Exponential stability of discrete‐time delayed neural networks with saturated impulsive control

Funding information National Key Research and Development Project, Grant/Award Number: 2018AAA0100101; National Natural Science Foundation of China, Grant/Award Numbers: 61633011, 61873213 Abstract This paper examines the problem of the locally exponentially stability for impulsive discrete-time delayed neural networks (IDDNNs) with actuator saturation. By fully considering the delay information of the state of the considered system, a new delay-dependent polytopic representation within a discrete-time framework is obtained. Based on the delayindependent polytopic representation approach, the saturation term is expressed as a delaydependent convex combination. In order to obtain some less conservative stability conditions and estimate a larger of the domain of attraction, a novel type of Lyapunov– Krasovskii function (LKF) dependent on the delay information and the impulses instant is proposed, which is called time-dependent LKF. Then, by combining with the proposed LKF, a discrete Wirtinger-based inequality, an extended reciprocally convex matrix inequality and some novel analysis techniques, several new exponential stability criteria dependent on the bounds of the delay are presented. Moreover, when saturation constraints are not considered in the impulsive controller, the stability of the system is also discussed. Finally, two examples are given to confirm the applicability of the proposed results.


INTRODUCTION
Neural networks (NNs) have gained considerable research attention in the last few decades because of its wide application in many disciplines including associative memories, pattern recognition, optimization algorithms, and other scientific [1][2][3][4][5][6][7][8]. In addition to this, time delay is inevitable in practical network communication owing to the limited speed of the processor/amplifier [11]. Common types of time delays mainly include discrete delays, distributed delays and neutral delays, and they are generally regarded as main factors causing leading even to its instability. Consequently, it is more meaning to consider delayed NNs. Currently, the research on the dynamics of time-delayed neural networks has received extensive attention, some selected works on NNs with delays are [12][13][14][15][16][17]. Note that the aforementioned NNs are continuous-time dynamical system described by non-linear functional differential equations. However, when the damper subjected to the percussive effects, the change of the shutter speed during the transition of the valve from open to closed state, the fluctuations of the pendulum system under the external impulsive effects, the use of radial acceleration to control the satellite orbit, the changes of chemical reaction rate when adding or removing catalyst, the disturbances in cellular neural networks, impulsive external intervention and optimization problems in population dynamics, the death in the populations as a result of impulsive effects, 'Shock' changes of the prices in the completely free market [24][25][26][27] etc. Note that this short-term sudden change or disturbance may be a feature of the system itself, or it may be an intervention from the external, such as impulsive control [28]. In order to describe this kind of phenomenon more accurately and seek the law of future evolution, the impulsive differential equation is usually used as an appropriate mathematical tool to establish the mathematical model of this kind of problem. Impulsive control, as a kind of discontinuous control strategy, has been widely used in some practical applications, such as ecosystems management, currency supply control in the financial market, stabilization and synchronization of neural networks, and other fields [29][30][31][32][33][34][35][36]. Compared with continuous control strategies such as adaptive control and sampled-data control, impulsive control only needs to be implemented at discrete instants rather than input the control signal at every moment. Hence, impulsive control has the distinctive superiorities of reducing control cost, improving confidentiality and robustness, and being easy to realize in practical operation. By now, impulsive control as an effective control strategy to obtain stability and synchronization of dynamic network has been researched by many scholars [37][38][39][40][41][42][43]. Nevertheless, it is find that the above results did not consider the constraint of actuator saturation in the corresponding impulsive control strategy.
In the actual control problems, actuator saturation is a nonnegligible problem due to the actuators are physically constrained by energy, space or specific actuator structure. Otherwise, ignoring actuator saturation when designing the controllers may lead to control failure or difficult to achieve the desired control goal [44,45]. The performance analysis of dynamic systems subject to saturated actuators has become a very important research hotpot in the field of control engineering, many interesting results have been published [46][47][48][49][50][51][52][53]. In the current work, there exist two main methods to deal with the non-linear saturation term, that is the polytopic representation approach and the generalized sector non-linearity model approach. In order to obtain some less conservative stability criteria, the authors proposed the delay-dependent polytopic approaches and the mixed-delay dependent sector conditions to deal with non-linear saturation term by considering the state information of time delays of the system saturation non-linearities in [54][55][56][57][58][59]. For example, in [56,58], the authors investigated the local stabilization problem for DDNNs subject to saturated actuators by applying delay-dependent polytopic method, and demonstrated that the polytopic approach can reduce the conservativeness more effectively than delayindependent polytopic method. Different from the delaydependent polytopic method given in [56], the authors studied the synchronization problem for a class of DDNNs with saturated actuators by applying the proposed delay-dependent sector conditions in [54]. However, the authors did not consider the stability of system under the impulsive control of actuator saturations. Based on the constraints of actuator saturation and the advantages of impulsive control, the synchronization problem of delayed coupled discrete-time networks via partial input-saturated impulsive control have been addressed by using the delay-independent sector conditions in [60]. Ouyang et al. [61] dealt with the problem of impulsive synchronization for coupled delayed NNs with actuator saturation by applying the sector non-linearity model method and its application to image encryption. By using the polytopic representation approach, the authors studied the impulsive stabilization problem of NNs with actuator saturation in [62]. Up to now, it is still a challenging topic to consider the actuator saturation into impulsive controller design.
So far, to the extent of author's knowledge, no results exist to study the exponential stability problem for IDDNNs with actuator saturations. The authors have proposed a novel delaydependent polytopic representation, by using some summation inequalities and novel time-dependent Lyapunov-Krasovskii function (LKF), some delay-dependent exponential stability criteria of the IDDNNs with actuator saturations are obtained. In comparison with the existing works, the contributions are as follows: 1) A novel impulsive controller with actuator saturations is developed, and an IDDNNs with input saturation is established. Compared with the IDDNNs without input constraints proposed in [32,37,63], the considered system has more practical significance. 2) To gain some less conservative results, the time delay information of the consider system is fully considered, and a novel delay-dependent polytopic representation within a discrete-time framework is proposed. Compared with the delay-independent polytopic representation approach, some looser stability conditions and lager estimation of the domain of attraction can be obtained by using the delay-dependent polytopic representation method. 3) Delay-dependent exponential stability conditions are proposed for IDDNNs with actuator saturation, which are less conservative as shown in the numerical examples. It is worth mentioning that an impulsive controller without actuator saturation and under same assumptions can achieve exponential stabilization of DDNNs with time-varying and constant delays.
The remainder of this paper is framed as follows. In Section 2, some preliminaries and the necessary lemmas are presented, and the IDDNNs with input saturation introduced. In Section 3, some stability conditions on the basis of linear matric inequalities (LMIs) are established. Moreover, some important corollaries are obtained. Then, in Section 4, two numerical examples are presented to verify the feasibility and correctness of the obtained results. Finally, the main conclusions of this article are summarized, and the upcoming research work is prospected in Section 5.
Notations: For any matrix ,  −1 and A T denote the inverse and transpose, and  > 0 or  < 0 indicates that  is positive definite or negative definite. Smy() is defined as the matrix  +  T . min () and max () mean separately the minimal and maximal eigenvalue of matrix .  (l ) denotes the l th row of the matrix . ℝ n denotes the n-dimensional linear vector space over the field of real number ℝ. ℝ n×n denotes the set of all n × n real matrices. n and + n denote the sets of n × n real symmetric matrices and n × n real positive definite symmetric matrices, respectively. The symmetric block in a matrix is rep- T . I and 0 denote identity and zero matrices with compatible dimension. ℕ represents the set of positive integers. ℕ 0 = {0} ∪ ℕ. ∈ ℝ n×n denotes a diagonal matrix with diagonal elements either 1 or 0. For given two integers r and s, I[r, s] is expressed as the set {r, r + 1, … , s}.
In order to deal with the saturation terms Sat(K (n k − 1)), we will use the polytopic representation approach. Thus, an important lemma can be stated as follows: Lemma 1 [44]. Let u ∈ ℝ n and v ∈ ℝ n be two given vectors. If ‖v‖ ∞ ≤ū, then where 'co' denotes the convex hull, For a given matrix H ∈ ℝ n×n , we define a polyhedron set According to Lemma 1, if (n k − 1) ∈ ℰ holds, there exist positive constants ℏ ∈ [0, 1] satisfying where 'co' denotes the convex hull, In what follows, we will improve the condition (10) based on Lemma 1 by considering the information about the time delays and the current state of (6) within the interval By Lemma 1, if the following constraint conditions hold: Then, the saturation non-linearity Sat(K (n k − 1)) can be rewritten as Remark 1. Based on the existing results [54,57], by fully utilizing the time-delay information of the DDNNs (1), the distributed-delay-dependent terms E ∑ n k −2 ) are additionally introduced in the linear convex combination (10). Different from the existing techniques [55,56,62,64], the condition (10) deals with discrete delay in a relatively simplified framework, and it should be more suitable for analyzing large-scale networks. In addition, the introduction of slack variables G and J can further improve the conservativeness of the results.
To derive the main results, the following important definition and lemmas are needed in the sequel.

MAIN RESULTS
In this section, some new stability criteria for the system (6) are derived by using suitable LKF.
In order to estimate the upper bound of ΔV (s), we introduce the following auxiliary vector: Then, one could derive that (s + 1) = 1 (s), (s) =  2 (s).
Remark 4. For sampled-data systems, there is a special approach named time-dependent Lyapunov functional approach which has widely used [73][74][75] because of its effectiveness for reducing conservatism. Moreover, some complex augmented-based LKFs containing multiple summation terms are constructed in [22, 56 67, 76, 77] in order to obtain the delayed-dependent stability criteria. Inspired by these viewpoints, in order to better deal with the estimation of (n k ) and the processing of saturation terms, we established a new time-dependent LKF (42) by introducing (s) = (s − n k )(s − n k + 1), so that it could obtain more relaxed stability conditions. Remark 5. When E = F = 0, if the constraint condition (n k − 1) ∈ ℰ holds, the polytopic representation (13) is transformed into (10). In the numerical example, we will check that the delay-dependent polytopic representation (13) can more effectively reduce the conservativeness of the stability criterion and obtain greater the domain of attraction compared with the delay-independent polytopic representation (10).
For the case of Sat(u(s)) = u(s), which implies that the impulsive controller (5) without input saturation. Let the function 1 (s) in LKF (42) be updated tō1(s) = (s) T P (s), where P ∈ + n . Based on the LKF̄(s) =̄1(s) + 2 (s) + 3 (s), similar to the proof of Theorem 1, we can obtain the following corollary.
Proof. If the LMI (57) is feasible, it can be deduced that the matrix G 2 is invertible. Letting G 2 = G 1 and X = G −1 1 , and introducing the following new variables: Left-multiplying and right-multiplying the LMIs (56) and (57) by X T and X , and noting (62), the LMIs (59) and (60) can be obtained. Left-multiplying and right-multiplying the LMI (58) by X T and X −1 , the LMIs (61) hold. □ It needs to be pointed out that for the case of constant delay, that is (s) ≡ , inequality (29) is no longer true, so the stability condition of Theorem 1 cannot be directly applied to the constant delay case. Therefore, it is necessary to further analyze the stability of the system (6) with constant delay. Similar to (11), we define the functional Under the constraint conditions the delay-dependent polytopic representation (13) can be reexpressed as follows: Moreover, we choose the following LKF:
For the case of Sat(u(s)) = u(s), according to Theorem 2, the following result can be deduced.

Corollary 3. For given constants
}, and < exp{− 2 ln }. If there exist some matrices P ∈ + n , U , W ∈ + n , G i ∈ ℝ n×n , and n × n positive diagonal matrices Q i , i = 1, 2, such that the LMI (58) and the following LMI holds: Then, the constant delay system (6) without actuator saturation is exponentially stable with a decay rate − Based on Corollary 3, the following corollary can be obtained by performing some congruence transformations to LMIs (58) and (69).
Remark 6. Theorem 1 and Theorem 2 achieve the reduction of conservatism with the cost of an increase in computational complexity. In terms of reducing the computational burden, this paper uses the extended reciprocally convex matrix inequality (15) to estimate the summation term appearing in the forward difference of the Lyapunov function, and reduces the computational complexity of the widely used reciprocally convex matrix inequality without requiring any extra decision variable.

NUMERICAL EXAMPLES
In this section, two numerical examples are given to exhibit the correctness of the proposed results, and to discuss the advantages and effectiveness of the method proposed.

Example 1.
Consider the following two-neuron DDNNs The activation functions f 0 ( (s)) and f 1 ( (s)) satisfy Assumption A1 with the parameters 0 = 1 = diag(0, 0) and  0 = 1 = diag(1, 1). It is clear that m = 2 and M = 4. Let n k = 4k, which implies 1 = 2 = = 4. Suppose that = 0.3, = 0.7, = 0.8,ū i = 1 (i = 1, 2), it is easy to judge that > max{ , It can be seen from Figure 1 that the state evolution of the network (71) in the absence of impulsive control cannot converge to the origin. In order to stabilize the network (71) to the origin, we choose the impulsive control gain matrix  (5) is locally exponentially stable. Figure 2 shows the state evolutions of the network (71) are exponentially convergent to origin under saturated impulsive control (5). In the case of 1 = 2 = , by solving the LMIs (17)- (20) in Theorem 1, the estimation of the domain of attraction can be obtained as which is shown by the solid red ellipse in Figure 3. When  Figure 3. For the two cases of E ≠ 0 or F ≠ 0 and E = F = 0, it can be seen from Figure 3 that the delay-dependent polytopic representation approach provides a lager estimation of the domain of attraction ( , ) in Theorem 1 than the one in Theorem 1 with E = F = 0, which implies that the delay-dependent polytopic representation (13) is more effective in reducing the conservatism than the polytopic representation (10). When we do not consider the factor of actuator saturation, according to Corollary 1, we can know that the system (71) is exponentially stable under the pulse controller (5) as long as there is a feasible solution for LMIs (56)- (58). Solving LMIs (56)-(58) by the LMI Toolbox in Matlab show that there are feasible solutions, which means that the system (71) can achieved exponential stability under the non-saturated impulsive controller (5), as shown in Figure 4. From Figures 3  and 4, we can see that actuator saturation will inhibit the convergence speed of the system (71) under impulsive control, which means that the actuator saturation reduces the performance of the impulsive controller (5).

Example 2.
Consider the following three-neuron discrete-time DDNNs where (s) = [ 1 (s), 2 (s), 3  Suppose that n k = 3k, = 3, = 0.3, = 0.5, = 0.8,ū i = 2 (i = 1, 2), it is easy to know that > max{ , According to Theorem 2 and Matlab LMI Toolbox, one could easily see that there do exist the feasible solution to the LIMs (66)- (67), which means that the network (72) is exponentially stable under the saturated impulsive control (5). Figure 5 shows the state evolution of the network (71) without impulsive control. Figure 6 shows the state evolution of the network (72) with saturated impulsive control (5). By the feasible solution of the LIMs (66)-(67), the initial conditions for guaranteeing the stability of the network (72) with saturated impulses is bounded by When E = 0, max is estimated as 35.7944. Figure 7 shows that the attraction domain ( , ) in Theorem 2 is larger than the one in Theorem 2 with E = 0. This means that the delaydependent delay-dependent polytopic representation is more effective in reducing the conservatism than the polytopic representation (10).

CONCLUSION
This work has addressed the local exponential stability analysis for IDDNNs with actuator saturations. By using a novel polytopic representation approach, a new time-dependent LKF and some summation inequalities, some delay-dependent sufficient conditions have been obtained to ensure exponential stability of the considered systems. Moreover, under impulsive controller without actuator saturation, the exponential stabilization problem of DDNNs with time-varying and constant delays has also been studied. Finally, two numerical examples have verified the validity and low conservativeness of the obtained results relative to the existing results. Although the importance of actuator saturation has been valued by more and more scholars, the results of saturated impulsive control are very rare. This is because it is very challenging to deal with saturation non-linearity at the moment of impulse. Therefore, it is still an open question to consider the actuator saturation into impulsive controller design. For the first time, the factor of actuator saturation is considered in the discrete-time delay neural network with impulsive input, and the delay-dependent stability criterion is given by constructing a new time-dependent Lyapunov function, using delaydependent polytopic representation and some matrix inequalities. This paper considers time-varying time delay and actuator saturation, which is more general than the results of [32]. In addition, we use the LKF method to obtain the system's delaydependent stability criterion, which is less conservative than [32].
For the saturated impulsive systems, in the future work, we explore some new research methods to further reduce the conservativeness of the conditions, reduce the computational burden caused by additional decision variables, and expand the estimation of the attraction domain. Moreover, state-dependent impulsive system with input saturation is also our next research work.