Adaptive smooth sampled-data control for synchronization of T–S fuzzy reaction-diffusion neural networks with actuator saturation

1.   Introduction

In recent years, Takagi–Sugeno fuzzy neural networks (TSFNNs), which are hybrid systems combining fuzzy logic and neural networks (NNs), have garnered considerable research interest because of their applications in nonlinear systems ^[1,2,3,4]. TSFNNs define input and output variables using fuzzy sets and fuzzy logic-based rules to establish their relationship, effectively capturing dynamic information in physical systems ^[5,6,7,8]. In many domains, ranging from intelligent transportation systems, financial risk analysis, medical diagnostics, and intelligent robotics, TSFNNs have been widely applied. Synchronization, a critical dynamic behavior in TSFNNs, continues to be a significant research focus. Synchronization can occur naturally or be achieved by design and has critical applications in communication systems, image encryption, and power systems^[9,10,11,12]. In ^[13], TFSNN synchronization was used for image encryption, effectively concealing and restoring the original image. Thus, the study of synchronization holds significant importance.

Various factors influence the states of TSFNNs. Accurately describing these states requires considering the effects of time delays and reaction–diffusion phenomena ^{[14,15,16,17]}. Appropriately timed delays can reduce oscillatory tendencies in a system and promote faster convergence. However, time delays can also result in the degradation of the system performance or even destabilization ^[18,19,20]. Therefore, it is crucial to account for the effects of delays. The study in ^[19] focused on observer-based dissipativity control in TSFNNs with distributed delays. Li et al. ^[20] addressed the event-triggered stabilization of a novel T–S fuzzy complex-valued memristive NN with mixed time-varying delays. Additionally, TSFNNs inherently exhibit the reaction–diffusion phenomenon due to environmental influences. This suggests that changes in the system's state depend not only on time but also on the spatial context. TFRNNs can better characterize the evolution of neurons during time and spatial changes. TFRNNs are more realistic than traditional TSFNNs. The study in ^[21] addressed the outlier-resistant nonfragile control problem in TFRDNNs. Liu et al. ^[22] explored the $H_\infty$ state estimation problem in TFRNNs, considering gain uncertainties and semi-Markov jump parameters.

In modern control systems, digital computers are usually used for data signal acquisition and measurement analysis, with the help of discrete-time SD controllers for controlling continuous-time objects to realize the control function of the system ^{[23,24,25,26,27]}. This control method is called SD control and is widely used in solving synchronization problems. You et al. ^[25] explored the issue of exponential synchronization in inertial NN examined within the context of aperiodic sampling and state quantization. The study in ^[26] addressed exponential synchronization in TFRNNs with additive time-varying delays using both time SD control and time–space SD control. The study in reference ^[27] employed a memory-based SD controller to investigate the synchronization of NNs in the presence of parameter uncertainties. Most SD controls are built using data either at the current moment or from previous moments. In reality, measured data can exhibit random fluctuations or anomalies. These errors can manifest in various forms, including loss or anomalies in data from node task conflicts and sampling jitter caused by hardware aging. Consequently, these issues may affect sampling measurement results, potentially causing redundant data transmission. Moreover, memory-based SD controllers inherently consume storage space. A very small sampling interval may deplete storage space, rendering the system inoperable. In summary, we propose using an ASSD controller to mitigate these challenges. This approach also provides further motivation for the research presented in this paper.

Inspired by the discussions above, we aim to explore the synchronization of TFRNNs with time-varying delays through the design of ASSD control. Table 1 lists fundamental mathematical symbols, and the primary contributions of this study are summarized as follows:

(1) This study offers a comprehensive view of the synchronization of TFRNNs, considering factors such as ASSD, reaction–diffusion and actuator saturation, thereby enhancing the applicability of the results.

(2) The improved ASSD controller uses the dynamic forgetting factor to allocate current and historical measurements, effectively countering data distortion and improving the system's control performance.

(3) The LKFs with membership function and time-varying delays correlation capture system information comprehensively. Furthermore, the compensation LKF $V_{3}(t)$ and the fuzzy zero equation are employed to address the integral term with the membership function's derivatives, increasing the feasible range of matrix variables.

(4) Unlike the traditional LKF method, the semi-looped-functional method does not require the sum of its derivatives to be negative definite. This results in the derivation of a synchronization criterion with more relaxed constraints.

2.   Preliminaries and problem formulation

The TFRNNs described by IF–THEN rules are considered as follows:

Plant rule $i$: IF ${\mathscr{C}_{1}}(t)$ is $\mathfrak{V} _{1}^{i}$, ${{\mathscr{C} }_{2}}(t)$ is $\mathfrak{V} _{2}^{i}$, ..., and ${\mathscr{C} _{l}}(t)$ is $\mathfrak{V} _{l}^{i}$, then

where for each $i \in \mathfrak{F}\triangleq \{1, 2, ..., p \}$, ${\mathscr{C} }_{\hbar}$ and $M_{\hbar}^{i}$ $(\hbar = 1, ..., p)$ denote the $\hbar$th premise variable and its corresponding fuzzy set. $x = \textbf{col}\{x_1, x_2, \ldots, x_{\omega}\}$ is the spatial variable within $\Gamma = [{\underline{\alpha}}_1, {\overline{\alpha}}_1]\times \ldots\times [{\underline{\alpha}}_\omega, {\overline{\alpha}}_\omega]$, and $\partial \Gamma$ denotes its boundary. The state vector $\varsigma(t, x) = \textbf{col}\{\varsigma_1(t, x), \varsigma_2(t, x), \ldots, \varsigma_n(t, x)\}$ represents the system state. ${{A}_{l}}$ represents the transmission diffusion coefficient; ${B}_{i}$ indicates the neuron charging time constant; ${C}_{i}$ and ${D}_{i}$ represent the connection weight matrix, $\mathfrak{T} (t)$ is the external input, and $\sigma (t)$ denotes the time-varying delays that satisfy

Utilizing the weighted average fuzzy blending approach, the TFRNNs are described as follows:

where $\mathscr{C} (t) = \textbf{col}\{{\mathscr{C} }_{1}(t), \, {\mathscr{C} }_{2}(t), ..., \, {\mathscr{C} }_{l}(t)\}$ and $\mathfrak{h} _i(\mathscr{C} (t))$ is the normalized membership function, and it satisfies

and the term $\mathfrak{V} _{j}^{i}\big(\mathscr{C} _{j}(t)\big)$ represents the membership grade of $\mathscr{C} _{j}(t)$ in $\mathfrak{V} _{j}^{i}$.

Considering system (2.3) as the master system, the corresponding slave system is defined as follows:

The initial values and boundary conditions associated with systems (2.3) and (2.5) are specified as follows:

Define the error vector as $\psi(t, x) = {\varsigma }(t, x) - \bar{\varsigma }(t, x)$. Combining (2.3) and (2.5), the resulting error system is as follows:

where $g(t, x) = f({\varsigma } (t, x))-f(\bar{\varsigma }(t, x))$ satisfies

and $\Omega_{1}$, $\Omega_{2}$ are constant matrices and $\Omega_{2}\ge\Omega_{1}$.

Control rule $j$: IF ${\mathscr{C}_{1}}(t)$ is $\mathfrak{V}_{1}^{j}$, ${\mathscr{C}}_{2}(t)$ is $\mathfrak{V}_{2}^{j}$, ..., and ${{\mathscr{C}}_{q}}(t)$ is $\mathfrak{V}_{q}^{j}$, then

where $\mathcal{K}_j$ are fuzzy the controller gains. By adopting a similar approach to that used for system (2.3), we obtain

Furthermore, considering the effects of SD, the states $\psi(t, x)$ can be measured at discrete sampling points $t_k$. Moreover, these samplings adhere to the assumption: $0\le{d}_{k}\triangleq{{t}_{k+1}}-{{t}_{k}}\le{d}_M, (k = 1, 2, \ldots, \infty)$. Subsequently, an ASSD controller is employed to conserve communication resources and improve anti-interference capabilities. The weighted measurement is expressed as:

where $\alpha (t_{k+1})(0 < \alpha (t_{k+1})\le 1)$ denotes the adaptive forgetting factor, starting with an initial value of ${{\alpha }_{0}}$, which assigns weights to historical measurement data.

Remark 2.1. Data accuracy is crucial for system performance and safety in control systems, with severe data distortion potentially causing system failure. The dynamic forgetting factor $\alpha(t_{k+1})$ in the ASSD mechanism assigns decay weights to historical measurements, smoothing sampling values and mitigating the impact of data distortion. A larger difference between the current and previous measurements results in a smaller forgetting factor $\alpha(t_{k+1})$, and vice versa. Furthermore, a larger deviation from the current measurement results in a smaller weight, while a smaller deviation leads to a larger weight. As $\tilde{\psi}(t_{k+1}, x)$ is calculated iteratively, the ASSD mechanism requires storing only one historical data $\tilde{\psi}(t_{t}, x)$, thus utilizing less storage space compared to memory-based SD control ^[27]. When $\alpha(t_{k+1}) = 1$, the ASSD control simplifies to the conventional SD control outlined in ^[25,26].

Considering the effect of actuator saturation, the saturation function $\sigma (u(t, x))$ satisfies the following conditions:

where $\sigma (u_{i}(t, x))$ = $sign(u_{i}(t, x)) min\left\{ \hat{u}_{i}(t, x), |u_{i}(t, x)|\right\}$ and $\hat{u}_{i}(t, x)$ represents the upper bound of the controller.

Furthermore, by defining $\varphi ({{\mathcal{K}}_{j}}\tilde{\psi}({{t}_{k}}, x))$ as the dead-zone nonlinearity function, the saturation function $\sigma (u(t, x))$ is divided into two parts ^[28]:

and the following holds for a real number $\bar{\varphi}$ within the interval $(0, 1)$:

Remark 2.2. Actuator saturation is a critical factor that must be considered in controller design ^[28,29]. Actuator saturation must be considered for two primary reasons: the physical limitations of the actuator and the impact of complex network conditions (such as external disturbances, packet loss, and delay) on system performance. To better capture the impact of actuator saturation, this paper models the saturation function $\sigma(u(t, x))$ as a combination of the linear segment $u(t, x)$ and the dead-zone nonlinear segment $\varphi(u(t, x))$, with their relationship defined in inequality (2.11). Additionally, the actuator saturation can be ignored when the saturation upper bound $\hat{u}_{i}(t, x)$ approaches infinity.

By substituting the aforementioned equations into the error system (2.6), we obtain:

Assumption 2.1. ^[30] Assume that ${{{\mathfrak{h} }_{p}}}(\mathscr{C}(t))$ is a continuously differentiable function, and $\left| {{{\dot{\mathfrak{h} }}}_{p}}(\mathscr{C}(t)) \right|\le {{\lambda }_{p}}$ with ${{\lambda }_{p}} > 0$, $p\in N$.

Lemma 2.1. ^[31] If $\epsilon(x)$ is a continuously differentiable real-valued function defined on the set $\Gamma$ and $\epsilon(x)/\partial{ \Gamma} = 0$, then

Lemma 2.2. ^[32] Consider $\kappa(s)$ as the differentiable function mapping from $[r_1, r_2]\to {{R}^{n}}$. Given symmetric matrices $X\in R^{n\times n} > 0$ and matrices $N_1, N_2, \in R ^{3n\times n}$, the subsequent inequality is satisfied:

where $\delta = r_2 -r_1$, $\vartheta = \left[ {{\kappa}^{T}}(r_2)\ {{\kappa}^{T}}(r_1)\ \frac{1}{\delta } {\int}_{r_1 }^{r_2 }{{{\kappa}^{T}}}(s)ds \right]^{T}$, ${{\Pi }_{1}} = {{\kappa}^{T}}(r_2)-{{\kappa}^{T}}(r_1)$ and ${{\Pi }_{2}} = {{\kappa}^{T}}(r_2)+{{\kappa}^{T}}(r_1)-\frac{2}{\delta } {\int}_{r_1}^{r_2}{{{\kappa}^{T}}}(s)ds$.

3.   Main results

For ease of presentation, the relevant notations are defined in Appendix A.

Theorem 3.1. For given scalars $\bar{\sigma } > 0$, $\eth$, $\bar{\varphi}$, ${o}_{1}$, ${o}_{2}$, and controller gain matrix ${{\mathcal{K}}_{j}}$, the master–slave systems are asymptotically synchronized if there exist symmetric positive-definite matrices ${P}_{1i}$, ${P}_{2i}$, ${R}_{1i}$, ${R}_{2i}$, ${T}_{r}$, $S$, ${\Lambda }_{3}$, ${U}_{r}$, ${W}_{r}$, $NA_{l}$, positive-definite diagonal matrices ${\Lambda }_{r}$, appropriate dimensional matrices $Y_q$, $H_p$, $G_p$, $M_q$, $N$, where $r = 1, 2, p = 1, 2, 3, q = 1, 2, ..., 4$ and $i, j\in\mathbb{N}$, such that the following LMIs hold:

where the other notations are listed in Appendix B.

Proof.

where we choose LKFs candidates are defined in Appendix C. □

Next, taking the derivative of $V(t)$ for $t \in (t_k, t_{k+1})$ yields

Considering the characteristic described in (2.4) and using symmetric TFRNN weighting matrices $G_1$, $G_2$, and $G_3$, the fuzzy zero equations are derived as follows:

For the fuzzy term $\mathfrak{L}_{1}$ presented in Eq (3.8), given Assumption 2.1 and considering condition (3.1), it can be deduced that:

An analogous analysis applied to $\mathfrak{L}_{2}$, $\mathfrak{L}_{3}$, $\mathfrak{L}_{4}$, and $\mathfrak{L}_{5}$ in Eq (3.8) yields the following inequality from conditions (3.2)–(3.4):

By Lemma 2.2, processing the integral term in (3.8) yields the following inequality:

where ${{\Upsilon }_{1}} = [\Phi _{7}^{T}{{M}_{1}}\ \Phi _{7}^{T}{{M}_{2}}], {{\Upsilon}_{2}} = [\Phi _{9}^{T}{{M}_{3}}\ \Phi _{9}^{T}{{M}_{4}}], {{\mathcal{U}}_{1}} = \textbf{diag}\{{{U}_{1}}, 3{{U}_{1}}\}, {{\mathcal{U}}_{2}} = \textbf{diag}\{{{U}_{2}}, 3{{U}_{2}}\}$.

Additionally, to improve the design flexibility of the proposed method, the matrix $N$ and parameters ${{o}_{1}}$ and ${{o}_{2}}$ are introduced through the following equations:

where $\mathcal{B} = {{o}_{1}}{\psi^{T}}(t, x)+{{o}_{2}}\frac{\partial \psi^{T}(t, x)}{\partial t}$.

Using Green's formula and the specified boundary condition, the following equalities are established:

Following the procedure in Eq (3.15) and applying Lemma 2.1, we obtain

where ${{A}_{\pi }} = \textbf{diag}\left\{ \sum\limits_{l = 1}^{\omega}{{{\big(\frac{\pi }{\bar{\alpha }_{l} -\underline{\alpha }_{l}}\big)}^{2}}{{A}_{1l}}, ..., \sum\limits_{l = 1}^{\omega}{{{\big(\frac{\pi }{\bar{\alpha }_{l} -\underline{\alpha }_{l}}\big)}^{2}}{{A}_{nl}}}} \right\}$.

From (2.7) and (2.11), it is clear that the following inequalities hold:

Combining (3.8)–(3.17), we deduce

where $\mathfrak{G}[t, \sigma (t)] = \sum\limits_{i = 1}^{\iota }{\sum\limits_{j = 1}^{\iota }{{{\mathfrak{h} }_{i}}}(\mathscr{C}(t)){{\mathfrak{h} }_{j}}}(\mathscr{C}(t_k))\mathfrak{C}[t, \sigma (t)]+{{d}_{1}}(t){{\Upsilon }_{1}}\mathcal{U}_{1}^{-1}\Upsilon_{1}^{T}+{{d}_{2}}(t){{\Upsilon }_{2}}\mathcal{U}_{2}^{-1}\Upsilon _{2}^{T}$.

In addition, from $V_6(t_k^{+}) = V_7(t_{k+1}^{-}) = 0$ and $\sum\limits_{\alpha = 1}^{5}{{{V}_{\alpha }}(t)}$ being continuous, we can obtain

Based on the methodology in reference ^[33] and the condition $\mathfrak{A}\big[t, \sigma (t)\big] = \mathfrak{C} \big[t, \sigma (t)\big] - \mathfrak{F} < 0$, the following inequality holds:

By employing the Schur complement, the inequality $\mathfrak{A}\big[t, \sigma (t)\big] < 0$ remains valid for $t\in ({{t}_{k}}, {{t}_{k+1}})$ and $\sigma(t) \in [0, \bar{\sigma} ]$ given that it satisfy (3.5) and (3.6).

Remark 3.1. By introducing LKF, ${{V}_{4}}(t) = {\int}_{\Gamma} ({{t}_{k+1}}-t){{{\tilde{\psi}}}^{T}}({{t}_{k}}, x)$ $S\tilde{\psi}({{t}_{k}}, x)dx$, we obtain ${{\dot{V}}_{4}}(t) = - {\int}_{\Gamma } \zeta^{T} (t, x) \mathcal{E}_{12}^{T}S\mathcal{E}_{12}\zeta(t, x)dx$, which leads to the sufficient condition in Theorem 3.1 for synchronizing the master–slave system. The primary purpose of $V_4(t)$ is to facilitate finding a solution to the feasible LMIs. Removing $V_4(t)$ renders the LMIs in Theorem 3.1 infeasible, as it sets the matrix diagonal element $-\mathcal{E}_{12}^{T}S\mathcal{E}_{12}$ to zero. Consequently, without $V_4(t)$, the synchronization of the master–slave system under ASSD control cannot be guaranteed.

Remark 3.2. Based on prior experience, capturing more system information in the constructed LKF yields less conservative results. To achieve the larger MASI, $V_1(t)$ and $V_2(t)$ incorporate the properties of the membership function and the time-varying delays. $V_5(t),$$V_6(t)$, and $V_7(t)$ consider more detailed system state information at the sampling point. In addition, when $P_{1n} = P_{2m}\ (n, m\in\mathbb{N})$ holds, LKF $V_{1}(t)$ simplifies to the basic quadratic form ${\int}_{\Gamma }{{e}^{T}}(t, x){P}{e}(t, x)dx$ ^[21,26].

Remark 3.3. The integral terms $\mathfrak{L}_1$ and $\mathfrak{L}_2$, which are related to the derivatives of the membership function, are handled using the fuzzy zero equations and the compensation LKF method. First, by using the membership function's property in (2.4), fuzzy zero equations are applied to introduce additional TFRNN-weighting matrices via matrices $G_2$ and $G_3$. Then, the compensation LKF $V_3(t)$ introduces the integral terms $\mathfrak{L}_4$ and $\mathfrak{L}_5$, which are independent of tcompensation LKF relaxed the initial constraints of the membership function. Finally, the fuzzy zero equation and $\sum\limits_{i = 1}^{\iota }{{{\lambda }_{i}}} ({{R}_{1i}}+{{G}_{2}})\le0$ and $\sum\limits_{i = 1}^{\iota }{{{\lambda }_{i}}}({{R}_{2i}}+{{G}_{3}})\le0$ to conditions (3.4) and (3.5).

Remark 3.4. The looped-functional method is commonly used to reduce conservatism in SD control ^[24,34,35]. The advantage of this method is that it does not require the restriction matrix to be positive definite in function but ensures that ${{V}_{l}}({{t}_{k}}) = {{V}_{l}}(t_{k+1}) = 0$ at the terminal points of the sampling interval. The semi-looped-functional method improves upon the looped-functional method. In this paper, the terms $V_6(t)$ and $V_7(t)$ are referred to as semi-looped-functionals because ${V}_{6}({t_{k}^{+}}) = 0, V_6(t_{k+1}^{-}) \neq 0$ and ${V}_{7}({t_{k}^{+}})\neq0, {V}_{7}(t_{k+1}^{-}) = 0$. Therefore, semi-looped-functionals offer more flexibility than looped-functionals. In contrast, the sum of the derivatives of LKFs is typically required to be negative definite in the traditional LKF approach ^{[13,17,21,24]}. The discontinuities of semi-looped-functionals enable the removal of the negative definite restriction. We ensure that $\dot{V}(t)$ meets the condition ${d}_{M}\dot{V}(t)\le {{V}_{6}}(t_{k+1}^{-})-{{V}_{7}}(t_{k}^{+})$, resulting in $V(t_k) > V({t}_{k+1})$. It is evident that the semi-looped-functional method has more relaxed constraints and thus results in less conservative outcomes.

Corollary 3.1. For given scalars $\bar{\sigma } > 0$, $\eth$, $\backepsilon$, $\bar{\varphi}$, ${o}_{1}$, and ${o}_{2}$, the master–slave systems are asymptotically synchronized if there exist symmetric positive-definite matrices ${P}_{1i}$, ${P}_{2i}$, ${R}_{1i}$, ${R}_{2i}$, ${T}_{r}$, $S$, ${\Lambda }_{3}$, ${U}_{r}$, ${W}_{r}$, $NA_{l}$, positive-definite diagonal matrices ${\Lambda }_{r}$; and appropriate dimensional matrices $Y_q$, $H_p$, $G_p$, $M_q$, $N$, $\hat{N}_r$, where $r = 1, 2, p = 1, 2, 3, q = 1, 2, ..., 4$ and $i, j\in\mathbb{N}$, such that the following LMIs hold:

where the relevant notations are given as follows:

and the controller gains are given as follows:

Proof. To address this within the LMI framework, the nonlinear component of Theorem 3.1 has been linearized as described in Corollary 3.1. Conducting a congruence transformation using

into (3.5) and (3.6), substituting ${\hat{N}_{j}} = {N}{\mathcal{K}}_{j}$ and $-\bar{\varphi }{{\hat{N}}_{j}}{{\Lambda }^{-1}_{3}}{{\hat{N}}_{j}}\le -2{{\backepsilon }_{1}}{{\hat{N}}_{j}}+\backepsilon _{1}^{2}\bar{\varphi }{{\Lambda }_{3}}$ one obtains (3.25) and (3.26). This completes the proof. □

4.   Simulation results

Example 4.1. Consider system parameters of the TFRNNs as follows ^[26]:

From these conditions, we deduce that $\bar{\sigma} = \sqrt{2}$, $\eth = \frac{\sqrt{2}}{10}$, $\Omega_{1} = 0_2$, and $\Gamma_2 = I_2$. For the master system (2.3), the initial conditions are set as ${\mathfrak{g}}_1(s, x) = 1.5 \rho(x)$ and ${\mathfrak{g}}_2(s, x) = -2 \rho(x)$. For the slave system (2.5), the initial conditions are defined as ${\bar{\mathfrak{g}}_1}(s, x) = 1.425 \rho(x)$ and ${\bar{\mathfrak{g}}_2}(s, x) = -2.1 \rho(x)$, where $\rho(x) = \cos \left(\frac{\pi (x - 2)}{6} \right)$.

This paper employs three-dimensional surface plots to visually represent the system's state trajectories across time and space. These plots feature the spatial variable $x$ and the time variable $t$ as independent variables, with the system state value ${\psi_i}(t, x)$ as the dependent variable. When $u(t, x) = 0$, Figure 1 shows the trajectories of states $\psi_{i}(t, x)$ and $\left\|\psi(t, x)\right\|$. Figure 1(a) and (b) show that the surface fluctuates continuously, indicating that the system state trajectories oscillate across the entire spatial range. Additionally, Figure 1(c) shows the evolution of $\left\|\psi(t, x)\right\|$. It quantitatively represents the synchronization error, showing how close or far the master–slave system is from the desired state. A smaller error value indicates a higher level of synchronization between the master–slave system. From Figure 1(c), the $\left\|\psi(t, x)\right\|$ is divergent and cannot converge to 0. In summary, this implies that the TFSRDNN master–slave system cannot achieve synchronization without ASSD control inputs.

Using Algorithm 1 and the solving parameters ${o}_{1} = -0.1943$ and ${o}_{2} = 0.4210$, we obtain the MASI $d_M = 0.0767$. Utilizing the matrix relations presented in (3.27), the corresponding controller gains are solved as $\mathcal{K}_1 = \mathcal{K}_2 = [-6.9728\ -2.9979;3.0663\ -6.8996]$. We then analyzed the system's motion trajectory images to evaluate the controller's effectiveness. Figure 2 illustrates the controlled trajectories of states $\psi_{i}(t, x)$ and $\left\|\psi(t, x)\right\|$ using these controller gains. Figure 2(a) and (b) show the original fluctuating surface gradually becoming the plane at $\psi_{i}(t, x) = 0$. In simpler terms, the system's states $\psi_{i}(t, x)$ approach 0 across all spatial dimensions. Figure 3(c) depicts the evolution of the controlled $\left\|\psi(t, x)\right\|$, demonstrating its rapid convergence under ASSD control inputs. Notably, around $t = 2$, $\left\|\psi(t, x)\right\|$ reaches 0 and is not fluctuating. Comparing Figures 1 and 2 shows that the ASSD controller and Corollary 3.1, as designed in this paper, effectively synchronize the master–slave system.

To demonstrate the ASSD controller's effectiveness in complex real-world scenarios, we examine actuator saturation. Figure 3(a) and (b) illustrate the effects of actuator saturation on control inputs. At $u_{1}(t, x) = 0.60$ and $u_{2}(t, x) = 0.32$, the originally curved boundary transitions to a straight one. This indicates that the saturation upper limits are $0.60$ and $0.32$, respectively. The ASSD controller effectively synchronizes the master–slave system, even under actuator saturation, demonstrating its reliability.

In SD control, MASI is a key performance indicator. A larger MASI indicates the use of fewer communication resources to achieve the desired control effect. Table 2 compares the MASI values for Corollary 3.1 in this paper with those of existing methods, highlighting the superiority of our method. Under identical conditions, Corollary 3.1 in this paper is less conservative. It achieves the 12.2% and 35.9% improvement in MASI values compared with Corollary 1 ^[26] and Theorem 1 ^[26], which MASI values are 0.0683 and 0.0564, respectively.

Example 4.2. Consider the TFRNNs with the following parameters:

from which we obtain $\bar{\sigma } = 2$, $\eth = {4}/{9}$, $\Omega_{1} = 0_{2}$, and $\Omega_{2} = {I_{2}}$. For the master system (2.3), set initial conditions as ${{\mathfrak{g} }_{1}}(s, x) = 2.8\rho (x)$ and ${{\mathfrak{g} }_{2}}(s, x) = -1.3\rho (x)$. For the slave system (2.5), set them as ${\bar{\mathfrak{g} }_{1}}(s, x) = 2.7\rho (x)$ and ${{\bar{\mathfrak{g} } }_{2}}(s, x) = -1.4\rho (x)$, and $\rho (x) = \cos (\pi x)$.

For $u(t, x) = 0$, Figure 4 illustrates the trajectories of states $\psi_{i}(t, x)$ and $\left\|\psi(t, x)\right\|$. Observations from Figure 4(a) and (b) reveal that the system's state continuously oscillates. Furthermore, Figure 4(c) indicates that $\left\|\psi(t, x)\right\|$ never converges to 0. This means that without control inputs, the master-slave system cannot achieve synchronization. Based on Algorithm 1 and the choice of solving parameters ${o}_{1} = -2.4019$ and ${o}_{2} = -1.9352$, we can obtain the MASI $d_M = 0.2753$. Utilizing the matrix relations presented in (3.27), the corresponding controller gains are solved as $\mathcal{K}_1 = \mathcal{K}_2 = [-1.8731\ 0.2639;0.4181\ -3.3359]$. Figure 5 shows the controlled trajectories of states $\psi_{i}(t, x)$ and $\left\|\psi(t, x)\right\|$ using these controller gains. Figure 5(a) and (b) illustrate that the system state effectively converges to 0 under ASSD control. In Figure 5(c), the $\left\|\psi(t, x)\right\|$ rapidly converges to 0 after control is applied, around $t = 5$. The master–slave system achieves synchronization in Figure 5, demonstrating the effectiveness of Corollary 3.1 and the ASSD controller. Figure 6(a) and (b) show the impact of actuator saturation on control inputs. These reveal that actuator saturation affects control input values, setting upper limits at $0.10$ and $0.20$.

5.   Conclusions

The synchronization issue of TFRNNs with time-varying delays and actuator saturation has been investigated. Then, the ASSD controller has been utilized for synchronization to conserve communication resources and enhance system resilience. Subsequently, we have constructed the LKFs that incorporate more system information. Moreover, fuzzy zero equations and compensation LKF have been used to handle terms with derivatives of the membership function, relaxing the constraints. The semi-looped-functional terms have been constructed to enlarge the feasible space further. Furthermore, sufficient conditions for synchronizing TFRNNs and a search algorithm for MASI have been established. Finally, two simulations have been exemplified to demonstrate the approach's feasibility.

Author contributions

Yuchen Niu: Investigation, software, validation, writing – original draft; Kaibo Shi: Conceptualization, writing – review & editing; Xiao Cai: Data curation, writing – review & editing; Shiping Wen: Writing – review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that there are no conflicts of interest.

A.   Appendix-A

B.   Appendix-B

C.   Appendix-C