Finite-Time State Estimation of Switched Neural Networks with Both Time-Varying Delays and Leakage Delay

: In this thesis


Introduction
Neural networks are the nonlinear macroscale adaptive dynamic system composed of many processing units.A lot of engineering applications of neural networks have been founded, for instance, image reparation, automation control and system identification [1][2][3].Since the practical setting is fully complex and changeable, the structure of the networks may change unpredictably.Therefore, the idea of switching system is developed to solve the problem that the structure of neural networks may change due to volatile external factors.In recent years, SNNs have been extensively used in the field of high-speed signal processing, gene selection for DNA microarray analysis and artificial intelligence [4][5][6].In addition, due to the limited switching speed of amplifiers or the limited propagation time of signals in biological networks, there are always inevitable delays in actual SNNs, which may give rise to undesirable dynamic behaviors, for instance, chaos, oscillation and multiple cycles [7][8][9][10].Recently, the dynamic behaviors of delayed SNNs have attracted a lot of research efforts [11][12][13].
In practical engineering, the state of the system may not be completely determined by the output of the networks.It is often necessary to design observers to estimate each neuron with the output of the neural networks and achieve certain design goals by estimating the state of the neurons.At present, there are many results about state estimation of SNNs.For example, under the available output measurements and the multiple L-K functional technique, the exponential state estimation of switched interval neural networks with mixed delays has been carried out [14].Based on a proper L-K functional, the author has derived a new sufficient condition making the estimation error system of SNNs be exponentially stable [15].The state estimation of discrete-time SNNs with mode-dependent time-varying delays has been investigated in H ∞ sense [16].
What should be noticed is that the design results of the above state estimator for SNNs are defined in an infinite time range.However, in practice, calculating the value of the Both Time-Varying Delays and Leakage Delay system state beforehand in order to remain within the specified range during the relevant time interval provides better transient performance and in turn prevents saturation from occurring [17][18][19].Although some significant work has been done for the FTSE of non-switched delayed neural networks [20][21][22], few studies focused on the corresponding research for SNNs.Additionally, leakage delay existing in a negative feedback term is hard to process since it has quick tendency to destroy the system performance [23][24][25].However, the leakage delay is often ignored in most past modeling of SNNs [26].Therefore, it is of great theoretical significance to investigate the design of finite-time estimator for SNNs when the combined effects of time-varying delays and leakage delay are taken fully into account.For all we know, there is no relevant results on this issue yet, which motivates our study.
In this thesis, we explore the issue of FTSE of the SNNs with time-varying delays and leakage delay.By constructing the appropriate piecewise L-K functional, employing ADT method and free-weighting matrix technique, some sufficient conditions are educed to make sure the FTB of the corresponding estimation error system.The existence and the characterization of the desired estimator can be achieved based on a series of LMIs.The results we obtained rely on both time-varying delays and leakage delay.Moreover, our results do not require the time delays to be differentiable, which cannot be applicable to the systems with inestimable or unknown time delays.The remaining part is arranged as follows.The model description and pre-knowledge are introduced in Section 2. The primary theorems are obtained in Section 3.An example is performed with its numerical simulations to corroborate the efficiency of the developed theoretical results in Section 4. The summary of our work is given in Section 5.
Notations.Let ℝ , + ℝ , n ℝ , n m × ℝ be the set of real numbers, the set of all positive real numbers, the n -dimensional real space, the n m × -dimensional real space, respectively.
. Notation ⊻ is the symmetric block in a matrix.
For SNNs (1), the following full-order state estimator is proposed as follows where ˆ( ) is the state estimation of ℝ is an initial function, (t) L σ is the estimator gain matrix that needs to be designed.Let the estimation error be and define functions as . Hence, the estimation error system can be expressed in the following form by combining system (1) and ( 2) To prove the results, we need to draw into the below definitions and lemma., , , , ) and the switching signal ( ) holds for 0 a τ > and 1 0 N ≥ , then the a τ and 1 N is said to be average dwell time and chattering bound, respectively.) ( )

Main Results
In this section, by constructing suitable piecewise L-K functional and using ADT method, we now present the analysis results for the estimation error system (3) to be FTB.
Theorem 1.Under Assumption 1, the estimation error systems (3) is FTB with respect to 1 2 0 ( , , , , ) such that the following inequalities hold for any i ∈ ℕ , and the ADT a τ satisfies 0 where , Proof.Construct L-K functional Both Time-Varying Delays and Leakage Delay 4 i j ∈ ℕ .Along the trajectories of system (3), the derivative of ki V ( ) By Lemma 1, there is In addition, note that .

PC f t t t T Q S S t t S L E t t t S A t t S B f t t S C f t t t t
where ( ) Integrating both sides of (15) from k t to t , it can be obtained that On the other hand, Using similarity estimation, the following inequality can be obtained Hence, for , substituting ( 17) into ( 16), we get from the definition of ADT that 0 0 Furthermore, it is clear that and By combining ( 18)-( 20), what can be concluded is that From Definition 1, it follows From ( 22), we can get Hence, for 1 h ≥ , according to ( 24) and ( 25), it follows 2 ( ) ( ) Therefore, the estimation error system (3) is FTB with respect to ( ) , , , , c c T R σ , which completes the proof.Remark 1. Until now, a lot of interesting results concerning the state estimation of SNNs with time-varying delays have been reported [30][31][32].Note that all the above results focus on the systems with differentiable time delays ( ) t τ , which cannot be applied to the systems with uncharted or inestimable time delays.In our work, by using the appropriate L-K functional related to the derivative of the variable of state and leading into the auxiliary equation in (11), the differentiability restriction of the time delays imposed in the above literatures is eliminated completely.
It should be noted that Theorem 1 is obtained when the gain matrix of estimator is known.Actually, ( ) to know in advance and needs to be determined in this circumstance.Therefore, the method to design the state estimator gain is given in the following.
Theorem 2. Under Assumption 1, the estimation error system (3) is FTB with respect to 1 2 0 ( , , , , ) such that the following inequalities hold for any i ∈ ℕ , , and the ADT a τ satisfies 0 where , Furthermore, the applicable state estimator gain is in Theorem 1 are reduced to (26).Then the state estimator gain can be designed as The proof is done.Remark 2. The FTSE issue for various kind of delayed neural networks have been studied in recent achievements [20][21][22].Compared with them, very little attention was paid to the corresponding researches for delayed SNNs.The authors studied the FTSE in H ∞ sense for SNNs with time-varying delays by making use of L-K functional coupled with ADT method [32].However, leakage delay is excluded, and time delays are required to be differentiable in their results.In our work, the conditions established in Theorem 2 rely both on the upper bound of time-varying delays and leakage delay.The estimator gain can be obtained by solving a family of LMIs without requiring time-varying delays to be differentiable.
Especially, when 0 δ = , the estimation error system (3) reduces to Then the following corollary can be obtained.Corollary 1.Under Assumption 1, the estimation error system (3) is FTB with respect to 1 2 0 ( , , , , ) such that the following inequalities hold for any i ∈ ℕ , where , and other parameters are the same as in Theorem 2.

Illustrative Example
A numerical example with its simulation results is supplied to prove the validity and applicability of the raised results.Consider estimation error system (3) with two neurons characterized by the following parameters 1 0.24 0 0.9 0.5 0.9 0.5 1.2 0.9 0.1 1.4 1.35 0.8 0.21 1.25 Using the MATLAB LMI Toolbox to solve the LMIs in Theorem 2, we come up with the following feasible solutions:

Conclusion
This article has gone into the issue of FTSE for a set of SNNs with leakage delay and time-varying delays.The L-K functional method and ADT switching laws have been employed to deduce some sufficient criteria, which can make the estimation error system (3) be FTB.The characterization of the estimator gain has been realized by solving certain LMIs.The obtained sufficient conditions are delay-dependent and do not require the time delays to be differentiable.Finally, an example is performed with its numerical simulations to corroborate the efficiency of the developed As we all know, SNNs have been studied extensively in various fields including biology, sociology and physics.Therefore, searching other analysis technique and other control methods for SNNs to acquire the less conservative results in finite-time sense will be our next work.
matrix A is a symmetric and negative definite (positive definite) matrix.and transpose of matrix A , respectively.If A and B are symmetric matrices, the maximum (minimum) eigenvalue of matrix A .Unless otherwise specified, I is the identity matrix with appropriate dimensions.set formed by the set of natural numbers.a b ∨ denotes the maximum value of a and b . 1 ( , ) { : where such that the associated integrals are well defined, then

(
= .In order to solve the unknown matrices conveniently, the following positive definite diagonal matrices are

.T
gain matrices of the state estimator are designed as By Theorem 2, the estimation error system (3) is FTB with respect to(1, 30,10, , )    I σ .When the initial functions are given by 0. Figure4depicts the switching signal ( ) t σ and switching instants.The simulation results indicate that the state estimator can track the state of delayed SNNs, which verifies the validity and correctness of the developed method.

Figure 1 .
Figure 1.The state trajectory of 1 ( ) x t and its estimation 1 ˆ( ) x t .

Figure 2 .
Figure 2. The state trajectory of 2 ( ) x t and its estimation 2 ˆ( ) x t .