.2 can be eaily obtained in a imilar way to the mean of normal ditribution. From the following correpondence: T a (I Y T S k 1, b β U T k 1 R 1 G B T R 1 G R 1 C B T R 1 C x T k W T k 1, α, γ R 1 W1,1 W 1,2, W 2,2 R 1 B B T R 1 B, 2.21 we have Exk U k 1,Y k 1 EW k 1 U k 1,Y k 1 1 W1,1 W 1,2 C W T 1,2 W 2,2 G R 1 ((I S Y k 1 B U k 1, 2.22

7 Mathematical Problem in Engineering 7 where W 1,1, W 1,2,andW 2,2 are given by It i noted that if A, C of the ytem 2.1 i obervable, W 1,1 i poitive definite and W 2,2 W T 1,2 W 1 1,1 W 1,2 i alo poitive definite, which implie that the block matrix including W 1,1, W 1,2,andW 2,2 i guaranteed to be poitive definite and hence noningular. Subtituting 2.22 into 2.11, we can obtain Ex k h U k 1,Y k 1. Through a long and tediou algebraic calculation, we have the covariance matrix of the mean-quare-error E x k h k x k h x k h k x k h T a follow: 1 W1,1 W 1,2 P Ξ W T 1,2 W Ξ T, 2, where Ξ i defined by Ξ A h A h 1 G A h 2 G G h {}}{ OO O, 2.24 for h. What we have done until now can be ummarized in the following theorem. Theorem 2.1. Suppoe that A, C of the ytem 2.1 i obervable. Then, the LMS RH etimator 1.1 i given by 1 W1,1 W 1,2 C x k h k Ξ W T 1,2 W 2,2 G L b, U k 1 L, Y k 1, R 1 ((I S Y k 1 B U k where x k h k denote Ex k h U k 1,Y k 1 and W 1,1, W 1,2, and W 2,2 are defined in The correponding covariance matrix i given a It i noted that the propoed LMS RH etimator 2.25 i deigned without requirement of the removal of ome noie and the noningular ytem matrix A. In previou work, thoe requirement are adopted to olve the problem more eaily. If contraint or aumption of exiting reult are applied to the propoed reult, the latter reduce to the former. For example, Q and B can be et to zero for comparion with the reult on ytem without ytem noie and input, repectively. A een in 2.25, the LMS RH etimator i linear with repect to input and output on the recent finite horizon k, k 1. So, finite memory may be called finite impule repone that i often ued in linear ignal proceing ytem. In addition, the deadbeat property i guaranteed in the following theorem. Theorem 2.2. If no noie i applied, the LMS RH etimator 2.25 i a deadbeat etimator, that i, x k h k x k h at the fixed-lag (h 1 or the current (h time.

8 8 Mathematical Problem in Engineering Proof. The LMS RH etimator 2.25 can be rearranged in term of Y k 1 and U k 1,thati, HY k 1 LU k 1 for ome gain matrice H and L. Subtituting Y k 1 in 2.12 into HY k 1 LU k 1, uing 2.9, and removing all noie, we have ( HY k 1 LU k 1 H C A h x k x k h (H S L, Y k 1. ( H B L b, L U k It can be eaily een that H C A h and term aociated with input U k 1 and output Y k 1 become zero. It follow then that we have HY k 1 LU k 1 x k h. Thi complete the proof. According to the deadbeat property, the LMS RH etimator track down the real tate exactly if no noie i applied. The prediction problem for h 1 can be eaily olved from the previou reult for h. Since x k h i repreented a h 1 x k h A h x k A i Bu ki i h 1 A i Gw ki, i 2.27 we have h 1 Ex k h U k 1,Y k 1 A h Ex k U k 1,Y k 1 A i Bu ki, i 2.28 which mean that Ex k h U k 1,Y k 1 can be obtained from Ex k U k 1,Y k 1. Setting h in 2.25 to zero, we can compute Ex k U k 1,Y k 1 eaily. 3. Iterative Computation The LMS FM etimator 2.25 i of the compact and imple form. However, thi form require the invere of big matrice, which may lead to long computation time and large numerical error. To overcome thee weak point, in thi ection, we provide an effective iterative form of Firt, we repreent 2.25 in another form for getting recurive equation. Recalling the following fact: ( C T Π 1 C 1 ( 1, W 1,1 W 1,2 W 1 2,2 1,2 W T C T Π 1 C T R 1 W 1,2W 1 G T 2,2 R 1, 3.1

9 Mathematical Problem in Engineering 9 we obtain C T Π 1 C T G T R 1 2 C T R 1 W 1,2W 1 G T 2,2 R 1 G T R 1 W 1,1 C T Π 1 C 1 W 2,2 2I W1,2 W 1 W T 2,2 1,2 W 2,2 I 1 2 I 1 2 W 1,2W 1 2,2 I 2I W1,2 W 1 2,2 I W 1,1 C T Π 1 W 1,2 W T 1,2 W 2,2 C T 1 G T R 1, 1 W1,1 W 1,2. W T 1,2 W 2,2 3.2 ote that noningularity of C T Π 1 C i guaranteed for the obervability of A, C. From 3.2, it can be eaily een that the LMS RH 2.25 can be repreented a W x k h k Ξ 1,1 C T Π 1 C 1 W 1,2 W T 1,2 W 2,2 ( C T Π 1 ((I S Y k 1 B U k 1 L, Y k 1 L b, U k 1. C T G T R 1 ((I S Y k 1 B U k ow, we conider two recurive equation for the batch form 3.3. One i for obtaining C T Π 1 C and C T Π 1 I S Y k 1 B U k 1. The other i for 3.3 given C T Π 1 C and C T Π 1 I S Y k 1 B U k 1. The firt one i computed in a backward time and the econd one in a forward time. ext ubection deal with each recurive equation Recurive Equation for Backward Computation Here, how to obtain C T Π 1 C and C T Π 1 I S Y k 1 B U k 1 in 3.3 will be dicued. Thee value will be computed in a backward time. C T Π 1 C 1 C T Π 1 I S Y k 1 B U k 1 and C T Π 1 C 1 will be denoted by β k and P, repectively, for conitency with the next ection. The following theorem provide the main reult. Theorem 3.1. P C T Π 1 C 1 and β k P C T Π 1 I S Y k 1 B U k 1 can be computed recurively a follow: P ( C T R 1 C P 1, 3.4 β k P ( C T R 1 y k α k, 3.5

10 1 Mathematical Problem in Engineering where P and α k are obtained from P 1 A T C T R 1 CA A T P A A T (C T R 1 C P G { Q 1 G T( } 1G C T R 1 T C P G ( C T R 1 C P A, { { α 1 k A T A T (C T R 1 C P G Q 1 G T( } 1G C T R 1 C P G T} { ( } α k C T R 1 y k P C T R 1 C (Bu k 1 GSR 1 y k 1, for 1 1, Q Q SR 1 S T, and P 1 and α 1 k are zero matrice with appropriate dimenion. Proof. Before going into a main proof, we introduce ome variable Ĉ,, Π,and P a Ĉ CA. CA 3 CA 2 CA 1 C A, 3.8 Ĉ 1 CG CA G CG.... G CA 3 G CG CA 2 CG, 3.9 Ĉ 1 G 1 Π Q 1 T R 1 R q 1 q 1, 3.1 P Ĉ T 1 Π Ĉ, 3.11 for 2. In term of Π in 3.1 and P in 3.11, β k and P can be repreented a β k P ( C T R 1 y k Ĉ T (( 1 Π I o S o Ŷ B Û o P (C T R 1 y k α k, 1, 3.12 P (C T R 1 C P where I o, S o, and B o are obtained by removing the firt row block of I, S, and B, repectively, and Ŷ, Û,andα k are given by Ŷ y T T, y T k k 1 Û u T k T, uk 1 T 3.13 (( α k Ĉ T I o S o Ŷ B o Û, 3.14 Π 1

11 Mathematical Problem in Engineering 11 for 2. In order to obtain P and β k, we have only to know P and α k.ow,in order to get P and α k, we try to find recurive equation for P and α k on 1. By uing recurion in 3.8 and 3.9, we have the following equality: C Π 1 1 Δ 1 Δ 1 G Ĉ Q 1 G T C Ĉ T Δ 1 C G Ĉ 1 G T C Ĉ T Δ 1, 3.15 where Δ i given by R Δ Π Pre- and potmultiplying 3.15 by T C C Ĉ T 1 AT, Ĉ 1 A, Ĉ Ĉ 3.17 repectively, we have 3.6. Pre- and potmultiplying 3.15 by T C Ĉ T 1 AT, Ĉ 3.18 ( I o S o 1Ŷ1 B 1Û1 o ( k C ( I o S y o Ŷ B o Bu k 1 GSR 1 y k 1, Û 3.19 Ĉ repectively, we have 3.7.otethat3.6 and 3.7 hold for i 2. From 3.11 and 3.14, P 2 and α 2 k can be written a P 2 Ĉ T 1 2 Π 2 Ĉ2 A T C T( 1CA CGQ G T C T T R, 3.2 (( α 2 k Ĉ T 1 2 Π 2 I o S o 2Ŷ2 B 2Û2 o A T C T( CGQ G T C T R (y k 1 CGSR 1 y k 2 CBu k 2. If P 1 and α 1 k are et to zero matrice with appropriate dimenion, P 2 in 3.2 and α 2 k in 3.21 can be calculated from 3.6 and 3.7. Thu, we can ay that 3.6 and 3.7 hold for i 1 and are initiated with P 1 andα 1 k. After obtaining P from 3.6, we can calculate P from 3.4. β k in 3.5 can be obtained from α k that come from 3.7. Thi complete the proof.

12 12 Mathematical Problem in Engineering 3.2. Recurive Equation for Forward Computation Here, the recurive equation for 3.3 on the horizon i derived under the aumption that P C T Π 1 C 1 and β k P C T Π 1 I S Y k 1 B U k 1 are given. P and β k in Section 3.1 are computed in a backward time while variable introduced in thi ection are computed in a forward time. Before proceeding to a main reult, we introduce ome variable and the neceary lemma. L G,i, M i,and i are defined a L G,i A i A i 1 G A i 2 G A G G, 3.22 C T M i i R 1 i C i P 1 C T i R 1 i G o i G ot i R 1 i C i G ot i R 1 i G o i Q 1, i Mi C T i Q 1 i R 1 i C i P 1 C T i R 1 i G i G T i R 1 i C i G T i R 1 i G i Q 1, i 3.24 where 2 i and G o i i the matrix obtained by removing the lat zero column block from G i,thati, G i G o i. In particular, L G,1, M 1,and 1 are defined a A G, C T R 1 CP 1 and M 1 Q 1. The following lemma how how variable L G,i, i,andm i are related to one another. Lemma 3.2. The following relation i atified: i L T G,i CT R 1 CL G,i M i1, 3.25 where L G,i, i, and M i are defined in , repectively. Proof. If Γ i i defined a Γ i A i 1 G A i 2 G A G G, 3.26 L G,i can be repreented a L G,i A i Γ i, 3.27 from which we have L T G,i CT R 1 CL G,i A Ti C T R 1 CA i A Ti C T R 1 CΓ i Γ T i CT R 1 CA i Γ T i CT R CΓ i

13 Mathematical Problem in Engineering 13 The four block element in M i1 can be expreed recurively a C T i1 R 1 C i1 i1 C T i R 1 i C i A Ti C T R 1 CA i, C T i1 R 1 G o i1 i1 C T i R 1 i G o i A Ti C T R 1 CΓ i, G ot G ot i1 R 1 G o i1 i1 i R 1 i G o i Γ T i CT R 1 CΓ i Uing 3.28 and 3.29, we have i L T G,i CT R 1 CL G,i M i1. Thi complete the proof. Lemma 3.2 i ueful for breaking up big matrice of 3.3 into mall matrice. We now exploit the recurive equation for two following quantitie: β k L G, 1 ( C T Π 1 Ỹm, 1 C T G T R 1 Ỹ m, 1 L B, Ũ 1 L S, Ỹ 1, 3.3 for 1 γ k Ξ 1 ( C T Π 1 Ỹm, 1 C T G T R 1 Ỹ m, 1 L b, Ũ 1 L, Ỹ 1, 3.31 for h, where L b, and L, are defined in a form 2.24, andỹ 1, Ũ 1, L B,, L S,, and Ỹ m, 1 are given by T, T, Ỹ 1 y T k k 1 yt Ũ 1 u T k k 1 ut L B, A 1 B A 2 B B, L S, A 1 GSR 1 A 2 GSR 1 GSR, 1 ( Ỹ m, 1 I S Ỹ 1 B Ũ ote that β k C T Π 1 I S Y k 1 B U k 1, γ h k β h k,and x k h k γ k. However, β k i recurively computed from to and γ k from h to with the initial value γ h k β h k. Even though β k doe not look ueful on k h, k, it i till ued to compute γ k on that horizon. ow, we try to find out recurive equation for β k and γ k in what follow Recurive Equation for β k on Uing Lemma 3.2, we will obtain a recurive equation for β k, which i introduced in the following theorem.

14 14 Mathematical Problem in Engineering Theorem 3.3. On, β k in 3.3 can be computed a follow: β 1 k A β k Bu k GSR 1 y k A P C T( R CP C T 1( yk Cβ k, 3.33 where P i given by P 1 A P A T GQ G T A P C T( R CP C T 1 CP A T, 3.34 and initial condition are et to P Section 3.1. C T Π 1 C 1 and β k P C T Π 1 Ỹm, 1 computed in Proof. Firt, we will obtain a cloed-form of P in 3.34 and then, uing the cloed-form of P, we how that β k in 3.3 can be computed recurively from Uing the defined variable 3.22 and 3.24, we aume that P in 3.34 i of the form: P L G, 1 L T G, By an induction method, we will prove For the firt tep, we check 3.35 for 1. Given the initial value P, we tranform P 1 to the form 3.35 by 3.34 a follow: P 1 A G P 1 C T R 1 1 C Q 1 A G T Equation 3.38 can be written in term of L G,i and i a P 1 L G,1 1 1 LT G,1. ow, we check P 1 under the aumption that P L G, 1 L T G,.From3.34, we have P 1 A L G, M 1 1 LT G, AT GQ G T L G,1 1 1 LT G, Thu, we can ee that P in 3.34 can be repreented a the form 3.35 in term of L G, and. Uing thi reult, we are in a poition to how that β k in 3.3 can be computed recurively from We can rewrite β k in 3.3 a β k L G, 1 T L B, Ũ 1 L S, Ỹ 1, 3.38 where T i given by P 1 C T T β k G T R 1 Ỹ m,

15 Mathematical Problem in Engineering 15 A in a derivation of P in a batch form, we how by an induction method that β k in 3.38 i equivalent to the one in Firt,wehowthatβ 1 k can be obtained from P C T Π 1 C 1 and β k P C T Π 1 Ỹm, 1: ( 1 ( β 1 k A P 1 C T R 1 C P 1 β k C T R 1 y k Bu k GSR 1 y k, 3.4 where, in term of L G,i, i,andt i, β 1 k can be written a β 1 k L G,1 1 1 T 1 L B,1 Ũ L S,1 Ỹ. ext, we how that β 1 k in the form 3.38 can be obtained from β k of the batch form, that i, β k L G, 1 T L B, Ũ 1 L S, Ỹ 1. Firt, note that we have the following relation: {A A P C T( R CP C T } 1 C L G, 1 T A L G, M 1 1 T, 3.41 A P C T( R CP C T 1 yk A L G, M 1 1 LT G, CT R 1 y k. Subtituting 3.41 into 3.33 yield β 1 k A L G, M 1 1 T A L G, M 1 1 LT G, CT R 1 y k A L G, M 1 1 LT G, CT R 1 C (L B, Ũ 1 L S, Ỹ 1 L B,1 Ũ L S,1 Ỹ, A L G, G M 1 1 P 1 C T 1 β Q k T R 1 G o 1 1Ỹm, 3.42 L B,1 Ũ L S,1 Ỹ L G,1 1 1 T 1 L B,1 Ũ L S,1 Ỹ, where G and Q in the firt and econd matrix block on the right-hand ide of the firt equality have no effect on the equation. Thi complete the proof. It i oberved that the recurive equation with 3.33 and 3.34 are the ame a the Kalman filter with initial condition P C T Π 1 C 1 and β k P C T Π 1 Ỹm, 1. β k on 1 provide initial value and input for a recurive equation of γ k 3.31, which will be invetigated in what follow Recurive Equation for γ k on h Here, we dicu the recurive equation for γ k on h. A mentioned before, the recurive equation for γ k tart from h with the initial value γ h k β h k.on h, γ k can be recurively computed with the help of β k. However, γ k x k h k i what we want to find out finally. The recurive equation for γ k i given in the following theorem. Theorem 3.4. On h 1, γ k in 3.31 can be computed a follow: γ 1 k γ k K C T( CP C T R 1(yk Cβ k, 3.43

16 16 Mathematical Problem in Engineering Backward computing u k y k u k +1 y k +1 u k +2 y k +2 u k 3 y k 3 u k 2 y k 2 u k 1 y k 1 α /k α 1/k α 2/k α 3/k α 2/k α 1/k = ꉱP ꉱP 1 ꉱP 2 ꉱP 3 ꉱP 2 ꉱP 1 = Forward computing u k y k u k +1 y k +1 u k +2 y k +2 u k h 1 u k h u k h+1 y k h 1 y k h y k h+1 u k 1 y k 1 β /k β 1/k β 2/k β h 1/k β β h+1/k h/k β 1/k γ h+1/k γ 1/k P P 1 P 2 P h 1 P h P h+1 P 1 γ /k = ꉱx k h/k Figure 2: An iterative form of the LMS RH etimator. where γ h k β h k, β k and P are obtained from 3.33 and 3.34, repectively, and K i given by K K 1 (I C T( 1CP 1 CP 1 C T R A T, 3.44 with the initial condition K h P h. Proof. Uing Lemma 3.2, γ 1 k in 3.31 for h can be repreented a 1(yk γ 1 k γ k Ξ 1 L T G, CT( CP C T R Cβ k If we denote Ξ 1 L T G, by K, we have only to prove Since Ξ h L G, h, K h i equal to P h.uinglemma 3.2, we can repreent K in a recurive form a follow: K K 1 (I C T( 1CP 1 CP C T R A T Thi complete the proof. It i noted that the recurive equation 3.43 in Theorem 3.4 i a fixed-point moother of the tate x k h, which run with a recurive equation 3.33 in Theorem 3.3. Variable for recurive equation in Section 3.1 and 3.2 are viualized in Figure 2. Startingfrom P 1 and α 1 k, we compute P and α k recurively in a backward time. From P and α k, we compute P and β k, from which we drive the forward recurive equation for β k to get γ k x k h k.

17 Mathematical Problem in Engineering 17 2 Etimation error of receding horizon and Kalman etimator 1 LMS RH etimator Etimation error Kalman etimator Time Figure 3: Etimation error of LMS RH and Kalman etimator when temporary uncertaintie exit. 4. Simulation In thi ection, a numerical example i given to demontrate the performance of the propoed LMS RH etimator. Suppoe that we have a tate pace model repreented a x i δi.7379 x.7379 δ i i y i 1 δ i 1 δ i xi v i, u.7594 i w 1 i, 4.1 where δ i i an uncertain model parameter given a {.1, 2 i 22, δ i, otherwie. 4.2 The ytem noie covariance Q and the meaurement noie covariance R are et to.1 2 and.27 2, repectively. The memory ize and the fixed-lag ize are taken a 1 and h 3, repectively. A inuoidal input i applied a an input. We carry out a imulation for the ytem 4.1 with temporary modeling uncertaintie 4.2. InFigure 3, we compare the etimation error of the LMS RH etimator with the fixedlag Kalman etimator 18. When uncertaintie do not exit, the fixed-lag Kalman etimator ha the maller etimation error than the propoed LMS RH etimator. It can, however, be een that the etimation error of the LMS RH etimator i coniderably maller than that of the fixed-lag Kalman etimator when modeling uncertaintie exit. Actually, one of pole of the fixed-lag Kalman etimator i o cloe to a unit that even mall uncertaintie have a good chance of divergence. Additionally, we can ee that the etimation error of the LMS RH etimator converge much more rapidly than that of the fixed-lag Kalman etimator after temporary modeling uncertainty diappear. The low repone i alo related to the pole

18 18 Mathematical Problem in Engineering near a unit. To be ummarized, we can ay that the propoed LMS RH etimator i more robut than other etimator with infinite memory when applied to ytem with modeling uncertaintie. 5. Concluion In thi paper, we propoed a receding horizon RH etimator baed on the mean-quareerror criterion for a dicrete-time tate pace model, called a leat-mean-quare LMS RH etimator. An unknown tate wa etimated by making ue of the finite number of input and output over the recent finite horizon without any arbitrary aumption and any a priori tate information. The propoed LMS RH etimator wa obtained from the conditional expectation of the initial tate and the ytem noie on the correponding horizon. It wa hown that the LMS RH etimator ha a deadbeat property and ha good robut performance through a numerical example. To the bet of author knowledge, the propoed LMS RH etimator would be the mot general verion among exiting RH or finite-memory etimator in the mean-quare-error ene. Furthermore, the LMS RH etimator could be extended to other tochatic ytem with imperfect communication, uncertaintie, and o on 19, 2. Acknowledgment Thi work wa upported by the Minitry of Knowledge Economy, Rebublic of Korea ID: The material in thid paper wa preented in part at the International Sympoium on Intelligent Informatic, Qingdao, China, May 211. Reference 1 J. B. Rawling and K. R. Muke, The tability of contrained receding horizon control, IEEE Tranaction on Automatic Control, vol. 38, no. 1, pp , G. De icolao, L. Magni, and R. Scattolini, Stabilizing receding-horizon control of nonlinear timevarying ytem, IEEE Tranaction on Automatic Control, vol. 37, no. 9, pp , B.K.Kwon,S.Han,O.K.Kwon,andW.H.Kwon, MinimumvarianceFIRmootherfordicretetime tate pace model, IEEE Signal Proceing Letter, vol. 14, no. 8, pp , W. H. Kwon, P. S. Kim, and P. Park, A receding horizon Kalman FIR filter for dicrete time-invariant ytem, IEEE Tranaction on Automatic Control, vol. 44, no. 9, pp , A. Oppenheim and R. Schafer, Dicrete-Time Signal Proceing, Prentice-Hall, Englewood Cliff, J, USA, G. J. Bierman, Fixed memory leat quare filtering, IEEE Tranaction on Information Theory, vol. 21, no. 6, pp , P. J. Buxbaum, Fixed-memory recurive filter, IEEE Tranaction on Information Theory, vol. 2, no. 1, pp , A. H. Jazwinki, Limited memory optimal filtering, IEEE Tranaction on Automatic Control, vol. 13, pp , F. Schweppe, Uncertain Dynamic Sytem, Prentice-Hall, Englewood Cliff, J, USA, K. V. Ling and K. W. Lim, Receding horizon recurive tate etimation, IEEE Tranaction on Automatic Control, vol. 44, no. 9, pp , A. Aleandri, M. Baglietto, and G. Battitelli, Receding-horizon etimation for dicrete-time linear ytem, IEEE Tranaction on Automatic Control, vol. 48, no. 3, pp , A. Aleandri, M. Baglietto, and G. Battitelli, Receding-horizon etimation for witching dicretetime linear ytem, IEEE Tranaction on Automatic Control, vol. 5, no. 11, pp , 25.

19 Mathematical Problem in Engineering A. M. Brucktein and T. Kailath, Recurive limited memory filtering and cattering theory, IEEE Tranaction on Information Theory, vol. 31, no. 3, pp , W. H. Kwon and A. E. Pearon, A modified quadratic cot problem and feedback tabilization of a linear ytem, IEEE Tranaction on Automatic Control, vol. 22, no. 5, pp , W. H. Kwon, P. S. Kim, and S. H. Han, A receding horizon unbiaed FIR filter for dicrete-time tate pace model, Automatica, vol. 38, no. 3, pp , S. Han and W. H. Kwon, L 2 -E FIR moother for determinitic dicrete-time tate-pace ignal model, IEEE Tranaction on Automatic Control, vol. 52, no. 5, pp , S. Han, B. K. Kwon, and W. H. Kwon, L 2 -E FIR moother for determinitic continuou-time tate pace ignal model, Automatica, vol. 45, no. 6, pp , T. Kailath, A. H. Sayed, and B. Haibi, Linear Etimation, Prentice Hall, B. Shen, Z. Wang, H. Shu, and G. Wei, Robut H finite-horizon filtering with randomly occurred nonlinearitie and quantization effect, Automatica, vol. 46, no. 11, pp , B. Shen, Z. Wang, H. Shu, and G. Wei, On nonlinear H filtering for dicrete-time tochatic ytem with miing meaurement, IEEE Tranaction on Automatic Control, vol. 53, no. 9, pp , 28.

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