Dissipativity analysis of Markovian Switched Neural Networks using Extended Reciprocally Convex Matrix Inequality

This paper aims to present improved result of dissipativity analysis in Markovian switched neural networks with time-delays. By applying the novel extended reciprocally convex matrix inequality with the construction of Lyapunov- Kravsovskii functional and by using the linear matrix inequality technique, a set of sufficient conditions has been established to guarantee the existence of analysis. Numerical example with simulation is illustrated for the effectiveness of the proposed method.


INTRODUCTION
There has been increasing research interests in the dynamical analysis of neural networks (NNs) such as pattern recognition, associative memories, optimization and in the field of data mining, medical diagnosis and so on [1][2][3][4]. It is worthwhile to note that the majority of the research in todays' digital world is concerned with discrete-time systems than the continuous-time systems. This is due to the reason that discretization dynamics of the continuous-time cannot be preserved when compared with the dynamics of the discrete-time systems. Therefore, more emphasis has taken to concentrate on the discrete-time switched NNs.
Moreover, time delays are inevitably encountered as the source of oscillation and instability in the feature of regulatory transmissions between the neurons. Hence, great deal of research is focussed towards the study the robust asymptotic and exponential stability of NNs with time-delay [1,2,5,7,9]. In order to achieve the stability criteria among the NN's, the dissipativity analysis has been used. The concept of dissipativity in dynamical system has found a lot of applications in chaos, synchronization, robust control and in stability theory [8].
On the other hand, Markovian jump systems have attracted may researchers from Mathematics and control communities. These Markovian jump systems are more appropriate to model practical systems that is found in the power systems, network control systems and manufacturing systems etc [6]. A Markovian jump system is governed by a Markov process where a Transition Probability Matrix (TPM) is required. Here, we apply the concept of Markovian process to the switched systems that are also called the hybrid systems containing finite number of subsystems with divergent trajectories [4]. The concept of dissipative analysis in dynamical system has found a lot of applications in chaos, synchronization, robust control and in stability theory [5]. We apply the extended convex approach lemma so as to reduce the conservatism. It is to be noticed that this lemma holds good for discrete-time NNs.
On the basis of the above discussions, the objective of this paper is to study the problem on the dissipative analysis for Markovian switched NNs with time varying delays and generalized activation functions. By employing appropriate Lyapunov-Krasovskii functionals and using stochastic analysis method with respect to LMI technique, we obtain a new sufficient condition for checking the global dissipativity of the addressed NNs. is a probability space, : is the sample space, F is the sigma algebra of subsets of the sample space and P is the probability measure on F , Pr {} represents the probability. E{ } denotes the expectational operator with respect to some probability measure P and | . | refers to the Euclidean vector norm. The superscript " T " represents the transpose and "*" denotes the term that is induced by symmetry.

PROBLEM DESCRIPTION AND PRELIMINARIES
is the neural state vector and the activation functions of the neuron are given as is the output of the system.
The discrete-time homogeneous Markov chain is given by the parameter ) )( ( Z t t 9 that takes value in the finite state space S={1, 2,…, S} with the transition probability matrix are known constant matrices with appropriate dimensions and The activation functions in (1) are continuous and bounded and satisfies the following condition as

MAIN RESULTS
In this section, we apply the extended reciprocally convex matrix inequality to handle the delay product type of LKF.
when the LMI's given in based on the convex combination technique, Defining, , By differencing the above inequalities based on the expectation, we get Then by applying Lemma 1, the R1 dependent term is estimated as, Combining the right side of equations after evaluations, Considering the activation functions which satisfies (4), and  (1), the rate function of the quadratic supply is given as Combining all the inequalities from (10) to (18), we get, The time-varying delay ) (t d is satisfied with equations (1) and (2) and the inequality 0 )) ( , provided all the equations in (8) The following inequality holds good if LMI (9) is true, We observe that from the equation (21), it follows that, , which therefore implies that system (1) is globally asymptotically stable. Hence, the proof is complete.