Journal of Applied Mathematics

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 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 filtering problem is investigated for linear continuous time-varying systems with uncertain observations and -norm bounded noises.

The -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 -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 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 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 filtering will be investigated for linear continuous time-varying systems with uncertain observations and -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 filtering problem for linear continuous time-varying systems with uncertain observations. It will be shown that the filtering problem for linear continuous time-varying systems with uncertain observations is partially equivalent to an filtering problem for a certain Krein space state-space model. Through employing projection theory, both the existence condition and a solution of the 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 filtering problem for systems with uncertain observations can be converted into an 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 “” and “” stand for the inverse and complex conjugation of a matrix, respectively. for and for . denotes the -dimensional Euclidean space. is the identity matrix with appropriate dimensions. For a real matrix, (resp., ) means that is symmetric and positive (resp., negative) definite. denotes the inner product in Krein space. denotes a block-diagonal matrix. means . denotes the linear space spanned by sequence . An abstract vector space that satisfies the following requirements is called a Kreinspace [1].(i) is a linear space over , the field of complex numbers.(ii) There exists a bilinear form on such that(a),(b), for any , , , and where denotes complex conjugation.(iii) The vector space admits a direct orthogonal sum decomposition  such that and are Hilbert spaces, and  for any and .

2. System Model and Problem Formulation

In this paper, we consider the following linear continuous time-varying system with uncertain observations where is the state vector, is an exogenous disturbance belonging to , is the observation, is the observation noise belonging to , is the signal to be estimated, and , , , and are known real time-varying matrices with appropriate dimensions.

The stochastic variable takes the values of 0 and 1 with [11]. Note that many literatures associated with observer design are based on the assumption that [1–4], it can be unreasonable in many practical applications [9, 10, 13]. In this paper, we assume that is a known positive scalar.

The finite-time filtering problem under investigation is stated as follows: given a scalar , a matrix , and the observation , find an estimate of the signal , denoted by , such that where .

Thus, the finite-time filtering problem can be equivalent to the following:(I) has a minimum with respect to ;(II) can be chosen such that the value of 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 filter and its existence condition.

3.1. 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

Thus, we can rewrite as where

Moreover, we introduce the following Krein space stochastic system where , , , , and are mutually uncorrelated white noises with zero means and known covariance matrices as

Let then it follows from (3.1), (3.3), (3.4), and (3.7) that

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

Lemma 3.1. Consider system (2.1), given a scalar and a matrix , then in (2.3) has the minimum over if and only if the innovation exists for , where , and denote the projection of onto . In this case the minimum value of is where is obtained from the Krein space projection of onto .

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

3.2. Solution of the Finite-Time 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 is calculated by where and is computed by

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 and a matrix , and suppose is the bounded positive definite solution to Riccati differential equation (3.14). Then, one possible level- finite-time filter that achieves (2.3) is given by where with , .

Proof. It follows from Lemma 3.3 that if is a bounded positive definite solution to Riccati differential equation (3.14), then the projection exists. According to Lemma 3.1, it is obvious that the filter that achieves (2.3) exists. If this is the case, the minimum value of is given by (3.11). In order to achieve , one natural choice is to set thus the finite-time 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 it follows from (2.1) and (3.16) that where Unlike [14–16], the parameter matrices in the filtering error equation (3.19) do not contain the stochastic variable , which is an interesting phenomenon. As mentioned in Definition 1 in [19], it is obvious that, if 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 filter design in this paper:(i) is detectable,(ii).

4. A Numerical Example

We consider system (2.1) with the following parameters: and set , , , and . In addition, we suppose that the noises and are generated by Gaussian with zero means and covariances , , the sampling time is 0.02 s, and the stochastic variable is simulated as in Figure 1. Based on Theorem 3.4, we design the finite-time filter. Figure 2 shows the true value of signal and its filtering estimate, and Figure 3 shows the estimation error . It can be observed from the simulation results that the finite-time filter has good tracking performance.

Figure 1: Stochastic variable .

Figure 2: True value of signal (solid line) and its filtering estimate (dashed line).

Figure 3: Estimation error .

5. Conclusions

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