Quantized Control for Local Synchronization of Fractional-Order Neural Networks with Actuator Saturation

Abstract

This brief discusses the use of quantized control with actuator saturation to achieve the local synchronization of master–slave fractional-order neural networks (FONNs). A refined sector condition (RSC) is proposed that addresses the issue of the simultaneous quantizer effects and actuator constraints. The RSC is used in the theoretical analysis of local synchronization in drive-response systems. The analysis employs inequality techniques on the Mittag–Leffler function and fractional-order Lyapunov theory. Additionally, this paper presents two convex optimization algorithms that aim to minimize the actuator’s costs and expand the admissible initial area (AIA). Finally, this paper employs a three-neuron FONN to demonstrate the efficacy of the proposed methods.

1. Introduction

Synchronization is a significant dynamic behavior of coupled systems that has garnered considerable attention due to its successful applications in various fields such as information sciences [1], secure communication [2], and combinatorial optimization [3]. In secure communication, chaotic drive and response systems are employed to generate pseudorandom numbers that serve as encryption and decryption keys, respectively. It is important to note that even a minor error between the decryption and encryption keys can result in decryption failure due to the sensitivity of the keys. Recently, there has been growing interest in using fractional calculus to model and analyze various systems, including neural networks. The benefit of fractional calculus lies in its ability to describe the fundamental characteristics and retention of the system model more precisely than conventional integer-order calculus [4,5,6,7,8,9]. One of the main advantages of using fractional-order derivatives is that they possess infinite memory, which is particularly important in neural network models. Conventional integer-order neural networks have constraints in representing the enduring memory of neuron synapses, which is a crucial feature of biological neural networks [10,11,12]. To address this limitation, the development of fractional-order neural networks (FONNs) based on fractional-order differential equations is crucial. This is because the electroconductibility of the plasma membrane exhibits fractional-order properties, and real capacitors have been shown to be “fractional” in practical applications [12,13]. Therefore, adding fractional capacitors to the large-scale integrated circuits of FONNs can improve the efficiency of information manipulation and the precision of the model [10]. As a result, FONNs have received substantial attention, and remarkable outcomes such as synchronization have been achieved [14,15,16,17].

With the growth of information and communication technology, networked control methods, such as event-triggered control [18], sampled-data control [19,20], intermittent control [17,21], and quantized control [15,22,23,24], have become increasingly popular in engineering applications. These methods share the common feature of using digital and isolated control intelligence. In contrast to conventional regulation approaches [14,16,25,26], networked control schemes can decrease control costs and network-introduced loads within a restricted and crowded communication capacity. When implementing extensive FONNs, where factors such as computing power, storage capacity, and communication resources are limited, alleviating these constraints has become a top priority. The quantizer used in quantized controllers can act as an information coder that converts continuous signals into piecewise constant ones, transmitting incomplete data sent to the CPU, thus saving energy and releasing congested I/O ports. However, most relevant results have primarily focused on integer-order systems, as classical Lyapunov techniques and inequality analysis methods cannot be applied to fractional-order systems. Therefore, the primary objective of this brief is to examine the stabilization of FONNs when subjected to quantized control.

It is important to note that all the aforementioned results assume that there are no constraints on the control input [14,15,16,25,26,27,28]. However, in practical applications, actuators often experience saturation owing to the constrained amplification ability of operational amplifiers in the physical configuration, resulting in control input with an amplitude constraint. This saturation phenomenon can degrade system performance or cause the instability of closed-loop systems. The sector condition is commonly used to address the saturation nonlinearity issue [21,29,30,31]. For instance, the problem of analyzing the $L_2$-gain and exponential synchronization of delayed chaotic neural networks has been investigated through the utilization of saturated intermittent control with the sector condition [31]. Furthermore, the study of exponential synchronization in dynamical networks has been conducted through the application of saturated intermittent control with the sector condition that depends on the delay, as described in [21]. However, the previous sector conditions and analysis methods cannot handle the coexistence of quantizer and actuator saturation due to the inability to obtain practical stability criteria. Thus, addressing this issue is a crucial aspect. To the best of our knowledge, the scarcity of outcomes is attributable to the challenges associated with handling quantized information that is saturated, especially in the case of fractional-order systems. These factors lead to intriguing questions, such as how to construct a stability criterion capable of managing the impact of actuator saturation and quantization? How do the parameters of the quantizer affect the actuator’s costs and AIA? And how to use Lyapunov theory and inequality methods to address this problem. This constitutes the secondary incentive behind this brief.

To address the difficulties discussed earlier, new analysis techniques are required, as solving these problems is not straightforward. This brief focuses on achieving local synchronization of FONNs under quantized control with actuator saturation. The key contributions can be summarized as follows: (a) An RSC is developed to tackle the challenge of dealing with both the quantizer effect and control actuator saturation simultaneously; (b) By utilizing Lyapunov stability theory, the RSC, and some inequality methods on the Mittag–Leffler function, a novel local synchronization criterion is derived for drive-response systems; and (c) Two optimization strategies are proposed, which aim to increase the AIA and minimize the actuator’s costs, respectively. Additionally, a qualitative relation between the quantizer parameter and the actuator’s costs is uncovered. Simulation results demonstrate the effectiveness of the proposed approach.

The structure of this brief is as follows. Section 2 introduces the necessary background information, including the RSC and the system model. Section 3 presents the main synchronization results and two optimization algorithms. A three-neuron FONN is used to demonstrate the effectiveness of the proposed method in Section 4. Finally, Section 5 provides a summary of the findings and the conclusions.

Notation 1.

${\mathbb{N}}_m$, ${\mathbb{R}}^{m\times m}$, and ${\mathbb{S}}^{m\times m}+$ represent the sets of non-negative integers not exceeding m, $m\times m$ real matrices, and $m\times m$ positive definite and symmetric matrices, respectively. $U\left(j\right)$ denotes the jth row of matrix U. $diag(\dots )$ is a block diagonal matrix. $U>0$ indicates that U is symmetric and positive definite. $U^T$$\left(U^{-1}\right)$ represents the transpose (inverse) of matrix U. ${\lambda }_{max}\left(U\right)$ represents the maximum eigenvalue of matrix U. $\mathsf{\Gamma }(·)$ represents the Gamma function, which is defined as $\mathsf{\Omega }\left(\gamma \right)={\int }_0^{+\infty }{e}^{-t}{t}^{\gamma -1}dt$. $\left|\right|·{\left|\right|}_p$, $p=1,2,\infty$ represents the p-norm. $\mathcal{L}$ represents the Laplace transform.

2. Preliminaries

Firstly, we utilize the Caputo fractional derivative to examine the stabilization of a specific group of FONNs through the use of quantized control and actuator saturation:

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }{\beta }_1\left(t\right)& =-C_{}^{}{\beta }_1\left(t\right)+V{\sigma }_{}\left({\beta }_1\left(t\right)\right),\hfill \end{array}$$

(1)

where ${\beta }_1\left(t\right)\in {\mathbb{R}}^n$ denotes the state vector of the drive system. $C=diag(c_1,c_2,\dots ,c_n)>0$ and $V={\left(v_{ij}\right)}_{n\times n}$ denote the connection weight matrices. $\sigma \left({\beta }_1\left(t\right)\right)={[\sigma \left({\beta }_{11}\left(t\right)\right),\sigma \left({\beta }_{12}\left(t\right)\right),\dots ,\sigma \left({\beta }_{1n}\left(t\right)\right)]}^T$ denotes the activation function.

By considering (1) as the drive system, the response system can be represented as follows:

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }{\beta }_2\left(t\right)& =-C_{}^{}{\beta }_2\left(t\right)+V{\sigma }_{}\left({\beta }_2\left(t\right)\right)+u\left(t\right),\hfill \end{array}$$

(2)

where ${\beta }_2\left(t\right)\in {\mathbb{R}}^n$ denotes the state vector of the drive system, and $u\left(t\right)\in {\mathbb{R}}^n$ is the control input to be designed later.

Denote $\alpha \left(t\right)={\beta }_2\left(t\right)-{\beta }_1\left(t\right)$, then the error system can be written as

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }{\alpha }_{}\left(t\right)& =-C_{}^{}{\alpha }_{}\left(t\right)+V{\sigma }_{}\left({\alpha }_{}\left(t\right)\right)+u\left(t\right),\hfill \end{array}$$

(3)

where $u\left(t\right)=\mathrm{sat}\left(Gp\right(\alpha \left(t\right)\left)\right)$, $p\left(\alpha \left(t\right)\right)=\in {\mathbb{R}}^n$ is the quantized state vector. $sat(·):{\mathbb{R}}^n\to {\mathbb{R}}^n$ denotes the saturation function with $sat\left(W_{\left(j\right)}p\left(\alpha \left(t\right)\right)\right)=\mathrm{sgn}\left(W_{\left(j\right)}p\left(\alpha \left(t\right)\right)\right)min\{{\iota }_j,|G_{\left(j\right)}p\left(\alpha \left(t\right)\right)\left|\right\}$, $j\in \mathbb{N}$. $G\in {\mathbb{R}}^{n\times n}$ represents the gain matrix. ${\iota }_i>0$ is the saturation level that is already known.

In addition, ${\sigma }_j(·)$ are assumed to have Lipschitz constants present in ${\mathsf{\Sigma }}_j$ ($j\in {\mathbb{N}}_n^{+}$), such that

$$0\le \frac{{\sigma }_j\left({\gamma }_1\right)-{\sigma }_j\left({\gamma }_2\right)}{{\gamma }_1-{\gamma }_2}\le {\mathsf{\Sigma }}_j,$$

for ${\gamma }_1\ne {\gamma }_2$. The number of quantization levels is

$${\mathsf{\Omega }}^{\left(j\right)}=\{\pm v_{\rho }^{\left(j\right)}:v_{\rho }^{\left(j\right)}={ϵ}_j^{\rho }{v}_0^{\left(j\right)},\rho =0,\pm 1,\pm 2,\dots \}\cup \left\{0\right\},$$

$$0<{ϵ}_j<1,v_0^{\left(j\right)}>0,j\in {\mathbb{N}}^{+},$$

where $v_0^{\left(j\right)}$ and ${\rho }_j$ represent the initial quantization and the quantizer density, respectively. The logarithmic quantizer $p_j\left({\alpha }_j\left(t\right)\right)$ is defined as

$$\begin{array}{c}\hfill p_j\left({\alpha }_j\left(t\right)\right)=\left\{\begin{array}{cc}{v}_{ϵ}^{\left(j\right)},\hfill & if\frac{1}{1+{\varpi }_j}{v}_{\rho }^{\left(j\right)}<{\alpha }_j\left(t\right)\le \frac{1}{1-{\varpi }_j}{v}_{\rho }^{\left(j\right)}\hfill \\ 0,\hfill & if{\alpha }_j\left(t\right)=0\hfill \\ -p_j(-{\alpha }_j\left(t\right)),\hfill & if{\alpha }_j\left(t\right)<0\hfill \end{array}\right.\end{array}$$

where ${\varpi }_j=\frac{1-{ϵ}_j}{1+{ϵ}_j}$ stands for the quantization parameter with $0<{\varpi }_j<1$. According to $p_j\left({\alpha }_j\left(t\right)\right)$, the quantizer has the characteristic

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(1-{\varpi }_j){\alpha }_j\left(t\right)\le v_{\rho }^{\left(j\right)}<(1+{\varpi }_j){\alpha }_j\left(t\right),{\alpha }_j\left(t\right)\ge 0,\hfill \\ (1+{\varpi }_j){\alpha }_j\left(t\right)<-v_{\rho }^{\left(j\right)}\le (1-{\varpi }_j){\alpha }_j\left(t\right),{\alpha }_j\left(t\right)<0.\hfill \end{array}\right.\end{array}$$

(4)

Then, for a diagonal matrix ${\mathsf{\Gamma }}_1>0$, one has

$$\begin{array}{c}\hfill -2{[p\left(\alpha \left(t\right)\right)-(I+\tilde{\varpi })\alpha \left(t\right)]}^T{\mathsf{\Gamma }}_1[p\left(\alpha \left(t\right)\right)-(I-\varpi )\alpha \left(t\right)]\ge 0,\end{array}$$

(5)

where $\tilde{\varpi }=diag({\varpi }_1,{\varpi }_2,\dots ,{\varpi }_n)$.

We define a dead-zone nonlinear function $\varphi \left(Gp\right(\alpha \left(t\right)\left)\right)$ [30] as

$$\varphi \left(Gp\right(\alpha \left(t\right)\left)\right)=Gp\left(\alpha \right(t\left)\right)-\mathrm{sat}\left(Gp\right(\alpha \left(t\right)\left)\right).$$

According to the dead-zone nonlinear function, we can rewrite System (1) as:

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }{\alpha }_{}\left(t\right)=& -C_{}^{}{\alpha }_{}\left(t\right)+V{\sigma }_{}\left({\alpha }_{}\left(t\right)\right)+Gp\left(\alpha \left(t\right)\right)-\varphi \left(Gp\left(\alpha \left(t\right)\right)\right).\hfill \end{array}$$

(6)

In the following, an RSC is introduced.

Lemma 1.

(RSC [32]) Given matrices $W,G\in {\mathbb{R}}^{n\times n}$, a matrix $Q>0$, and a positive definite diagonal matrix $\mathsf{\Lambda }\in {\mathbb{S}}_{+}^{n\times n}$, an ellipsoidal set is defined as $\varrho \left(Q\right)=\{\alpha \left(t\right)\in {\mathbb{R}}^n:{\alpha }^T\left(t\right)Q\alpha \left(t\right)\le 1\}$, and a polyhedral set is given as

$$\mathsf{\Pi }\triangleq \{p\left(\alpha \left(t\right)\right)\in {\mathbb{R}}^n;|(G_{\left(j\right)}-W_{\left(j\right)})p\left(\alpha \left(t\right)\right)|\le {\iota }_j\}.$$

Suppose that $\alpha \left(t\right)\in \varrho \left(Q\right)$ and the following matrix inequality is satisfied

$$\left[\begin{array}{cc}\frac{Q}{{(1+{\varpi }_{max})}^2}& {(G_{\left(j\right)}-W_{\left(j\right)})}^T\\ *& {\iota }_j^2\end{array}\right]\ge 0,$$

(7)

where ${\varpi }_{max}={max}_{1\le j\le n}\left\{{\varpi }_j\right\}$, $j\in {\mathbb{N}}_n^{+}$. Then, we have $p\left(\alpha \right(t\left)\right)\in \mathsf{\Pi }$, and furthermore, an RSC can be derived as

$$\begin{array}{c}\hfill {\varphi }^T\left(Gp\left(\alpha \left(t\right)\right)\right)\mathsf{\Lambda }(\varphi \left(Gp\left(\alpha \left(t\right)\right)\right)-Wp\left(\alpha \left(t\right)\right))\le 0.\end{array}$$

(8)

The subsequent definitions and lemmas are provided prior to introducing the main results.

Assumption A1.

The closed-loop systems (6) are assumed to be controllable.

Definition 1

([32]). The definition of the Caputo fractional derivative of a function $f\left(t\right)$ with order $0<\gamma <1$ is given as follows:

$${}_0{D}_t^{\gamma }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\gamma )}{\int }_0^t{(t-\tau )}^{-\gamma }{f}^{\prime }\left(\tau \right)d\tau .$$

where $t\ge 0$ and $\mathsf{\Gamma }\left(\gamma \right)={\int }_0^{+\infty }{e}^{-t}{t}^{\gamma -1}dt$.

Definition 2

([32]). The Laplace transform of the Mittag–Leffler function with two parameters for $\vartheta \in R$ is expressed as follows:

$$\mathcal{L}\left\{t^{{\chi }_2-1}{E}_{{\chi }_1,{\chi }_2}(-\vartheta t_1^{{\chi }_1})\right\}=\frac{s^{{\chi }_1-{\chi }_2}}{s^{{\chi }_1}+\vartheta }Re\left(s\right)>{\left|\vartheta \right|}^{1/{\chi }_1}.$$

Lemma 2

([25]). Provided that $\alpha \left(t\right)$ is a differentiable function that returns a vector in $R^n$, and Q is a positive definite matrix, the subsequent inequality is valid

$${}_0{D}_t^{\gamma }{\alpha }^T\left(t\right)Q\alpha \left(t\right)\le 2{\alpha }^T\left(t\right)Q_0{D}_t^{\gamma }\alpha \left(t\right),\forall \alpha \in (0,1].$$

Lemma 3

([32]). If we have a real number $\mu \in R$ and a complex number β, with $0<\gamma <2$ and $0.5\pi \gamma <\mu <min\{\pi ,\pi \gamma \}$, then it follows that

$$E_{\gamma ,\beta }\left(ϵ\right)=-\sum _{k=1}^r\frac{{ϵ}^{-k}}{\mathsf{\Omega }(\beta -\gamma k)}+I_r\left(ϵ\right),ϵ\in G^{-}(1,\varpi )$$

in which $\pi \gamma /2<\varpi <\mu <min\{\pi ,\pi \gamma \}$, $r\ge 1$, $I_r\left(ϵ\right)=\frac{\varphi }{2\pi \gamma i{ϵ}^r}$, $\varphi ={\int }_{\gamma (1,\varpi )}\frac{exp\left({\varsigma }^{1/\gamma }\right){\varsigma }^{(1-\beta )/\gamma +p}}{\varsigma -ϵ}d\varsigma$, $\mu \le \left|arg\right(ϵ\left)\right|\le \pi$, $\left|ϵ\right|\to \infty$. The complex plane can be split into two parts, namely $G^{-}(1,\varpi )$ and $G^{+}(1,\varpi )$, by the line $\gamma (1,\varpi )$. As $\left|ϵ\right|$ grows large, the following condition holds:

$$min\varsigma \in \gamma (1,\varpi )|\varsigma -ϵ|=\left|ϵ\right|sin(\mu -\varpi ).$$

Therefore, when $\left|ϵ\right|$ is sufficiently large, we can obtain the following approximation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |I_r\left(ϵ\right)|& =\left|\frac{\varphi }{2\pi \gamma i{ϵ}^r}\right|\le M{\left|ϵ\right|}^{-1-r}.\hfill \end{array}\end{array}$$

with $M=\frac{1}{2\pi \gamma sin(\mu -\varpi )}{\int }_{\gamma (1,\varpi )}|e^{{\varsigma }^{1/\gamma }}\left|\right|{\varsigma }^r|d\varsigma$. For $\left|\varsigma \right|\ge 1$, $arg\left(\varsigma \right)=\pm \varpi$, one has $|e^{{\varsigma }^{1/\gamma }}|=e^{{\left|\varsigma \right|}^{1/\gamma }cos\left(\frac{\varpi }{\gamma }\right)}$ with $cos\left(\frac{\varpi }{\gamma }\right)<0$. Thus, the integral along the contour on the right side is converging.

Lemma 4

([30]). Suppose that $Q>0$ and let $\tilde{Q}=Y^{T}QY$. If the subsequent inequality is satisfied:

$$\left[\begin{array}{cc}{Y}_0& I\\ *& Y+Y^T-\tilde{Q}\end{array}\right]>0,$$

it follows that $Q<Q_0$, where Y is a nonsingular matrix.

3. Local Synchronization Results

Next, we focus on investigating the issue of achieving local synchronization for FONNs with actuator saturation under quantized control. Firstly, a sufficient criterion is established for the closed-loop drive-response FONNs using the RSC. Subsequently, two optimization algorithms are developed to increase the AIA and reduce the actuator’s costs. Finally, some additional remarks are included to emphasize the contributions and challenges of this study.

3.1. Main Theorem

Theorem 1.

If a non-negative scalar ϑ and $n\times n$ matrices G and W are given, and if there exist positive diagonal matrices $\tilde{\varpi }$, Λ, ${\mathsf{\Gamma }}_1$, and ${\mathsf{\Gamma }}_2$, satisfying the LMIs (7) and

$$\mathsf{\Xi }=\left[\begin{array}{cccc}{\mathsf{\Xi }}_{11}& {\mathsf{\Xi }}_{12}& -Q& QV+\mathsf{\Sigma }{\mathsf{\Gamma }}_2\\ *& -2{\mathsf{\Gamma }}_1& W^T\mathsf{\Lambda }& 0\\ *& *& -2\mathsf{\Lambda }& 0\\ *& *& *& -2{\mathsf{\Gamma }}_2\end{array}\right]\le 0,$$

(9)

are satisfied, where ${\mathsf{\Xi }}_{11}=-QC-CQ-2(I+\tilde{\varpi }){\mathsf{\Gamma }}_1(I-\tilde{\varpi })+\vartheta Q$, $\mathsf{\Sigma }=diag({\mathsf{\Sigma }}_1,{\mathsf{\Sigma }}_2,\dots ,{\mathsf{\Sigma }}_n)$, ${\mathsf{\Xi }}_{12}=QG+(I+\tilde{\varpi }){\mathsf{\Gamma }}_1+(I-\tilde{\varpi }){\mathsf{\Gamma }}_1$, $\tilde{\varpi }=diag({\varpi }_1,{\varpi }_2,\dots ,{\varpi }_n)$. Thus, for any initial conditions $\alpha \left(0\right)$ within the set $\varrho \left(Q\right)=\alpha \left(t\right)\in {\mathbb{R}}^n:{\alpha }^T\left(t\right)Q\alpha \left(t\right)\le 1$, the trajectories of the closed-loop system (6) will converge asymptotically to zero.

Proof. 

Construct the subsequent auxiliary function.

$$W\left(t\right)={\alpha }^T\left(t\right)Q\alpha \left(t\right).$$

According to Lemma 2, we obtain

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }W\left(t\right)& \le 2{\alpha }^T\left(t\right)Q_0{D}_t^{\gamma }\alpha \left(t\right).\hfill \end{array}$$

(10)

Considering the characteristics of $\sigma \left(\alpha \right(t\left)\right)$ and a positive diagonal matrix ${\mathsf{\Gamma }}_2$, we have that

$$\begin{array}{cc}\hfill 0& \le 2\left[\begin{array}{cc}{\alpha }^T\left(t\right)& {\sigma }^T\left(\alpha \left(t\right)\right)\end{array}\right]\left[\begin{array}{cc}0& \frac{\mathsf{\Sigma }{\mathsf{\Gamma }}_2}{2}\\ *& -{\mathsf{\Gamma }}_2\end{array}\right]\left[\begin{array}{c}\alpha \left(t\right)\\ \sigma \left(\alpha \right(t\left)\right)\end{array}\right].\hfill \end{array}$$

(11)

By using (5), (6), (8), (10), and (11), we can derive

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }W\left(t\right)& \le 2{\alpha }^T\left(t\right)Q[-C_{}^{}{\alpha }_{}\left(t\right)+V{\sigma }_{}\left({\alpha }_{}\left(t\right)\right)+Gp\left(\alpha \left(t\right)\right)\hfill \\ \hfill & -\varphi \left(Gp\left(\alpha \left(t\right)\right)\right)]+2{\varphi }^T\left(Gp\left(\alpha \left(t\right)\right)\right)\mathsf{\Lambda }(Gp\left(\alpha \left(t\right)\right)-\hfill \\ \hfill & {\varphi \left(Gp\left(\alpha \left(t\right)\right)\right))-2[p\left(\alpha \left(t\right)\right)-(I+\varpi )\alpha \left(t\right)]}^T{\mathsf{\Gamma }}_1\hfill \\ \hfill & \times [p\left(\alpha \left(t\right)\right)-(I-\varpi )\alpha \left(t\right)]+2{\sigma }^T\left(\alpha \left(t\right)\right){\mathsf{\Gamma }}_{2}F\alpha \left(t\right)\hfill \\ \hfill & -2{\sigma }^T\left(\alpha \left(t\right)\right){\mathsf{\Gamma }}_2\sigma \left(\alpha \left(t\right)\right)+\vartheta {\alpha }^T\left(t\right)Q\alpha \left(t\right)-\vartheta W\left(t\right)\hfill \\ \hfill & ={\eta }^T\left(t\right)\mathsf{\Xi }\eta \left(t\right)-\vartheta V\left(t\right),\hfill \end{array}$$

where $\eta \left(t\right)={[\alpha \left(t\right),p\left(\alpha \left(t\right)\right),\varphi \left(Gp\left(\alpha \left(t\right)\right)\right),\sigma \left(\alpha \left(t\right)\right)]}^T$.

One can deduce from Condition (9) stated in Theorem 1 that

$${}_0{D}_t^{\gamma }W\left(t\right)\le -\vartheta W\left(t\right).$$

Therefore, a non-negative function $\zeta \left(t\right)$ exists such that

$$\begin{array}{c}\hfill {}_0{D}_t^{\gamma }W\left(t\right)+\zeta \left(t\right)=-\vartheta W\left(t\right).\end{array}$$

(12)

By taking the Laplace transform of (12), we obtain

$$s^{\gamma }W\left(s\right)-s^{\gamma -1}W\left(0\right)+\zeta \left(s\right)=-\vartheta W\left(s\right),$$

where $W\left(s\right)=\mathcal{L}\left\{W\right(t\left)\right\}$, $\zeta \left(s\right)=\mathcal{L}\left\{\zeta \right(t\left)\right\}$, and

$$W\left(s\right)=\frac{s^{\gamma -1}W\left(0\right)-\zeta \left(s\right)}{s^{\gamma }+\vartheta }.$$

Next, by applying the inverse Laplace transform, we obtain

$$W\left(t\right)=W\left(0\right)E_{\gamma ,1}(-\vartheta t^{\gamma })-\zeta \left(t\right)\ast t^{\gamma -1}{E}_{\gamma ,\gamma }(-\vartheta t^{\gamma }).$$

Here, ∗ denotes the convolution operator. It should be noted that $E_{\gamma ,\gamma }(-\vartheta t^{\gamma })$ is non-negative for $t>0$ [33,34], which implies that we can obtain

$$W\left(t\right)\le W\left(0\right)E_{\gamma ,1}(-\vartheta t^{\gamma }).$$

Consequently, we can conclude that

$${\left|\right|\gamma \left(t\right)\left|\right|}_2^2\le \frac{{\lambda }_{max}\left(Q\right)}{{\lambda }_{min}\left(Q\right)}{\left|\right|\gamma \left(0\right)\left|\right|}_2^2\left|E_{\gamma ,1}(-\vartheta t^{\gamma })\right|.$$

Denote $ϵ=-\vartheta t^{\gamma }$. Using Lemma 3 as a basis, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |E_{\gamma ,1}\left(ϵ\right)|& =\left|-\sum _{k=1}^r\frac{{ϵ}^{-k}}{\mathsf{\Gamma }(1-k\gamma )}+I_r\left(ϵ\right)\right|\hfill \\ & \le \left|\sum _{k=1}^r\frac{{ϵ}^{-k}}{\mathsf{\Omega }(1-k\gamma )}\right|+\left|I_r\left(ϵ\right)\right|\hfill \\ & \le \sum _{k=1}^r\frac{|{ϵ}^{-k}|}{\mathsf{\Omega }(1-k\gamma )}+\left|I_r\left(ϵ\right)\right|\hfill \\ & \le \sum _{k=1}^r\frac{{\left|ϵ\right|}^{-k}}{\mathsf{\Omega }(1-k\gamma )}+M{\left|ϵ\right|}^{-1-r}\hfill \end{array}\end{array}$$

Here, M is the constant defined in Lemma 3.

Moreover, we have

$$\begin{array}{c}\hfill \begin{array}{ccc}& |E_{\gamma ,1}\left(ϵ\right)|\hfill & \\ \hfill & \le \frac{{\left|ϵ\right|}^{-1}}{\left|\mathsf{\Omega }\right(1-\gamma \left)\right|}+\frac{{\left|ϵ\right|}^{-2}}{\left|\mathsf{\Omega }\right(1-2\gamma \left)\right|}+\dots +\frac{{\left|ϵ\right|}^{-r}}{\left|\mathsf{\Omega }\right(1-r\gamma \left)\right|}\hfill \\ & +{N\left|ϵ\right|}^{-r-1}\hfill \\ & =\frac{{\left|ϵ\right|}^r\left|\mathsf{\Omega }(1-2\gamma )\right|\dots \left|\mathsf{\Omega }(1-r\gamma )\right|}{{\left|ϵ\right|}^{r+1}\left|\mathsf{\Omega }(1-\gamma )\right|\left|\mathsf{\Omega }(1-2\gamma )\right|\dots \left|\mathsf{\Gamma }(1-r\gamma )\right|}\hfill \\ & \frac{+{\left|ϵ\right|}^{r-1}\left|\mathsf{\Omega }(1-\gamma )\right|\left|\mathsf{\Omega }(1-3\gamma )\right|\dots \left|\mathsf{\Omega }(1-r\gamma )\right|+\dots }{{\left|ϵ\right|}^{r+1}\left|\mathsf{\Omega }(1-\gamma )\right|\left|\mathsf{\Omega }(1-2\gamma )\right|\dots \left|\mathsf{\Omega }(1-r\gamma )\right|}\hfill \\ & \frac{+{\left|ϵ\right|}^1\left|\mathsf{\Omega }(1-\gamma )\right|\left|\mathsf{\Omega }(1-2\gamma )\right|\dots \left|\mathsf{\Omega }(1-(r-1)\gamma )\right|}{{\left|ϵ\right|}^{r+1}\left|\mathsf{\Omega }(1-\gamma )\right|\left|\mathsf{\Omega }(1-2\gamma )\right|\dots \left|\mathsf{\Omega }(1-r\gamma )\right|}\hfill \\ & \frac{+M\left|\mathsf{\Omega }\right(1-\gamma \left)\right|\left|\mathsf{\Omega }\right(1-2\gamma \left)\right|\dots \left|\mathsf{\Omega }\right(1-r\gamma \left)\right|}{{\left|ϵ\right|}^{r+1}\left|\mathsf{\Omega }(1-\gamma )\right|\left|\mathsf{\Omega }(1-2\gamma )\right|\dots \left|\mathsf{\Omega }(1-r\gamma )\right|}\hfill \\ & =\frac{{\left|ϵ\right|}^r{\displaystyle \prod _{k=2}^r}\left|\mathsf{\Omega }(1-k\gamma )\right|+{\left|ϵ\right|}^{r-1}{\displaystyle \prod _{k=1,k\ne 2}^r}\left|\mathsf{\Omega }(1-k\gamma )\right|}{{\left|ϵ\right|}^{r+1}{\displaystyle \prod _{k=1}^r}\left|\mathsf{\Omega }(1-k\gamma )\right|}\hfill \\ & \frac{+\dots +{\left|ϵ\right|}^1{\displaystyle \prod _{k=1}^{r-1}}\left|\mathsf{\Omega }(1-k\gamma )\right|+M{\displaystyle \prod _{k=1}^r}\left|\mathsf{\Omega }(1-k\gamma )\right|}{{\left|ϵ\right|}^{r+1}{\displaystyle \prod _{k=1}^r}\left|\mathsf{\Omega }(1-k\gamma )\right|}.\hfill \end{array}\end{array}$$

When $t\to \infty$, it holds that $\left|ϵ\right|=|-\vartheta t^{\gamma }|\to \infty$, and then, we have

$$\underset{t\to \infty }{lim}\left|E_{\gamma ,1}(-\vartheta t^{\gamma })\right|=0.$$

Namely, it yields

$$\underset{t\to \infty }{lim}{\left|\right|\alpha \left(t\right)\left|\right|}_2^2=0.$$

Thus, for any initial conditions $\alpha \left(0\right)$ within the set $\varrho \left(Q\right)=\alpha \left(t\right)\in {\mathbb{R}}^n:{\alpha }^T\left(t\right)Q\alpha \left(t\right)\le 1$, the trajectories of the closed-loop system (6) will converge asymptotically to zero.

It is worth noting that if the quantization parameter $\varpi$ is equal to zero, $sat\left(Gp\right(\alpha \left(t\right)\left)\right)$ will simplify to $sat\left(G\alpha \right(t\left)\right)$. Consequently, (6) and (8) can be expressed as follows:

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }{\alpha }_{}\left(t\right)=& -C_{}^{}{\alpha }_{}\left(t\right)+V{\sigma }_{}\left({\alpha }_{}\left(t\right)\right)+G\alpha \left(t\right)-\varphi \left(G\alpha \left(t\right)\right).\hfill \end{array}$$

(13)

$$\begin{array}{c}\hfill {\varphi }^T\left(G\alpha \left(t\right)\right)\mathsf{\Lambda }(\varphi \left(G\alpha \left(t\right)\right)-W\alpha \left(t\right))\le 0.\end{array}$$

(14)

□

Corollary 1 can be obtained for System (13) based on Theorem 1.

Corollary 1.

If there are $n\times n$ matrices G and W, as well as scalar $\vartheta \ge 0$, and diagonal matrices Λ and ${\mathsf{\Gamma }}_2$ that are positive, then the LMIs hold:

$$\left[\begin{array}{cc}Q& {(G_{\left(j\right)}-W_{\left(j\right)})}^T\\ *& {\iota }_j^2\end{array}\right]\ge 0,$$

(15)

$$\tilde{\mathsf{\Xi }}=\left[\begin{array}{ccc}{\tilde{\mathsf{\Xi }}}_{11}& -Q+W^T\mathsf{\Lambda }& QG+\mathsf{\Sigma }{\mathsf{\Gamma }}_2\\ *& -2\mathsf{\Lambda }& 0\\ *& *& -2{\mathsf{\Gamma }}_2\end{array}\right]\le 0,$$

(16)

are satisfied, where ${\tilde{\mathsf{\Xi }}}_{11}=-QC-CQ+QG+G^{T}Q+\vartheta Q$, $\mathsf{\Sigma }=diag({\mathsf{\Sigma }}_1,{\mathsf{\Sigma }}_2,\dots ,{\mathsf{\Sigma }}_n)$. Then, the closed-loop system (13) has asymptotic convergence to the origin for all initial conditions $\alpha \left(0\right)$ within the ellipsoidal set $\varrho \left(Q\right)=\alpha \left(t\right)\in {\mathbb{R}}^n:{\alpha }^T\left(t\right)Q\alpha \left(t\right)\le 1$.

Proof. 

In a similar fashion, the following function is constructed:

$$W\left(t\right)={\alpha }^T\left(t\right)Q\alpha \left(t\right).$$

By combining Equations (10), (11), (13), and (14), we can derive:

$$\begin{array}{cc}\hfill {}_0{D}_t^{\gamma }W\left(t\right)& \le 2{\alpha }^T\left(t\right)P[-C_{}^{}{\alpha }_{}\left(t\right)+V{\sigma }_{}\left({\alpha }_{}\left(t\right)\right)+G\alpha \left(t\right)\hfill \\ \hfill & -\varphi \left(G\alpha \left(t\right)\right)]+2{\varphi }^T\left(G\alpha \left(t\right)\right)\mathsf{\Lambda }(W\alpha \left(t\right)-\varphi \left(G\alpha \left(t\right)\right))\hfill \\ \hfill & +2{\sigma }^T\left(\alpha \left(t\right)\right){\mathsf{\Gamma }}_{2}F\alpha \left(t\right)-2{\sigma }^T\left(\alpha \left(t\right)\right){\mathsf{\Gamma }}_2\sigma \left(\alpha \left(t\right)\right)\hfill \\ \hfill & +\vartheta {\alpha }^T\left(t\right)Q\alpha \left(t\right)-\vartheta W\left(t\right)\hfill \\ \hfill & ={\tilde{\eta }}^T\left(t\right)\tilde{\mathsf{\Xi }}\tilde{\eta }\left(t\right)-\vartheta W\left(t\right),\hfill \end{array}$$

where $\tilde{\eta }\left(t\right)={[\alpha \left(t\right),\varphi \left(G\alpha \left(t\right)\right),\sigma \left(\alpha \left(t\right)\right)]}^T$.

By applying Condition (16) from Corollary 1, we obtain:

$${}_0{D}_t^{\gamma }W\left(t\right)\le -\vartheta W\left(t\right).$$

The remaining part of the proof follows the same procedure as in Theorem 1. Hence, we can conclude that the proof is complete. □

Remark 1.

The main technical challenge in this work is to establish an RSC that enables the derivation of feasible LMI-based conditions. To investigate systems under actuator saturation [30,31], the traditional sector condition (14) is derived. However, this traditional sector condition is not useful in dealing with the problem presented herein when considering the effects of quantized control. This is because the quantized state, i.e., $p\left(\alpha \left(t\right)\right)=(I+{\mathsf{\Delta }}_{\varpi })\alpha \left(t\right)$, does not belong to the open-loop system state $\alpha \left(t\right)$ because of the existence of the quantizer. As a result, the feedback signal is $Gp\left(\alpha \right(t\left)\right)$ rather than $G\alpha \left(t\right)$. In order to address this issue, the set is refined as $\mathsf{\Pi }\triangleq p\left(\alpha \left(t\right)\right)\in {\mathbb{R}}^n;\left|(G_{\left(j\right)}-W_{\left(j\right)})p\left(\alpha \left(t\right)\right)\right|\le {\iota }_j.$ Combined with $\varrho \left(Q\right)$, this yields Inequality (5) and the RSC (8), which allows one to derive feasible stability criteria.

3.2. Optimization Algorithms

In order to account for actuator saturation and actuator costs, two optimization algorithms are presented. The first scheme aims to enlarge the AIA, whereas the second scheme aims to minimize the actuator’s costs.

Algorithm 1.

When dealing with systems with actuator saturation, it becomes necessary to increase the AIA. Given matricesGandW, and scalars${\varpi }_{max}$ and ${\iota }_i$, our objective is to estimate the largest AIA. It is worth noting that maximizing the minimal axis of $\varrho \left(Q\right)$ is equivalent to minimizing ${\lambda }_{max}\left(Q\right)$. According to Lemma 4, this can be achieved by utilizing the following optimization algorithm:

$$\begin{gathered} \mathit{min}\theta \\ \mathit{subject}\mathit{to} \\ \left(7\right)\mathit{and}\left(9\right)\mathit{as}\mathit{well}\mathit{as} \\ \left[\begin{array}{cc}\theta I& I\\ *& 2I-Q\end{array}\right]>0, \end{gathered}$$

which ensures that ${\lambda }_{max}\left(Q\right)<\theta$, and $\varpi \left(\theta I\right)\subset \varrho \left(Q\right)$.

Algorithm 2.

Given the AIA $\varrho \left(Q\right)=\alpha \in {\mathbb{R}}^n:{\alpha }^{T}Q\alpha \le 1$, a matrix $\mathsf{\Lambda }$, and a scalar ${\varpi }_{max}$, our primary objective is to obtain the gain G while minimizing the actuator’s costs. This cost can typically be weighed using saturation levels ${\iota }_j$, where $i\in {\mathrm{N}}_n^{+}$. We denote ${\overline{\iota }}_j={\iota }_j^2$ and $\iota ={\sum }_{j=1}^n{e}_j{\overline{\iota }}_j$. The following optimization scheme is considered:

$$\begin{gathered} min\sum _{j=1}^n{e}_j{\overline{\iota }}_j \\ \mathit{subject}\mathit{to} \\ \left(9\right)\mathit{and} \\ \left[\begin{array}{cc}\frac{Q}{{(1+{\varpi }_{max})}^2}& {(G_{\left(j\right)}-W_{\left(j\right)})}^T\\ *& {\overline{\iota }}_j\end{array}\right]>0. \end{gathered}$$

Here, the positive weights in the construction of the cost function are denoted by $e_j$.

Remark 2.

The local synchronization problem of Systems (1) and (2) under actuator saturation and quantized control is challenging to investigate using traditional Lyapunov–Krasovskii stability theory [18,20] and matrix measure techniques due to the presence of fractional-order dynamics. Furthermore, the previous sector condition [30,31] is not available in this case either due to the effects of saturation nonlinearity and quantization. To tackle these difficulties, we propose the use of a fractional-order function, an RSC, and many inequality methods to derive a local synchronization criterion for the drive-response FONNs under saturated quantized control. Additionally, we develop two optimization algorithms to, respectively, maximize the AIA and minimize the actuator’s costs.

Remark 3.

It is worth noting that the results derived in this paper are independent of the fractional order γ. In addition, the MATLAB toolbox and LMI-based stabilization conditions make it easy to handle the computational burden and optimization algorithms.

4. Numerical Simulations

4.1. Simulation Based on Algorithm 1

Example 1.

As described in [14], a three-neuron FONN is defined by $\gamma =0.99$, $\sigma \left(\alpha \right)=tanh\left(\alpha \right)$, and $C=I$,

$$V=\left[\begin{array}{ccc}2& -1.2& 0\\ 1.8& 1.71& 1.15\\ -4.75& 0& 1.1\end{array}\right],$$

thus, one can obtain $\mathsf{\Sigma }=diag(1,1,1)$. Choose ${\beta }_1\left(0\right)={(1.4,-0.65,1.4)}^T$ and ${\beta }_2\left(0\right)={(1.18,-0.25,1.98)}^T$. Then, the trajectories of the drive system and the response system can be depicted, as shown in Figure 1. It can be seen that this example is controllable.

Set ${\varpi }_i={\varpi }_{max}=0.06$, ${\iota }_i=3.873$, $i=1,2,3$, $\vartheta =0.001$, and

$$G=\left[\begin{array}{ccc}-4.4500& -0.1000& -0.1000\\ -0.0100& -3.5000& -3.3000\\ -0.1000& -0.0100& -4.1800\end{array}\right],$$

and

$$W=\left[\begin{array}{ccc}-0.1100& -0.1010& -0.1000\\ -0.2200& -0.0100& -0.0100\\ -0.0200& -0.1100& 0.1100\end{array}\right].$$

Then, it yields ${\xi }_{min}=4.8148$,

$$\mathsf{\Lambda }=\left[\begin{array}{ccc}124.6593& 0& 0\\ 0& 19.4431& 0\\ 0& 0& 21.3281\end{array}\right],$$

$${\mathsf{\Gamma }}_1=\left[\begin{array}{ccc}12.8616& 0& 0\\ 0& 8.5337& 0\\ 0& 0& 7.9718\end{array}\right],$$

$${\mathsf{\Gamma }}_2=\left[\begin{array}{ccc}10.7341& 0& 0\\ 0& 2.4596& 0\\ 0& 0& 1.3151\end{array}\right],$$

and the AIA

$$\varrho \left(Q\right)=\left\{\alpha \in {\mathbb{R}}^3:{\alpha }^T\left[\begin{array}{ccc}1.7522& 0.0139& -0.0173\\ 0.0139& 1.5437& 0.2644\\ -0.0173& 0.2644& 1.5108\end{array}\right]\alpha \le 1\right\}.$$

Figure 2a shows the set of the AIA utilizing Algorithm 1. Since $\alpha \left(0\right)={(-0.22,0.40,0.58)}^T\in \varrho \left(Q\right)$, Figure 2b indicates that System (6) can be locally stabilized via the quantized control with actuator saturation. That is, the local synchronization of Systems (1)–(2) can be successfully achieved.

4.2. Simulation Based on Algorithm 2

Example 2.

A chaotic three-neuron FONN is proposed with $\sigma \left(\alpha \right)=0.5\left(\right|\alpha +1|-|\alpha -1\left|\right)$, $C=I$, $\gamma =0.97$, and

$$V=\left[\begin{array}{ccc}1.2& -1.6& 0\\ 1.24& 1& 0.9\\ 0& 2.2& 1.5\end{array}\right],$$

thus, it can be calculated as $\mathsf{\Sigma }=diag(1,1,1)$. Take ${\beta }_1\left(0\right)={(-0.8,0.65,-1.2)}^T$ in the simulation. Then, the chaotic behavior of the drive system can be depicted, as shown in in Figure 3a,b.

Choose $\mathsf{\Lambda }=diag(1,1,1)$, ${\varpi }_i={\varpi }_{max}=0.1$, $\vartheta =0.1$, $e_i=1$, $i=1,2,3$, where AIA represents the set $\varpi \left(Q\right)=\{\alpha \in {\mathbb{R}}^3:5{\alpha }^T\alpha \le 1\}$. Through simulation, we can obtain ${\overline{u}}_{0\left(1\right)}=0.1804$, ${\overline{u}}_{0\left(2\right)}=0.9465$, ${\overline{u}}_{0\left(3\right)}=1.0782$, $\iota =2.2051$,

$${\mathsf{\Gamma }}_1=\left[\begin{array}{ccc}9.9949& 0& 0\\ 0& 21.9430& 0\\ 0& 0& 22.7100\end{array}\right],$$

$${\mathsf{\Gamma }}_2=\left[\begin{array}{ccc}9.9953& 0& 0\\ 0& 18.5372& 0\\ 0& 0& 11.4177\end{array}\right],$$

and the feedback control gain

$$G=\left[\begin{array}{ccc}-7.0368& 0.1906& 0.8460\\ 0.1345& -7.6965& -2.4338\\ 0.5629& -2.3901& -7.9822\end{array}\right],$$

and

$$W=\left[\begin{array}{ccc}-6.2500& 0.1184& 0.4975\\ 0.1174& -6.2128& -1.1263\\ 0.3136& -1.0984& -6.3317\end{array}\right].$$

In the simulation, we choose ${\beta }_2\left(0\right)$ to be ${(-0.55,0.40,-0.95)}^T$, such that $\alpha \left(0\right)={(0.25,-0.25,0.25)}^T\in \varpi \left(Q\right)$. Theorem 1 guarantees local stability for closed-loop FONNs, as shown in Figure 4. We denote the interval parameters as ${\varpi }_1^{*}=0.15$, ${\varpi }_2^{*}=0.35$, and ${\varpi }_{step}=0.05$, respectively.

Table 1. Minimum size of $\iota$ for different $\overline{\varpi }$.

$\overline{\mathit{\varpi }}$	0.15	0.2	0.25	0.3	0.35
$\iota$	3.0547	4.2048	5.8199	8.1187	11.4276
${\iota }_1$	0.2694	0.4014	0.5933	0.8725	1.2808
${\iota }_2$	1.3021	1.7813	2.4525	3.4088	4.7876
${\iota }_3$	1.4831	2.0221	2.7740	3.8374	5.3592

presents the related minimal actuator size $\iota$ with the above ${\varpi }_{max}$. It is observed that as ${\varpi }_{max}$ increases, $\iota$ also increases. However, a larger ${\varpi }_{max}$ can help save energy resources by minimizing the amount of quantitative data transmitted to the controller. Therefore, a compromise can be achieved by selecting appropriate parameters that satisfy the desired requirements in engineering applications.

5. Conclusions

In this paper, we have investigated the local synchronization problem of FONNs with actuator saturation within the framework of quantized control. An RSC was established based on the features of quantizers and actuator saturation. Using fractional-order theory, we developed a sufficient condition for addressing the local synchronization issue. Two optimization algorithms were presented to maximize the AIA and minimize the actuator’s costs, respectively. Finally, we provided a demonstration of the proposed control scheme using a three-neuron FONN. The results show the validity and effectiveness of the proposed approach.

In the future, we will focus on the semi-global synchronization issue of FONNs under quantized control with actuator saturation. This is a challenging problem that will require further investigation in our future research.