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

Abstract

This paper is concerned with the finite-time 𝐻_∞ filtering problem for linear continuous time-varying systems with uncertain observations and ℒ₂-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 ℒ₂-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 ℒ₂-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 “−1” and “∗” stand for the inverse and complex conjugation of a matrix, respectively. 𝛿(𝑡−𝜏)=0 for 𝑡≠𝜏 and 𝛿(𝑡−𝜏)=1 for 𝑡=𝜏. ℝ^𝑛 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. ⟨⋅,⋅⟩ denotes the inner product in Krein space. diag{⋯} denotes a block-diagonal matrix. 𝜃(𝑡)∈ℒ₂[0,𝑇] means ∫^𝑇𝑡=0𝜃^∗(𝑡)𝜃(𝑡)𝑑𝑡<∞. ℒ{⋯} denotes the linear space spanned by sequence {⋯}. An abstract vector space {𝒦,⟨⋅,⋅⟩} that satisfies the following requirements is called a Kreinspace [1].(i)𝒦 is a linear space over 𝒞, the field of complex numbers.(ii) There exists a bilinear form ⟨⋅,⋅⟩∈𝒞 on 𝒦 such that(a)⟨𝐲,𝐱⟩=⟨𝐱,𝐲⟩^∗,(b)⟨𝑎𝐱+𝑏𝐲,𝐳⟩=𝑎⟨𝐱,𝐳⟩+𝑏⟨𝐲,𝐳⟩, for any 𝐱,𝐲,𝐳∈𝒦, 𝑎, 𝑏∈𝒞, and where ∗ denotes complex conjugation.(iii) The vector space 𝒦 admits a direct orthogonal sum decomposition 𝒦=𝒦₊⊕𝒦₋(1.1) such that {𝒦₊,⟨⋅,⋅⟩} and {𝒦₋,−⟨⋅,⋅⟩} are Hilbert spaces, and ⟨𝐱,𝐲⟩=0(1.2) for any 𝐱∈𝒦₊ and 𝐲∈𝐾₋.

2. System Model and Problem Formulation

In this paper, we consider the following linear continuous time-varying system with uncertain observations 𝑥𝑦̇((𝑡𝑡))==𝑧𝐴𝑟(𝑡(()𝑥𝑡𝑡))=(𝐶𝑥0𝐿)((𝑡=𝑡)()𝑡+𝑥)𝑥𝑥₀(𝐵𝑡()𝑡()+𝑡),𝑣𝑤((𝑡𝑡)),,,(2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑤(𝑡)∈ℝ^𝑝 is an exogenous disturbance belonging to ℒ₂[0,𝑇], 𝑦(𝑡)∈ℝ^𝑚 is the observation, 𝑣(𝑡)∈ℝ^𝑚 is the observation noise belonging to ℒ₂[0,𝑇], 𝑧(𝑡)∈ℝ^𝑞 is the signal to be estimated, and 𝐴(𝑡), 𝐵(𝑡), 𝐶(𝑡), and 𝐿(𝑡) are known real time-varying matrices with appropriate dimensions.

The stochastic variable 𝑟(𝑡)∈ℝ takes the values of 0 and 1 with ProbP{ro𝑟b(𝑡{)𝑟=(𝑡0)}==1}1=−𝐸𝐸_𝑟𝑟{𝑟(𝑡)}=𝑝(𝑡),𝐸{𝑟(𝑡)}=1−𝑝(𝑡),_𝑟{𝑟(𝑡)𝐸𝑟(𝑠)}=𝑝(𝑡)𝑝(𝑠),𝑡≠𝑠,_𝑟𝑟²(𝑡)=𝑝(𝑡)(2.2) [11]. Note that many literatures associated with observer design are based on the assumption that 𝑝(𝑡)=1 [1–4], it can be unreasonable in many practical applications [9, 10, 13]. In this paper, we assume that 𝑝(𝑡) 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≤𝑡}}, find an estimate of the signal 𝑧(𝑡), denoted by 𝑧(̆𝑡)=ℱ{𝑦(𝑠)|_{0≤𝑠≤𝑡}}, such that 𝐽_ℱ=𝐸_𝑟‖‖𝑥⁰‖‖^2𝑃₀₋₁+^𝑇0‖𝑤(𝑡)‖²𝑑𝑡+^𝑇0‖𝑣(𝑡)‖²𝑑𝑡−𝛾⁻²^𝑇0‖‖𝑒_𝑓(𝑡)‖‖𝑑𝑡>0,(2.3) where 𝑒_𝑓(𝑡)=𝑧(̆𝑡)−𝑧(𝑡).

Thus, the finite-time 𝐻_∞ filtering problem can be equivalent to the following:(I)𝐽_ℱ has a minimum with respect to {𝑥₀,𝑤(𝑡)|_{0≤𝑡≤𝑇}};(II)𝑧(̆𝑡) can be chosen such that the value of 𝐽_ℱ 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: 𝑑𝑑𝑡Φ(𝑡,𝜏)=𝐴(𝑡)Φ(𝑡,𝜏),Φ(𝜏,𝜏)=𝐼,(3.1) it follows from the state-space model (2.1) that 𝑦(𝑡)𝑧=̆(𝑡)𝑟(=𝑡)𝐿𝐶((𝑡𝑡))ΦΦ((𝑡𝑡,,00))𝑥𝑥₀₀+𝑟(𝑡)𝐶(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏+𝑣(𝑡),(3.2)+𝐿(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏+𝑒_𝑓(𝑡).(3.3)

Thus, we can rewrite 𝐽_ℱ as 𝐽_ℱ=𝐸_𝑟‖‖𝑥⁰‖‖^2𝑃₀₋₁+^𝑇0‖𝑤(𝑡)‖²𝑑𝑡+^𝑇0‖𝑣(𝑡)‖²𝑑𝑡−𝛾⁻²^𝑇0=𝑥_∗0‖‖𝑒_𝑓(𝑡)‖‖²𝑑𝑡+𝑃𝐸⁰⁻¹_𝑟𝑥₀+^𝑇0𝑤^∗(𝑡)𝑤(𝑡)𝑑𝑡^𝑇0𝑦(𝑡)−𝑟(𝑡)𝐶(𝑡)Φ(𝑡,0)𝑥⁰−𝑟(𝑡)𝐶(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏^∗−𝛾₋₂×𝑦(𝑡)−𝑟(𝑡)𝐶(𝑡)Φ(𝑡,0)𝑥₀−𝑟(𝑡)𝐶(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏𝑑𝑡^𝑇0𝑧̆(𝑡)−𝐿(𝑡)Φ(𝑡,0)𝑥⁰−𝐿(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏^∗=𝑥×_∗0𝑧̆(𝑡)−𝐿(𝑡)Φ(𝑡,0)𝑥₀−𝐿(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏𝑑𝑡+𝑃⁰⁻¹𝑥₀+^𝑇0𝑤^∗(𝑡)𝑤(𝑡)𝑑𝑡_𝑇0𝑦⁰(𝑡)−𝐶₁(𝑡)Φ(𝑡,0)𝑥₀−𝐶₁(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏^∗+×𝑦₀(𝑡)−𝐶₁(𝑡)Φ(𝑡,0)𝑥₀−𝐶₁(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏𝑑𝑡_𝑇0𝑦^𝑠(𝑡)−𝐶₂(𝑡)Φ(𝑡,0)𝑥₀−𝐶₂(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏^∗−×𝛾𝑦_𝑠−2(𝑡)−𝐶₂(𝑡)Φ(𝑡,0)𝑥₀−𝐶₂(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏𝑑𝑡^𝑇0𝑧̆(𝑡)−𝐿(𝑡)Φ(𝑡,0)𝑥⁰−𝐿(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏^∗×𝑧̆(𝑡)−𝐿(𝑡)Φ(𝑡,0)𝑥₀−𝐿(𝑡)^𝑡0Φ(𝑡,𝜏)𝐵(𝜏)𝑤(𝜏)𝑑𝜏𝑑𝑡,(3.4) where 𝐶₁(𝑡)=𝑝(𝑡)𝐶(𝑡),𝐶₂(𝑡)=√𝑝(𝑡)(1−𝑝(𝑡))𝐶(𝑡),𝑦₀(𝑡)=𝑦(𝑡),𝑦_𝑠(𝑡)≡0.(3.5)

Moreover, we introduce the following Krein space stochastic system 𝐱̇(𝐲𝑡)=𝐴(𝑡)𝐱(𝑡)+𝐵(𝑡)𝐰(𝑡),₀𝐲(𝑡)=𝐶₁(𝑡)𝐱(𝑡)+𝐯(𝑡),_𝑠𝐳̆((𝑡𝑡))==𝐿𝐶₂(𝑡()𝑡)𝐱𝐱(𝑡()𝑡)++𝐞𝐯_𝑠𝑓(𝑡),𝐱(0)=𝐱₀(𝑡),,(3.6) where 𝐱₀, 𝐰(𝑡), 𝐯(𝑡), 𝐯_𝑠(𝑡), and 𝐞_𝑓(𝑡) are mutually uncorrelated white noises with zero means and known covariance matrices as ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐱⁰𝐯𝐰𝐯((𝑡𝑡)_𝑠)𝐞(𝑡)_𝑓(𝑡)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐱⁰𝐯𝐰𝐯((𝜏𝜏)_𝑠)𝐞(𝜏)_𝑓(𝜏)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑃⁰0000𝐼𝛿(𝑡0000−𝜏)𝐼𝛿(𝑡0000−𝜏)𝐼𝛿(𝑡0000−𝜏)−𝛾₂0000𝐼𝛿(𝑡−𝜏)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(3.7)

Let 𝑦𝑦₀(𝑡)=𝐶₁(𝑡)𝑥(𝑡)+𝑣(𝑡),_𝑠(𝑡)=𝐶₂(𝑡)𝑥(𝑡)+𝑣_𝑠(𝑡),(3.8) then it follows from (3.1), (3.3), (3.4), and (3.7) that 𝐽_ℱ=𝑥+^∗0⟨𝐱₀,𝐱₀⟩⁻¹𝑥₀+^𝑇0𝑤^∗(𝑡)⟨𝐰(𝑡),𝐰(𝑡)⟩⁻¹𝑤(𝑡)𝑑𝑡+^𝑇0𝑣^∗(𝑡)⟨𝐯(𝑡),𝐯(𝑡)⟩⁻¹𝑣(𝑡)𝑑𝑡_𝑇0𝑣^∗𝑠(𝑡)⟨𝐯_𝑠(𝑡),𝐯_𝑠(𝑡)⟩⁻¹𝑣_𝑠(𝑡)𝑑𝑡+^𝑇0𝑒^∗𝑓(𝑡)𝐞_𝑓(𝑡),𝐞_𝑓(𝑡)⁻¹𝑒_𝑓(𝑡)𝑑𝑡.(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 𝐽_ℱ in (2.3) has the minimum over {𝑥₀,𝑤(𝑡)|_{0≤𝑡≤𝑇}} if and only if the innovation 𝐲̃_𝑧(𝑡) exists for 0≤𝑡≤𝑇, where 𝐲̃_𝑧(𝑡)=𝐲_𝑧(𝑡)−𝐲̂_𝑧(𝑡),(3.10)𝐲_𝑧(𝑡)=[𝐲^∗0(𝑡)𝐲^∗𝑠(𝑡)𝐳̆^∗(𝑡)]^∗, and 𝐲̂_𝑧(𝑡) denote the projection of 𝐲_𝑧(𝑡) onto ℒ{{𝐲_𝑧(𝜏)}|_{0≤𝜏<𝑡}}. In this case the minimum value of 𝐽_ℱ is min𝐽_ℱ=+^𝑇0𝑦₀(𝑡)−𝐶₁(𝑡)𝑥̂(𝑡)^∗𝑦₀(𝑡)−𝐶₁(𝑡)𝑥̂(𝑡)𝑑𝑡_𝑇0−𝛾₋₂𝑦_𝑠(𝑡)−𝐶₂(𝑡)𝑥̂(𝑡)^∗𝑦_𝑠(𝑡)−𝐶₂(𝑡)𝑥̂(𝑡)𝑑𝑡^𝑇0(𝑧̆(𝑡)−𝐿(𝑡)𝑥̂(𝑡))^∗(𝑧̆(𝑡)−𝐿(𝑡)𝑥̂(𝑡))𝑑𝑡,(3.11) where 𝑥̂(𝑡) is obtained from the Krein space projection of 𝐱(𝑡) onto ℒ{{𝐲_𝑧(𝑗)}|_{0≤𝜏<𝑡}}.

Remark 3.2 3.2. By analyzing the indefinite quadratic form 𝐽_ℱ 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 𝐱̂(𝑡) is calculated by 𝐱̇̂(𝑡)=𝐴(𝑡)𝐱̂(𝑡)+𝐾(𝑡)𝐲̃_𝑧(𝑡),(3.12) where 𝐻𝐲(𝑡̃_𝑧)(=𝑡)=𝐶𝐲_𝑧∗1(𝑡)−𝐻(𝑡)𝐱̂(𝑡),𝐾(𝑡)=𝑃((𝑡𝑡))𝐻𝐶^∗2_∗(𝑡)𝐿^∗(𝑡)^∗,𝑅(𝑡)𝑅^{−1̃𝑦𝑧}(𝑡),_̃𝑦𝑧(𝑡)=diag𝐼,𝐼,−𝛾²𝐼,(3.13) and 𝑃(𝑡) is computed by 𝑃̇(𝑡)=𝐴(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴^∗𝑃(𝑡()0+)=𝐵𝑃(𝑡₀)𝐵^∗(𝑡)−𝐾(𝑡)𝑅𝑦_𝑧̃(𝑡)𝐾^∗(𝑡),.(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(𝑡) 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 𝑧̆(𝑡)=𝐿(𝑡)𝑥̂(𝑡),0≤𝑡≤𝑇,(3.15) where 𝑥̇̂(𝑡)=𝐴(𝑡)𝑥̂(𝑡)+𝑃(𝑥𝑡̂)(0𝐶)^∗𝑓=(𝑡)0𝑦_𝑓(𝑡)−𝐶_𝑓(𝑡)𝑥̂(𝑡),(3.16) with 𝑦_𝑓(𝑡)=[𝑦^∗0(𝑡)𝑦^∗𝑠(𝑡)]^∗, 𝐶_𝑓(𝑡)=[𝐶^∗1(𝑡)𝐶^∗2(𝑡)]^∗.

Proof. It follows from Lemma 3.3 that if 𝑃(𝑡) is a bounded positive definite solution to Riccati differential equation (3.14), then the projection 𝐱̂(𝑡) 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 𝐽_ℱ is given by (3.11). In order to achieve min𝐽_ℱ>0, one natural choice is to set 𝑧̆(𝑡)−𝐿(𝑡)𝑥̂(𝑡)=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(𝑡)=𝑥(𝑡)−𝑥̂(𝑡),(3.18) it follows from (2.1) and (3.16) that 𝑒̇(𝑡)=(𝐴(𝑡)−Γ(𝑡)𝐶(𝑡))𝑒(𝑡)+𝐵(𝑡)𝑤(𝑡)−𝑃(𝑡)𝐶^∗𝑓(𝑡)𝑣_𝑧(𝑡),(3.19) where Γ(𝑡)=𝑃(𝑡)𝐶^∗𝑓(𝑡)⎡⎢⎢⎣√𝑝(𝑡)𝐼𝑝(𝑡)(1−𝑝(𝑡))𝐼⎤⎥⎥⎦,𝑣^𝑧(𝑡)=⎡⎢⎢⎣𝑣𝑣(𝑡)_𝑠(𝑡)⎤⎥⎥⎦.(3.20) Unlike [14–16], the parameter matrices in the filtering error equation (3.19) do not contain the stochastic variable 𝑟(𝑡), which is an interesting phenomenon. As mentioned in Definition 1 in [19], it is obvious that, if (𝐶(𝑡),𝐴(𝑡)) 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)(𝐶(𝑡),𝐴(𝑡)) is detectable,(ii)𝑤(𝑡),𝑣(𝑡)∈ℒ₂[0,𝑇].

4. A Numerical Example

We consider system (2.1) with the following parameters: 𝐴(𝑡)=⎡⎢⎢⎣−210−65⎤⎥⎥⎦,𝐵(𝑡)=⎡⎢⎢⎣21..86⎤⎥⎥⎦,𝐶(𝑡)=[189.5],𝐿(𝑡)=[11](4.1) and set 𝛾=1.1, 𝑥(0)=[00]^∗, 𝑝(𝑡)=0.8, and 𝑃₀=𝐼. In addition, we suppose that the noises 𝑤(𝑡) and 𝑣(𝑡) are generated by Gaussian with zero means and covariances 𝑄_𝑤=1, 𝑄_𝑣=0.02, the sampling time is 0.02 s, and the stochastic variable 𝑟(𝑡) 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(𝑡) and its 𝐻_∞ filtering estimate, and Figure 3 shows the estimation error 𝑧(̃𝑡)=𝑧(𝑡)−𝑧(̆𝑡). It can be observed from the simulation results that the finite-time 𝐻_∞ filter has good tracking performance.

Figure 1 

Stochastic variable 𝑟(𝑡).

Figure 2 

True value of signal 𝑧(𝑡) (solid line) and its 𝐻∞ filtering estimate (dashed line).

Figure 3 

Estimation error 𝑧(̃𝑡).

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).