Recent Developments on the Stability and Control of Stochastic SystemsView this Special Issue
Robust Filtering for Uncertain Neutral Stochastic Systems with Markovian Jumping Parameters and Time Delay
This paper deals with the robust filter design problem for a class of uncertain neutral stochastic systems with Markovian jumping parameters and time delay. Based on the Lyapunov-Krasovskii theory and generalized Finsler Lemma, a delay-dependent stability condition is proposed to ensure not only that the filter error system is robustly stochastically stable but also that a prescribed performance level is satisfied for all admissible uncertainties. All obtained results are expressed in terms of linear matrix inequalities which can be easily solved by MATLAB LMI toolbox. Numerical examples are given to show that the results obtained are both less conservative and less complicated in computation.
Time delay exists extensively in chemical process systems, communication systems, economic systems, microwave oscillator, and networked control systems. Meanwhile, because of the modeling inaccuracies and changes in the environment of the model parameter, uncertainties are unavoidable in the process of modeling. The appearance of time delay and uncertainties in many systems can often bring instability, oscillation, and poor performance; considerable attention has been focused on the stability analysis of uncertain time delays systems; see [1–6] and the references therein. On the other hand, a real system is often affected by external disturbances such as stochastic perturbations. The stochastic effects can also lead to oscillation, divergence, or other poor performances. Therefore, the stability study of stochastic systems with time delay has been paid great attention and a lot of significant results have been reported in the literature; see [7–12] and the references therein.
In the past few decades, filtering problem has been a hot issue in the fields of signal processing. Kalman filtering has been successfully applied in many fields such as manufacturing systems, economic systems, and maneuvered target tracking. However, the exact requirement of known dynamics system and precise noise statistics have restricted its practical application. In such a case, we can resort to filtering [13–22] and - filtering [23–25]; see the references therein.
Markovian jump systems, originally raised by Krasovskii and Lidskii , are famous for the description of many dynamical practical systems whose structure and parameters are subject to random changes. Therefore, the stability analysis and filtering problem for Markovian jump systems have been studied [27–34]. For example, the stability analysis of impulsive stochastic neural networks with Markovian jump are studied in [27, 29, 31, 34]; control and mode-dependent filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities are, respectively, investigated in [28, 30, 32]. The design of reduced-order filter for Markovian jumping systems with time delay is studied in . The filter design for stochastic time-varying delay systems with Markovian jumping parameters is considered in .
Many methods are proposed in the process of robust stochastic stability analysis and filtering design, which develops from the early solving Riccati equation to model transformation method and cross-terms bounding technique , free-weighting matrices method [11, 32], slack matrix variables [17–19, 24, 28], and delay-partitioning method . However, model transformations may give rise to additional dynamics of the original systems , and cross-terms bounding techniques can bring conservatism. Moreover, as pointed out in , under certain circumstance, free matrix variables may not lessen the conservatism. In recent years, another popular method called Finsler Lemma is carried out so as to decrease the computational cost as well , and the Finsler Lemma in deterministic setting is extended to generalized Finsler Lemma in stochastic systems in [5, 33, 34].
On the other hand, many dynamical systems can be modeled by neutral systems which are organized by neutral functional differential equations. Other than retarded time delay systems containing delays only in states, a neutral time delay system contains delays in both its state and its derivatives of state. Recently, for neutral stochastic time delay systems, the stability analysis and filter design problem are mainly addressed in [4, 9, 19]. It is necessary to point out that the delay-dependent robust filtering for uncertain neutral stochastic time delay system is studied in [19, 24]. Robust filter design for neutral stochastic uncertain systems with time-varying delay is studied in . To the best of the authors’ knowledge, the filtering problem has not been reported about uncertain neutral stochastic systems with Markovian jumping parameters and time delay, which motivates the present study.
Motivated by the works in [5, 13, 17, 18], the robust filtering for uncertain neutral stochastic systems with Markovian jumping parameters and time delay is considered in this paper. By generalized Finsler Lemma, the robust stochastic stability condition is obtained. The presented condition is simple and efficient. Finally, the effectiveness of the approach is verified by illustrative examples including a comparison with some recent results.
Throughout this paper, denotes the -dimensional Euclidean space. is the set of real matrices. is the identity matrix. denotes Euclidean norm for vectors and denotes the spectral norm of matrices. denotes the set of all natural numbers; that is, . is a complete probability space with filtration satisfying the usual conditions. stands for the transpose of the matrix . For symmetric matrices and , the notation (resp., ) means that the is positive definite (resp., positive semidefinite). denotes a block that is readily inferred by symmetry. stands for the mathematical expectation operator with respect to the given probability measure .
2. Problem Description
Consider the following uncertain neutral stochastic systems with Markovian jumping parameters and time delay: where is the state vector, is the measured output, is the disturbance input in , and is the signal to be estimated. , , , , , , , , , , , , , , and are the matrix functions of the random jumping process , where is a finite-state Markovian jump process representing the system mode; that is, takes discrete values in given finite set . Here is the initial condition and is assumed to be continuously differentiable on . Consider indicates the time delay. is a scalar Brownian motion (Wiener process) defined on the complete probability space satisfyingThe transition probability matrix of systems (1) is given by where , , , , is the transition rate from mode at time to mode at time , and
For the purpose of simplicity, in this paper, for each , , , and are denoted by , , , and so on. In systems (1), and , , , , , , , , , , , , , , and are known real constant matrices with appropriate dimensions. , , , and are unknown matrices representing norm-bounded parameter uncertainties, which are assumed to satisfywhere , , , and are known real constant matrices and is an unknown time-varying matrix function satisfying The parameter uncertainties , , , and are said to be admissible if both (6) and (7) hold.
In this paper, we make the following assumption on the matrix in systems (1).
Assumption 1. The matrix in systems (1) satisfies where the notation denotes the spectral radius of .
We now consider a full-order filter for systems (1) with the following form: where is the filter state, is the estimation of , and , , and are the filter parameters with compatible dimensions to be determined.
Define Then the filtering error systems can be obtained as follows: where with Then the problem of robust filtering to be addressed in this paper is formulated as follows: given the uncertain stochastic delay systems (1) and a prescribed attenuation level , design linear stochastic filter () as the form of (9) such that the filtering error systems (11) are robustly stochastically stable and under zero initial conditions, the following inequality holds: for all nonzero and all admissible uncertainties.
Before concluding this section, we introduce the following Lemmas, which will be used in the derivation of our main results in the next section.
Lemma 2. For any vectors and any scalar , matrices , , are real matrices of appropriate dimensions with , then the following inequality hold:
Proposition 3 (, generalized Finsler Lemma (GFL)). Consider stochastic vector , symmetric and positive matrix , and matrix with rank . Let represent the right orthogonal complement of , that is, , then the following four statements are equivalent: , , , ;;;.
Remark 4. Based on generalized Finsler Lemma, the stability of neutral stochastic systems with time delay has been studied in [5, 34]. In addition, it should be noted that the stochastic systems in [5, 34] are not Markovian jump systems. And the filtering problem for stochastic time delay systems with Markovian jumping parameters is considered in . However, it should be pointed out that the systems in  do not include any analysis of neutral phenomena. So filtering for neutral stochastic systems with Markovian jumping parameters and time delay is considered in this paper.
3. Main Results
Theorem 5. Consider the uncertain neutral stochastic Markovian jump systems (1). For given scalars , , systems (1) are robustly stochastically stable for all admissible uncertainties satisfying (6) and (7), if there exist symmetric positive matrices , , , and scalar satisfying where
Proof. For the purpose of convenience, the following notations are adopted: and then the filtering error systems (11) become Choose the Lyapunov-Krasovskii functional candidate as follows: where . According to Itô’s differential formula, the stochastic differential along systems (11) is where Firstly, we show that the filtering error systems (11) with are robustly stochastically stable.
Taking mathematical expectation on both sides of system (19) and by virtue of , we obtain Integrating both sides of (23) from to , we have From the definition of with and (24), it is easy to obtain where The right orthogonal complements of is Taking mathematical expectation on both sides of (21) and then substituting (22) into (21), we have where In order to prove the robust stochastic stability of the filtering error systems (11) with , it suffices to showBy virtue of Proposition 3, (30) is equivalent to where It is obvious that are implied by (16) according to Schur complements. Therefore, if (16) is feasible, then filtering error systems (11) with are robustly stochastically stable.
Next, we will establish the performance for the filtering error systems (11) under the zero initial condition.
From the definition of and together with (24), it implies where The right orthogonal complements of are Noticing (21)-(22) together with , we can obtain where and set Adding the right side of (38) to both sides of (36) and integrating both sides of (36) from 0 to and then taking the zero initial condition into account, we can acquire where and .
According to Proposition 3, is equivalent to where According to Schur complement and (12), we can obtain where Then by Lemma 2, it can be seen that where , .
By (41), (46), and Schur complements, holds if and only if . This completes the proof.
Remark 6. Theorem 5 is established based on GFL. For the sake of reducing the computational complexity, similar to [6, 8, 10], the first two equivalent conditions of Proposition 3 are adopted in this paper.
Now we are in a position to present the filter design for uncertain neutral stochastic system with Markovian jumping parameters and time delay based on Theorem 5.
Theorem 7. Consider systems (1), for given scalars , ; then there exists a linear stochastic full-order filter with the form (9), such that filter error systems (11) are robustly stochastically stable and satisfy prescribed disturbance attenuation level for all admissible uncertainties (6) and (7) if there exist symmetric positive matrices , , , and and matrices , , , scalars , such that the following LMI holds: where Meanwhile, the filter parameters are given by
Proof. We note that, from (48), it is easy to see , and . Define then applying Schur complement, guarantees . Let where . Substituting and (12) into (16), then pre- and postmultiplying (16) by and , respectively, and using (51), the desired result (48) follows immediately. This completes the proof.
Remark 8. Theorem 7 considers the filtering problem for uncertain neutral stochastic time delay systems with Markovian jumping parameters. It should be noted that the proposed conditions are formulated in terms of LMIs. Therefore, by MATLAB LMI toolbox, for given different or , the lower bound of performance index and the upper bound of can be efficiently obtained by solving a generalized eigenvalue problem.
Now we would like to proceed to present the filtering for uncertain neutral stochastic time delay systems without Markovian jumping parameters. Considering the system without the Markovian jumping parameters, the following systems can be obtained: where and , , , and are unknown matrices satisfying We can obtain the following Corollary 9 for system ().
Corollary 9. Consider the system , for given scalars , ; then there exists a linear stochastic full-order filter with the form (9), if there exist symmetric positive matrices , , , , matrices , , , and scalar , such that the following LMI holds: where then the robust filtering problem is solvable. Furthermore, the parameters of the desired robust filter can be given as
Remark 10. The proof of Corollary 9 follows the same lines as that in the proof of Theorem 7, so the detailed procedure is omitted here. When , , , and , systems (52)–(55) in this paper reduces to systems in . It is noticed that the filtering problem studied in  is a special case of this paper. For comparisons of our results with that in , see Example 1 in detail.
4. Numerical Examples
In this section, numerical examples and simulations are given to illustrate the validity and benefits of the proposed approach.
Example 1. Consider systems (52)–(55) without Markovian jumping parameters as follows: For different given noise attenuation levels , the upper bounds of delay for systems (52)–(55) in Corollary 9 of this paper are presented in Table 1. For different given time delays, the lower bounds of noise attenuation level for systems (52)–(55) in Corollary 9 of this paper are provided in Table 2.
In particular, when , , , and , systems (52)–(55) in this paper reduce to systems in . For different given noise attenuation levels , the comparison of the upper bounds of delay in  with our results is presented in Table 3, and, for different given time delays, the lower bounds of noise attenuation level for systems in  and the same systems in this paper are provided in Table 4.
Besides, Theorem 2 in  fails to give a feasible solution. The number of decision variables of Theorem 3 in  is , which is the same as that in Corollary 9 of this paper. From Tables 3 and 4, we can see that our proposed method is less conservative than that in .
Now in the case when , , we resort to the MATLAB LMI control toolbox to solve the LMI (59), and the feasible solution can be obtained as follows: The initial conditions are also taken as and . The simulation results of the state response of the system are plotted in Figures 1–3. The filter state and its estimation and state and its estimation are given in Figures 1 and 2, respectively. Figure 3 depicts the estimation error . The simulation results demonstrate that the designed filters are feasible, effective and the stochastic stability of the error systems is ensured.
Example 2. Consider systems (1) with Markovian jumping parameters as follows.
Mode 1. ConsiderMode 2. Consider