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

This paper is concerned with the finite-time H∞ filtering problem for linear continuous timevarying systems with uncertain observations and L2-norm bounded noise. The design of finitetimeH∞ 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 statespace 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.


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 L 2 -norm bounded noises.
The H 2 -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 fixedpoint 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 timevarying systems with uncertain observations and L 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 H 2 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 H 2 optimal estimation problem subject to a fictitious Krein space stochastic systems; ii it develops a Kalman-like robust estimator for linear continuous timevarying 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.iii The vector space K admits a direct orthogonal sum decomposition • } are Hilbert spaces, and x, y 0 1.2 for any x ∈ K and y ∈ K − .

System Model and Problem Formulation
In this paper, we consider the following linear continuous time 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 P 0 > 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 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 {x 0 , w t | 0≤t≤T }; II z t can be chosen such that the value of J F at its minimum is positive.

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.

Construct a Partially Equivalent Krein Space Problem
To begin with, we introduce the following state transition matrix: it follows from the state-space model 2.1 that Φ t, τ B τ w τ dτ e f t .

3.3
Thus, we can rewrite J F as where Let 3.8 then it follows from 3.1 , 3.3 , 3.4 , and 3.7 that −1 e f t dt.

3.9
According to 1 and 3 , we have the following results.
where x t is obtained from the Krein space projection of x t onto L{{y z j }| 0≤τ<t }.
Remark 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 H 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 H ∞ estimator for linear systems with uncertain observations, which is more simple and intuitive than previous versions.

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 t is calculated by where

A Numerical Example
We consider system 2.1 with the following parameters: and set γ 1.1, x 0 0 0 * , p t 0.8, and P 0 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.

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 statespace model, it is shown that the H ∞ filtering problem can be partially equivalent to a Krein space H 2 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.
denotes the linear space spanned by sequence {• • • }.An abstract vector space {K, •, • } that satisfies the following requirements is called a Krein space 1 .iK is a linear space over C, the field of complex numbers.iiThere 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.

4 Figure 2 : 8 Figure 3 :
Figure 2: True value of signal z t solid line and its H ∞ filtering estimate dashed line .
, v t , v s t , and e f t are mutually uncorrelated white noises with zero means and known covariance matrices as Consider system 2.1 , given a scalar γ > 0 and a matrix P 0 > 0, then J F in 2.3 has the minimum over {x 0 , w t | 0≤t≤T } if and only if the innovation y z t exists for 0 ≤ t ≤ T , where , and y 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 * Consider system 2.1 , given a scalar γ > 0 and a matrix P 0 > 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 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 x 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 min J F > 0, one natural choice is to set ∞ 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.
i C t , A t is detectable, ii w t , v t ∈ L 2 0, T .Figure 1: Stochastic variable r t .