Robust Stabilization and H ∞ Control for Uncertain Neural Networks with Mixed Time Delays

This paper is concerned with the problem of robust stabilization andH ∞ control for a class of uncertain neural networks. For the robust stabilization problem, sufficient conditions are derived based on the quadratic convex combination property together with Lyapunov stability theory. The feedback controller we design ensures the robust stability of uncertain neural networks with mixed time delays.We further design a robustH ∞ controller which guarantees the robust stability of the uncertain neural networks with a givenH ∞ performance level.The delay-dependent criteria are derived in terms of LMI (linearmatrix inequality). Finally, numerical examples are provided to show the effectiveness of the obtained results.


Introduction
Neural networks have received a great deal of attention due to their successful applications in various engineering fields such as associative memory [1], pattern recognition [2], adaptive control, and optimization.When designing or implementing a neural network such as Hopfield neural networks and cellular neural networks, the occurrence of time delays is unavoidable in the processing of storage and transmission.Since the existence of time delays is usually one of the main sources of instability and oscillations, the stability problem of neural networks with time delays has been widely considered by many researchers (see [3][4][5][6][7][8][9][10][11][12][13]).Generally speaking, stability criteria of neural networks with time delays are classified into two categories: delay-independent stability criteria and delay-dependent stability criteria.Delaydependent stability criteria are less conservative than delayindependent ones.Therefore, people always consider the delay-dependent stability criteria.Neural networks usually have a spatial extent due to the presence of many parallel pathways of a variety of axon sizes and lengths [7].Thus, there will be a distribution of conduction velocities along these pathways and a distribution of propagation delays [14], and both the discrete and the distributed delays should be considered in the neural network model [6,7,[15][16][17][18].
However, in practical application of neural networks, uncertainties are inevitable in neural networks because of the existence of modeling errors and external disturbances.Parameter uncertainties will destroy the stability, so that taking uncertainty into account is important when studying the dynamical behaviors of neural networks (see [12,[19][20][21]).To facilitate the design of neural networks, it is important to consider neural networks with various activation functions, because the conditions to be imposed on the neural network are determined by the characteristics of various activation functions as well as network parameters [22].The generalization of activation functions will provide a wider scope for neural network designs and applications [23].Stability and stabilization results for delayed neural networks with various activation functions can be found in [22][23][24][25][26]. References [24,25] investigated the stability problem of neural networks with various activation functions.Phat and Trinh [23] dealt with the exponential stabilization problem for neural networks with various activation functions via the Lyapunov-Krasovskii functional.Nevertheless, the results reported therein do not consider the parameter uncertainties and disturbances.Sakthivel et al. [26] studied the problem of robust stabilization and  ∞ control for a class of uncertain neural networks with various activation functions and mixed time delays by employing the Lyapunov functional method and the matrix inequality technique.In recent years, control of time-delay systems is a subject of both practical and theoretical importance.The performance of a neural control system is influenced by external disturbances.Thus, it is important to use the  ∞ robust technique to eliminate the effect of external disturbances.The  ∞ control problem for time-delay systems has been addressed in [6,[26][27][28][29][30][31][32][33][34].However, to the best of our knowledge, the robust stabilization and  ∞ control for uncertain systems with time-varying delays have not yet been fully investigated.
In this paper, we consider the problem of robust stabilization and  ∞ control for a class of uncertain neural networks by employing a new augmented Lyapunov-Krasovskii functional and estimating its derivative from a novel viewpoint.Our aim is to obtain a  ∞ control law to guarantee the robust stability of the closed-loop system with parameter uncertainties and a given disturbance attenuation level  > 0. The results employ the quadratic convex combination technique, which is different from the linear convex combination and inverse convex combination techniques extensively used in other literature studies.The criteria are derived with the framework of LMIs, which can be easily calculated by the MATLAb LMI control toolbox.Numerical examples are provided to illustrate the effectiveness of the results.
Notations.The notations used throughout the paper are fairly standard.  denotes the -dimensional Euclidean space;  × is the set of all  ×  real matrices; the notation  > 0 (<0) means  is a symmetric positive (negative) definite matrix;  −1 and   denote the inverse of matrix  and the transpose of matrix ;  represents the identity matrix with proper dimensions, respectively; a symmetric term in a symmetric matrix is denoted by ( * ); sym() represents (+  ); diag{⋅} stands for a block-diagonal matrix.Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
In order to conduct the analysis, the following assumptions are necessary.

Robust Stabilization
We use the following control law to tackle the robust stabilization problem in this paper: where  1 are the gain matrix of the controller.
Proof.Construct a new class of Lyapunov-Krasovskii functional as follows: where where We define a vector   as ( Remark 1.Our paper uses the idea of second-order convex combination, and the property of quadratic convex function is given in Lemma 7. Remark 2. We fully consider the various activation functions in constructing the Lyapunov-Krasovskii functional.So the augmented vector   uses more information about (()), (()), and ℎ(()) than in [26].The Lyapunov functional in our paper is more general than that in [26], and the criteria in our paper may be more applicable.
Remark 3. In our paper, the augmented vector   utilizes more information on state variables than in [26], such as ẋ ( − ).This leads to reducing the conservatism of stabilization condition.
The time derivative of (, ()) along the trajectory of system is given by where It is easy to obtain the following identities: Therefore, we can disassemble the integral into two parts as follows: It is easy to show the following relation: ( Applying Lemma 5 to  1 (, ()), we get According to Assumption 2, we have which is equivalent to where   is the unit column vector. Let ] ⩽ 0,  = 1, 2, . . ., , and it is equivalent to Similarly, we obtain By using ( 31)-( 32), we have The following equality holds: which is equivalent to where  is any matrix.

Numerical Examples
In this section, numerical examples are provided to illustrate effectiveness of the developed method for uncertain neural networks with discrete and distributed time-varying delays.
Example 1.We consider the neural networks (12) when the disturbance input V() = 0.The parameters are as follows: and the activation functions are so that Figures 1 and 2 present the state responses of the considered neural networks.Figure 1 shows the time response of the state variables  1 () and  2 () of the open-loop system from initial values (1, −1). Figure 2 shows the time response of the state variables  1 () and  2 () of the closed-loop system from initial values (1, −1).The open-loop system means the system without feedback control, and the closed-loop system means the system with the feedback control.It is clear that  1 () and  2 () converge rapidly to zero under the feedback control law and they cannot converge to zero without the feedback control.The simulation results reveal that the considered system with discrete and distributed time-varying delays is robustly asymptotically stable under the feedback control law.they cannot converge to zero without the feedback control.
The simulation results reveal that the considered system with discrete and distributed time-varying delays is robustly asymptotically stable under the feedback control law.

Conclusions
In this paper, we investigated the robust stabilization problem and  ∞ control for a class of uncertain neural networks.By implementing the quadratic convex combination technique together with Lyapunov-Krasovskii functional approach, new delay-dependent conditions were established.The stabilization criterion was derived by the augmented Lyapunov-Krasovskii functional, which ensures the robust stability of the considered uncertain neural networks with various activation functions.Furthermore, our result was extended to the design of a robust  ∞ controller, which guarantees the closed-loop system robustly asymptotically stable with a prescribed  ∞ performance level.The criteria are derived in terms of LMIs, which can be easily calculated by the MATLAB toolbox.Numerical examples are provided to illustrate the effectiveness of the obtained results.

Figure 1 :Figure 2 :
Figure 1: State responses of the open-loop system.

Figures 3 and 4
Figures3 and 4present the state responses of the considered neural networks with the disturbance input V() = [1/(0.5+ ), 1/(1 +  2 )]  .Figure3shows the time response of the state variables  1 () and  2 () of the open-loop system from initial values (0.5, −0.5).Figure4shows the time response of the state variables  1 () and  2 () of the closed-loop system from initial values (0.5, −0.5).It is clear that  1 () and  2 () converge rapidly to zero under the feedback control law and