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Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems
Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems Huihong Zhao a,n, Chenghui Zhang b a b
Clean Energy Research and Technology Promotion Center, Dezhou University, No. 566 University Rd. West, Dezhou 253023, PR China School of Control Science and Engineering, Shandong University, 17923 Jingshi Road, Jinan 250061, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 July 2015 Received in revised form 26 September 2015 Accepted 2 October 2015 Communicated by Ma Lifeng Ma
This study deals with the input noise quadratic polynomial estimation problem for linear discrete-time non-Gaussian systems. The design of the non-Gaussian noise quadratic deconvolution filter and fixed-lag smoother is firstly converted into a linear estimation problem in a suitable second-order polynomial extended system. By employing the Kronecker algebra rules, the stochastic characteristics of the augmented noise in the augmented system are discussed. Then a solution to the non-Gaussian noise quadratic estimator is obtained through applying the projection formula in Kalman filtering theory. In addition, the stability is proved by constructing an equivalent state-space model with uncorrelated noises. Finally, a numerical example is given to show the effectiveness of the proposed method. & 2015 Elsevier B.V. All rights reserved.
Keywords: Input noise quadratic polynomial estimation Kronecker algebra Deconvolution filter Fixed-lag smoother
1. Introduction The input noise estimation (also known as deconvolution) has a rich history and a wide range of applications in image restoration, oil exploration, speech signal processing, fault detection and so on [1–4]. The task of the deconvolution problem is to estimate the intended unknown input noise of a system by utilizing the obtainable outputs. For the first time, an optimal white noise smoother with application to seismic data processing in oil exploration was presented in [2]. Applying the polynomial approach in frequency domain, the optimal deconvolution estimator was derived based on spectral factorization in [5]. Later, both input and measurement white noise estimators were designed by using the modern time series analysis method in [6]. Recently, the deconvolution theory was successfully applied to the multi-sensor linear discrete time systems [7,8] and the systems with packet dropouts [9–11]. Note that the above results were obtained based on the input Gaussian noise assumption, however, in many important technical areas the input noise is non-Gaussian (see for instance [12–14]). This is the motivation to develop a new algorithm which permits us to find a satisfactory non-Gaussian noise estimator for linear discrete-time time-varying systems.
n
Corresponding author. E-mail addresses: [email protected] (H. Zhao), [email protected] (C. Zhang).
The estimation problem for non-Gaussian systems has received more and more attention and some fundamental results have been developed, refer to [12,15] and the references therein. For linear non-Gaussian systems, the conditional expectation giving the minimum mean square error estimate is an infinite dimensional problem, and its solution cannot be easily numerically computed [15]. Although the Kalman filter is the best affine estimator for the non-Gaussian case, its estimated accuracy is inadequate in some cases. Note that the polynomial filtering algorithm [12,15], which employ both the observations of the original system and their Kronecker products, is more accurate than the classical Kalman filter, while maintaining the characteristics of easy calculability and recursivity. Therefore, an increasing number of authors have focussed on the polynomial estimator design for the non-Gaussian systems. The pioneer work can be traced back to the recursive arbitrary-degree finite-memory polynomial estimator design via the classical Kalman filtering theory [12]. Later, the result was successfully extended to polynomial filter for stochastic bilinear systems [16] and polynomial extended Kalman filter [17]. When the state-space model was unknown, the fixed-point, fixedinterval and fixed-lag smoothers from uncertain observations were presented based on the covariance information of the processes in [18]. Recently, this method was applied to the study of multi-sensor information fusion quadratic filter for linear systems with uncertain observations [19]. However, these works have a limitation that the Non-Gaussian noise polynomial estimator was not investigated.
http://dx.doi.org/10.1016/j.neucom.2015.10.015 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
H. Zhao, C. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
In this paper, we will investigate the non-Gaussian noise quadratic estimation problem for linear discrete-time time-varying systems. The linear recursive estimator, the non-Gaussian noise quadratic deconvolution filter and the non-Gaussian noise quadratic fixed-lag smoother are proposed. The stability of the non-Gaussian noise quadratic estimator is also discussed. Although the deconvolution estimation has been well studied, the non-Gaussian noise quadratic estimation is still difficult since the stochastic characteristics analysis problem for the second-order polynomial extended system involves the Kronecker product. To solve this problem, some Kronecker algebra rules constituting a powerful tool in treating vector polynomials are adopted in this paper. The main contribution of this paper can be summarized as follows: (i) it extends the polynomial filtering methodology to the input noise estimation of the linear discrete-time time-varying systems and (ii) it develops a recursive Kalman-like input noise quadratic estimator with more accurately. The remainder of this paper is arranged as follows. The linear discrete-time non-Gaussian systems and the least mean-squared error second-order polynomial estimation problem are introduced in Section 2. In Section 3, the linear recursive estimator is developed by using Kalman filtering theory. In Section 4, the nonGaussian noise quadratic deconvolution filter and the nonGaussian noise quadratic fixed-lag smoother are derived by calculating the extended Riccati difference equation. The stability analysis of the non-Gaussian noise quadratic estimator is proposed in Section 5. And an example is provided to prove the effect of the presented estimator in Section 6. Finally, the conclusions are proposed in Section 7.
2. System model and problem formulation We consider the following class of linear discrete-time systems:
noise quadratic filtering estimate and non-Gaussian noise quadratic fixed-lag smoothing estimate, respectively.
3. The linear recursive estimator Let us find the linear recursive estimator for system (1)–(3). By using the classical Kalman filtering theory, we state the linear deconvolution filtering of z(k) in the following lemma. Lemma 1. Consider the system (1)–(3) under the assumptions (4)– (7). Then the linear deconvolution filter z^ ðkj kÞ of z(k) is given by 1 ~ z^ ðkj kÞ ¼ LðkÞðst 1 Ψ N;2 ÞGT ðkÞRy ~ ðkÞyðkÞ
where ~ ^ k 1Þ yðkÞ ¼ yðkÞ CðkÞxðkj
ð9Þ
^ þ 1j kÞ ¼ AðkÞxðkj ^ k 1Þ þ K 0 ðkÞyðkÞ ~ xðk
ð11Þ
Ry~ ðkÞ ¼ CðkÞP 0 ðkÞC T ðkÞ þ GðkÞðst 1 Ψ N;2 ÞGT ðkÞ
ð12Þ
P 0 ðk þ 1Þ ¼ AðkÞP 0 ðkÞAT ðkÞ þ FðkÞðst 1 Ψ N;2 ÞF T ðkÞ K 0 ðkÞRy~ ðkÞK T0 ðkÞ P 0 ð0Þ ¼ st 1 Ψ x;2
ð13Þ
Furthermore, we present the linear fixed-lag smoother of z(k) in the following lemma. Lemma 2. Consider system (1)–(3) under the assumptions (4)–(7). Then, for a given integer d 4 0, a linear fixed-lag smoother z^ ðkj k þ dÞ of z(k) is given by z^ ðkj k þ dÞ ¼ z^ ðkj kÞ þ LðkÞ
d X
~ þjÞ Γ kk þ j C T ðk þ jÞRy~ 1 ðk þ jÞyðk
ð1Þ
yðkÞ ¼ CðkÞxðkÞ þ GðkÞNðkÞ
ð2Þ
Γ kk þ j þ 1 ¼ Γ kk þ j ½Aðk þ jÞ K 0 ðk þ jÞCðk þ jÞT
ð3Þ
j ¼ 1; 2; …; d 1
n
m
where xðkÞ A R is the state, yðkÞ A R is the measurement output, zðkÞ A Rq is the signal to be estimated, and the noise NðkÞ A Rr forms a sequence of non-Gaussian random vector variables, with all moments up to the fourth order finite and known: EðNðkÞÞ ¼ 0;
EðN ðkÞÞ ¼ Ψ N;i ; ½i
i ¼ 2; 3; 4:
ð4Þ
ð10Þ
K 0 ðkÞ ¼ AðkÞP 0 ðkÞC T ðkÞ þ FðkÞðst 1 Ψ N;2 ÞGT ðkÞ Ry~ 1 ðkÞ
xðk þ 1Þ ¼ AðkÞxðkÞ þ FðkÞNðkÞ; xð0Þ ¼ x
zðkÞ ¼ LðkÞNðkÞ
ð8Þ
ð14Þ
j¼1
where
ð15Þ
with
Γ kk þ 1 ¼ ðst 1 Ψ N;2 ÞF T ðkÞ ðst 1 Ψ N;2 ÞGT ðkÞK T0 ðkÞ
ð16Þ
~ þ jÞ, K 0 ðk þ jÞ and Ry~ ðk þ jÞ are computed by (8), Besides, z^ ðkj kÞ, yðk (9), (11) and (12), respectively.
Moreover, without loss of generality, we assume that st 1 Ψ N;2 ¼ EðNðkÞ NT ðkÞÞ
ð5Þ
The sequence fNðkÞg forms, with the initial state random vector x, a family of independent random variables. Also, the initial state x is endowed with statical moments, namely EðxÞ ¼ 0 Eðx ½i Þ ¼ Ψ x;i ;
ð6Þ i ¼ 2; 3; 4:
ð7Þ
4. The quadratic recursive estimator 4.1. The extended state-space model In order to obtain the estimator for system (1)–(3), let us define the following extended vector: " # xðkÞ X e ðkÞ ¼ ½2 ð17Þ x ðkÞ
The non-Gaussian noise quadratic estimation problem for the system model (1)–(3) can be stated as
According to Eqs. (1), (4) and (5), we have ½2
Problem 1. Given an integer d Z 0 and the observation sequence d ffyðsÞgks þ ¼ 0 g, find a least mean-squared error second-order polynomial estimator z^ ðkj k þdÞ of z(k). Note that the above estimation problem includes two cases, i.e. d ¼ 0 and d 4 0 which correspond to the cases of non-Gaussian
x ðk þ 1Þ ¼ xðk þ1Þ xðk þ 1Þ ¼ ½AðkÞxðkÞ þ FðkÞNðkÞ ½AðkÞxðkÞ þ FðkÞNðkÞ ¼ A½2 ðkÞx½2 ðkÞ þ ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ þ ðFðkÞNðkÞÞ ðAðkÞxðkÞÞ þF ½2 ðkÞN ½2 ðkÞ ¼ A½2 ðkÞx½2 ðkÞ þ dðkÞ þ f ðkÞ ð18Þ
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
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with dðkÞ ¼ F ðkÞEðN ðkÞÞ ¼ F ðkÞΨ N;2 : ½2
½2
½2
ð19Þ
3
Eðf ðkÞ f ðjÞÞ ¼ 0;
T
kaj
EðgðkÞ g T ðjÞÞ ¼ 0;
kaj
and f ðkÞ ¼ ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ þ ðFðkÞNðkÞÞ ðAðkÞxðkÞÞ þ F ðkÞðN ðkÞ Ψ ½2
½2
T N;2 Þ ¼ ðI þ C n;n Þ½ðAðkÞxðkÞÞ T N;2 ÞðI þ C n;n ÞððAðkÞxðkÞÞ
ðFðkÞNðkÞÞ þF ½2 ðkÞðN ½2 ðkÞ Ψ
IÞðFðkÞNðkÞÞ þ F ½2 ðkÞðN ½2 ðkÞ Ψ N;2 Þ
ð20Þ
where we have used the property: ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ ¼ ðAðkÞxðkÞ 1Þ ðI FðkÞNðkÞÞ ¼ ðAðkÞxðkÞ IÞ ðFðkÞNðkÞÞ:
ð21Þ
In order to define the quadratic optimal filter, let us introduce the following post-processed quadratic output: ~ yðkÞ ¼ y½2 ðkÞ G½2 ðkÞΨ N;2
ð22Þ
which depends on the extended state by the equation ~ yðkÞ ¼ C ½2 ðkÞx½2 ðkÞ þ gðkÞ
ð23Þ
with
From (20), we have nh T Eðf ðkÞ f ðkÞÞ ¼ E ðI þ C Tn;n Þ½ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ i þF ½2 ðkÞðN ½2 ðkÞ Ψ N;2 Þ h ðI þ C Tn;n Þ½ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ þ F ½2 ðkÞðN½2 ðkÞ n T o ¼ E ðI þ C Tn;n Þ½ðAðkÞxðkÞÞ Ψ N;2 Þ o ðFðkÞNðkÞÞ½ðAðkÞxðkÞÞ ðFðkÞNðkÞÞT ðI þ C Tn;n ÞT n o T þE F ½2 ðkÞðN½2 ðkÞ Ψ N;2 ÞðN ½2 ðkÞ Ψ N;2 ÞT F ½2 ðkÞ n ¼ E ðI þ C Tn;n ÞðAðkÞxðkÞxT ðkÞAT ðkÞÞ o ðFðkÞNðkÞN T ðkÞF T ðkÞÞ ðI þC Tn;n ÞT n þE F ½2 ðkÞst 1 ½ðN½2 ðkÞ Ψ N;2 Þ ðN ½2 ðkÞ Ψ N;2 Þ o T F ½2 ðkÞ ¼ ðI þ C Tn;n Þ½ðAðkÞΨ x ðkÞAT ðkÞÞ
gðkÞ ¼ ðCðkÞxðkÞÞ ðGðkÞNðkÞÞ þ ðGðkÞNðkÞÞ ðCðkÞxðkÞÞ þG ðkÞ½N ðkÞ Ψ N;2 ½2
½2
ðFðkÞðst 1 Ψ N;2 ÞF T ðkÞÞðI þC Tn;n Þ n þE F ½2 ðkÞst 1 ½N ½4 ðkÞ N ½2 ðkÞ Ψ N;2 Ψ N;2 o T ½2 N ½2 ðkÞ þ Ψ N;2 F ½2 ðkÞ ¼ ðI þ C Tn;n Þ½ðAðkÞΨ x ðkÞAT ðkÞÞ
ð24Þ
Now, we can state the following lemma. Lemma 3. The sequences ff ðkÞg and fgðkÞg are zero-mean white sequences with Eðf ðkÞÞ ¼ EðgðkÞÞ ¼ 0; ( T
Eðf ðkÞ f ðjÞÞ ¼
0;
( EðgðkÞ g ðjÞÞ ¼
kaj
0;
( Eðf ðkÞ g ðjÞÞ ¼
0;
kaj
ð27Þ
kaj
ð28Þ
Ψ fg ðkÞ; k ¼ j (
0;
ð29Þ
where
Ψ f ðkÞ ¼
Ψ x ðkÞA ðkÞÞ ðFðkÞðst T
ðI þ C Tn;n Þ þ F ½2 ðkÞfst 1 ð
Thus, the equality (30) is proved. And, the equalities (31) and (32) can be proved by employing the similar line as the proof of (30). This completes the proof.□ Furthermore, we give the following lemma which summarizes the results of this subsection. Lemma 4. The extended state evolves according to the system equations
kaj
Ψ Tfg ðkÞ; k ¼ j
ðI þ C Tn;n Þ½ðAðkÞ
Ψ N;4 Ψ
1
Ψ N;2 ÞF ðkÞÞ
ð34Þ
Y e ðkÞ ¼ C e ðkÞX e ðkÞ þ g e ðkÞ
ð35Þ
where
T
½2 ½2 T ðkÞ N;2 ÞgF
X e ðk þ1Þ ¼ Ae ðkÞX e ðkÞ þ de ðkÞ þf e ðkÞ
ð30Þ
½2
T
ðI þ C Tm;m Þ þ G½2 ðkÞfst 1 ðΨ N;4 Ψ N;2 ÞgG½2 ðkÞ
" ð31Þ
Ae ðkÞ ¼
ð32Þ
de ðkÞ ¼
Ψ fg ðkÞ ¼ ðI þ C Tn;n Þ½ðAðkÞΨ x ðkÞC T ðkÞÞ ðFðkÞðst 1 Ψ N;2 ÞGT ðkÞÞ ðI þ C Tm;m Þ þ F ½2 ðkÞfst 1 ð
Ψ N;4 Ψ
½2 ½2 T ðkÞ N;2 ÞgG
Ψ x ðk þ 1Þ ¼ Eðxðk þ 1Þ xT ðk þ 1ÞÞ ¼ AðkÞΨ x ðkÞAT ðkÞ þFðkÞðst 1 Ψ N;2 ÞF T ðkÞ
"
ð33Þ Proof. Notice that N(k) is a sequence of independent zero mean non-Gaussian random variables, independent of the initial state, characterized by the knowledge of all moments up to the fourth one: EðN ½i ðkÞÞ ¼ Ψ N;i ;
i ¼ 2; 3; 4:
it follows from (20) and (24) that Eðf ðkÞÞ ¼ EðgðkÞÞ ¼ 0;
8 k ¼ 0; 1; 2; ⋯
"
Y e ðkÞ ¼
Ψ g ðkÞ ¼ ðI þ C Tm;m Þ½ðCðkÞΨ x ðkÞC T ðkÞÞ ðGðkÞðst 1 Ψ N;2 ÞGT ðkÞÞ
EðNðkÞÞ ¼ 0;
T
½2
Ψ N;2 ÞgF ½2 ðkÞ
ð26Þ
Ψ g ðkÞ; k ¼ j
T
T
ð25Þ
Ψ f ðkÞ; k ¼ j
T
EðgðkÞ f ðjÞÞ ¼
8 k ¼ 0; 1; 2; …
ðFðkÞðst 1 Ψ N;2 ÞF T ðkÞÞðI þC Tn;n Þ þ F ½2 ðkÞfst 1 ðΨ N;4
# ð36Þ
AðkÞ
0
0
A½2 ðkÞ
0
# ð37Þ
# ð38Þ
dðkÞ "
f e ðkÞ ¼
FðkÞNðkÞ
# ð39Þ
f ðkÞ "
C e ðkÞ ¼ " g e ðkÞ ¼
yðkÞ ~ yðkÞ
CðkÞ
0
0
C ½2 ðkÞ
GðkÞNðkÞ
# ð40Þ
#
gðkÞ
ð41Þ
and ff e ðkÞg, fg e ðkÞg are zero-mean white sequences such that
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
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4
Eðf e ðkÞg Te ðjÞÞ ¼ 0;
kaj
ð42Þ
Moreover, the auto-covariances and the cross-covariance of the noises ff e ðkÞg and fg e ðkÞg are, respectively, given by 2 3 FðkÞðst 1 Ψ N;2 ÞF T ðkÞ st 1 ½F ½3 ðkÞΨ N;3 6 7 T T Eðf e ðkÞf e ðkÞÞ ¼ Q ðkÞ ¼ 4 1 ½3 5 Ψ f ðkÞ st ½F ðkÞΨ N;3 ð43Þ 2
Eðg e ðkÞg Te ðkÞÞ
GðkÞðst 1 Ψ N;2 ÞGT ðkÞ 6 T ¼ RðkÞ ¼ 4 st 1 ½G½3 ðkÞΨ N;3
st 1 ½G½3 ðkÞΨ N;3
Ψ g ðkÞ
3 7 5 ð44Þ
" Eðf e ðkÞg Te ðkÞÞ ¼ SðkÞ ¼
FðkÞðst 1 Ψ N;2 ÞGT ðkÞ S21 ðkÞ
S12 ðkÞ
Ψ fg ðkÞ
#
X^ e ð0j 1Þ ¼ EðX e ð0ÞÞ and for (54) it is Pð0Þ ¼ EfðX e ð0Þ EðX e ð0ÞÞÞðX e ð0Þ EðX e ð0ÞÞÞT g
ð46Þ
n oT S21 ðkÞ ¼ st 1 f½F ½2 ðkÞ GðkÞΨ N;3 g
ð47Þ
Proof. Only (43)–(45) require a proof. It can be obtained by taking into account the initial conditions of system (1)–(3), assumptions (4)–(7) and using the Kronecker algebra rules. For simplicity, we T only give the proof of EfFðkÞNðkÞ f ðkÞg ¼ st 1 ½F ½3 ðkÞΨ N;3 . SubT stituting (20) into EfFðkÞNðkÞ f ðkÞg and using the property that N (k) is independent of the initial state, we have EfFðkÞNðkÞ f ðkÞg ¼ EfFðkÞNðkÞ½F ½2 ðkÞðN ½2 ðkÞ Ψ N;2 ÞT g n o ¼ E st 1 f½F ½2 ðkÞðN½2 ðkÞ Ψ N;2 Þ ðFðkÞNðkÞÞg n o ¼ E st 1 f½F ½2 ðkÞ FðkÞ½ðN ½2 ðkÞ Ψ N;2 Þ NðkÞg
Theorem 1. Consider the system (1)–(3) with conditions (4)–(7). Then a non-Gaussian noise quadratic deconvolution filter z^ ðkj kÞ of z (k) is given by z^ ðkj kÞ ¼ TðkÞeðkÞ
Thus, it is obviously that (43) is established. Moreover, by employing the similar line, we can give the proof of (44) and (45). The proof is completed.□ 4.2. The non-Gaussian noise quadratic estimate In this section, we will give the non-Gaussian noise quadratic estimator for the extended system (34)–(35). First, we introduce the following innovation sequence and the corresponding covariance matrix: eðkÞ ¼ Y e ðkÞ C e ðkÞX^ e ðkj k 1Þ
ð49Þ
Re ðkÞ ¼ EðeðkÞeT ðkÞÞ
ð50Þ
Applying projection formula in the classical Kalman filtering, X^ e ðkj k 1Þ is computed recursively as k X X^ e ðk þ1j kÞ ¼ E X e ðk þ 1ÞeT ðjÞ Re 1 ðjÞeðjÞ j¼0
¼ Ae ðkÞX^ e ðkj k 1Þ þ de ðkÞ þ KðkÞeðkÞ
ð57Þ
T 1 TðkÞ ¼ LðkÞðst Ψ N;2 ÞG ðkÞ
ð51Þ
where KðkÞ ¼ ½Ae ðkÞPðkÞC Te ðkÞ þ SðkÞRe 1 ðkÞ
ð52Þ
Re ðkÞ ¼ C e ðkÞPðkÞC Te ðkÞ þ RðkÞ
ð53Þ
st 1
G½2 ðkÞ LðkÞ Ψ N;3 Re 1 ðkÞ ð58Þ
and Re(k) is calculated by (53). Proof. According to projection formula in Hilbert space, we have z^ ðkj kÞ ¼
k X E zðkÞeT ðjÞ Re 1 ðjÞeðjÞ ¼ E zðkÞeT ðkÞ Re 1 ðkÞeðkÞ j¼0
n T o 1 Re ðkÞeðkÞ ¼ E ðLðkÞNðkÞÞ C e ðkÞex ðkÞ þ g e ðkÞ n T o 1 Re ðkÞeðkÞ ¼ E ðLðkÞNðkÞÞ g e ðkÞ h i T 1 ¼ LðkÞðst Ψ N;2 ÞG ðkÞ E LðkÞNðkÞg T ðkÞ Re 1 ðkÞeðkÞ
T
ð48Þ
ð56Þ
which can be easily calculated by using (6)–(7). We are now in a position to provide a non-Gaussian noise quadratic deconvolution filtering for z(k), which is summarized into the following theorem.
where ð45Þ
S12 ðkÞ ¼ st 1 f½G½2 ðkÞ FðkÞΨ N;3 g
¼ st 1 ½F ½3 ðkÞΨ N;3
ð55Þ
ð59Þ
Note that E LðkÞNðkÞg T ðkÞ can be obtained by applying the similar line of (48). This completes the proof of Theorem 1.□ Moreover, we present the non-Gaussian noise quadratic fixedlag smoother of z(k) in the following Theorem 2. For the convenience of discussion, we denote Rkk þ i ¼ EðNðkÞ; ex ðk þ iÞÞ;
i ¼ 1; …; d
ð60Þ
Theorem 2. Consider the system (1)–(3) with conditions (4)–(7). Then, for a given integer d 40, a non-Gaussian noise quadratic fixedlag smoother z^ ðkj k þ dÞ of z(k) is given by z^ ðkj k þ dÞ ¼ TðkÞeðkÞ þ LðkÞ
d X
Rkk þ i C Te ðk þiÞRe 1 ðk þ iÞeðk þ iÞ
ð61Þ
i¼1
where Rkk þ i is calculated recursively as Rkk þ i þ 1 ¼ Rkk þ i ½Ae ðk þ iÞ Kðk þ iÞC e ðk þ iÞT i ¼ 1; 2; …; d 1
ð62Þ
with
T 1 1 F ½2 ðkÞ I Ψ N;3 Rkk þ 1 ¼ ðst Ψ N;2 ÞF ðkÞ st T 1 1 G½2 ðkÞ I Ψ N;3 K T ðkÞ ðst Ψ N;2 ÞG ðkÞ st
ð63Þ
In addition, T(k), Re ðk þ iÞ and Kðk þ iÞ are respectively as in (58), (53) and (52).
and PðkÞ ¼ Eðex ðkÞeTx ðkÞÞ; ðex ðkÞ ¼ X e ðkÞ X^ e ðkj k 1ÞÞ is the solution to the following standard Riccati equation:
Proof. Note that z^ ðkj k þ dÞ is the projection of z(k) onto the linear space Lfeð0Þ; eð1Þ; …; eðk þ dÞg and that z(k) is uncorrelated with Lfeð0Þ; eð1Þ; …; eðk 1Þg. By using the projection formulation, z^ ðkj k þdÞ is given by
Pðk þ 1Þ ¼ Ae ðkÞPðkÞATe ðkÞ þ Q ðkÞ KðkÞRe ðkÞK T ðkÞ
z^ ðkj k þ dÞ ¼
Besides, the initial condition for (51) is
ð54Þ
kX þd
E zðkÞeT ðjÞ Re 1 ðjÞeðjÞ
j¼0
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H. Zhao, C. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
¼
d X E zðkÞeT ðk þ jÞ Re 1 ðk þ jÞeðk þ jÞ
T
ð64Þ
j¼0
From Theorem 1, it is easy to know that d X E zðkÞeT ðk þ jÞ Re 1 ðk þ jÞeðk þ jÞ
z^ ðkj k þ dÞ ¼ TðkÞeðkÞ þ
ð65Þ
j¼1
Now we calculate E zðkÞeT ðk þ jÞ . From the definition (49) and (60), it follows that E zðkÞeT ðk þ jÞ ¼ LðkÞRkk þ j C Te ðk þ jÞ ð66Þ Rkk þ j
where is as in (60). With the considering of (34), (35), (49) and (51), we obtain ex ðk þ i þ1Þ ¼ ½Ae ðk þ iÞ Kðk þ iÞC e ðk þ iÞex ðk þ iÞ þ f e ðk þ iÞ Kðk þ iÞg e ðk þ iÞ
5
ð78Þ
eðkÞ ¼ Y e ðkÞ C e ðkÞX^ e ðkj k 1Þ
ð79Þ
Next, the above one-step predictor will be proved to be theoretically equivalent to the one-step predictor given in Section 4.2. of this paper. Theorem 3. The one-step predictor (49), (51)–(54) for system (34)– (35) is equivalent to the one-step predictor (75)–(79) for system (71) and (35). Proof. According to (35), (69) and (79), we have X^ e ðk þ 1j kÞ ¼ A e ðkÞX^ e ðkj k 1Þ þ de ðkÞ þ JðkÞY e ðkÞ þ K ðkÞeðkÞ ¼ A ðkÞX^ ðkj k 1Þ þ d ðkÞ þ K ðkÞeðkÞ e
ð67Þ
Since z(k) is uncorrelated with f e ðk þ iÞ and g e ðk þ iÞ. From the definition (60), (62) follows directly from (67). Furthermore, (61) follows directly from (65) and (66). Thus the proof is completed.□
T
P ðk þ 1Þ ¼ A e ðkÞP ðkÞA e ðkÞ þQ ðkÞ K ðkÞR e ðkÞK ðkÞ
e
e
ð80Þ
1
where 1
K 1 ðkÞ ¼ K ðkÞ þSðkÞR 1 ðkÞ ¼ A e ðkÞP ðkÞC Te ðkÞR e ðkÞ þ SðkÞR 1 ðkÞ 1
¼ Ae ðkÞP ðkÞC Te ðkÞR e ðkÞ þSðkÞ½R 1 ðkÞ R 1 ðkÞ 1
C e ðkÞP ðkÞC Te ðkÞR e ðkÞ
ð81Þ
Note that 5. The stability analysis of the non-Gaussian noise quadratic estimator
1
½R 1 ðkÞ R 1 ðkÞC e ðkÞP ðkÞC Te ðkÞR e ðkÞR e ðkÞ ¼ I
ð82Þ
Then Inspired by [20], the stability analysis for the non-Gaussian noise quadratic estimator is presented in this section. Noticing that the input noise and the measurement noise in this paper are correlated while the ones are assumed uncorrelated in [20], we first convert the system (34)–(35) to be one with uncorrelated noise. First, by introducing an item equaling zero into the right side of state Eq. (34), we have X e ðk þ 1Þ ¼ Ae ðkÞX e ðkÞ þ de ðkÞ þ f e ðkÞ þ JðkÞ Y e ðkÞ C e ðkÞX e ðkÞ g e ðkÞ ð68Þ
ð83Þ
On the other hand, it follows from (34) and (80) that e x ðk þ 1Þ ¼ X e ðk þ 1Þ X^ e ðk þ1j kÞ ¼ Ae ðkÞe x ðkÞ þ f e ðkÞ K 1 ðkÞeðkÞ ð84Þ Moreover, according to (84), the Riccati equation P ðkÞ ¼ Eðe x ðkÞ e Tx ðkÞÞ given by (78) can be re-expressed as T
P ðk þ 1Þ ¼ Ae ðkÞP ðkÞATe ðkÞ þ Q ðkÞ K 1 ðkÞR e ðkÞK 1 ðkÞ
Define A e ðkÞ ¼ Ae ðkÞ JðkÞC e ðkÞ
ð69Þ
f e ðkÞ ¼ f e ðkÞ JðkÞg e ðkÞ
ð70Þ
Then, Eq. (34) can be re-expressed as X e ðk þ 1Þ ¼ A e ðkÞX e ðkÞ þde ðkÞ þ JðkÞY e ðkÞ þ f e ðkÞ
ð71Þ
And the observation equation is still (35). Furthermore, it is readily obtained from (43)–(45) that Eðf e ðkÞÞ ¼ 0 E½f e ðkÞg Te ðjÞ ¼ ðSðkÞ JðkÞRðkÞÞδkj
ð72Þ
Let JðkÞ ¼ SðkÞR 1 ðkÞ
ð73Þ T e ðkÞg e ðjÞ ¼
0, i.e. f e ðkÞ and ge(k) are uncorreThen, we have E½f lated. Moreover, the auto-covariance of the noise f e ðkÞ is given by E½f e ðkÞf
1
K 1 ðkÞ ¼ ½Ae ðkÞP ðkÞC Te ðkÞ þ SðkÞR e ðkÞ
T e ðkÞ
¼ Q ðkÞ ¼ Q ðkÞ SðkÞR
1
T
ðkÞS ðkÞ
ð74Þ
For the system (35) and (71), the one-step predictor of X e ðk þ 1Þ is determined by the following equations: X^ e ðk þ 1j kÞ ¼ A e ðkÞX^ e ðkj k 1Þ þde ðkÞ þ JðkÞY e ðkÞ þK ðkÞeðkÞ 1
ð75Þ
ð85Þ
Thus, X^ e ðk þ 1j kÞ, K 1 ðkÞ, R e ðkÞ, P ðk þ 1Þ and eðkÞ have the same structure with X^ e ðk þ 1j kÞ, K(k), Re(k), Pðk þ 1Þ and e(k), respectively. And, the one-step predictor (75)–(79) has the same initial values and observations with the one-step predictor (49), (51)– (54). Naturally, we can readily get the conclusion of this theorem.□ In the light of Theorem 3 in this paper and Theorem 5.3 in [20], we have the following results. Theorem 4. Consider system (1)–(3) under the assumptions (4)–(7). If ½A e ðkÞ, R 1=2 ðkÞC e ðkÞ is uniformly detectable, then one-step predictor error covariance P(k) in (54) is bounded. If, in addition, ½A e ðkÞ 1=2 ; Q ðkÞ is uniformly stabilizable, then the non-Gaussian noise quadratic deconvolution filter and the non-Gaussian noise quadratic fixed-lag smoother are exponentially stable. Proof. Noticing that the one-step predictor X^ e ðk þ 1j kÞ given by (51) can be considered as the state equation of the deconvolution filter given in (57) and the fixed-lag smoother given in (61), it follows that if the one-step predictor X^ e ðk þ 1j kÞ is exponentially Table 1 The probability distributions of N(k) and x.
K ðkÞ ¼ A e ðkÞP ðkÞC Te ðkÞR e ðkÞ
ð76Þ
N(k)
1
2
2
x
0.8
0.4
R e ðkÞ ¼ C e ðkÞP ðkÞC Te ðkÞ þ RðkÞ
ð77Þ
pðNðkÞÞ
6/9
2/9
1/9
p ðx Þ
1/3
2/3
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H. Zhao, C. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
2
4 True value Linear filtering Quadratic filtering
2
1
1
0.5
Estimation error
Signal, linear filter and quadratic filter
3
0 −1
0 −0.5
−2
−1
−3
−1.5
−4
−2
0
10
Linear smoothing Quadratic smoothing
1.5
20
30
40
50
0
10
30
40
50
Time k
Time k
Fig. 1. Signal (solid line), linear filtering (dashed line) and quadratic filtering (thick line).
Fig. 4. Linear three fixed-lag smoothing error (solid line) and quadratic three fixedlag smoothing error (dashed line).
exponentially stable. By using the system equations (71) and (35), the results given in this theorem follow directly from Theorem 5.3 in [20].□
4 Linear filtering Quadratic filtering
3
20
2 Estimation error
6. A numerical example 1
Consider system (1)–(3) with the following parameters: 0
AðkÞ ¼ 0:8e 0:1k ; FðkÞ ¼ 0:5; CðkÞ ¼ 0:3; GðkÞ ¼ 0:1 þ 0:2 cos ð2kÞ; LðkÞ ¼ 0:8
−1 −2 −3 −4
0
10
20
30
40
50
Time k
Fig. 2. Linear filtering error (solid line) and quadratic filtering error (dashed line).
Signal, linear smoother and quadratic smoother
4 True value Linear smoothing Quadratic smoothing
3
where the non-Gaussian noise N(k) and the initial value of the state x are independent, zero mean random sequences with the distributions found in Table 1. The linear recursive estimation algorithm and the quadratic recursive estimation algorithm have been implemented. In order to compare the results of the two methods, numerical simulations have been performed under the same conditions in both cases. The results are displayed in Figs. 1–4 for 50 iterations. It is obvious that the tracking performance of the quadratic recursive estimator is better than the linear recursive estimator.
2
7. Conclusions
1
In this paper, by employing the classical Kalman filtering theory, we first give the non-Gaussian noise linear filter and fixed-lag smoother for a class of linear discrete-time systems. Then, the non-Gaussian noise least-squares second-order filter and fixed-lag smoother are derived by applying the innovation technique to suitably defined augmented state-space model. With the introduction of an equivalent state-space model for the second-order polynomial extended system, the asymptotic stability of the nonGaussian noise quadratic estimators is analyzed. Finally, a numerical example demonstrates the effectiveness and accuracy of the proposed algorithm.
0 −1 −2 −3 −4
0
10
20
30
40
50
Time k
Fig. 3. Signal (solid line), linear three fixed-lag smoothing (dashed line) and quadratic three fixed-lag smoothing (thick line).
Acknowledgments
stable then the deconvolution filter and the fixed-lag smoother are exponentially stable. In view of Theorem 3 in this paper, X^ e ðk þ 1j kÞ given by (51) is exponentially stable if and only if X^ e ðk þ 1j kÞ given by (75) is
This work was supported in part by the National Natural Science Foundation of China (Nos. 61403061, 61503171, 61273097), the Science and Technology Development Project of Dezhou, China (No. 2012B05), the Talent Introduction Project of Dezhou
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
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University, China (No. 311432), and the Solar Energy Special Project of Dezhou University, China (No. 14ZX03).
References [1] X.G. Yu, C.S. Hsu, R.H. Bamberger, H1 deconvolution filter design and its application in image restoration, in: Proceedings of the 35th Conference on Decision and Control, Japan, 1996, pp. 4802–4807. [2] J.M. Mendel, White-noise estimators for seismic data processing in oil exploration, IEEE Trans. Autom. Control 22 (5) (1977) 694–706. [3] W. Yuan, J. Lin, W. An, Y. Wang, N. Chen, Noise estimation based on time– frequency correlation for speech enhancement, Appl. Acoust. 74 (5) (2013) 770–781. [4] F. Shi, R.J. Patton, Fault estimation and active fault tolerant control for linear parameter varying descriptor systems, Int. J. Robust Nonlinear 25 (5) (2015) 689–706. [5] A. Ahlen, M. Sternad, Optimal deconvolution based on polynominal method, IEEE Trans. Acoust. Speech 37 (2) (1989) 217–226. [6] Z.L. Deng, H.S. Zhang, S.J. Liu, L. Zhou, Optimal and self-tuning white noise estimators with application to deconvolution and filtering problem, Automatica 32 (20) (1996) 199–216. [7] X.J. Sun, Y. Gao, Z.L. Deng, C. Li, J.W. Wang, Multi-model information fusion Kalman filtering and white noise deconvolution, Inf. Fusion 11 (2) (2010) 163–173. [8] X.J. Sun, G.M. Yan, Self-tuning weighted measurement fusion white noise deconvolution estimator and its convergence analysis, Digit. Signal Process. 23 (2013) 38–48. [9] J.C. Hung, Fixed-order optimal deconvolution filter with irregular missing data, Int. J. Control Signal Process. 24 (2010) 311–321. [10] C. Han, W. Wang, Y. Zhang, White noise estimators for networked systems with packet dropouts, Int. J. Control Autom. 11 (6) (2013) 1187–1195. [11] W. Wang, C. Han, H. Fang, White noise estimation for discrete-time systems with random delay and packet dropout, J. Syst. Sci. Complex. 27 (2014) 476–493. [12] F. Carravetta, A. Germani, M. Raimondi, Polynomial filtering for linear discrete time non-Gaussian systems, SIAM J. Control Optim. 34 (5) (1996) 1666–1690. [13] N. Towghi, B. Javidi, Image recognition in the presence of non-Gaussian noise with unknown statistics, J. Opt. Soc. Am. A 18 (11) (2001) 2744–2753. [14] L. Guo, H. Wang, T. Chai, Fault detection for non-linear non-Gaussian stochastic systems using entropy optimization principle, Trans. Inst. Meas. Control 28 (2) (2006) 145–161. [15] A. Fasano, A. Germani, A. Monteriu, Reduced-order quadratic Kalman-like filtering of non-Gaussian systems, IEEE Trans. Autom. Control 58 (7) (2013) 1744–1757.
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[16] F. Carravetta, A. Germani, M. Raimondi, Polynomial filtering of discrete-time stochastic linear systems with multiplicative state noise, IEEE Trans. Autom. Control 42 (8) (1997) 1106–1126. [17] A. Germani, C. Manes, P. Palumbo, Polynomial extended Kalman filter, IEEE Trans. Autom. Control 50 (12) (2005) 2059–2064. [18] S. Nakamori, R. Caballero, A. Hermoso, J.D. Jimenez, J. Linares, Signal polynomial smoothing from correlated interrupted observations based on covariances, Math. Methods Appl. Sci. 30 (2007) 1645–1665. [19] R. Caballero, A. Hermoso, J. Linares, Linear and quadratic estimation using uncertain observations from multiple sensors with correlated uncertainty, Signal Process. 91 (2011) 330–337. [20] B.D.O. Anderson, J.B. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Control Optim. 19 (1) (1981) 20–32.
Huihong Zhao received his Ph.D. degree in Control Theory and Control Engineering from Shandong University in 2011. He is currently a lecturer at Dezhou University, Dezhou Shandong, China. His research interest covers optimal control and estimation, network control system, and time-delay systems.
Chenghui Zhang received his Bachelor and Master degrees in Automation Engineering from Shandong University of Technology, Jinan, China, in 1985 and 1988, and the Ph.D. degree in Control Theory and Operational Research from Shandong University, Jinan, in 2001, respectively. In 1988, he joined Shandong University, where he is currently a Professor of School of Control Science and Engineering, the chief manager of Power Electronic Energy-saving Technology & Equipment Research Center of Education Ministry, and a Specially Invited Cheung Kong Scholars Professor by China Ministry of Education. He is also the chief expert of the National “863” high technological planning. Professor Zhang's research interests include optimal control of engineering, power electronics and motor drives, energy-saving techniques and time-delay systems.
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems Huihong Zhao a,n, Chenghui Zhang b a b
Clean Energy Research and Technology Promotion Center, Dezhou University, No. 566 University Rd. West, Dezhou 253023, PR China School of Control Science and Engineering, Shandong University, 17923 Jingshi Road, Jinan 250061, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 July 2015 Received in revised form 26 September 2015 Accepted 2 October 2015 Communicated by Ma Lifeng Ma
This study deals with the input noise quadratic polynomial estimation problem for linear discrete-time non-Gaussian systems. The design of the non-Gaussian noise quadratic deconvolution filter and fixed-lag smoother is firstly converted into a linear estimation problem in a suitable second-order polynomial extended system. By employing the Kronecker algebra rules, the stochastic characteristics of the augmented noise in the augmented system are discussed. Then a solution to the non-Gaussian noise quadratic estimator is obtained through applying the projection formula in Kalman filtering theory. In addition, the stability is proved by constructing an equivalent state-space model with uncorrelated noises. Finally, a numerical example is given to show the effectiveness of the proposed method. & 2015 Elsevier B.V. All rights reserved.
Keywords: Input noise quadratic polynomial estimation Kronecker algebra Deconvolution filter Fixed-lag smoother
1. Introduction The input noise estimation (also known as deconvolution) has a rich history and a wide range of applications in image restoration, oil exploration, speech signal processing, fault detection and so on [1–4]. The task of the deconvolution problem is to estimate the intended unknown input noise of a system by utilizing the obtainable outputs. For the first time, an optimal white noise smoother with application to seismic data processing in oil exploration was presented in [2]. Applying the polynomial approach in frequency domain, the optimal deconvolution estimator was derived based on spectral factorization in [5]. Later, both input and measurement white noise estimators were designed by using the modern time series analysis method in [6]. Recently, the deconvolution theory was successfully applied to the multi-sensor linear discrete time systems [7,8] and the systems with packet dropouts [9–11]. Note that the above results were obtained based on the input Gaussian noise assumption, however, in many important technical areas the input noise is non-Gaussian (see for instance [12–14]). This is the motivation to develop a new algorithm which permits us to find a satisfactory non-Gaussian noise estimator for linear discrete-time time-varying systems.
n
Corresponding author. E-mail addresses: [email protected] (H. Zhao), [email protected] (C. Zhang).
The estimation problem for non-Gaussian systems has received more and more attention and some fundamental results have been developed, refer to [12,15] and the references therein. For linear non-Gaussian systems, the conditional expectation giving the minimum mean square error estimate is an infinite dimensional problem, and its solution cannot be easily numerically computed [15]. Although the Kalman filter is the best affine estimator for the non-Gaussian case, its estimated accuracy is inadequate in some cases. Note that the polynomial filtering algorithm [12,15], which employ both the observations of the original system and their Kronecker products, is more accurate than the classical Kalman filter, while maintaining the characteristics of easy calculability and recursivity. Therefore, an increasing number of authors have focussed on the polynomial estimator design for the non-Gaussian systems. The pioneer work can be traced back to the recursive arbitrary-degree finite-memory polynomial estimator design via the classical Kalman filtering theory [12]. Later, the result was successfully extended to polynomial filter for stochastic bilinear systems [16] and polynomial extended Kalman filter [17]. When the state-space model was unknown, the fixed-point, fixedinterval and fixed-lag smoothers from uncertain observations were presented based on the covariance information of the processes in [18]. Recently, this method was applied to the study of multi-sensor information fusion quadratic filter for linear systems with uncertain observations [19]. However, these works have a limitation that the Non-Gaussian noise polynomial estimator was not investigated.
http://dx.doi.org/10.1016/j.neucom.2015.10.015 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
H. Zhao, C. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
In this paper, we will investigate the non-Gaussian noise quadratic estimation problem for linear discrete-time time-varying systems. The linear recursive estimator, the non-Gaussian noise quadratic deconvolution filter and the non-Gaussian noise quadratic fixed-lag smoother are proposed. The stability of the non-Gaussian noise quadratic estimator is also discussed. Although the deconvolution estimation has been well studied, the non-Gaussian noise quadratic estimation is still difficult since the stochastic characteristics analysis problem for the second-order polynomial extended system involves the Kronecker product. To solve this problem, some Kronecker algebra rules constituting a powerful tool in treating vector polynomials are adopted in this paper. The main contribution of this paper can be summarized as follows: (i) it extends the polynomial filtering methodology to the input noise estimation of the linear discrete-time time-varying systems and (ii) it develops a recursive Kalman-like input noise quadratic estimator with more accurately. The remainder of this paper is arranged as follows. The linear discrete-time non-Gaussian systems and the least mean-squared error second-order polynomial estimation problem are introduced in Section 2. In Section 3, the linear recursive estimator is developed by using Kalman filtering theory. In Section 4, the nonGaussian noise quadratic deconvolution filter and the nonGaussian noise quadratic fixed-lag smoother are derived by calculating the extended Riccati difference equation. The stability analysis of the non-Gaussian noise quadratic estimator is proposed in Section 5. And an example is provided to prove the effect of the presented estimator in Section 6. Finally, the conclusions are proposed in Section 7.
2. System model and problem formulation We consider the following class of linear discrete-time systems:
noise quadratic filtering estimate and non-Gaussian noise quadratic fixed-lag smoothing estimate, respectively.
3. The linear recursive estimator Let us find the linear recursive estimator for system (1)–(3). By using the classical Kalman filtering theory, we state the linear deconvolution filtering of z(k) in the following lemma. Lemma 1. Consider the system (1)–(3) under the assumptions (4)– (7). Then the linear deconvolution filter z^ ðkj kÞ of z(k) is given by 1 ~ z^ ðkj kÞ ¼ LðkÞðst 1 Ψ N;2 ÞGT ðkÞRy ~ ðkÞyðkÞ
where ~ ^ k 1Þ yðkÞ ¼ yðkÞ CðkÞxðkj
ð9Þ
^ þ 1j kÞ ¼ AðkÞxðkj ^ k 1Þ þ K 0 ðkÞyðkÞ ~ xðk
ð11Þ
Ry~ ðkÞ ¼ CðkÞP 0 ðkÞC T ðkÞ þ GðkÞðst 1 Ψ N;2 ÞGT ðkÞ
ð12Þ
P 0 ðk þ 1Þ ¼ AðkÞP 0 ðkÞAT ðkÞ þ FðkÞðst 1 Ψ N;2 ÞF T ðkÞ K 0 ðkÞRy~ ðkÞK T0 ðkÞ P 0 ð0Þ ¼ st 1 Ψ x;2
ð13Þ
Furthermore, we present the linear fixed-lag smoother of z(k) in the following lemma. Lemma 2. Consider system (1)–(3) under the assumptions (4)–(7). Then, for a given integer d 4 0, a linear fixed-lag smoother z^ ðkj k þ dÞ of z(k) is given by z^ ðkj k þ dÞ ¼ z^ ðkj kÞ þ LðkÞ
d X
~ þjÞ Γ kk þ j C T ðk þ jÞRy~ 1 ðk þ jÞyðk
ð1Þ
yðkÞ ¼ CðkÞxðkÞ þ GðkÞNðkÞ
ð2Þ
Γ kk þ j þ 1 ¼ Γ kk þ j ½Aðk þ jÞ K 0 ðk þ jÞCðk þ jÞT
ð3Þ
j ¼ 1; 2; …; d 1
n
m
where xðkÞ A R is the state, yðkÞ A R is the measurement output, zðkÞ A Rq is the signal to be estimated, and the noise NðkÞ A Rr forms a sequence of non-Gaussian random vector variables, with all moments up to the fourth order finite and known: EðNðkÞÞ ¼ 0;
EðN ðkÞÞ ¼ Ψ N;i ; ½i
i ¼ 2; 3; 4:
ð4Þ
ð10Þ
K 0 ðkÞ ¼ AðkÞP 0 ðkÞC T ðkÞ þ FðkÞðst 1 Ψ N;2 ÞGT ðkÞ Ry~ 1 ðkÞ
xðk þ 1Þ ¼ AðkÞxðkÞ þ FðkÞNðkÞ; xð0Þ ¼ x
zðkÞ ¼ LðkÞNðkÞ
ð8Þ
ð14Þ
j¼1
where
ð15Þ
with
Γ kk þ 1 ¼ ðst 1 Ψ N;2 ÞF T ðkÞ ðst 1 Ψ N;2 ÞGT ðkÞK T0 ðkÞ
ð16Þ
~ þ jÞ, K 0 ðk þ jÞ and Ry~ ðk þ jÞ are computed by (8), Besides, z^ ðkj kÞ, yðk (9), (11) and (12), respectively.
Moreover, without loss of generality, we assume that st 1 Ψ N;2 ¼ EðNðkÞ NT ðkÞÞ
ð5Þ
The sequence fNðkÞg forms, with the initial state random vector x, a family of independent random variables. Also, the initial state x is endowed with statical moments, namely EðxÞ ¼ 0 Eðx ½i Þ ¼ Ψ x;i ;
ð6Þ i ¼ 2; 3; 4:
ð7Þ
4. The quadratic recursive estimator 4.1. The extended state-space model In order to obtain the estimator for system (1)–(3), let us define the following extended vector: " # xðkÞ X e ðkÞ ¼ ½2 ð17Þ x ðkÞ
The non-Gaussian noise quadratic estimation problem for the system model (1)–(3) can be stated as
According to Eqs. (1), (4) and (5), we have ½2
Problem 1. Given an integer d Z 0 and the observation sequence d ffyðsÞgks þ ¼ 0 g, find a least mean-squared error second-order polynomial estimator z^ ðkj k þdÞ of z(k). Note that the above estimation problem includes two cases, i.e. d ¼ 0 and d 4 0 which correspond to the cases of non-Gaussian
x ðk þ 1Þ ¼ xðk þ1Þ xðk þ 1Þ ¼ ½AðkÞxðkÞ þ FðkÞNðkÞ ½AðkÞxðkÞ þ FðkÞNðkÞ ¼ A½2 ðkÞx½2 ðkÞ þ ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ þ ðFðkÞNðkÞÞ ðAðkÞxðkÞÞ þF ½2 ðkÞN ½2 ðkÞ ¼ A½2 ðkÞx½2 ðkÞ þ dðkÞ þ f ðkÞ ð18Þ
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with dðkÞ ¼ F ðkÞEðN ðkÞÞ ¼ F ðkÞΨ N;2 : ½2
½2
½2
ð19Þ
3
Eðf ðkÞ f ðjÞÞ ¼ 0;
T
kaj
EðgðkÞ g T ðjÞÞ ¼ 0;
kaj
and f ðkÞ ¼ ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ þ ðFðkÞNðkÞÞ ðAðkÞxðkÞÞ þ F ðkÞðN ðkÞ Ψ ½2
½2
T N;2 Þ ¼ ðI þ C n;n Þ½ðAðkÞxðkÞÞ T N;2 ÞðI þ C n;n ÞððAðkÞxðkÞÞ
ðFðkÞNðkÞÞ þF ½2 ðkÞðN ½2 ðkÞ Ψ
IÞðFðkÞNðkÞÞ þ F ½2 ðkÞðN ½2 ðkÞ Ψ N;2 Þ
ð20Þ
where we have used the property: ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ ¼ ðAðkÞxðkÞ 1Þ ðI FðkÞNðkÞÞ ¼ ðAðkÞxðkÞ IÞ ðFðkÞNðkÞÞ:
ð21Þ
In order to define the quadratic optimal filter, let us introduce the following post-processed quadratic output: ~ yðkÞ ¼ y½2 ðkÞ G½2 ðkÞΨ N;2
ð22Þ
which depends on the extended state by the equation ~ yðkÞ ¼ C ½2 ðkÞx½2 ðkÞ þ gðkÞ
ð23Þ
with
From (20), we have nh T Eðf ðkÞ f ðkÞÞ ¼ E ðI þ C Tn;n Þ½ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ i þF ½2 ðkÞðN ½2 ðkÞ Ψ N;2 Þ h ðI þ C Tn;n Þ½ðAðkÞxðkÞÞ ðFðkÞNðkÞÞ þ F ½2 ðkÞðN½2 ðkÞ n T o ¼ E ðI þ C Tn;n Þ½ðAðkÞxðkÞÞ Ψ N;2 Þ o ðFðkÞNðkÞÞ½ðAðkÞxðkÞÞ ðFðkÞNðkÞÞT ðI þ C Tn;n ÞT n o T þE F ½2 ðkÞðN½2 ðkÞ Ψ N;2 ÞðN ½2 ðkÞ Ψ N;2 ÞT F ½2 ðkÞ n ¼ E ðI þ C Tn;n ÞðAðkÞxðkÞxT ðkÞAT ðkÞÞ o ðFðkÞNðkÞN T ðkÞF T ðkÞÞ ðI þC Tn;n ÞT n þE F ½2 ðkÞst 1 ½ðN½2 ðkÞ Ψ N;2 Þ ðN ½2 ðkÞ Ψ N;2 Þ o T F ½2 ðkÞ ¼ ðI þ C Tn;n Þ½ðAðkÞΨ x ðkÞAT ðkÞÞ
gðkÞ ¼ ðCðkÞxðkÞÞ ðGðkÞNðkÞÞ þ ðGðkÞNðkÞÞ ðCðkÞxðkÞÞ þG ðkÞ½N ðkÞ Ψ N;2 ½2
½2
ðFðkÞðst 1 Ψ N;2 ÞF T ðkÞÞðI þC Tn;n Þ n þE F ½2 ðkÞst 1 ½N ½4 ðkÞ N ½2 ðkÞ Ψ N;2 Ψ N;2 o T ½2 N ½2 ðkÞ þ Ψ N;2 F ½2 ðkÞ ¼ ðI þ C Tn;n Þ½ðAðkÞΨ x ðkÞAT ðkÞÞ
ð24Þ
Now, we can state the following lemma. Lemma 3. The sequences ff ðkÞg and fgðkÞg are zero-mean white sequences with Eðf ðkÞÞ ¼ EðgðkÞÞ ¼ 0; ( T
Eðf ðkÞ f ðjÞÞ ¼
0;
( EðgðkÞ g ðjÞÞ ¼
kaj
0;
( Eðf ðkÞ g ðjÞÞ ¼
0;
kaj
ð27Þ
kaj
ð28Þ
Ψ fg ðkÞ; k ¼ j (
0;
ð29Þ
where
Ψ f ðkÞ ¼
Ψ x ðkÞA ðkÞÞ ðFðkÞðst T
ðI þ C Tn;n Þ þ F ½2 ðkÞfst 1 ð
Thus, the equality (30) is proved. And, the equalities (31) and (32) can be proved by employing the similar line as the proof of (30). This completes the proof.□ Furthermore, we give the following lemma which summarizes the results of this subsection. Lemma 4. The extended state evolves according to the system equations
kaj
Ψ Tfg ðkÞ; k ¼ j
ðI þ C Tn;n Þ½ðAðkÞ
Ψ N;4 Ψ
1
Ψ N;2 ÞF ðkÞÞ
ð34Þ
Y e ðkÞ ¼ C e ðkÞX e ðkÞ þ g e ðkÞ
ð35Þ
where
T
½2 ½2 T ðkÞ N;2 ÞgF
X e ðk þ1Þ ¼ Ae ðkÞX e ðkÞ þ de ðkÞ þf e ðkÞ
ð30Þ
½2
T
ðI þ C Tm;m Þ þ G½2 ðkÞfst 1 ðΨ N;4 Ψ N;2 ÞgG½2 ðkÞ
" ð31Þ
Ae ðkÞ ¼
ð32Þ
de ðkÞ ¼
Ψ fg ðkÞ ¼ ðI þ C Tn;n Þ½ðAðkÞΨ x ðkÞC T ðkÞÞ ðFðkÞðst 1 Ψ N;2 ÞGT ðkÞÞ ðI þ C Tm;m Þ þ F ½2 ðkÞfst 1 ð
Ψ N;4 Ψ
½2 ½2 T ðkÞ N;2 ÞgG
Ψ x ðk þ 1Þ ¼ Eðxðk þ 1Þ xT ðk þ 1ÞÞ ¼ AðkÞΨ x ðkÞAT ðkÞ þFðkÞðst 1 Ψ N;2 ÞF T ðkÞ
"
ð33Þ Proof. Notice that N(k) is a sequence of independent zero mean non-Gaussian random variables, independent of the initial state, characterized by the knowledge of all moments up to the fourth one: EðN ½i ðkÞÞ ¼ Ψ N;i ;
i ¼ 2; 3; 4:
it follows from (20) and (24) that Eðf ðkÞÞ ¼ EðgðkÞÞ ¼ 0;
8 k ¼ 0; 1; 2; ⋯
"
Y e ðkÞ ¼
Ψ g ðkÞ ¼ ðI þ C Tm;m Þ½ðCðkÞΨ x ðkÞC T ðkÞÞ ðGðkÞðst 1 Ψ N;2 ÞGT ðkÞÞ
EðNðkÞÞ ¼ 0;
T
½2
Ψ N;2 ÞgF ½2 ðkÞ
ð26Þ
Ψ g ðkÞ; k ¼ j
T
T
ð25Þ
Ψ f ðkÞ; k ¼ j
T
EðgðkÞ f ðjÞÞ ¼
8 k ¼ 0; 1; 2; …
ðFðkÞðst 1 Ψ N;2 ÞF T ðkÞÞðI þC Tn;n Þ þ F ½2 ðkÞfst 1 ðΨ N;4
# ð36Þ
AðkÞ
0
0
A½2 ðkÞ
0
# ð37Þ
# ð38Þ
dðkÞ "
f e ðkÞ ¼
FðkÞNðkÞ
# ð39Þ
f ðkÞ "
C e ðkÞ ¼ " g e ðkÞ ¼
yðkÞ ~ yðkÞ
CðkÞ
0
0
C ½2 ðkÞ
GðkÞNðkÞ
# ð40Þ
#
gðkÞ
ð41Þ
and ff e ðkÞg, fg e ðkÞg are zero-mean white sequences such that
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H. Zhao, C. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Eðf e ðkÞg Te ðjÞÞ ¼ 0;
kaj
ð42Þ
Moreover, the auto-covariances and the cross-covariance of the noises ff e ðkÞg and fg e ðkÞg are, respectively, given by 2 3 FðkÞðst 1 Ψ N;2 ÞF T ðkÞ st 1 ½F ½3 ðkÞΨ N;3 6 7 T T Eðf e ðkÞf e ðkÞÞ ¼ Q ðkÞ ¼ 4 1 ½3 5 Ψ f ðkÞ st ½F ðkÞΨ N;3 ð43Þ 2
Eðg e ðkÞg Te ðkÞÞ
GðkÞðst 1 Ψ N;2 ÞGT ðkÞ 6 T ¼ RðkÞ ¼ 4 st 1 ½G½3 ðkÞΨ N;3
st 1 ½G½3 ðkÞΨ N;3
Ψ g ðkÞ
3 7 5 ð44Þ
" Eðf e ðkÞg Te ðkÞÞ ¼ SðkÞ ¼
FðkÞðst 1 Ψ N;2 ÞGT ðkÞ S21 ðkÞ
S12 ðkÞ
Ψ fg ðkÞ
#
X^ e ð0j 1Þ ¼ EðX e ð0ÞÞ and for (54) it is Pð0Þ ¼ EfðX e ð0Þ EðX e ð0ÞÞÞðX e ð0Þ EðX e ð0ÞÞÞT g
ð46Þ
n oT S21 ðkÞ ¼ st 1 f½F ½2 ðkÞ GðkÞΨ N;3 g
ð47Þ
Proof. Only (43)–(45) require a proof. It can be obtained by taking into account the initial conditions of system (1)–(3), assumptions (4)–(7) and using the Kronecker algebra rules. For simplicity, we T only give the proof of EfFðkÞNðkÞ f ðkÞg ¼ st 1 ½F ½3 ðkÞΨ N;3 . SubT stituting (20) into EfFðkÞNðkÞ f ðkÞg and using the property that N (k) is independent of the initial state, we have EfFðkÞNðkÞ f ðkÞg ¼ EfFðkÞNðkÞ½F ½2 ðkÞðN ½2 ðkÞ Ψ N;2 ÞT g n o ¼ E st 1 f½F ½2 ðkÞðN½2 ðkÞ Ψ N;2 Þ ðFðkÞNðkÞÞg n o ¼ E st 1 f½F ½2 ðkÞ FðkÞ½ðN ½2 ðkÞ Ψ N;2 Þ NðkÞg
Theorem 1. Consider the system (1)–(3) with conditions (4)–(7). Then a non-Gaussian noise quadratic deconvolution filter z^ ðkj kÞ of z (k) is given by z^ ðkj kÞ ¼ TðkÞeðkÞ
Thus, it is obviously that (43) is established. Moreover, by employing the similar line, we can give the proof of (44) and (45). The proof is completed.□ 4.2. The non-Gaussian noise quadratic estimate In this section, we will give the non-Gaussian noise quadratic estimator for the extended system (34)–(35). First, we introduce the following innovation sequence and the corresponding covariance matrix: eðkÞ ¼ Y e ðkÞ C e ðkÞX^ e ðkj k 1Þ
ð49Þ
Re ðkÞ ¼ EðeðkÞeT ðkÞÞ
ð50Þ
Applying projection formula in the classical Kalman filtering, X^ e ðkj k 1Þ is computed recursively as k X X^ e ðk þ1j kÞ ¼ E X e ðk þ 1ÞeT ðjÞ Re 1 ðjÞeðjÞ j¼0
¼ Ae ðkÞX^ e ðkj k 1Þ þ de ðkÞ þ KðkÞeðkÞ
ð57Þ
T 1 TðkÞ ¼ LðkÞðst Ψ N;2 ÞG ðkÞ
ð51Þ
where KðkÞ ¼ ½Ae ðkÞPðkÞC Te ðkÞ þ SðkÞRe 1 ðkÞ
ð52Þ
Re ðkÞ ¼ C e ðkÞPðkÞC Te ðkÞ þ RðkÞ
ð53Þ
st 1
G½2 ðkÞ LðkÞ Ψ N;3 Re 1 ðkÞ ð58Þ
and Re(k) is calculated by (53). Proof. According to projection formula in Hilbert space, we have z^ ðkj kÞ ¼
k X E zðkÞeT ðjÞ Re 1 ðjÞeðjÞ ¼ E zðkÞeT ðkÞ Re 1 ðkÞeðkÞ j¼0
n T o 1 Re ðkÞeðkÞ ¼ E ðLðkÞNðkÞÞ C e ðkÞex ðkÞ þ g e ðkÞ n T o 1 Re ðkÞeðkÞ ¼ E ðLðkÞNðkÞÞ g e ðkÞ h i T 1 ¼ LðkÞðst Ψ N;2 ÞG ðkÞ E LðkÞNðkÞg T ðkÞ Re 1 ðkÞeðkÞ
T
ð48Þ
ð56Þ
which can be easily calculated by using (6)–(7). We are now in a position to provide a non-Gaussian noise quadratic deconvolution filtering for z(k), which is summarized into the following theorem.
where ð45Þ
S12 ðkÞ ¼ st 1 f½G½2 ðkÞ FðkÞΨ N;3 g
¼ st 1 ½F ½3 ðkÞΨ N;3
ð55Þ
ð59Þ
Note that E LðkÞNðkÞg T ðkÞ can be obtained by applying the similar line of (48). This completes the proof of Theorem 1.□ Moreover, we present the non-Gaussian noise quadratic fixedlag smoother of z(k) in the following Theorem 2. For the convenience of discussion, we denote Rkk þ i ¼ EðNðkÞ; ex ðk þ iÞÞ;
i ¼ 1; …; d
ð60Þ
Theorem 2. Consider the system (1)–(3) with conditions (4)–(7). Then, for a given integer d 40, a non-Gaussian noise quadratic fixedlag smoother z^ ðkj k þ dÞ of z(k) is given by z^ ðkj k þ dÞ ¼ TðkÞeðkÞ þ LðkÞ
d X
Rkk þ i C Te ðk þiÞRe 1 ðk þ iÞeðk þ iÞ
ð61Þ
i¼1
where Rkk þ i is calculated recursively as Rkk þ i þ 1 ¼ Rkk þ i ½Ae ðk þ iÞ Kðk þ iÞC e ðk þ iÞT i ¼ 1; 2; …; d 1
ð62Þ
with
T 1 1 F ½2 ðkÞ I Ψ N;3 Rkk þ 1 ¼ ðst Ψ N;2 ÞF ðkÞ st T 1 1 G½2 ðkÞ I Ψ N;3 K T ðkÞ ðst Ψ N;2 ÞG ðkÞ st
ð63Þ
In addition, T(k), Re ðk þ iÞ and Kðk þ iÞ are respectively as in (58), (53) and (52).
and PðkÞ ¼ Eðex ðkÞeTx ðkÞÞ; ðex ðkÞ ¼ X e ðkÞ X^ e ðkj k 1ÞÞ is the solution to the following standard Riccati equation:
Proof. Note that z^ ðkj k þ dÞ is the projection of z(k) onto the linear space Lfeð0Þ; eð1Þ; …; eðk þ dÞg and that z(k) is uncorrelated with Lfeð0Þ; eð1Þ; …; eðk 1Þg. By using the projection formulation, z^ ðkj k þdÞ is given by
Pðk þ 1Þ ¼ Ae ðkÞPðkÞATe ðkÞ þ Q ðkÞ KðkÞRe ðkÞK T ðkÞ
z^ ðkj k þ dÞ ¼
Besides, the initial condition for (51) is
ð54Þ
kX þd
E zðkÞeT ðjÞ Re 1 ðjÞeðjÞ
j¼0
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¼
d X E zðkÞeT ðk þ jÞ Re 1 ðk þ jÞeðk þ jÞ
T
ð64Þ
j¼0
From Theorem 1, it is easy to know that d X E zðkÞeT ðk þ jÞ Re 1 ðk þ jÞeðk þ jÞ
z^ ðkj k þ dÞ ¼ TðkÞeðkÞ þ
ð65Þ
j¼1
Now we calculate E zðkÞeT ðk þ jÞ . From the definition (49) and (60), it follows that E zðkÞeT ðk þ jÞ ¼ LðkÞRkk þ j C Te ðk þ jÞ ð66Þ Rkk þ j
where is as in (60). With the considering of (34), (35), (49) and (51), we obtain ex ðk þ i þ1Þ ¼ ½Ae ðk þ iÞ Kðk þ iÞC e ðk þ iÞex ðk þ iÞ þ f e ðk þ iÞ Kðk þ iÞg e ðk þ iÞ
5
ð78Þ
eðkÞ ¼ Y e ðkÞ C e ðkÞX^ e ðkj k 1Þ
ð79Þ
Next, the above one-step predictor will be proved to be theoretically equivalent to the one-step predictor given in Section 4.2. of this paper. Theorem 3. The one-step predictor (49), (51)–(54) for system (34)– (35) is equivalent to the one-step predictor (75)–(79) for system (71) and (35). Proof. According to (35), (69) and (79), we have X^ e ðk þ 1j kÞ ¼ A e ðkÞX^ e ðkj k 1Þ þ de ðkÞ þ JðkÞY e ðkÞ þ K ðkÞeðkÞ ¼ A ðkÞX^ ðkj k 1Þ þ d ðkÞ þ K ðkÞeðkÞ e
ð67Þ
Since z(k) is uncorrelated with f e ðk þ iÞ and g e ðk þ iÞ. From the definition (60), (62) follows directly from (67). Furthermore, (61) follows directly from (65) and (66). Thus the proof is completed.□
T
P ðk þ 1Þ ¼ A e ðkÞP ðkÞA e ðkÞ þQ ðkÞ K ðkÞR e ðkÞK ðkÞ
e
e
ð80Þ
1
where 1
K 1 ðkÞ ¼ K ðkÞ þSðkÞR 1 ðkÞ ¼ A e ðkÞP ðkÞC Te ðkÞR e ðkÞ þ SðkÞR 1 ðkÞ 1
¼ Ae ðkÞP ðkÞC Te ðkÞR e ðkÞ þSðkÞ½R 1 ðkÞ R 1 ðkÞ 1
C e ðkÞP ðkÞC Te ðkÞR e ðkÞ
ð81Þ
Note that 5. The stability analysis of the non-Gaussian noise quadratic estimator
1
½R 1 ðkÞ R 1 ðkÞC e ðkÞP ðkÞC Te ðkÞR e ðkÞR e ðkÞ ¼ I
ð82Þ
Then Inspired by [20], the stability analysis for the non-Gaussian noise quadratic estimator is presented in this section. Noticing that the input noise and the measurement noise in this paper are correlated while the ones are assumed uncorrelated in [20], we first convert the system (34)–(35) to be one with uncorrelated noise. First, by introducing an item equaling zero into the right side of state Eq. (34), we have X e ðk þ 1Þ ¼ Ae ðkÞX e ðkÞ þ de ðkÞ þ f e ðkÞ þ JðkÞ Y e ðkÞ C e ðkÞX e ðkÞ g e ðkÞ ð68Þ
ð83Þ
On the other hand, it follows from (34) and (80) that e x ðk þ 1Þ ¼ X e ðk þ 1Þ X^ e ðk þ1j kÞ ¼ Ae ðkÞe x ðkÞ þ f e ðkÞ K 1 ðkÞeðkÞ ð84Þ Moreover, according to (84), the Riccati equation P ðkÞ ¼ Eðe x ðkÞ e Tx ðkÞÞ given by (78) can be re-expressed as T
P ðk þ 1Þ ¼ Ae ðkÞP ðkÞATe ðkÞ þ Q ðkÞ K 1 ðkÞR e ðkÞK 1 ðkÞ
Define A e ðkÞ ¼ Ae ðkÞ JðkÞC e ðkÞ
ð69Þ
f e ðkÞ ¼ f e ðkÞ JðkÞg e ðkÞ
ð70Þ
Then, Eq. (34) can be re-expressed as X e ðk þ 1Þ ¼ A e ðkÞX e ðkÞ þde ðkÞ þ JðkÞY e ðkÞ þ f e ðkÞ
ð71Þ
And the observation equation is still (35). Furthermore, it is readily obtained from (43)–(45) that Eðf e ðkÞÞ ¼ 0 E½f e ðkÞg Te ðjÞ ¼ ðSðkÞ JðkÞRðkÞÞδkj
ð72Þ
Let JðkÞ ¼ SðkÞR 1 ðkÞ
ð73Þ T e ðkÞg e ðjÞ ¼
0, i.e. f e ðkÞ and ge(k) are uncorreThen, we have E½f lated. Moreover, the auto-covariance of the noise f e ðkÞ is given by E½f e ðkÞf
1
K 1 ðkÞ ¼ ½Ae ðkÞP ðkÞC Te ðkÞ þ SðkÞR e ðkÞ
T e ðkÞ
¼ Q ðkÞ ¼ Q ðkÞ SðkÞR
1
T
ðkÞS ðkÞ
ð74Þ
For the system (35) and (71), the one-step predictor of X e ðk þ 1Þ is determined by the following equations: X^ e ðk þ 1j kÞ ¼ A e ðkÞX^ e ðkj k 1Þ þde ðkÞ þ JðkÞY e ðkÞ þK ðkÞeðkÞ 1
ð75Þ
ð85Þ
Thus, X^ e ðk þ 1j kÞ, K 1 ðkÞ, R e ðkÞ, P ðk þ 1Þ and eðkÞ have the same structure with X^ e ðk þ 1j kÞ, K(k), Re(k), Pðk þ 1Þ and e(k), respectively. And, the one-step predictor (75)–(79) has the same initial values and observations with the one-step predictor (49), (51)– (54). Naturally, we can readily get the conclusion of this theorem.□ In the light of Theorem 3 in this paper and Theorem 5.3 in [20], we have the following results. Theorem 4. Consider system (1)–(3) under the assumptions (4)–(7). If ½A e ðkÞ, R 1=2 ðkÞC e ðkÞ is uniformly detectable, then one-step predictor error covariance P(k) in (54) is bounded. If, in addition, ½A e ðkÞ 1=2 ; Q ðkÞ is uniformly stabilizable, then the non-Gaussian noise quadratic deconvolution filter and the non-Gaussian noise quadratic fixed-lag smoother are exponentially stable. Proof. Noticing that the one-step predictor X^ e ðk þ 1j kÞ given by (51) can be considered as the state equation of the deconvolution filter given in (57) and the fixed-lag smoother given in (61), it follows that if the one-step predictor X^ e ðk þ 1j kÞ is exponentially Table 1 The probability distributions of N(k) and x.
K ðkÞ ¼ A e ðkÞP ðkÞC Te ðkÞR e ðkÞ
ð76Þ
N(k)
1
2
2
x
0.8
0.4
R e ðkÞ ¼ C e ðkÞP ðkÞC Te ðkÞ þ RðkÞ
ð77Þ
pðNðkÞÞ
6/9
2/9
1/9
p ðx Þ
1/3
2/3
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
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2
4 True value Linear filtering Quadratic filtering
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0.5
Estimation error
Signal, linear filter and quadratic filter
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0 −1
0 −0.5
−2
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Linear smoothing Quadratic smoothing
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Time k
Fig. 1. Signal (solid line), linear filtering (dashed line) and quadratic filtering (thick line).
Fig. 4. Linear three fixed-lag smoothing error (solid line) and quadratic three fixedlag smoothing error (dashed line).
exponentially stable. By using the system equations (71) and (35), the results given in this theorem follow directly from Theorem 5.3 in [20].□
4 Linear filtering Quadratic filtering
3
20
2 Estimation error
6. A numerical example 1
Consider system (1)–(3) with the following parameters: 0
AðkÞ ¼ 0:8e 0:1k ; FðkÞ ¼ 0:5; CðkÞ ¼ 0:3; GðkÞ ¼ 0:1 þ 0:2 cos ð2kÞ; LðkÞ ¼ 0:8
−1 −2 −3 −4
0
10
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30
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Time k
Fig. 2. Linear filtering error (solid line) and quadratic filtering error (dashed line).
Signal, linear smoother and quadratic smoother
4 True value Linear smoothing Quadratic smoothing
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where the non-Gaussian noise N(k) and the initial value of the state x are independent, zero mean random sequences with the distributions found in Table 1. The linear recursive estimation algorithm and the quadratic recursive estimation algorithm have been implemented. In order to compare the results of the two methods, numerical simulations have been performed under the same conditions in both cases. The results are displayed in Figs. 1–4 for 50 iterations. It is obvious that the tracking performance of the quadratic recursive estimator is better than the linear recursive estimator.
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7. Conclusions
1
In this paper, by employing the classical Kalman filtering theory, we first give the non-Gaussian noise linear filter and fixed-lag smoother for a class of linear discrete-time systems. Then, the non-Gaussian noise least-squares second-order filter and fixed-lag smoother are derived by applying the innovation technique to suitably defined augmented state-space model. With the introduction of an equivalent state-space model for the second-order polynomial extended system, the asymptotic stability of the nonGaussian noise quadratic estimators is analyzed. Finally, a numerical example demonstrates the effectiveness and accuracy of the proposed algorithm.
0 −1 −2 −3 −4
0
10
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Time k
Fig. 3. Signal (solid line), linear three fixed-lag smoothing (dashed line) and quadratic three fixed-lag smoothing (thick line).
Acknowledgments
stable then the deconvolution filter and the fixed-lag smoother are exponentially stable. In view of Theorem 3 in this paper, X^ e ðk þ 1j kÞ given by (51) is exponentially stable if and only if X^ e ðk þ 1j kÞ given by (75) is
This work was supported in part by the National Natural Science Foundation of China (Nos. 61403061, 61503171, 61273097), the Science and Technology Development Project of Dezhou, China (No. 2012B05), the Talent Introduction Project of Dezhou
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i
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University, China (No. 311432), and the Solar Energy Special Project of Dezhou University, China (No. 14ZX03).
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[16] F. Carravetta, A. Germani, M. Raimondi, Polynomial filtering of discrete-time stochastic linear systems with multiplicative state noise, IEEE Trans. Autom. Control 42 (8) (1997) 1106–1126. [17] A. Germani, C. Manes, P. Palumbo, Polynomial extended Kalman filter, IEEE Trans. Autom. Control 50 (12) (2005) 2059–2064. [18] S. Nakamori, R. Caballero, A. Hermoso, J.D. Jimenez, J. Linares, Signal polynomial smoothing from correlated interrupted observations based on covariances, Math. Methods Appl. Sci. 30 (2007) 1645–1665. [19] R. Caballero, A. Hermoso, J. Linares, Linear and quadratic estimation using uncertain observations from multiple sensors with correlated uncertainty, Signal Process. 91 (2011) 330–337. [20] B.D.O. Anderson, J.B. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Control Optim. 19 (1) (1981) 20–32.
Huihong Zhao received his Ph.D. degree in Control Theory and Control Engineering from Shandong University in 2011. He is currently a lecturer at Dezhou University, Dezhou Shandong, China. His research interest covers optimal control and estimation, network control system, and time-delay systems.
Chenghui Zhang received his Bachelor and Master degrees in Automation Engineering from Shandong University of Technology, Jinan, China, in 1985 and 1988, and the Ph.D. degree in Control Theory and Operational Research from Shandong University, Jinan, in 2001, respectively. In 1988, he joined Shandong University, where he is currently a Professor of School of Control Science and Engineering, the chief manager of Power Electronic Energy-saving Technology & Equipment Research Center of Education Ministry, and a Specially Invited Cheung Kong Scholars Professor by China Ministry of Education. He is also the chief expert of the National “863” high technological planning. Professor Zhang's research interests include optimal control of engineering, power electronics and motor drives, energy-saving techniques and time-delay systems.
Please cite this article as: H. Zhao, C. Zhang, Non-Gaussian noise quadratic estimation for linear discrete-time time-varying systems, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.10.015i