This paper is concerned with the finite-time š»āˆž filtering problem for linear continuous time-varying systems with uncertain observations and ā„’2-norm bounded noise. The design of finite-time š»āˆž filter is equivalent to the problem that a certain indefinite quadratic form has a minimum and the filter is such that the minimum is positive. The quadratic form is related to a Krein state-space model according to the Krein space linear estimation theory. By using the projection theory in Krein space, the finite-time š»āˆž filtering problem is solved. A numerical example is given to illustrate the performance of the š»āˆž filter.

1. Introduction

Most of the literatures on estimation problem always assume the observations contain the signal to be estimated [1ā€“8]. In [5], the linear matrix inequality technique was applied to solve the finite-time š»āˆž filtering problem of singular Markovian jump systems. In [6], new stability and robust stability results for 2D discrete stochastic systems were proposed based on weaker conservative assumptions. In [7], an observer was incorporated to the vaccination control rule for an SEIR propagation disease model. In [8], two linear observer prototypes for a class of linear hybrid systems were proposed based on the prediction error. However, in practice, the observation may contain the signal in a random manner, that is, the observation consists of noise alone in a nonzero probability, and it is commonly called uncertain observations or missing measurements [9, 10]. In this paper, the finite-time š»āˆž filtering problem is investigated for linear continuous time-varying systems with uncertain observations and ā„’2-norm bounded noises.

The š»2-based optimal filtering has been well studied for linear systems with uncertain observations [9ā€“13]. In [9], the recursive least-squares estimator was proposed for linear discrete-time systems with uncertain observations. The robust optimal filter for discrete time-varying systems with missing measurements and norm-bounded parameter uncertainties was designed by optimizing the upper bound of the state estimation error variance in [10]. Using the covariance information, the recursive least-squares filtering and fixed-point smoothing algorithms for linear continuous-time systems with uncertain observations were proposed in [11]. Linear and nonlinear one-step prediction algorithms for discrete-time systems with uncertain observations were presented from a covariance assignment viewpoint in [12]. The statistical convergence properties of the estimation error covariance were studied, and the existence of a critical value for the arrival rate of the observations was shown in [13]. In recent years, due to the fact that the š»āˆž-based estimation approach does not require the information on statistics of input noise, it has received more and more attention for linear systems with uncertain observations [14ā€“16]. Using Lyapunov function approach, the š»āˆž filtering algorithms in terms of linear matrix inequalities were proposed for systems with missing measurements in [14ā€“16]. To authorsā€™ best knowledge, research on finite-time š»āˆž filtering for linear continuous time-varying systems with uncertain observations has not been fully investigated and remains to be challenging, which motivates the present study.

Although the Krein space linear estimation theory [1, 3] has been applied to fault detection and nonlinear estimation [17, 18], no results have been developed for systems with uncertain observations, which will be an interesting research topic in the future. In this paper, the problem of finite-time š»āˆž filtering will be investigated for linear continuous time-varying systems with uncertain observations and ā„’2-norm bounded input noise. Based on the knowledge of Krein space linear estimation theory [1, 3], a new approach in Krein space will be developed to handle the š»āˆž filtering problem for linear continuous time-varying systems with uncertain observations. It will be shown that the š»āˆž filtering problem for linear continuous time-varying systems with uncertain observations is partially equivalent to an š»2 filtering problem for a certain Krein space state-space model. Through employing projection theory, both the existence condition and a solution of the š»āˆž filtering can be obtained in terms of a differential Riccati equation. The major contribution of this paper can be summarized as follows: (i) it shows that the š»āˆž filtering problem for systems with uncertain observations can be converted into an š»2 optimal estimation problem subject to a fictitious Krein space stochastic systems; (ii) it develops a Kalman-like robust estimator for linear continuous time-varying systems with uncertain observations.

Notation.ā€‰Elements in a Krein space will be denoted by boldface letters, and elements in the Euclidean space of complex numbers will be denoted by normal letters. The superscripts ā€œāˆ’1ā€ and ā€œāˆ—ā€ stand for the inverse and complex conjugation of a matrix, respectively. š›æ(š‘”āˆ’šœ)=0 for š‘”ā‰ šœ and š›æ(š‘”āˆ’šœ)=1 for š‘”=šœ. ā„š‘› denotes the š‘›-dimensional Euclidean space. š¼ is the identity matrix with appropriate dimensions. For a real matrix, š‘ƒ>0 (resp., š‘ƒ<0) means that š‘ƒ is symmetric and positive (resp., negative) definite. āŸØā‹…,ā‹…āŸ© denotes the inner product in Krein space. diag{ā‹Æ} denotes a block-diagonal matrix. šœƒ(š‘”)āˆˆā„’2[0,š‘‡] means āˆ«š‘‡š‘”=0šœƒāˆ—(š‘”)šœƒ(š‘”)š‘‘š‘”<āˆž. ā„’{ā‹Æ} denotes the linear space spanned by sequence {ā‹Æ}. An abstract vector space {š’¦,āŸØā‹…,ā‹…āŸ©} that satisfies the following requirements is called a Kreinspace [1].(i)š’¦ is a linear space over š’ž, the field of complex numbers.(ii) There exists a bilinear form āŸØā‹…,ā‹…āŸ©āˆˆš’ž on š’¦ such that(a)āŸØš²,š±āŸ©=āŸØš±,š²āŸ©āˆ—,(b)āŸØš‘Žš±+š‘š²,š³āŸ©=š‘ŽāŸØš±,š³āŸ©+š‘āŸØš²,š³āŸ©,ā€‰for any š±,š²,š³āˆˆš’¦, š‘Ž, š‘āˆˆš’ž, and where āˆ— denotes complex conjugation.(iii) The vector space š’¦ admits a direct orthogonal sum decomposition š’¦=š’¦+āŠ•š’¦āˆ’(1.1)ā€‰such that {š’¦+,āŸØā‹…,ā‹…āŸ©} and {š’¦āˆ’,āˆ’āŸØā‹…,ā‹…āŸ©} are Hilbert spaces, and āŸØš±,š²āŸ©=0(1.2)ā€‰for any š±āˆˆš’¦+ and š²āˆˆš¾āˆ’.

2. System Model and Problem Formulation

In this paper, we consider the following linear continuous time-varying system with uncertain observations š‘¦Ģ‡š‘„(š‘”)=š“(š‘”)š‘„(š‘”)+šµ(š‘”)š‘¤(š‘”),(š‘”)=š‘Ÿ(š‘”)š¶(š‘”)š‘„(š‘”)+š‘£(š‘”),š‘§(š‘”)=šæ(š‘”)š‘„(š‘”),š‘„(0)=š‘„0,(2.1) where š‘„(š‘”)āˆˆā„š‘› is the state vector, š‘¤(š‘”)āˆˆā„š‘ is an exogenous disturbance belonging to ā„’2[0,š‘‡], š‘¦(š‘”)āˆˆā„š‘š is the observation, š‘£(š‘”)āˆˆā„š‘š is the observation noise belonging to ā„’2[0,š‘‡], š‘§(š‘”)āˆˆā„š‘ž is the signal to be estimated, and š“(š‘”), šµ(š‘”), š¶(š‘”), and šæ(š‘”) are known real time-varying matrices with appropriate dimensions.

The stochastic variable š‘Ÿ(š‘”)āˆˆā„ takes the values of 0 and 1 with Prob{š‘Ÿ(š‘”)=1}=šøš‘Ÿ{š‘Ÿ(š‘”)}=š‘(š‘”),Prob{š‘Ÿ(š‘”)=0}=1āˆ’šøš‘Ÿšø{š‘Ÿ(š‘”)}=1āˆ’š‘(š‘”),š‘Ÿšø{š‘Ÿ(š‘”)š‘Ÿ(š‘ )}=š‘(š‘”)š‘(š‘ ),š‘”ā‰ š‘ ,š‘Ÿī€½š‘Ÿ2ī€¾(š‘”)=š‘(š‘”)(2.2) [11]. Note that many literatures associated with observer design are based on the assumption that š‘(š‘”)=1 [1ā€“4], it can be unreasonable in many practical applications [9, 10, 13]. In this paper, we assume that š‘(š‘”) is a known positive scalar.

The finite-time š»āˆž filtering problem under investigation is stated as follows: given a scalar š›¾>0, a matrix š‘ƒ0>0, and the observation {š‘¦(š‘ )|0ā‰¤š‘ ā‰¤š‘”}, find an estimate of the signal š‘§(š‘”), denoted by Ģ†š‘§(š‘”)=ā„±{š‘¦(š‘ )|0ā‰¤š‘ ā‰¤š‘”}, such that š½ā„±=šøš‘Ÿī‚»ā€–ā€–š‘„0ā€–ā€–2š‘ƒ0āˆ’1+ī€œš‘‡0ā€–š‘¤(š‘”)ā€–2ī€œš‘‘š‘”+š‘‡0ā€–š‘£(š‘”)ā€–2š‘‘š‘”āˆ’š›¾āˆ’2ī€œš‘‡0ā€–ā€–š‘’š‘“ā€–ā€–ī‚¼(š‘”)š‘‘š‘”>0,(2.3) where š‘’š‘“(š‘”)=Ģ†š‘§(š‘”)āˆ’š‘§(š‘”).

Thus, the finite-time š»āˆž filtering problem can be equivalent to the following:(I)š½ā„± has a minimum with respect to {š‘„0,š‘¤(š‘”)|0ā‰¤š‘”ā‰¤š‘‡};(II)Ģ†š‘§(š‘”) can be chosen such that the value of š½ā„± at its minimum is positive.

3. Main Results

In this section, through introducing a fictitious Krein space-state space model, we construct a partially equivalent Krein space projection problem. By using innovation analysis approach, we derive the finite-time š»āˆž filter and its existence condition.

3.1. Construct a Partially Equivalent Krein Space Problem

To begin with, we introduce the following state transition matrix: š‘‘š‘‘š‘”Ī¦(š‘”,šœ)=š“(š‘”)Ī¦(š‘”,šœ),Ī¦(šœ,šœ)=š¼,(3.1) it follows from the state-space model (2.1) that š‘¦(š‘”)=š‘Ÿ(š‘”)š¶(š‘”)Ī¦(š‘”,0)š‘„0ī€œ+š‘Ÿ(š‘”)š¶(š‘”)š‘”0Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœ+š‘£(š‘”),(3.2)Ģ†š‘§(š‘”)=šæ(š‘”)Ī¦(š‘”,0)š‘„0ī€œ+šæ(š‘”)š‘”0Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœ+š‘’š‘“(š‘”).(3.3)

Thus, we can rewrite š½ā„± as š½ā„±=šøš‘Ÿī‚»ā€–ā€–š‘„0ā€–ā€–2š‘ƒ0āˆ’1+ī€œš‘‡0ā€–š‘¤(š‘”)ā€–2ī€œš‘‘š‘”+š‘‡0ā€–š‘£(š‘”)ā€–2š‘‘š‘”āˆ’š›¾āˆ’2ī€œš‘‡0ā€–ā€–š‘’š‘“ā€–ā€–(š‘”)2ī‚¼š‘‘š‘”=š‘„āˆ—0š‘ƒ0āˆ’1š‘„0+ī€œš‘‡0š‘¤āˆ—(š‘”)š‘¤(š‘”)š‘‘š‘”+šøš‘Ÿī‚»ī€œš‘‡0ī‚µš‘¦(š‘”)āˆ’š‘Ÿ(š‘”)š¶(š‘”)Ī¦(š‘”,0)š‘„0ī€œāˆ’š‘Ÿ(š‘”)š¶(š‘”)š‘”0ī‚¶Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœāˆ—Ć—ī‚µš‘¦(š‘”)āˆ’š‘Ÿ(š‘”)š¶(š‘”)Ī¦(š‘”,0)š‘„0ī€œāˆ’š‘Ÿ(š‘”)š¶(š‘”)š‘”0ī‚¶ī‚¼Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœš‘‘š‘”āˆ’š›¾āˆ’2ī€œš‘‡0ī‚µĢ†š‘§(š‘”)āˆ’šæ(š‘”)Ī¦(š‘”,0)š‘„0ī€œāˆ’šæ(š‘”)š‘”0ī‚¶Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœāˆ—Ć—ī‚µĢ†š‘§(š‘”)āˆ’šæ(š‘”)Ī¦(š‘”,0)š‘„0ī€œāˆ’šæ(š‘”)š‘”0ī‚¶Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœš‘‘š‘”=š‘„āˆ—0š‘ƒ0āˆ’1š‘„0+ī€œš‘‡0š‘¤āˆ—+ī€œ(š‘”)š‘¤(š‘”)š‘‘š‘”š‘‡0ī‚µš‘¦0(š‘”)āˆ’š¶1(š‘”)Ī¦(š‘”,0)š‘„0āˆ’š¶1ī€œ(š‘”)š‘”0ī‚¶Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœāˆ—Ć—ī‚µš‘¦0(š‘”)āˆ’š¶1(š‘”)Ī¦(š‘”,0)š‘„0āˆ’š¶1ī€œ(š‘”)š‘”0ī‚¶+ī€œĪ¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœš‘‘š‘”š‘‡0ī‚µš‘¦š‘ (š‘”)āˆ’š¶2(š‘”)Ī¦(š‘”,0)š‘„0āˆ’š¶2ī€œ(š‘”)š‘”0ī‚¶Ī¦(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœāˆ—Ć—ī‚µš‘¦š‘ (š‘”)āˆ’š¶2(š‘”)Ī¦(š‘”,0)š‘„0āˆ’š¶2ī€œ(š‘”)š‘”0Ī¦ī‚¶(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœš‘‘š‘”āˆ’š›¾āˆ’2ī€œš‘‡0ī‚µĢ†š‘§(š‘”)āˆ’šæ(š‘”)Ī¦(š‘”,0)š‘„0ī€œāˆ’šæ(š‘”)š‘”0Ī¦ī‚¶(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœāˆ—Ć—ī‚µĢ†š‘§(š‘”)āˆ’šæ(š‘”)Ī¦(š‘”,0)š‘„0ī€œāˆ’šæ(š‘”)š‘”0Ī¦ī‚¶(š‘”,šœ)šµ(šœ)š‘¤(šœ)š‘‘šœš‘‘š‘”,(3.4) where š¶1(š‘”)=š‘(š‘”)š¶(š‘”),š¶2āˆš(š‘”)=š‘(š‘”)(1āˆ’š‘(š‘”))š¶(š‘”),š‘¦0(š‘”)=š‘¦(š‘”),š‘¦š‘ (š‘”)ā‰”0.(3.5)

Moreover, we introduce the following Krein space stochastic system Ģ‡š²š±(š‘”)=š“(š‘”)š±(š‘”)+šµ(š‘”)š°(š‘”),0(š‘”)=š¶1š²(š‘”)š±(š‘”)+šÆ(š‘”),š‘ (š‘”)=š¶2(š‘”)š±(š‘”)+šÆš‘ Ģ†(š‘”),š³(š‘”)=šæ(š‘”)š±(š‘”)+šžš‘“(š‘”),š±(0)=š±0,(3.6) where š±0, š°(š‘”), šÆ(š‘”), šÆš‘ (š‘”), and šžš‘“(š‘”) are mutually uncorrelated white noises with zero means and known covariance matrices as ī„”āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š±0šÆš°(š‘”)šÆ(š‘”)š‘ šž(š‘”)š‘“āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦,āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š±(š‘”)0šÆš°(šœ)šÆ(šœ)š‘ šž(šœ)š‘“āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦ī„•=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š‘ƒ(šœ)000000š¼š›æ(š‘”āˆ’šœ)00000š¼š›æ(š‘”āˆ’šœ)00000š¼š›æ(š‘”āˆ’šœ)00000āˆ’š›¾2āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦š¼š›æ(š‘”āˆ’šœ).(3.7)

Let š‘¦0(š‘”)=š¶1š‘¦(š‘”)š‘„(š‘”)+š‘£(š‘”),š‘ (š‘”)=š¶2(š‘”)š‘„(š‘”)+š‘£š‘ (š‘”),(3.8) then it follows from (3.1), (3.3), (3.4), and (3.7) that š½ā„±=š‘„āˆ—0āŸØš±0,š±0āŸ©āˆ’1š‘„0+ī€œš‘‡0š‘¤āˆ—(š‘”)āŸØš°(š‘”),š°(š‘”)āŸ©āˆ’1ī€œš‘¤(š‘”)š‘‘š‘”+š‘‡0š‘£āˆ—(š‘”)āŸØšÆ(š‘”),šÆ(š‘”)āŸ©āˆ’1+ī€œš‘£(š‘”)š‘‘š‘”š‘‡0š‘£āˆ—š‘ (š‘”)āŸØšÆš‘ (š‘”),šÆš‘ (š‘”)āŸ©āˆ’1š‘£š‘ ī€œ(š‘”)š‘‘š‘”+š‘‡0š‘’āˆ—š‘“ī«šž(š‘”)š‘“(š‘”),šžš‘“ī¬(š‘”)āˆ’1š‘’š‘“(š‘”)š‘‘š‘”.(3.9)

According to [1] and [3], we have the following results.

Lemma 3.1. Consider system (2.1), given a scalar š›¾>0 and a matrix š‘ƒ0>0, then š½ā„± in (2.3) has the minimum over {š‘„0,š‘¤(š‘”)|0ā‰¤š‘”ā‰¤š‘‡} if and only if the innovation Ģƒš²š‘§(š‘”) exists for 0ā‰¤š‘”ā‰¤š‘‡, where Ģƒš²š‘§(š‘”)=š²š‘§Ģ‚š²(š‘”)āˆ’š‘§(š‘”),(3.10)š²š‘§(š‘”)=[š²āˆ—0(š‘”)š²āˆ—š‘ Ģ†š³(š‘”)āˆ—(š‘”)]āˆ—, and Ģ‚š²š‘§(š‘”) denote the projection of š²š‘§(š‘”) onto ā„’{{š²š‘§(šœ)}|0ā‰¤šœ<š‘”}. In this case the minimum value of š½ā„± is minš½ā„±=ī€œš‘‡0ī€·š‘¦0(š‘”)āˆ’š¶1ī€ø(š‘”)Ģ‚š‘„(š‘”)āˆ—ī€·š‘¦0(š‘”)āˆ’š¶1ī€ø+ī€œ(š‘”)Ģ‚š‘„(š‘”)š‘‘š‘”š‘‡0ī€·š‘¦š‘ (š‘”)āˆ’š¶2ī€ø(š‘”)Ģ‚š‘„(š‘”)āˆ—ī€·š‘¦š‘ (š‘”)āˆ’š¶2ī€ø(š‘”)Ģ‚š‘„(š‘”)š‘‘š‘”āˆ’š›¾āˆ’2ī€œš‘‡0(Ģ†š‘§(š‘”)āˆ’šæ(š‘”)Ģ‚š‘„(š‘”))āˆ—(Ģ†š‘§(š‘”)āˆ’šæ(š‘”)Ģ‚š‘„(š‘”))š‘‘š‘”,(3.11) where Ģ‚š‘„(š‘”) is obtained from the Krein space projection of š±(š‘”) onto ā„’{{š²š‘§(š‘—)}|0ā‰¤šœ<š‘”}.

Remark 3.2 3.2. By analyzing the indefinite quadratic form š½ā„± in (3.4) and using the Krein space linear estimation theory [1], it has been shown that the š»āˆž filtering problem for linear systems with uncertain observations is equivalent to the š»2 estimation problem with respect to a Krein space stochastic system, which is new as far as we know. In this case, Krein space projection method can be applied to derive an š»āˆž estimator for linear systems with uncertain observations, which is more simple and intuitive than previous versions.

3.2. Solution of the Finite-Time š»āˆž Filtering Problem

By applying the standard Kalman filter formula to system (3.6), we have the following lemma.

Lemma 3.3. Consider the Krein space stochastic system (3.6), the prediction Ģ‚š±(š‘”) is calculated by Ģ‡Ģ‚š±Ģ‚š±Ģƒš²(š‘”)=š“(š‘”)(š‘”)+š¾(š‘”)š‘§(š‘”),(3.12) where Ģƒš²š‘§(š‘”)=š²š‘§Ģ‚ī€ŗš¶(š‘”)āˆ’š»(š‘”)š±(š‘”),š»(š‘”)=āˆ—1(š‘”)š¶āˆ—2(š‘”)šæāˆ—ī€»(š‘”)āˆ—,š¾(š‘”)=š‘ƒ(š‘”)š»āˆ—(š‘”)š‘…āˆ’1Ģƒš‘¦š‘§š‘…(š‘”),Ģƒš‘¦š‘§ī€½(š‘”)=diagš¼,š¼,āˆ’š›¾2š¼ī€¾,(3.13) and š‘ƒ(š‘”) is computed by Ģ‡š‘ƒ(š‘”)=š“(š‘”)š‘ƒ(š‘”)+š‘ƒ(š‘”)š“āˆ—(š‘”)+šµ(š‘”)šµāˆ—(š‘”)āˆ’š¾(š‘”)š‘…Ģƒš‘¦š‘§(š‘”)š¾āˆ—(š‘”),š‘ƒ(0)=š‘ƒ0.(3.14)

Now we are in the position to present the main results of this subsection.

Theorem 3.4 3.4. Consider system (2.1), given a scalar š›¾>0 and a matrix š‘ƒ0>0, and suppose š‘ƒ(š‘”) is the bounded positive definite solution to Riccati differential equation (3.14). Then, one possible level-š›¾ finite-time š»āˆž filter that achieves (2.3) is given by Ģ†š‘§(š‘”)=šæ(š‘”)Ģ‚š‘„(š‘”),0ā‰¤š‘”ā‰¤š‘‡,(3.15) where Ģ‡Ģ‚š‘„(š‘”)=š“(š‘”)Ģ‚š‘„(š‘”)+š‘ƒ(š‘”)š¶āˆ—š‘“ī€·š‘¦(š‘”)š‘“(š‘”)āˆ’š¶š‘“ī€ø,(š‘”)Ģ‚š‘„(š‘”)Ģ‚š‘„(0)=0(3.16) with š‘¦š‘“(š‘”)=[š‘¦āˆ—0(š‘”)š‘¦āˆ—š‘ (š‘”)]āˆ—, š¶š‘“(š‘”)=[š¶āˆ—1(š‘”)š¶āˆ—2(š‘”)]āˆ—.

Proof. It follows from Lemma 3.3 that if š‘ƒ(š‘”) is a bounded positive definite solution to Riccati differential equation (3.14), then the projection Ģ‚š±(š‘”) exists. According to Lemma 3.1, it is obvious that the š»āˆž filter that achieves (2.3) exists. If this is the case, the minimum value of š½ā„± is given by (3.11). In order to achieve minš½ā„±>0, one natural choice is to set Ģ†š‘§(š‘”)āˆ’šæ(š‘”)Ģ‚š‘„(š‘”)=0(3.17) thus the finite-time š»āˆž filter can be given by (3.15).
On the other hand, from (3.12) and (3.15), It is easy to verify that (3.16) holds.

Remark 3.5. Let š‘’(š‘”)=š‘„(š‘”)āˆ’Ģ‚š‘„(š‘”),(3.18) it follows from (2.1) and (3.16) that Ģ‡š‘’(š‘”)=(š“(š‘”)āˆ’Ī“(š‘”)š¶(š‘”))š‘’(š‘”)+šµ(š‘”)š‘¤(š‘”)āˆ’š‘ƒ(š‘”)š¶āˆ—š‘“(š‘”)š‘£š‘§(š‘”),(3.19) where Ī“(š‘”)=š‘ƒ(š‘”)š¶āˆ—š‘“āŽ”āŽ¢āŽ¢āŽ£āˆš(š‘”)š‘(š‘”)š¼āŽ¤āŽ„āŽ„āŽ¦š‘(š‘”)(1āˆ’š‘(š‘”))š¼,š‘£š‘§āŽ”āŽ¢āŽ¢āŽ£š‘£(š‘”)=š‘£(š‘”)š‘ (āŽ¤āŽ„āŽ„āŽ¦š‘”).(3.20) Unlike [14ā€“16], the parameter matrices in the filtering error equation (3.19) do not contain the stochastic variable š‘Ÿ(š‘”), which is an interesting phenomenon. As mentioned in Definition 1 in [19], it is obvious that, if (š¶(š‘”),š“(š‘”)) is detectable, then the filtering error equation (3.19) is exponentially stable. Based on the above analysis, it can be concluded that the following assumptions are necessary for the finite-time š»āˆž filter design in this paper:(i)(š¶(š‘”),š“(š‘”)) is detectable,(ii)š‘¤(š‘”),š‘£(š‘”)āˆˆā„’2[0,š‘‡].

4. A Numerical Example

We consider system (2.1) with the following parameters: āŽ”āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ¦[][]š“(š‘”)=āˆ’1062āˆ’5,šµ(š‘”)=2.81.6,š¶(š‘”)=189.5,šæ(š‘”)=11(4.1) and set š›¾=1.1, š‘„(0)=[00]āˆ—, š‘(š‘”)=0.8, and š‘ƒ0=š¼. In addition, we suppose that the noises š‘¤(š‘”) and š‘£(š‘”) are generated by Gaussian with zero means and covariances š‘„š‘¤=1, š‘„š‘£=0.02, the sampling time is 0.02ā€‰s, and the stochastic variable š‘Ÿ(š‘”) is simulated as in Figure 1. Based on Theorem 3.4, we design the finite-time š»āˆž filter. Figure 2 shows the true value of signal š‘§(š‘”) and its š»āˆž filtering estimate, and Figure 3 shows the estimation error Ģƒš‘§(š‘”)=š‘§(š‘”)āˆ’Ģ†š‘§(š‘”). It can be observed from the simulation results that the finite-time š»āˆž filter has good tracking performance.

5. Conclusions

In this paper, we have proposed a new finite-time š»āˆž filtering technique for linear continuous time-varying systems with uncertain observations. By introducing a Krein state-space model, it is shown that the š»āˆž filtering problem can be partially equivalent to a Krein space š»2 filtering problem. A sufficient condition for the existence of the finite-time š»āˆž filter is given, and the filter is derived in terms of a differential Riccati equation.

Future research work will extend the proposed method to investigate the š»āˆž multistep prediction and fixed-lag smoothing problem for linear continuous time-varying systems with uncertain observations.


The authors sincerely thank the anonymous reviewers for providing valuable comments and useful suggestions aimed at improving the quality of this paper. The authors also thank the editor for the efficient and professional processing of their paper. This work is supported by the National Natural Science Foundation of China (60774004, 61034007, and 60874016) and the Independent Innovation Foundation of Shandong University, China (2010JC003).