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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 710904, 11 pages
http://dx.doi.org/10.1155/2012/710904
Research Article

Finite-Time 𝐻∞ Filtering for Linear Continuous Time-Varying Systems with Uncertain Observations

1Department of Electromechanical Engineering, Dezhou University, No. 566 University Road West, Dezhou 253023, China
2School of Control Science and Engineering, Shandong University, 17923 Jingshi Road, Jinan 250061, China

Received 29 March 2012; Revised 3 June 2012; Accepted 6 June 2012

Academic Editor: QiangΒ Ling

Copyright Β© 2012 Huihong Zhao and Chenghui Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the finite-time 𝐻∞ filtering problem for linear continuous time-varying systems with uncertain observations and β„’2-norm bounded noise. The design of finite-time 𝐻∞ 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 𝐻∞ filtering problem is solved. A numerical example is given to illustrate the performance of the 𝐻∞ filter.

1. Introduction

Most of the literatures on estimation problem always assume the observations contain the signal to be estimated [1–8]. In [5], the linear matrix inequality technique was applied to solve the finite-time 𝐻∞ 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 𝐻∞ filtering problem is investigated for linear continuous time-varying systems with uncertain observations and β„’2-norm bounded noises.

The 𝐻2-based optimal filtering has been well studied for linear systems with uncertain observations [9–13]. 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 𝐻∞-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 [14–16]. Using Lyapunov function approach, the 𝐻∞ filtering algorithms in terms of linear matrix inequalities were proposed for systems with missing measurements in [14–16]. To authors’ best knowledge, research on finite-time 𝐻∞ 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 𝐻∞ filtering will be investigated for linear continuous time-varying systems with uncertain observations and β„’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 𝐻∞ filtering problem for linear continuous time-varying systems with uncertain observations. It will be shown that the 𝐻∞ filtering problem for linear continuous time-varying systems with uncertain observations is partially equivalent to an 𝐻2 filtering problem for a certain Krein space state-space model. Through employing projection theory, both the existence condition and a solution of the 𝐻∞ 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 𝐻∞ filtering problem for systems with uncertain observations can be converted into an 𝐻2 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. 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 β€œβˆ’1” and β€œβˆ—β€ stand for the inverse and complex conjugation of a matrix, respectively. 𝛿(π‘‘βˆ’πœ)=0 for π‘‘β‰ πœ and 𝛿(π‘‘βˆ’πœ)=1 for 𝑑=𝜏. ℝ𝑛 denotes the 𝑛-dimensional Euclidean space. 𝐼 is the identity matrix with appropriate dimensions. For a real matrix, 𝑃>0 (resp., 𝑃<0) means that 𝑃 is symmetric and positive (resp., negative) definite. βŸ¨β‹…,β‹…βŸ© denotes the inner product in Krein space. diag{β‹―} denotes a block-diagonal matrix. πœƒ(𝑑)βˆˆβ„’2[0,𝑇] means βˆ«π‘‡π‘‘=0πœƒβˆ—(𝑑)πœƒ(𝑑)𝑑𝑑<∞. β„’{β‹―} denotes the linear space spanned by sequence {β‹―}. An abstract vector space {𝒦,βŸ¨β‹…,β‹…βŸ©} that satisfies the following requirements is called a Kreinspace [1].(i)𝒦 is a linear space over π’ž, the field of complex numbers.(ii) There exists a bilinear form βŸ¨β‹…,β‹…βŸ©βˆˆπ’ž on 𝒦 such that(a)⟨𝐲,𝐱⟩=⟨𝐱,π²βŸ©βˆ—,(b)βŸ¨π‘Žπ±+𝑏𝐲,𝐳⟩=π‘ŽβŸ¨π±,𝐳⟩+π‘βŸ¨π²,𝐳⟩, for any 𝐱,𝐲,π³βˆˆπ’¦, π‘Ž, π‘βˆˆπ’ž, and where βˆ— denotes complex conjugation.(iii) The vector space 𝒦 admits a direct orthogonal sum decomposition 𝒦=𝒦+βŠ•π’¦βˆ’(1.1) such that {𝒦+,βŸ¨β‹…,β‹…βŸ©} and {π’¦βˆ’,βˆ’βŸ¨β‹…,β‹…βŸ©} are Hilbert spaces, and ⟨𝐱,𝐲⟩=0(1.2) for any π±βˆˆπ’¦+ and π²βˆˆπΎβˆ’.

2. System Model and Problem Formulation

In this paper, we consider the following linear continuous time-varying system with uncertain observations 𝑦̇π‘₯(𝑑)=𝐴(𝑑)π‘₯(𝑑)+𝐡(𝑑)𝑀(𝑑),(𝑑)=π‘Ÿ(𝑑)𝐢(𝑑)π‘₯(𝑑)+𝑣(𝑑),𝑧(𝑑)=𝐿(𝑑)π‘₯(𝑑),π‘₯(0)=π‘₯0,(2.1) where π‘₯(𝑑)βˆˆβ„π‘› is the state vector, 𝑀(𝑑)βˆˆβ„π‘ is an exogenous disturbance belonging to β„’2[0,𝑇], 𝑦(𝑑)βˆˆβ„π‘š is the observation, 𝑣(𝑑)βˆˆβ„π‘š is the observation noise belonging to β„’2[0,𝑇], 𝑧(𝑑)βˆˆβ„π‘ž is the signal to be estimated, and 𝐴(𝑑), 𝐡(𝑑), 𝐢(𝑑), and 𝐿(𝑑) are known real time-varying matrices with appropriate dimensions.

The stochastic variable π‘Ÿ(𝑑)βˆˆβ„ takes the values of 0 and 1 with Prob{π‘Ÿ(𝑑)=1}=πΈπ‘Ÿ{π‘Ÿ(𝑑)}=𝑝(𝑑),Prob{π‘Ÿ(𝑑)=0}=1βˆ’πΈπ‘ŸπΈ{π‘Ÿ(𝑑)}=1βˆ’π‘(𝑑),π‘ŸπΈ{π‘Ÿ(𝑑)π‘Ÿ(𝑠)}=𝑝(𝑑)𝑝(𝑠),𝑑≠𝑠,π‘Ÿξ€½π‘Ÿ2ξ€Ύ(𝑑)=𝑝(𝑑)(2.2) [11]. Note that many literatures associated with observer design are based on the assumption that 𝑝(𝑑)=1 [1–4], it can be unreasonable in many practical applications [9, 10, 13]. In this paper, we assume that 𝑝(𝑑) is a known positive scalar.

The finite-time 𝐻∞ filtering problem under investigation is stated as follows: given a scalar 𝛾>0, a matrix 𝑃0>0, and the observation {𝑦(𝑠)|0≀𝑠≀𝑑}, find an estimate of the signal 𝑧(𝑑), denoted by ̆𝑧(𝑑)=β„±{𝑦(𝑠)|0≀𝑠≀𝑑}, such that 𝐽ℱ=πΈπ‘Ÿξ‚»β€–β€–π‘₯0β€–β€–2𝑃0βˆ’1+ξ€œπ‘‡0‖𝑀(𝑑)β€–2ξ€œπ‘‘π‘‘+𝑇0‖𝑣(𝑑)β€–2π‘‘π‘‘βˆ’π›Ύβˆ’2ξ€œπ‘‡0‖‖𝑒𝑓‖‖(𝑑)𝑑𝑑>0,(2.3) where 𝑒𝑓(𝑑)=̆𝑧(𝑑)βˆ’π‘§(𝑑).

Thus, the finite-time 𝐻∞ filtering problem can be equivalent to the following:(I)𝐽ℱ has a minimum with respect to {π‘₯0,𝑀(𝑑)|0≀𝑑≀𝑇};(II)̆𝑧(𝑑) can be chosen such that the value of 𝐽ℱ 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 𝐻∞ filter and its existence condition.

3.1. Construct a Partially Equivalent Krein Space Problem

To begin with, we introduce the following state transition matrix: 𝑑𝑑𝑑Φ(𝑑,𝜏)=𝐴(𝑑)Ξ¦(𝑑,𝜏),Ξ¦(𝜏,𝜏)=𝐼,(3.1) it follows from the state-space model (2.1) that 𝑦(𝑑)=π‘Ÿ(𝑑)𝐢(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œ+π‘Ÿ(𝑑)𝐢(𝑑)𝑑0Ξ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœ+𝑣(𝑑),(3.2)̆𝑧(𝑑)=𝐿(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œ+𝐿(𝑑)𝑑0Ξ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœ+𝑒𝑓(𝑑).(3.3)

Thus, we can rewrite 𝐽ℱ as 𝐽ℱ=πΈπ‘Ÿξ‚»β€–β€–π‘₯0β€–β€–2𝑃0βˆ’1+ξ€œπ‘‡0‖𝑀(𝑑)β€–2ξ€œπ‘‘π‘‘+𝑇0‖𝑣(𝑑)β€–2π‘‘π‘‘βˆ’π›Ύβˆ’2ξ€œπ‘‡0‖‖𝑒𝑓‖‖(𝑑)2𝑑𝑑=π‘₯βˆ—0𝑃0βˆ’1π‘₯0+ξ€œπ‘‡0π‘€βˆ—(𝑑)𝑀(𝑑)𝑑𝑑+πΈπ‘Ÿξ‚»ξ€œπ‘‡0𝑦(𝑑)βˆ’π‘Ÿ(𝑑)𝐢(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œβˆ’π‘Ÿ(𝑑)𝐢(𝑑)𝑑0ξ‚ΆΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœβˆ—Γ—ξ‚΅π‘¦(𝑑)βˆ’π‘Ÿ(𝑑)𝐢(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œβˆ’π‘Ÿ(𝑑)𝐢(𝑑)𝑑0ξ‚Άξ‚ΌΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœπ‘‘π‘‘βˆ’π›Ύβˆ’2ξ€œπ‘‡0̆𝑧(𝑑)βˆ’πΏ(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œβˆ’πΏ(𝑑)𝑑0ξ‚ΆΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœβˆ—Γ—ξ‚΅Μ†π‘§(𝑑)βˆ’πΏ(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œβˆ’πΏ(𝑑)𝑑0ξ‚ΆΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœπ‘‘π‘‘=π‘₯βˆ—0𝑃0βˆ’1π‘₯0+ξ€œπ‘‡0π‘€βˆ—+ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑𝑇0𝑦0(𝑑)βˆ’πΆ1(𝑑)Ξ¦(𝑑,0)π‘₯0βˆ’πΆ1ξ€œ(𝑑)𝑑0ξ‚ΆΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœβˆ—Γ—ξ‚΅π‘¦0(𝑑)βˆ’πΆ1(𝑑)Ξ¦(𝑑,0)π‘₯0βˆ’πΆ1ξ€œ(𝑑)𝑑0ξ‚Ά+ξ€œΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœπ‘‘π‘‘π‘‡0𝑦𝑠(𝑑)βˆ’πΆ2(𝑑)Ξ¦(𝑑,0)π‘₯0βˆ’πΆ2ξ€œ(𝑑)𝑑0ξ‚ΆΞ¦(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœβˆ—Γ—ξ‚΅π‘¦π‘ (𝑑)βˆ’πΆ2(𝑑)Ξ¦(𝑑,0)π‘₯0βˆ’πΆ2ξ€œ(𝑑)𝑑0Ξ¦ξ‚Ά(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœπ‘‘π‘‘βˆ’π›Ύβˆ’2ξ€œπ‘‡0̆𝑧(𝑑)βˆ’πΏ(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œβˆ’πΏ(𝑑)𝑑0Ξ¦ξ‚Ά(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœβˆ—Γ—ξ‚΅Μ†π‘§(𝑑)βˆ’πΏ(𝑑)Ξ¦(𝑑,0)π‘₯0ξ€œβˆ’πΏ(𝑑)𝑑0Ξ¦ξ‚Ά(𝑑,𝜏)𝐡(𝜏)𝑀(𝜏)π‘‘πœπ‘‘π‘‘,(3.4) where 𝐢1(𝑑)=𝑝(𝑑)𝐢(𝑑),𝐢2√(𝑑)=𝑝(𝑑)(1βˆ’π‘(𝑑))𝐢(𝑑),𝑦0(𝑑)=𝑦(𝑑),𝑦𝑠(𝑑)≑0.(3.5)

Moreover, we introduce the following Krein space stochastic system ̇𝐲𝐱(𝑑)=𝐴(𝑑)𝐱(𝑑)+𝐡(𝑑)𝐰(𝑑),0(𝑑)=𝐢1𝐲(𝑑)𝐱(𝑑)+𝐯(𝑑),𝑠(𝑑)=𝐢2(𝑑)𝐱(𝑑)+𝐯𝑠̆(𝑑),𝐳(𝑑)=𝐿(𝑑)𝐱(𝑑)+πžπ‘“(𝑑),𝐱(0)=𝐱0,(3.6) where 𝐱0, 𝐰(𝑑), 𝐯(𝑑), 𝐯𝑠(𝑑), and πžπ‘“(𝑑) are mutually uncorrelated white noises with zero means and known covariance matrices as ξ„”βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π±0𝐯𝐰(𝑑)𝐯(𝑑)π‘ πž(𝑑)π‘“βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎒⎒⎒⎒⎣𝐱(𝑑)0𝐯𝐰(𝜏)𝐯(𝜏)π‘ πž(𝜏)π‘“βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ„•=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘ƒ(𝜏)000000𝐼𝛿(π‘‘βˆ’πœ)00000𝐼𝛿(π‘‘βˆ’πœ)00000𝐼𝛿(π‘‘βˆ’πœ)00000βˆ’π›Ύ2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦πΌπ›Ώ(π‘‘βˆ’πœ).(3.7)

Let 𝑦0(𝑑)=𝐢1𝑦(𝑑)π‘₯(𝑑)+𝑣(𝑑),𝑠(𝑑)=𝐢2(𝑑)π‘₯(𝑑)+𝑣𝑠(𝑑),(3.8) then it follows from (3.1), (3.3), (3.4), and (3.7) that 𝐽ℱ=π‘₯βˆ—0⟨𝐱0,𝐱0βŸ©βˆ’1π‘₯0+ξ€œπ‘‡0π‘€βˆ—(𝑑)⟨𝐰(𝑑),𝐰(𝑑)βŸ©βˆ’1ξ€œπ‘€(𝑑)𝑑𝑑+𝑇0π‘£βˆ—(𝑑)⟨𝐯(𝑑),𝐯(𝑑)βŸ©βˆ’1+ξ€œπ‘£(𝑑)𝑑𝑑𝑇0π‘£βˆ—π‘ (𝑑)βŸ¨π―π‘ (𝑑),𝐯𝑠(𝑑)βŸ©βˆ’1π‘£π‘ ξ€œ(𝑑)𝑑𝑑+𝑇0π‘’βˆ—π‘“ξ«πž(𝑑)𝑓(𝑑),πžπ‘“ξ¬(𝑑)βˆ’1𝑒𝑓(𝑑)𝑑𝑑.(3.9)

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

Lemma 3.1. Consider system (2.1), given a scalar 𝛾>0 and a matrix 𝑃0>0, then 𝐽ℱ in (2.3) has the minimum over {π‘₯0,𝑀(𝑑)|0≀𝑑≀𝑇} if and only if the innovation ̃𝐲𝑧(𝑑) exists for 0≀𝑑≀𝑇, where ̃𝐲𝑧(𝑑)=𝐲𝑧̂𝐲(𝑑)βˆ’π‘§(𝑑),(3.10)𝐲𝑧(𝑑)=[π²βˆ—0(𝑑)π²βˆ—π‘ Μ†π³(𝑑)βˆ—(𝑑)]βˆ—, and ̂𝐲𝑧(𝑑) denote the projection of 𝐲𝑧(𝑑) onto β„’{{𝐲𝑧(𝜏)}|0β‰€πœ<𝑑}. In this case the minimum value of 𝐽ℱ is min𝐽ℱ=ξ€œπ‘‡0𝑦0(𝑑)βˆ’πΆ1ξ€Έ(𝑑)Μ‚π‘₯(𝑑)βˆ—ξ€·π‘¦0(𝑑)βˆ’πΆ1ξ€Έ+ξ€œ(𝑑)Μ‚π‘₯(𝑑)𝑑𝑑𝑇0𝑦𝑠(𝑑)βˆ’πΆ2ξ€Έ(𝑑)Μ‚π‘₯(𝑑)βˆ—ξ€·π‘¦π‘ (𝑑)βˆ’πΆ2ξ€Έ(𝑑)Μ‚π‘₯(𝑑)π‘‘π‘‘βˆ’π›Ύβˆ’2ξ€œπ‘‡0(̆𝑧(𝑑)βˆ’πΏ(𝑑)Μ‚π‘₯(𝑑))βˆ—(̆𝑧(𝑑)βˆ’πΏ(𝑑)Μ‚π‘₯(𝑑))𝑑𝑑,(3.11) where Μ‚π‘₯(𝑑) is obtained from the Krein space projection of 𝐱(𝑑) onto β„’{{𝐲𝑧(𝑗)}|0β‰€πœ<𝑑}.

Remark 3.2 3.2. By analyzing the indefinite quadratic form 𝐽ℱ in (3.4) and using the Krein space linear estimation theory [1], it has been shown that the 𝐻∞ filtering problem for linear systems with uncertain observations is equivalent to the 𝐻2 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 𝐻∞ estimator for linear systems with uncertain observations, which is more simple and intuitive than previous versions.

3.2. Solution of the Finite-Time 𝐻∞ 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 ̂𝐱(𝑑) is calculated by ̇̂𝐱̂𝐱̃𝐲(𝑑)=𝐴(𝑑)(𝑑)+𝐾(𝑑)𝑧(𝑑),(3.12) where ̃𝐲𝑧(𝑑)=𝐲𝑧̂𝐢(𝑑)βˆ’π»(𝑑)𝐱(𝑑),𝐻(𝑑)=βˆ—1(𝑑)πΆβˆ—2(𝑑)πΏβˆ—ξ€»(𝑑)βˆ—,𝐾(𝑑)=𝑃(𝑑)π»βˆ—(𝑑)π‘…βˆ’1̃𝑦𝑧𝑅(𝑑),̃𝑦𝑧(𝑑)=diag𝐼,𝐼,βˆ’π›Ύ2𝐼,(3.13) and 𝑃(𝑑) is computed by ̇𝑃(𝑑)=𝐴(𝑑)𝑃(𝑑)+𝑃(𝑑)π΄βˆ—(𝑑)+𝐡(𝑑)π΅βˆ—(𝑑)βˆ’πΎ(𝑑)𝑅̃𝑦𝑧(𝑑)πΎβˆ—(𝑑),𝑃(0)=𝑃0.(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 𝛾>0 and a matrix 𝑃0>0, and suppose 𝑃(𝑑) is the bounded positive definite solution to Riccati differential equation (3.14). Then, one possible level-𝛾 finite-time 𝐻∞ filter that achieves (2.3) is given by ̆𝑧(𝑑)=𝐿(𝑑)Μ‚π‘₯(𝑑),0≀𝑑≀𝑇,(3.15) where Μ‡Μ‚π‘₯(𝑑)=𝐴(𝑑)Μ‚π‘₯(𝑑)+𝑃(𝑑)πΆβˆ—π‘“ξ€·π‘¦(𝑑)𝑓(𝑑)βˆ’πΆπ‘“ξ€Έ,(𝑑)Μ‚π‘₯(𝑑)Μ‚π‘₯(0)=0(3.16) with 𝑦𝑓(𝑑)=[π‘¦βˆ—0(𝑑)π‘¦βˆ—π‘ (𝑑)]βˆ—, 𝐢𝑓(𝑑)=[πΆβˆ—1(𝑑)πΆβˆ—2(𝑑)]βˆ—.

Proof. It follows from Lemma 3.3 that if 𝑃(𝑑) is a bounded positive definite solution to Riccati differential equation (3.14), then the projection ̂𝐱(𝑑) exists. According to Lemma 3.1, it is obvious that the 𝐻∞ filter that achieves (2.3) exists. If this is the case, the minimum value of 𝐽ℱ is given by (3.11). In order to achieve min𝐽ℱ>0, one natural choice is to set ̆𝑧(𝑑)βˆ’πΏ(𝑑)Μ‚π‘₯(𝑑)=0(3.17) thus the finite-time 𝐻∞ 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 𝑒(𝑑)=π‘₯(𝑑)βˆ’Μ‚π‘₯(𝑑),(3.18) it follows from (2.1) and (3.16) that ̇𝑒(𝑑)=(𝐴(𝑑)βˆ’Ξ“(𝑑)𝐢(𝑑))𝑒(𝑑)+𝐡(𝑑)𝑀(𝑑)βˆ’π‘ƒ(𝑑)πΆβˆ—π‘“(𝑑)𝑣𝑧(𝑑),(3.19) where Ξ“(𝑑)=𝑃(𝑑)πΆβˆ—π‘“βŽ‘βŽ’βŽ’βŽ£βˆš(𝑑)𝑝(𝑑)𝐼⎀βŽ₯βŽ₯βŽ¦π‘(𝑑)(1βˆ’π‘(𝑑))𝐼,π‘£π‘§βŽ‘βŽ’βŽ’βŽ£π‘£(𝑑)=𝑣(𝑑)𝑠(⎀βŽ₯βŽ₯βŽ¦π‘‘).(3.20) Unlike [14–16], the parameter matrices in the filtering error equation (3.19) do not contain the stochastic variable π‘Ÿ(𝑑), which is an interesting phenomenon. As mentioned in Definition 1 in [19], it is obvious that, if (𝐢(𝑑),𝐴(𝑑)) 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 𝐻∞ filter design in this paper:(i)(𝐢(𝑑),𝐴(𝑑)) is detectable,(ii)𝑀(𝑑),𝑣(𝑑)βˆˆβ„’2[0,𝑇].

4. A Numerical Example

We consider system (2.1) with the following parameters: ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦[][]𝐴(𝑑)=βˆ’1062βˆ’5,𝐡(𝑑)=2.81.6,𝐢(𝑑)=189.5,𝐿(𝑑)=11(4.1) and set 𝛾=1.1, π‘₯(0)=[00]βˆ—, 𝑝(𝑑)=0.8, and 𝑃0=𝐼. In addition, we suppose that the noises 𝑀(𝑑) and 𝑣(𝑑) are generated by Gaussian with zero means and covariances 𝑄𝑀=1, 𝑄𝑣=0.02, the sampling time is 0.02 s, and the stochastic variable π‘Ÿ(𝑑) is simulated as in Figure 1. Based on Theorem 3.4, we design the finite-time 𝐻∞ filter. Figure 2 shows the true value of signal 𝑧(𝑑) and its 𝐻∞ filtering estimate, and Figure 3 shows the estimation error ̃𝑧(𝑑)=𝑧(𝑑)βˆ’Μ†π‘§(𝑑). It can be observed from the simulation results that the finite-time 𝐻∞ filter has good tracking performance.

710904.fig.001
Figure 1: Stochastic variable π‘Ÿ(𝑑).
710904.fig.002
Figure 2: True value of signal 𝑧(𝑑) (solid line) and its 𝐻∞ filtering estimate (dashed line).
710904.fig.003
Figure 3: Estimation error ̃𝑧(𝑑).

5. Conclusions

In this paper, we have proposed a new finite-time 𝐻∞ filtering technique for linear continuous time-varying systems with uncertain observations. By introducing a Krein state-space model, it is shown that the 𝐻∞ filtering problem can be partially equivalent to a Krein space 𝐻2 filtering problem. A sufficient condition for the existence of the finite-time 𝐻∞ 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 𝐻∞ multistep prediction and fixed-lag smoothing problem for linear continuous time-varying systems with uncertain observations.

Acknowledgments

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).

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