Exponential synchronization control of delayed memristive neural network based on canonical Bessel-Legendre inequality

1.   Introduction

In recent years, MNNs has attracted extensive attention because of its broad application prospect in engineering fields such as associative memory, signal processing and pattern recognition ^[1,2,3,4]. Resistors are used in real systems to simulate the synapses of neural networks and store the historical state of the system. Due to the influence of large volume resistance on the integral density of neural network, the function of synapse cannot be completely simulated. This causes the calculation of neural network circuit to deviate from the real value ^[5]. Memristors were first proposed by Professor Chua in 1971, and the first physical memristors were developed by HP LABS in 2008. The value of the memristor is determined by the voltage applied to the memristor, its polarity, and the duration of the applied voltage ^[6]. In 2010, Professor Lu constructed the first electroneural circuit using memristors and verified the memory properties of memristors to the internal states of the system ^[7,8]. This characteristic suggests that memristors are the electronic components that most closely resemble the synapses of neurons. MNNs was developed by replacing the resistors in traditional neural networks with memristors.

Compared with traditional neural network, memristor neural network has more powerful information capacity and computing power, which enables it to better deal with problems related to state memory and information processing ^[9]. In most cases, the control signal is transmitted from one system to another in the wireless communication network, so the nonlinear dynamic behavior will inevitably occur in the control process and the system time delay will be generated ^[10,11]. When the time lag interval is very large, the design of the stability criterion with time-delay independent will be very conservative ^[12]. Time delay may also lead to the instability, shock and performance deterioration of the neural network system ^[13]. Therefore, in order to reduce the conservativeness of the results, the consideration of time delay is an indispensable condition for the design of controllers to keep the system stable.

Synchronization means that with the development of the network, the state of the driving system and the response system tend to be in a common state, which is the most basic and important dynamic characteristic of the neural network ^[14]. System synchronization control is often used in communication security, biological systems, signal processing and other fields ^[15]. Many researchers are studying synchronization control schemes for various neural networks ^[16,17,18]. In ^[16], a distributed event-triggered controller is designed to study fixed-time synchronization of coupled MNNs. In ^[17], the fixed time synchronization problem of coupled MNNs based on decentralized event triggering scheme is studied. In ^[18], the author designed an exponential attenuation switching event-trigger scheme to study the global stabilition of delayed MNNs. Inspired by ^[19], this paper designs a new discontinuous feedback control scheme, and a more strict inequality is used to reduce the conservatism. The upper bound of control gain can be reduced by adjusting two adjustable scalars of the controller. Therefore, the controller application here is more flexible.

In the past, Jensen's inequality has been widely applied as a method to reduce the conservatism of the stability criterion for systems with time delay ^[20]. However, the integral inequality based on Wirtinger proposed by Seuret and Gouaisbaut was considered to be less conservative than Jensen's inequality, and contains Jensen's inequality ^[21]. And, many researchers have improved the estimation methods of integral terms, such as Wirtinger inequalities ^[22], integral inequalities based on free matrices ^[23], integral inequalities based on auxiliary functions ^[24]. Recently, the B-L inequality has received more and more attention. It generalizes the above integral inequality and has smaller amplification for some integral terms ^[25]. However, the integral interval of this inequality is fixed $[-h, 0]$. Therefore, B-L inequality has a limited range of applications. The canonical B-L inequality transfers the integral interval to the general interval $[a_1, a_2]$ by introducing a canonical orthogonal polynomial ^[26]. At present, most results for the stability of delayed neural networks are to choose a special Bessel-Legendre inequality (N = 1 or N = 2), as mentioned in reference ^[27]. In order to further reduce the conservatism of the result, Legendre vector information is fully considered in the design of the augmented LKF. By using the canonical B-L inequality and anti-convex inequality lemma, the criterion of exponential synchronization of the system is obtained.

This paper deals with the exponential synchronization control of delayed MNNs based on discontinuous feedback controllers. We take the average of the maximum and minimum weights of the memristor synapse as the weight of the memristor synapse and transform the state parameters of the memristor neural network into the traditional neural network with uncertain parameters. The main contributions of this paper are as follows:

(1) In order to solve the parameter mismatch problem of MNNs, intermittent control scheme is adopted in the synchronization of master-slave neural network system to relax the strict assumptions. The designed discontinuous controller has two tunable scalars, which can adjust the upper limit of control gain flexibly and reduce control cost.

(2) In order to reduce the conservatism of the results, this paper chooses to use canonical B-L inequality to estimate the bounds of the integral term. It has been shown in ^[28] that if the LKF used is less relevant to the B-L inequality, the effect of obtaining tighter bounds of the inequality will be greatly reduced. Therefore, this paper constructs a suitable augmented LKF and fully considers the Legendre polynomial information.

Finally, the stability criterion with low conservatism is obtained by using canonical B-L inequality, and the conservatism decreases with the increase of N.

Notations: In this paper, the superscript $T$ of the matrix is the transpose of the matrix, and $-1$ is its inverse. The $\mathbb{R}^n$ means the $n$-dimensional Euclidean space. $\mathbb{S}^n_+$ means the set of the positive definite matrices of $\mathbb{R}^{n\times n}$. The symbol $\|\cdot\|$ refers to the Euclidean vector norm. There $\binom{j}{i} = \frac{j!}{(j-i)!i!}$.

2.   Problem formulation

The time-delay neural network can be realized by large-scale integrated circuits using memory resistors, which represent connection weights. According to Kirchhoff's current law, the equation of the pth subsystem can be described as follows:

where $p = 1, 2, ..., n$, $x_p(t)$ is the voltage of the capacitor $C_p$ at $t\geq 0$, and $R_p$ is resistance in parallel with $C_p$. $I_p$ is the external input or bias, and $l_q$ is the activation function. $h_q(t)$ is the time-varying delay of the transmission of the $qth$ neuron, and it satisfies

$M_{fpq}$ is the memristor connecting the activation function $l_q(x_q(t))$ to $x_p(t)$, $M_{gpq}$ is the memristor connecting the activation function $l_q(x_q(t-h_q(t)))$ to $x_p(t-h(t))$. The memductance of memristors $M_{fpq}$ and $M_{gpq}$ is expressed in terms of $W_{fpq}$ and $W_{gpq}$ respectively.

The initial conditions for system (1) are $x_p(t) = \phi_p(t)\in\mathcal{C}([-h, 0], \mathbb{R})$, $h = max_{1\leq q\leq n}\{h_q\}$, $\mu = max_{1\leq q\leq n}\{\mu_q\}$.

As described by Chua ^[29], depending on the computer information being encoded only in '0' and '1', the memristor only needs to display two different states. System (1) can be rewritten as

where

and the switching jumps $\mathcal{T}_p > 0$, $\acute{b}_{pq}$, $\grave{b}_{pq}$, $\acute{d}_{pq}$, and $\grave{d}_{pq}$, $p, q = 1, 2, ..., n$, are constants.

Consider system (3) is taken as the driving system, and the corresponding responsive system is:

where

where $u_p(t)$ is the appropriate control input to be designed, the initial condition of system (4) are $y_p(t) = \varphi_p(t)\in\mathcal{C}([-h, 0], \mathbb{R}), p = 1, 2, ..., n$.

It can be seen that $b_{pq}(x_p(t))$ and $d_{pq}(x_p(t))$ are discontinuous, and the system (3) is a discontinuous switching system. In this case, the classical solution is not available. Therefore, the solution of this system is processed in Filippov sense. Now, the following definition is given.

Definition 1. ^[30] Consider the system $\dot{x}(t) = F(x)$, $x\in{R}^n$ with discontinuous right-hand sides, a set-valued map is defined as

where $\overline{co}[G]$ is the closure of the convex hull of set $G, B(x, \mu) = \{y:\|y-x\|\leqslant \mu\}$, and $\delta(R)$ is the Lebesgue measure of set R. A solution in Filippov's sense of the Cauchy problem for the above system with initial condition $x(0) = x_0$ is an absolutely continuous function $x(t), t\in[0, T]$, which satisfies $x(0) = x_0$ and differential inclusion: $\dot{x}(t)\in\Phi(x)$, for a.e. $t\in [0, T]$

The research in this paper requires the following assumptions.

Assumption 1. There exist constants $m_p$, such that for any $u\in\mathbb{R}$, $|l_p(u)| \leq m_p$, $p = 1, 2, ..., n$ is true.

Assumption 2. The neuron activation function in system (3) satisfies $l_q(0) = 0$ and

where $\sigma^-_q, \sigma^+_q$ are some constants, and $K_1 = diag\{\sigma^-_1, ..., \sigma^-_n\}$, $K_2 = diag\{\sigma^+_1, ..., \sigma^+_n\}$

Remark 1. Obviously, system (3) is a discontinuous state-dependent switching system. "Filippov proposes that the solutions of discontinuous systems have the same set of solutions contained in the definite derivatives ^[30]." According to the theory of differential inclusion and the definition of Filippov solutions, system (3) can be rewritten as:

where $co\{\acute{b}_{pq}, \grave{b}_{pq}\} = [\underline{b}_{pq}, \overline{b}_{pq}]$, $co\{\acute{d}_{pq}, \grave{d}_{pq}\} = [\underline{d}_{pq}, \overline{d}_{pq}]$, $\underline{b}_{pq} = min\{\acute{b}_{pq}, \grave{b}_{pq}\}$, $\overline{b}_{pq} = max\{\acute{b}_{pq}, \grave{b}_{pq}\}$, $\underline{d}_{pq} = min\{\acute{d}_{pq}, \grave{d}_{pq}\}$, $\overline{d}_{pq} = max\{\acute{d}_{pq}, \grave{d}_{pq}\}$.

Denote the interval matrices $[\underline B, \overline B] = [\underline{b}_{pq}, \overline{b}_{pq}]_{n\times{n}}$, $[\underline D, \overline D] = [\underline{d}_{pq}, \overline{d}_{pq}]_{n\times{n}}$. The system (6) can be rewritten as the following matrix form

where $x(t) = [x_1(t), x_2(t), ..., x_n(t)]^T\in \mathbb{R}^n$, $l(x(t)) = [l_1(x_1(t)), l_2(x_2(t)), ..., l_n(x_n(t))]^T\in \mathbb{R}^n$, $l(x(t-h(t))) = [l_1(x_1(t-h_1(t))), l_2(x_2(t-h_2(t))), ..., l_n(x_n(t-h_n(t)))]^T\in \mathbb{R}^n$, $\tilde{L}(t) = [\tilde{L}_1(t), \tilde{L}_2(t), ..., \tilde{L}_n(t)]^T\in \mathbb{R}^n$, $K = diag\{k_1, k_2, ..., k_n\}$, there are measurable function $B(x(t))\in [\underline B, \overline B]$, $D(x(t))\in [\underline D, \overline D]$ such that system (8) can be of the form

Similarly, from system (4) we have

where $B(y(t))\in [\underline B, \overline B]$, $D(y(t))\in [\underline D, \overline D]$, $U(t) = [u_1(t), u_2(t), ..., u_n(t)]^T\in \mathbb{R}^n$.

Define the synchronization error $e(t) = y(t)-x(t)$. Then we can obtain the following synchronization error system

where $f(e(t)) = l(y(t))-l(x(t))$, $f(e(t-h(t))) = l(y(t-h(t)))-l(x(t-h(t)))$, $N(t) = [B(y(t))-B(x(t))]l(x(t))+[D(y(t))-D(x(t))]l(x(t-h(t)))$

After the state measurements of the master-slave system are transmitted to the processor, the synchronization error $e(z_k)$ is calculated and used to construct the controller

where $\tilde{K}$ is the controller gain to be determined. The updated control parameters are transmitted to the zero-order hold (ZOH) over the communication network.

Definition 2. ^[31] If the error system (15) is exponentially stable, then the primary system (9) and the slave system (10) are exponentially synchronized. There are two positive scalars $\alpha$, $\beta$ that satisfy

where $\alpha, \beta$ are the exponential decay coefficient and decay rate.

Lemma 1. ^[32] There exist matrix G, H and the time-varying matrix Z(t) with appropriate dimensions, for given time-varying matrix $B(t)\in [\underline{B}, \bar B]$, and $\underline{B}\in\mathbb{R}^{n\times n}$, $\bar B\in\mathbb{R}^{n\times n}$, such that

and $Z^T(t)Z(t)\leq I$.

Remark 2. Let,

according to Lemma 1, we have $B(x(t)) = \tilde{B}+G^bZ^1(t)E^b$, $D(x(t)) = \tilde{D}+G^dZ^2(t)E^d$, $B(y(t)) = \tilde{B}+G^bZ^3(t)E^b$, $D(y(t)) = \tilde{D}+G^dZ^4(t)E^d$, and $(Z^i(t))^TZ^i(t)\leq I$.

Then, the error system (11) can be written in the following form:

where $N(t) = (G^b(Z^3(t)-Z^1(t))E^b)l(x(t))+(G^d(Z^4(t)-Z^2(t))E^d)l(x(t-h(t)))$.

Lemma 2. ^[33] For a symmetric matrix $A = \begin{bmatrix}A_{11}&A_{12}\\ * &A_{22} \end{bmatrix}$, where $A_{11}\in\mathbb{R}^{n\times n}$, the following conditions are equivalent:

We introduce two lemmas that are critical to the results of this paper.

Lemma 3. ^[26] (Canonical Bessel-Legendre inequalities)For any n-dimensional positive definite matrix $P$ $(P\in\mathbb{R}^{n\times n})$, any positive integer $N\geq0$, $a_1 < a_2$, and $e\in\mathscr{L}_2([a_1, a_2] \rightarrow \mathbb{R}^n)$, the inequality

holds, where

with $\xi = col\{e(a_2), e(a_1), \dfrac{1}{c_2-c_1}\bar\xi\}$, $\bar\xi = \{\int_{c_1}^{c_2}e(s)ds, \cdots, \int_{c_1}^{c_2}(\dfrac{c_2-s}{c_2-c_1})^{N-1}e(s)ds\}$, and $p^i_k = (-1)\binom{i}{k}\binom{i+k}{k}$.

Lemma 4. ^[34] For $\omega_1, \omega_2\in \mathbb{R}^m$, $\alpha\in (0, 1)$, and given $m\times m$ constant real matrice $\aleph_1 > 0, \aleph_2 > 0$, the following inequality is satisfied for any $Y_1, Y_2\in\mathbb{R}^{m\times m}$

Lemma 5. ^[35] For any vector $x, y\in \mathbb{R}^n$, matrices G, E and Z which are real matrices of appropriate dimensions, $Z^T(t)Z(t)\leq I$, and scalar $\varepsilon > 0$, the following inequality holds:

Lemma 6. ^[36] For a matrix $R\in \mathbb{S}^n_+$, scalars $m$ and $n$ with $m < n$, and vector $x$, the following inequality holds

3.   Main results

To simplify the processing of problems, the following terms for vectors and matrices are defined as

Theorem 1. For given scalars $h > 0$, $v_{1}\geq0$, $v_{2}\geq0$, $\mu$, $\omega\geq0$, $N\in\mathbb{N}$. When the error system (15) under the control law (12) is stable at the attenuation rate exponential, this ensures exponential synchronization between the primary and secondary systems, if there exist matrices $P > 0$, $Q_i > 0, i = 1, 2$, $Q_{iN} > 0, i = 3, 4, 5$, $U > 0, U_{1} > 0$, diagonal matrices $\bar{H} > 0$, $\Lambda_i > 0, i = 1, 2, 3$, $\Upsilon_i > 0, i = 1, 2, 3$, $\mathcal{L}_i > 0, i = 1, 2, 3$ and appropriate dimensional matrices $\tilde{H}$, $Y_{1N}, Y_{2N}$ and $\Gamma_N$ such that

In addition, the expected controller gain is as follows

where

Proof. At first, the following augmented LKF candidate function is constructed:

where

and

Via differentiating Eq (27) along the trajectory of the system, we have

Combining with the Eq (29), we can get

Now, using Lemma 3, the integral term of the equation satisfies the following conditions:

Using Lemma 4, the following inequalities can be further calculated

Use Lemma 6 to deal with the following double integrals

Based on system (15), for any positive diagonal matrix $\bar{H}$, there are:

The equation mentioned above can also be written as

where $\tilde{H} = \bar{H}\tilde{K}$.

Form Lemma 5, for any scalar $\varepsilon_1 > 0$, $\varepsilon_2 > 0$, we have

Form Assumption 1 and Eq (25),

On the other hand, from Assumption 2, we can get

which is equivalent to

where $\varrho_p = col\{0, ...0, 1_p, 0, ..., 0\}$. So you can get

Similarly, we can get the following inequality

Form Eq (21), it is obvious that

Then for any appropriate dimension $\Gamma_N$,

Combined with the above analysis, the following conclusions can be drawn

where

Since $\Xi_N(h(t), \dot{h}(t))$ is linear on both $h(t)$ and $\dot{h}(t)$, $\Xi_N(h(t), \dot{h}(t)) < 0$ is satisfied for any $(h(t), \dot{h}(t))\in[0, h]\times[\mu_1, \mu_2]$ if it holds at the four vertices $(0, \mu_1), (0, \mu_2), (h, \mu_1), (h, \mu_2)$, which is

According to the Lemma 2, it can be obtained Eq (22) and (23), and $\Xi_N(h(t), \dot{h}(t)) < 0$, so we know from Eq (48) that

In addition, from Eq (27), we have

where

By Eq (27), (49), (50) we can obtain that

Therefore

Accoding to Definition 2, system (9) and system (10) is globally exponentially synchronized when the control law is (12). So that's the proof of the Theorem 1.

To prove the correctness of the theorem, we will give two numerical examples in the next section.

4.   Numerical examples

Example 1. Consider MRNN with the follows parameters: $k_1 = k_2 = 1$, $l_1 = l_2 = 0$, $l_i(x_i(t)) = tanh(x_i(t)), i = 1, 2$ where

The activation functions $l_i(x_i(t)) = tanh(x_i(t))$ satisfy Assumption 1 and 2 with $\sigma^-_1 = \sigma^-_2 = 0$, $\sigma^+_1 = \sigma^+_2 = 1$, $m_1 = m_2 = 1$. And, we have

As can be seen from Tables 1 and 2, when $N = 1$ $h = 1$, $\mu = 0.25$, $\lambda = 0.5$, $v_2 = 1$ and $v_1$ increases, the upper limit of control gain decreases, indicating that the controller Eq (12) is more flexible. When $N = 1$ $h = 1$, $\mu = 0.25$, $\lambda = 0.5$, $v_1 = 1$, the upper limit of control gain decreases when $v_2$ decreases. Different controller gain $\tilde{K}$ can be obtained by adjusting $v_1, v_2$, and the appropriate controller gain can be selected according to the control requirements. Table 3 shows some comparisons of control gains with different $h$ values. And it's worth noting that ^[4,37] requires $\mu < 1$. Therefore, compared with the existing synchronization standard in ^[4,37], our results are less conservative.

When the initial values of system (9) and (10) are set as $x(t) = [2\quad-1.2]^T$, $y(t) = [-0.5\quad 0.5]^T$, $v_1 = 1$, $v_2 = 1$, $\omega = 0$, $N = 1$, the synchronization trajectory of the master and slave system without controller is shown in Figure 1, the synchronization error is shown in Figure 2, and the state response of the master system is shown in Figure 3. From the figure, we can see that the driver system and the response system are non-synchronization.

When the initial values of system (9) and (10) are set to $x(t) = [2\quad -1.2]^T$, $y(t) = [-0.5\quad 0.5]^T$, $v_1 = 1$, $v_2 = 1$, $\omega = 0$, $N = 1$, the control law (12) is used to obtain the controller gain as show below,

The synchronous trajectories of the drive system and the response system are shown in Figure 4, the synchronization error of the driving system and the response system is shown in Figure 5.

As shown in Figure 6, the synchronization time is reduced by comparing the synchronization process when N = 1 in Figure 5 with that when N = 2 in Figure 6. By combining the values given in Figures 5 and 6 and Table 4, it can be seen that the conservatism of the results decreases with the increase of the order of the Bessel-Legendre inequality. Moreover, the synchronization time of the error system is reduced and the performance of the controller is better.

Example 2. Consider MRNN with the follows parameters: $k_1 = k_2 = k_3 = 1$, $l_1 = l_2 = l_3 = 0$, $l_i(x_i(t)) = tanh(x_i(t)), i = 1, 2, 3$ where

The activation functions $l_i(x_i(t)) = tanh(x_i(t))$ satisfy Assumption 1 and 2 with $\sigma^-_1 = \sigma^-_2 = \sigma^-_3 = 0$, $\sigma^+_1 = \sigma^+_2 = \sigma^+_3 = 1$, $m_1 = m_2 = m_3 = 1$.And, we have

Select the initial value $x(t) = [0.7\quad-1.2\quad0.3]^T$, $y(t) = [-0.5 \quad0.5\quad0.7]^T$, $v_1 = 1$, $v_2 = 1$, $\omega = 0$, $N = 1$. The control gain is obtained as follows

When $N = 1$, Figure 7 shows the synchronization trajectories of $x(t)$ and $y(t)$ in Example 2. Figure 8 shows the status of the error signal $e(t)$. It is obvious that the error signal state converges exponentially to zero, and the results conform to the conclusion of Theorem 1 in this paper.

5.   Conclusions

The exponential synchronization problem of a class of delayed memory neural networks is studied. By using The N-order Bessel-Legendre inequality, the exponential synchronization criterion of n-related memory neural networks is given, and a more flexible intermittent feedback controller is constructed by introducing two adjustable scalars. Finally, the criterion for the conservatism decreasing with the increase of the order of Bessel-Legendre inequality is given. The validity of the main results is verified by two simulation examples.

Acknowledgements

The project supported by National Natural Science Foundation of China under Grant No. 51807198, 41906154, 52101358.

Conflict of interest

The authors declare that they have no conflicts of interest.