Stability Analysis of Stochastic Markovian Jump Neural Networks with Different Time Scales and Randomly Occurred Nonlinearities Based on Delay-Partitioning Projection Approach
In this paper, the mean square asymptotic stability of stochastic Markovian jump neural networks with different time scales and randomly occurred nonlinearities is investigated. In terms of linear matrix inequality (LMI) approach and delay-partitioning projection technique, delay-dependent stability criteria are derived for the considered neural networks for cases with or without the information of the delay rates via new Lyapunov-Krasovskii functionals. We also obtain that the thinner the delay is partitioned, the more obviously the conservatism can be reduced. An example with simulation results is given to show the effectiveness of the proposed approach.
The human brain is made up of a large amount of neurons and their interconnections. An artificial neural network is an information processing system that has certain characteristics in common with biological neural networks. During the past decades, neural networks have been used for a wide variety of applications, for example, associative memories, pattern recognition, signal processing, and the other fields [1–4]. As is well known, the stability of neural networks plays an important role in modern control theories for these applications. However, time delays are often attributed as the major sources of instability in various engineering systems. Therefore, how to find sufficient conditions to guarantee the stability of neural networks with time delays is an important research topic [5–15].
Markovian jump system is an important class of stochastic models, which can be described by a set of nonlinear systems with the transitions between models determined by a Markovian chain in a finite mode set. This kind of system has been extensively applied to study the stability of neural networks. In real life, neural networks have a phenomenon of information latching, and it is recognized that the best way for modeling this class of neural networks is Markovian jump system [16–18]. Obviously, the Markovian jump system is more complex and challenging than the system without Markovian jump parameters, in which many authors are interested [12, 19–24].
On the other hand, it should be pointed out that lots of practical systems are influenced by additive randomly occurred nonlinear disturbances which are caused by environmental circumstances. In today’s networked environment, such nonlinear disturbances may be subject to random abrupt changes, which may result from abrupt phenomena, such as random failures and repairs of the components, environmental disturbance, and so forth. In other words, the nonlinear disturbances may occur in a probabilistic way, but they are randomly changeable in terms of their types and/or intensity. The stochastic nonlinearities, which are then named as randomly occurred nonlinearities (RONs), have recently attracted much attention [25–28].
Recently, by introducing free-weighting matrices [29, 30], model transformation method , linear matrix inequality (LMI) approach [32, 33], and adopting the concept of delay partitioning [9, 11, 34], stability criteria have been obtained for some neural networks. Sufficient conditions for the robust stability of uncertain stochastic system with interval time-varying delay were derived in  by employing the delay partitioning approach. Delay-dependent conditions on mean square asymptotic stability of stochastic neural networks with Markovian jumping parameters are presented by using the delay partitioning method which is different from the existing ones in the literature and convex combination method in . The RONs model and the sensor failure model were introduced in . In , better delay-dependent stability criteria for continuous systems with multiple delay components were established by utilizing a delay-partitioning projection approach. On the other hand, it is worth mentioning that neural networks on time scales have been presented and studied [35–37], which can unify the continuous and discrete situations. To the best of the authors’ knowledge, the delay-partitioning projection approach to stability analysis of stochastic Markovian jump neural networks with different time scales and randomly occurred nonlinearities has never been tackled in the previous literature. This motivates our research.
In this paper, the problem of stability analysis of stochastic Markovian jump neural networks with different time scales and RONs is considered. The paper is organised as follows. Section 2 introduces model description and preliminaries. RONs are introduced to model a class of sector-like nonlinearities whose occurrence is governed by a Bernoulli distributed white sequence with a known conditional probability. In Section 3, we derive the stability results based on delay-partitioning projection approach for stochastic Markovian jump neural networks with RONs. The results include two cases, one with a specified delay rates and the other with arbitrary delay rates. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size; we verify that the conservatism of the conditions is a nonincreasing function of the number of partitions. A numerical example is presented to illustrate the effectiveness of the obtained criteria in Section 4. And finally, conclusions are drawn in Section 5.
Notations. Throughout this paper, the notation is fairly standard. denotes the -dimensional Euclidean space. stands for real matrix of size (simply abbreviated when ). is used to define a real symmetric positive definite (negative definite) matrix. For real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). The symmetric terms in a symmetric matrix are denoted by and denotes a block-diagonal matrix. The superscripts and stand for the transpose and inverse of matrix . denotes a complete probability space with a filtration , where is a sample space, is the -algebra of subset of the sample space, and is the probability measure on . stands for the expectation of the stochastic variable .
2. Model Description and Preliminaries
In this paper, the stochastic Markovian jump neural networks with different time scales RONs are considered: where is the state vector. is a scalar zero mean Gaussian white noise process. is a real-valued initial condition. is the time scale. is a right-continuous Markov chain on the probability space taking values in a given finite set . The transition probability matrix of system (1) is given by where and , is the transition rate from mode at time to mode at time , and . is a positive diagonal matrix. , , , and are known matrices. For the sake of simplicity, in the sequel, each possible value of is denoted by and , , , are abbreviated as , , , , respectively. denotes the time-varying delay satisfying where and are constant real numbers.
Finally, stands for the mismatched external nonlinearity. The stochastic variable , which accounts for the phenomena of RONs, is a Bernoulli distributed white noise sequence specified by the following distribution laws: Furthermore, we can show that where is a constant.
Before proceeding, we make the following assumption.
Assumption 1. For all , the nonlinear function is assumed to satisfy the following sector-bounded condition: where and are known real matrices of appropriate dimensions and .
Remark 2. The nonlinear function satisfying (6) is customarily said to belong to sector . Because such a nonlinear condition is quite general which includes the usual Lipschitz condition as a special case, the systems with sector-bounded nonlinearities have been intensively studied in [25, 26].
Recall that the time derivative of a Wiener process is a white noise [27, 28]. We establish , where is a scalar Wiener process on a probability space , which is independent from the Markov chain . It is further assumed that and the stochastic variable are mutually independent. Besides, satisfies Hence, the network (1) is rewritten as the following stochastic differential equations:
Remark 3. The stochastic Markovian neural network with different time scales and RONs (8) is general enough to include many models as its special cases. For example, if we do not consider the RONs, time scales and remove the noise perturbations, then (8) is reduced to those studied in [9, 23, 38–41], whereas the authors in [9, 38–41] ignore the Markovian jump parameters. If we do not consider the RONs and time scales, then (8) is discussed in  and its references. If we do not take the noise perturbations and time scales into account and let , , (8) is just the one in . If we do not consider the RONs, the Markovian jump parameters, and the noise perturbations, then (8) is studied in . Furthermore, if we remove the time scales, (8) is the one in . Thus, our model generalizes and improves greatly many previous works and is therefore very significant.
Remark 4. In , the delay-partitioning approach was introduced to study the discrete-time recurrent neural networks. As an extension of the approach, we use two different partitions to deal with the continuous Markovian jump neural networks with different time scales in this paper.
Note that it is easy to prove that there exists at least one equilibrium point for (8) by employing the well-known Brouwer’s fixed point theorem. To end this section, the lemmas which are necessary for the proof of our main results are introduced as follows.
Lemma 5. For any constant matrix , any scalars and with , and a vector function such that the integrations concerned are well defined, the following inequality holds:
Lemma 6 (see ). Let and the bidiagonal upper triangular block matrix If with , , then
Lemma 7 (Finsler’s lemma). Suppose that , and such that has full row rank. Then, the following statements are equivalent:(1) there exists a vector such that and ,(2) there exists a scalar such that ,(3) such that ,(4), where is the orthogonal complement of .
Remark 8. It should be pointed out that various problems in control theory have been solved by combing Lyapunov control approach with Finsler’s lemma. In lots of applications, Finsler’s lemma is referred to Elimination lemma, which devotes to eliminate the redundant variables in matrix inequalities .
3. Main Results
In this section, we will give the mean square asymptotic stability for the system (8) in terms of LMI approach. The main results are stated as follows.
Theorem 9. For given constants and , and two positive integers and , the stochastic system (8) is mean square asymptotically stable if there exist matrices , , , , and and positive scalar such that the following LMI holds: where is all arbitrary but fixed matrix whose columns form a basis of the kernel space of ,
Proof. For presentation convenience, in the following, we denote
then, the system (8) becomes
By employing the idea of delay partitioning approach to the time-delay , we choose the following Lyapunov-Krasovskii functional candidate:
Let be the weak infinitesimal generator of the random process along the trajectory of the system (8). Then, by Itô differential formula, we have
By virtue of (3), (7), and (9), it can be verified that
On the other hand, note that (6) is equivalent to
where is a positive constant.
Substituting (5), (7), (15), and (22) into (20), we can obtain where In addition, it follows from the Newton-Leibniz formula that which are equivalent to The full column rank matrix representation of the right orthogonal complement of is denoted by , and Lemma 6 offers a computation method with , . According to Lemma 7, is negative as long as holds, which, in other words, can be expressed as (13). Hence, the stochastic system (8) is asymptotical stable in the mean square, and it completes the proof.
Remark 10. The delay-partitioning projection technique employed in this paper constitutes the major improvement from most existing results in the literature. Firstly, it should be pointed out that such technique is very rational. The reasons are twofold. The properties of subinterval delays may be sharply different in many practical situations. Thus, it is not reasonable to combine them together. When reaches its upper bound, we do not necessarily have every subinterval delay reaches its maximum at the same time. That is to say, if we use an upper bound to bound the delay we have to use the sum of the maxima of subinterval delays; however, does not achieve this maximum value usually. Therefore, by adopting the delay-partitioning projection approach, less conservative conditions can be proposed. Secondly, in , the central point of variation of the delay was introduced to study the stability for time-delay systems, which is called the DCP method. As an extension of the method, we divide the delay into several subintervals which permits employing one slightly different function for different subintervals. This treatment makes us utilize more information on the time delay and thus may present better criteria with less conservatism. Thirdly, two different partitions are made to deal with the time-varying delay in this paper, which are different from the approach in . The parameters and refer to the number of delay partitioning, and it indicates that the solution can be searched in a wider space, which leads to the reduction of conservatism. Finally, using the similar analysis method of , it is easy to verify that the conservatism of the conditions is a nonincreasing function of the number of delay partitions. However, as we all know, the computational complexity will be increased as the partitioning becomes thinner. Therefore, the delay-partitioning projection approach can provide the flexibility that allows us to trade off between complexity and performance of the stability analysis.
We now consider that the time-varying delay is nondifferentiable or the bound of the delay is unknown, which means that the restriction on the delay is removed. For this goal, we modify the Lyapunov-Krasovskii functional as The proof can then be derived by following a similar line of arguments as that in Theorem 9. Besides, in will turn out to be .
Theorem 11. For given constant and two positive integers and , the stochastic system (8) is mean square asymptotically stable if there exist matrices , , , , and and positive scalar such that the following LMI holds: where
and , , , , , , , are the same as in Theorem 9.
Remark 12. Obviously, the delay rate is not specified in Theorem 11, which includes the constant time delay as a special case. Therefore, Theorem 11 is applicable to the special case to derive some results.
Remark 13. In previous work as , the authors have investigated the stability of neutral type neural networks with discrete and distributed delays. The reason why we do not take these neural networks into account is to make our idea more lucid and to avoid complicated notations. However, it is not difficult to extend our results to the neutral type neural networks with mixed time-varying delays. The results will appear in our following study.
4. A Numerical Example
One illustrative example is presented to show the effectiveness of the theoretical results. For simplicity, the system (8) with Markovian switching between two modes is taken into consideration. In addition, the state vector of each node is of dimension two. In other words, , . The mode switching is governed by the rate matrix , and the other parameters are assumed as The sector-bounded nonlinear function is given as follows: which can be bounded by We choose the parameters and . It is easy to know and .
By using Matlab LMI control toolbox, we can find the feasible solution of the LMI (13) as follows: The simulation result is shown in Figure 1, which implies that all the expected system performance requirements are well achieved.
Remark 14. It is worth pointing out that if , then the function in (8) will be the neuron activation function. According to Assumption 1, satisfies the sector-bounded condition which is more general than the Lipschitz condition. Therefore, the stability criteria in  can not apply to our example.
Remark 15. In our example, many factors such as noise perturbations, Markovian jump parameters, RONs, and different time scales are considered and the delay-partitioning projection approach is employed. Therefore, the results reported in [9, 23, 38–41] do not hold in our example. Moreover, our results are expressed by LMIs, which can be easily checked by using the powerful Matlab LMI Toolbox. Thus, our stability criteria are more computationally efficient than those given in [13, 40, 41].
In this paper, we have dealt with the mean square asymptotic stability problem for stochastic Markovian jump neural networks with different time scales and RONs. By using new Lyapunov-Krasovskii functionals and delay-partitioning projection approach, the LMI-based criteria have been developed to guarantee stability of such systems. It is also shown that the delay-partitioning projection approach can achieve the aim of reducing conservativeness. An numerical example is exploited to illustrate the effectiveness of the theoretical results.
This work was supported by the Fundamental Research Funds for the Central Universities (JUSRP51317B) and the National Natural Science Foundation of China under Grant 10901073 and 11002061.
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