Neural Optimization Techniques for Noisy-Data Observer-Based Neuro-Adaptive Control for Strict-Feedback Control Systems: Addressing Tracking and Predefined Accuracy Constraints

Abstract

This research proposes a fractional-order adaptive neural control scheme using an optimized backstepping (OB) approach to address strict-feedback nonlinear systems with uncertain control directions and predefined performance requirements. The OB framework integrates both fractional-order virtual and actual controllers to achieve global optimization, while a Nussbaum-type function is introduced to handle unknown control paths. To ensure convergence to desired accuracy within a prescribed time, a fractional-order dynamic-switching mechanism and a quartic-barrier Lyapunov function are employed. An input-to-state practically stable (ISpS) auxiliary signal is designed to mitigate unmodeled dynamics, leveraging classical lemmas adapted to fractional-order systems. The study further investigates a decentralized control scenario for large-scale stochastic nonlinear systems with uncertain dynamics, undefined control directions, and unmeasurable states. Fuzzy logic systems are employed to approximate unknown nonlinearities, while a fuzzy-phase observer is designed to estimate inaccessible states. The use of Nussbaum-type functions in decentralized architectures addresses uncertainties in control directions. A key novelty of this work lies in the combination of fractional-order adaptive control, fuzzy logic estimation, and Nussbaum-based decentralized backstepping to guarantee that all closed-loop signals remain bounded in probability. The proposed method ensures that system outputs converge to a small neighborhood around the origin, even under stochastic disturbances. The simulation results confirm the effectiveness and robustness of the proposed control strategy.

1. Introduction

In order to accomplish stability, the unpredictable unidentified behavior of fractional-order nonlinear (FON) frameworks was handled by using the estimated effectiveness of neural networks (NNs) or fuzzy logic systems. In [1], an adaptive-control method based on NN properties was proposed for FON frameworks via an actuator malfunction. In [2], a fuzzy controller was designed for nonstrict-feedback FON frameworks. Liu et al. [3] presented an interdisciplinary backstepping control method for nonaffine FON frameworks exhibiting uncertain interactions resulting in consistency. In [4], a Lyapunov-based finite-time uncertain controller was designed for FON tumor structures undergoing treatment. The research and experiences listed earlier do not consider the impact of comprehensive system limitations on FON frameworks. It is worth emphasizing this. Recently, Chu et al. [5] investigated a modified technique for the electrical responses of discrete FON biophysical neural network models and their dynamical reactions. Chu et al. [5] examined the complex adaptive-learning cortical NN system for addressing time-fractional difference equations with bursting and mixed-mode oscillation patterns. Alzahrani and Rashid [6] proposed a comprehensive systematic review of the network design model for topology control and routing synergy in wireless sensor networks for prolonged longevity. Rashid et al. [7] expounded on the perspectives for mixed-mode oscillations of fractional neural network approaches to the analysis of neurophysiological data from the perspective of the observability of complex networks. Rashid et al. [8] derived new computations of the fractional worm transmission model in wireless sensor networks in view of a new integral transform with statistical analysis. Chu et al. [9] presented complex adaptive-learning cortical NN systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviors.

Network restrictions are a common issue in commercial and pragmatic mechanisms, reducing effectiveness and causing instabilities. The barrier-based Lyapunov functional is an advantageous instrument for preventing state restrictions from being violated in FON frameworks, and numerous limitation-control algorithms have been developed [10,11]. In [11], an observer-based adaptive NN control method was developed for nonstrict-feedback FON frameworks featuring an outcome requirement. This algorithm ensures that tracking errors fulfill limitations. In [12], an ensemble adaptive-control approach was constructed for many FON frameworks with limitations, and the barrier-based Lyapunov functional was used to regulate output within a predetermined region. In [13], an adaptive dynamic interface control strategy for FON frameworks employing asymmetrical status requirements was constructed utilizing an unbalanced barrier-based Lyapunov functional. However, the features of the controlling signal that enter ought to be avoided and disregarded.

Optimized control is an oversight strategy that maximizes or minimizes particular effectiveness measures for an unpredictable system [14]. However, we can find the optimum management technique by building and analyzing the Hamilton–Jacobi–Bellman (H–J–B) formula [15] for the mechanism under consideration. In fact, the scenario in real networks is complicated and changing; it is highly challenging to reach excellence in optimal oversight. Recently, an approach known as optimized backstepping (OB) was initially presented [16]. The fundamental concept is to address the respective subsystem’s H–J–B problem while constructing the optimized virtualized control and the optimized actual control in order to accomplish global optimization. Wen et al. [17] reduced the OB method construction approach for greater generalization. Several researchers have created multiple optimized control methods for a number of networks using the modified OB approach [18]. Li et al. [19] created a condition-constrained productivity measure for unpredictable processes and proposed an OB regulation approach to limit state variability. Li et al. [20] proposed a state-observer-based OB controlling approach for strict-feedback (SF) nonlinear structures with unnoticed conditions. Zhang et al. [21] proposed an observer-based OB controlling technique for SF structures involving configuration and source restrictions, building on [20]. However, the control mechanisms listed earlier only ensure system reliability.

In contemporary manufacturing environments, it is frequently essential to meet predetermined effectiveness specifications, which primarily comprise two aspects: (i) attaining control targets over a predetermined time frame and (ii) bringing the system’s monitoring or regulatory inaccuracies into conformity with predetermined precision. Several analogous findings have been documented and reported as the hypothesis of control has evolved [22,23]. Indeed, Chen et al. [24] employed Barbalat’s argument and three $C^n$-group mappings to create a universal regulator over SF nonlinear structures that permits the measurement inaccuracy to approach predetermined correctness. Li et al. [25] offered observer-based regulation techniques for probabilistic SF networks, resulting in mean-square stability within a particular time frame. Interestingly, these automation systems only provide predetermined duration and precision monitoring. Because of the intricate nature of unpredictable nonlinear structures, it is incredibly challenging to achieve sufficient deviation convergence values regarding the predetermined reliability in the duration specified. Zhou et al. [26] proposed a method of regulation for SF mechanisms involving a time-dependent transitioning function and the barrier Lyapunov function. This approach promises monitoring within the predefined degree of the convergence process and preciseness throughout the predefined time according to a single confined initial condition. While there has been significant success in controlling predefined functionality for numerous platforms, there has been insufficient investigation into unpredictable control approaches.

Control approaches for nonlinear structures address the issue of uncertain controlling paths, which can lead to network destabilization or implosion [27,28,29]. Regarding output-feedback structures featuring unidentified regulation instructions, the observation-based condition-control methodologies presented in [28,29,30] handle unanticipated perturbation and actuation failure issues, respectively. Wu et al. [31] suggested an approach to controlling for unpredictable SF networks involving state delays, which addresses the issue of uncertain controlling choices. Liu et al. [32] developed an algorithm for controlling unpredictable SF machines featuring undetermined regulation advances, allowing the error spectrum to approximate predetermined correctness. However, their monitoring method prevented predetermining the amount of time required to fulfill the oversight target.

Stability analysis and influence modeling of stochastic nonlinear phenomena have gained popularity in recent years due to their significance in various real-world applications, such as hydrogen, thermal energy, biochemical, financial, and immunological techniques. Itô stochastic formulations, along with parametric effective regulation frameworks and uncertain mechanism regulation schemes [33,34], have been employed to develop various techniques for stochastic nonlinear mechanism filtering and regulation, including $H_2/H_{\infty }$ filter development [35], sliding-mode controller conception [36], and T–S approaches involving uncertain influence or uncertain interpolating regulation conception [37]. Recently, the widely utilized OB methodology was exploited to address stochastic complex networks (see [38]). In [33], the OB strategy for unpredictable individual-input/output complex environments was examined using the quartic Lyapunov functional. In [39], an accurate adaptive OB development strategy was suggested for unpredictable individual-input/output complex assemblies exhibiting unmodeled interactions using the small-gain hypothesis. These findings have been adapted to probabilistic complex systems with massive scales, and dynamic decentralized backstepping controlling techniques were devised in [40,41]. Note that these outcomes only apply to complex systems when their nonlinear features have been determined or are capable of being accurately parameterized. Thus, approaches are unable to be employed in stochastic processes containing hierarchical uncertainty.

Intelligent control techniques have gained significant attention in recent years for managing dynamic and nonlinear systems, particularly in cases where conventional PID or model-based controllers prove inadequate. One such approach was presented in the study “Development of an Adaptive Fuzzy-Neural Controller for Temperature Control in a Brick Tunnel kiln”, where the authors [42] developed an adaptive fuzzy–neural controller. This research proposed a hybrid system that combines neural networks with fuzzy logic to address the challenges of temperature regulation in a brick tunnel kiln—an industrial process characterized by strong nonlinearity and time-delay. In this approach, the fuzzy inference system incorporates expert knowledge through a set of rules, while the neural network module adaptively tunes the fuzzy parameters to enhance performance under varying operating conditions. Yao et al. [43] investigated model reference adaptive tracking control for hydraulic servo systems using nonlinear NNs.

Numerous NN OB control development techniques and adapted uncertain techniques have been created over the past decade to address the previous problem for stochastic nonlinear networks containing entire undetermined exponential functionalities (for instance, see [44]). Wang et al. [45] examined control strategies regarding adaptive uncertainty or NN state feedback for a group of single-input/output unpredictable nonlinear mechanisms. Tong et al. [46] contemplated adapted feedback suggestions for regulators for individual-input/output stochastic nonlinear systems. Li et al. [47] established adapted uncertain and decentralized result control layout strategies based on NNs to stabilize a class of large-scale stochastic nonlinear networks. While adaptive-uncertain and NN OB approaches have made substantial advancements in stochastic complex oversight, the current solutions overlook unpredictable settings involving undetermined input directions. As pointed out in [47,48], unidentified control tendencies might occur in various operational mechanisms, making the control layout resilient and hard. Wang et al. [49] investigated the adaptive-uncertain controlling issue pertaining to unpredictable stochastic nonlinears involving uncertain input orientations. Yu et al. [50] explained a Razumikhin–Nussbaum argument that was developed for probabilistic dynamic time-delay networks involving undetermined regulation flow. This argument has been broadened to include probabilistic pure-feedback time-delay structures [51]. The findings presented in [49,50,51] require immediate assessment of controllable network states, making them ineffective for solving outcome control via feedback problems. The Nussbaum function paradigms suggested in [51,52] are limited to individual input/output stochastic nonlinear structures and are not applicable to probabilistic enormous-scale nonlinear networks.

The controlling procedure for practical architectural engineering typically requires effort to transfer information to controls, resulting in system-generated inefficiencies and serving as a primary driver of system effectiveness degradation. Several successful strategies for dealing with significant feedback latency have been discovered for FON systems [53,54]. Geng et al. [53] determined the feedback controller for an FON system with incoming latency, constructed employing the Smith prediction. Keighobadi et al. [54] detailed an enhanced adaptive-control approach using function estimation for FON systems, including feedback latency. In [55], a command-filter-based adaptive regulator for FONSs with signal latency was created utilizing the fractional integral. Tong and Li [56] investigated an adaptive NN controlling approach for FON systems with signal latency, devised utilizing the supplemental mechanism. Jian et al. [57] examined an adaptive neural OB approach for fuzzy systems involving unidentified control paths and their predefined accuracy. To date, no published studies have addressed adaptive neural-optimized control for fuzzy systems with unidentified regulatory paths and predefined performance requirements in FON systems and stochastic data-driven techniques.

Inspired by the aforesaid methodologies, this research proposes an adaptive NN controlling approach using the OB methodology for SF applications involving undetermined variable orientations and predetermined efficiency. The oversight approach ensures that the system’s tracking error settles to a predetermined precision in a predetermined time while maintaining a low effectiveness factor. We summarize the research’s significant achievements in three key areas:

• This work introduces a novel approach for predetermined-time and precision control within an SF framework using unknown control directions and the Mittag–Leffler input-to-state practical stability Lyapunov function (MLISPSLF). It establishes sufficient conditions for fractional-order systems to guarantee a memory-dependent signal with convergence of measurement error to a predefined accuracy within a fixed time.
• The proposed optimization method decomposes the performance index into distinct error terms using the OB methodology. This improves efficiency over the existing predetermined performance control techniques [58] by reducing control efforts. Unlike [59], generic-class ${\mathcal{K}}_{\infty }$ functions are employed for Lyapunov estimates, broadening applicability. The MLISPSLF framework extends classical input-to-state practical stability results [60] to fractional systems.
• A Nussbaum-type function is utilized to handle uncertain control directions in memory-dependent fractional systems, distinguishing the method from conventional OB control schemes [17]. The simulation results confirm the effective achievement of the control objectives.
• The adaptive decentralized stabilization scheme addresses fractional-order uncertain stochastic nonlinear large-scale systems with unknown dynamics, unrestricted states, and inputs. Fuzzy fractional-order logic systems are leveraged alongside the Mittag–Leffler function for modeling unknown nonlinearities, with an observer designed for estimating unmeasured disturbances.
• A novel Nussbaum function property is introduced to tackle uncertain fractional control directions in decentralized output-feedback loops. Combining OB techniques with this property yields an adaptive-uncertain decentralized control strategy ensuring stochastic boundedness of closed-loop signals and convergence of estimation errors to a small neighborhood of the origin.

2. Preliminaries and System Descriptions

Notations. In what follows, $\mathbb{N}$ represents the set of natural numbers, and ${\mathbb{R}}^{\mathrm{n}}$ has emerged as the Euclidean space with dimension $\mathrm{n}$. ${\mathbb{C}}^{\mathrm{q}}\left([0,+\infty ],\mathbb{R}\right)$ refers to a set of functions from $[0,\infty )$ to $\mathbb{R}$ with $\mathrm{q}\in \mathbb{N}$-order continuous derivatives. $\parallel .\parallel$ represents the absolute value of real/complex number, whereas $\parallel .\parallel$ and ${\parallel .\parallel }_{\infty }$ represent the Euclidean norm of vectors and ${\mathcal{L}}_{\infty }$ norm for temporal functions, respectively. Assuming two mappings, $\mathcal{F}\left(\mathrm{y}\right)$ and $\mathcal{G}\left(\mathrm{y}\right)$, are computable, and then $\mathcal{F}\circ \mathcal{G}\left(\mathrm{y}\right)\triangleq \mathcal{F}\left(\mathcal{G}\right(\mathrm{y}\left)\right)$. A function with class $\mathcal{K}$ is continuous, strictly monotone, and has $\tilde{\Xi }\left(0\right)=0$. When $\tilde{\Xi }\left(\mathrm{y}\right)\mapsto \infty$ as $\mathrm{y}\mapsto \infty$, then $[0,\infty )$ represents a function with class ${\mathcal{K}}_{\infty }$. Similarly, $\tilde{\Xi }:[0,\infty )\times [0,\infty )\mapsto [0,\infty )$ represents a collection of mappings ${\mathcal{K}}_{\mathcal{L}}$ if $\tilde{\Xi }(.,\tau )$ has emerged as a function with classification $\mathcal{K}$ with regard to the fixed $\tau$ and $\tilde{\Xi }(\mathrm{y},.)$ drops and anticipates 0 as $\tau \mapsto \infty$ for the specified $\mathrm{y}$. For vector $\mathrm{y}={({\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\mathrm{n}})}^{\mathbf{T}}\in {\mathbb{R}}^{\mathrm{n}},$ ${\overline{\mathrm{y}}}_{\sigma },$ ${\overline{\mathrm{y}}}_{\sigma }$ represents the initial $\sigma$ element in $\mathrm{y}$; that is, ${\overline{\mathrm{y}}}_{\sigma }\triangleq {({\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\sigma })}^{\mathbf{T}}.$

The following section introduces significant meanings and lemmas for Caputo fractional-order operators.

Definition 1 

([61]). The Caputo fractional-order derivative of $\mathcal{F}\left(\tau \right)\in {\mathbb{C}}^{\mathrm{q}+1}\left([0,+\infty )\right)$ with order $\alpha >00$ can be expressed as

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\mathcal{F}\left(\tau \right)={\displaystyle \frac{1}{\Gamma (\mathrm{q}-\alpha )}}\underset{0}{\overset{\tau }{\int }}{\displaystyle \frac{{\mathcal{F}}^{\left(\mathrm{q}\right)}\left(\varpi \right)}{{(\tau -\varpi )}^{\alpha +1-\mathrm{q}}}}d\varpi ,\end{array}$$

where $\mathrm{q}$ is a non-negative integer with $\alpha \in (\mathrm{q}-1,\mathrm{q}]$, and $\Gamma (.)$ defines the Gamma function.

Definition 2 

([61]). The Mittag–Leffler function, involving parameters α and $v_1>0$, is specified as

$$\begin{array}{c}\hfill {\mathcal{E}}_{\alpha ,v_1}=\sum _{\ell =0}^{\infty }{\displaystyle \frac{{\varsigma }^{\ell }}{\Gamma (\alpha \ell +v_1)}},\end{array}$$

where ς denotes a complex quantity. If $v_1=1,$ then ${\mathcal{E}}_{\alpha ,v_1}$ becomes a single-parameter Mittag–Leffler function, which can be expressed as

$$\begin{array}{c}\hfill {\mathcal{E}}_{\alpha }\left(\varsigma \right)={\mathcal{E}}_{\alpha ,1}\left(\varsigma \right)={\displaystyle \frac{{\varsigma }^{\ell }}{\Gamma (\alpha \ell +1)}}.\end{array}$$

Lemma 1 

([61]). For $\alpha \in (0,2)$ and $v_1\in \mathbb{R},$ the subsequent variant satisfies

$$\begin{array}{c}\hfill \left|{\mathcal{E}}_{\alpha ,v_1}\left(\varsigma \right)\right|\le {\displaystyle \frac{\mathcal{W}}{1+\left|\varsigma \right|}},\end{array}$$

(1)

where $\upsilon \le |arg\left(\varsigma \right)|\le \pi ,\left|\varsigma \right|\ge 0,\upsilon \in \left({\displaystyle \frac{\pi \alpha }{2}},min\{\pi ,\pi \alpha \}\right)$, and $\mathcal{W}$ is a constant.

In the 20th century, Jiang and Praly [62] suggested various stability analyses for classical systems $\dot{\mathrm{y}}=\mathcal{F}(\mathrm{y},\mathrm{s})$. These concepts are crucial for control concerns like stable control and nonlinear stabilization. Input-to-state stability [62] plays a crucial role in the design of controllers for systems with unmodeled behavior. It allows for the creation of a memorizing signal to prevent incorporating unidentified state factors instantly into the controller. Jiang and Praly’s [62] conclusions only apply to integer-order unmodeled phenomena and therefore are not applicable to FON ones. To address unmodeled FON behavior, we introduce the Mittag–Leffler input-to-state operational stabilization Lyapunov function and define an essential requirement for FON systems that possess it. Assume that the FON system is

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\mathrm{y}\left(\tau \right)=\mathcal{F}\left(\mathrm{y}\left(\tau \right),\mathrm{s}\left(\tau \right)\right),\alpha \in (0,1),\end{array}$$

(2)

where $\mathrm{y}\left(\tau \right)\in {\mathbb{R}}^{\mathrm{n}}$ indicates the state variable, $\mathrm{s}\left(\tau \right)\in {\mathbb{R}}^{\overline{n}}$ is considered to be external input, and $\mathcal{F}:{\mathbb{R}}^{\mathrm{n}}\times {\mathbb{R}}^{\overline{n}}\mapsto {\mathbb{R}}^{\mathrm{n}}$ is continuously differentiable function.

Definition 3 

([63]). The solution $\mathrm{y}\left(\tau \right)$ of system (2) is input-to-state operational stabilization concerning $\mathrm{s}\left(\tau \right).$ For a function ${\tilde{\Xi }}_1(.,.)$ of class $\mathcal{K}\mathcal{L}$, a mapping ${\tilde{\Xi }}_2(.)$ of class $\mathcal{K}$, and a fixed number ${\tilde{\Xi }}_0\ge 0$ such that, for locally bounded controlling input $\mathrm{s}\left(\tau \right)(\tau \ge 0)$ as well as for some initial requirement $\mathrm{y}\left(0\right),\mathrm{y}\left(\tau \right)$ fulfills

$$\begin{array}{c}\hfill \parallel \mathrm{y}\left(\tau \right)\parallel \le {\tilde{\Xi }}_1\left(\parallel \mathrm{y}\left(0\right)\parallel ,\tau \right)+{\tilde{\Xi }}_2\left(\parallel \mathrm{s}\left(\tau \right)\parallel \right)+{\tilde{\Xi }}_0.\end{array}$$

To prove Lemma 3, we start by introducing a helpful concept and specify complete monotonicity.

Lemma 2 

([63]). For $0<\alpha <1$ and for any fixed ${\tilde{\Xi }}_1>0,$ the following identity satisfies

$$\begin{array}{c}\hfill {\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })=-{\displaystyle \frac{1}{{\tilde{\Xi }}_1}}{\displaystyle \frac{d}{d\tau }}{\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha }),\tau >0.\end{array}$$

(3)

Definition 4 

([64]). Assume that there is a completely monotonic mapping $\mathcal{F}\left(\tau \right)$ defined on $\mathbb{I}\subset \mathbb{R},$ if $\mathcal{F}\left(\tau \right)$ possesses differentiation up to ℓ and hold

$$\begin{array}{c}\hfill {(-1)}^{\ell }{\mathcal{F}}^{(\ell )}\left(\tau \right)\ge 0,\forall \tau \in \mathbb{I},\ell =0,1,2,\dots .\end{array}$$

(4)

Lemma 3. 

Assume that there is an FON system (2) containing locally bounded control input $\mathrm{s}\left(\tau \right)(\tau \ge 0).$ Consider origin to be a steady state of (2) and ∃ continuously differentiable and locally Lipschitz mapping $\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right):[0,\infty )\times {\mathbb{R}}^{\mathrm{n}}\mapsto {\mathbb{R}}_{+}$ such that

$$\begin{array}{c}\hfill {\tilde{\Xi }}_3(\parallel {\mathrm{y}}_1\left(\tau \right)\parallel )\le \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)\le {\tilde{\Xi }}_4(\parallel {\mathrm{y}}_1\left(\tau \right)\parallel ),\end{array}$$

(5)

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)\le -{\tilde{\Xi }}_1\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)+{\tilde{\Xi }}_5(\parallel \mathrm{s}\parallel )+{\tilde{\Xi }}_2,\end{array}$$

(6)

the value of ${\mathrm{y}}_1\left(\tau \right)$ is input-state practically stable with regard to $\mathrm{s}$, while the function that results $\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)$ is referred to as the Mittag–Leffler input-state practically stable Lyapunov candidate, where ${\tilde{\Xi }}_{\ell }(.),(\ell =3,4,5)$ are ${\mathcal{K}}_{\infty }$ functions and ${\tilde{\Xi }}_1>0$ and ${\tilde{\Xi }}_2\ge 0$ constitute integers.

Proof. 

Under the assumption of (6), ∃ a mapping ${\rm Y}\left(\tau \right)\ge 0$ such that

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)+{\rm Y}\left(\tau \right)\le -{\tilde{\Xi }}_1\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)+{\tilde{\Xi }}_5(\parallel \mathrm{s}\parallel )+{\tilde{\Xi }}_2.\end{array}$$

(7)

Applying the Laplace transform in terms of Caputo sense [61] on both sides of (7), we get

$$\begin{array}{c}\hfill \mathrm{U}\left(\varrho \right)={\displaystyle \frac{1}{{\varrho }^{\alpha }+{\tilde{\Xi }}_1}}\left\{{\varrho }^{\alpha -1}\mathrm{U}\left(0\right)-{\rm Y}\left(\varrho \right)+{\tilde{\Xi }}_5\left(\varrho \right)+{\displaystyle \frac{{\tilde{\Xi }}_2}{\varrho }}\right\},\end{array}$$

(8)

where $\mathrm{U}\left(0\right)=\mathrm{U}\left(0,\mathrm{y}\left(0\right)\right),\mathrm{U}\left(\varrho \right)=\mathfrak{L}\mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right),{\rm Y}\left(\varrho \right)=\mathfrak{L}{\rm Y}\left(\tau \right)$, and ${\tilde{\Xi }}_5\left(\phi \right)=\mathfrak{L}{\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel ).$ Implementing the inverse Laplace transform yields

$$\begin{array}{ccc}\hfill \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)& & =\mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })+\left({\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel )-{\rm Y}\left(\tau \right))\right)★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })\right)\hfill \\ & & +{\tilde{\Xi }}_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\tilde{\Xi }}_1{\tau }^{\alpha }),\hfill \end{array}$$

(9)

where ★ serves as the convolution operator. Lemma 1 states that

$$\begin{array}{c}\hfill {\tilde{\Xi }}_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\tilde{\Xi }}_1{\tau }^{\alpha })\le {\displaystyle \frac{\mathcal{W}{\tilde{\Xi }}_2{\tau }^{\alpha }}{1+|{\tilde{\Xi }}_1{\tau }^{\alpha }|}}\le {\displaystyle \frac{\mathcal{W}{\tilde{\Xi }}_2}{{\tilde{\Xi }}_1}}.\end{array}$$

(10)

Considering the aforesaid variant, we can conclude that ${\tau }^{\alpha -1}$ and ${\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })$ are non-negative; we obtain

$$\begin{array}{c}\hfill \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)\le \mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })+\underset{0}{\overset{\tau }{\int }}{\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel )\varkappa (\tau -\varpi )d\varpi +{\displaystyle \frac{\mathcal{W}{\tilde{\Xi }}_2}{{\tilde{\Xi }}_1}},\end{array}$$

(11)

where $\varkappa \left(\varrho \right)={\varrho }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\varrho }^{\alpha }).$ Using the fact of Lemma 2, we find

$$\begin{array}{ccc}\hfill \underset{0}{\overset{\tau }{\int }}{\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel )\varkappa (\tau -\varpi )d\varpi & & =\underset{0}{\overset{\tau }{\int }}{\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel ){(\tau -\varpi )}^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{(\tau -\varpi )}^{\alpha })d\varpi \hfill \\ & & \le {\displaystyle \frac{1}{{\tilde{\Xi }}_1}}{\tilde{\Xi }}_5{(\parallel \mathrm{s}\left(\tau \right)\parallel }_{\infty }).\hfill \end{array}$$

(12)

So, (11) can be expressed as

$$\begin{array}{c}\hfill \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)\le \mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })+{\displaystyle \frac{1}{{\tilde{\Xi }}_1}}{\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right){\parallel }_{\infty })+{\displaystyle \frac{\mathcal{W}{\tilde{\Xi }}_2}{{\tilde{\Xi }}_1}},\end{array}$$

(13)

Implementing (5), we obtain

$$\begin{array}{c}\hfill \parallel {\mathrm{y}}_1\left(\tau \right)\parallel \le \underset{{\tilde{\Xi }}_1(\parallel {\mathrm{y}}_1\left(0\right)\parallel ,\tau )}{\underbrace{{\tilde{\Xi }}_3^{-1}\left(3\mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })\right)}}+\underset{{\tilde{\Xi }}_2{(\parallel \mathrm{s}\left(\tau \right)\parallel }_{\infty })}{\underbrace{{\tilde{\Xi }}_3^{-1}\left({\displaystyle \frac{3}{{\tilde{\Xi }}_1}}{\tilde{\Xi }}_5\left({\parallel \mathrm{s}\left(\tau \right)\parallel }_{\infty }\right)\right)}}+\underset{{\tilde{\Xi }}_0}{\underbrace{{\tilde{\Xi }}_3^{-1}\left({\displaystyle \frac{3\mathcal{W}{\tilde{\Xi }}_2}{{\tilde{\Xi }}_1}}\right)}}.\end{array}$$

(14)

${\mathcal{E}}_{\alpha ,v_1}(-\varrho )(\varrho \ge 0)$ is completely monotone for $0<\alpha \le$ and $v_1\ge \alpha .$ Thus, Definition 4 states that ${\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })$ is a monotonically decreasing function for $\tau \ge 0$. Furthermore, results from [65] lead to $\underset{\tau \mapsto \infty }{lim}{\mathcal{E}}_{\alpha }(-\tilde{\Xi }{\tau }^{\alpha })=0.$ Therefore, ${\tilde{\Xi }}_1(\parallel {\mathrm{y}}_1\left(0\right)\parallel ,\tau )$ represents a class of functions $\mathcal{K}\mathfrak{L}$. Given that ${\tilde{\Xi }}_5(.)$ is an expression of class ${\mathcal{K}}_{\infty }$, we may conclude that ${\tilde{\Xi }}_2{(\parallel \mathrm{s}\left(\tau \right)\parallel }_{\infty })$ is also a function of class $\mathcal{K}$. Based on Definition 3, ${\mathrm{y}}_1\left(\tau \right)$ is input-state practically stable concerning $\mathrm{s}$. The documentation is completed. □

Lemma 4. 

If framework (2) possesses a Mittag–Leffler input-state practically stable Lyapunov candidate, eventually, for every fixed value ${\overline{\tilde{\Xi }}}_1\in (0.{\tilde{\Xi }}_1)$ and some ${\mathrm{y}}_0={\mathrm{y}}_1\left(0\right)$, any continuous mapping ${\tilde{\Xi }}_5\left(\mathrm{s}\right)\ge 0{\tilde{\Xi }}_5(\parallel \mathrm{s}\parallel )$ and ${\mathrm{z}}_0>0$, and ∃ a finite ${\mathbf{T}}_0={\mathbf{T}}_0({\overline{\tilde{\Xi }}}_1,{\mathrm{z}}_0,{\mathrm{y}}_0)\ge 0,$ a positive bounded mapping $U_1(0,\tau )$ stated on $[0,+\infty )$ and a signal $\mathrm{z}\left(\tau \right)$ is described as

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\mathrm{z}\left(\tau \right)=-{\overline{\tilde{\Xi }}}_1\mathrm{z}\left(\tau \right)+{\overline{\tilde{\Xi }}}_5\left(\mathrm{s}\right)+{\tilde{\Xi }}_2,\mathrm{z}\left(0\right)={\mathrm{z}}_0,\end{array}$$

(15)

where $U_1(0,\tau )=0,\forall \tau \ge {\mathbf{T}}_0$ and

$$\begin{array}{c}\hfill \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)\le \mathrm{z}\left(\tau \right)+U_1(0,\tau ),\tau \ge 0.\end{array}$$

(16)

Proof. 

From (6) (analogous to the demonstration of Lemma 3), applying the Laplace and inverse Laplace transforms on both sides of (15) produces the outcome

$$\begin{array}{c}\hfill \mathrm{z}\left(\tau \right)=\mathrm{z}\left(0\right){\mathcal{E}}_{\alpha }(-{\overline{\tilde{\Xi }}}_1{\tau }^{\alpha })+{\overline{\tilde{\Xi }}}_5\left(\mathrm{s}\left(\tau \right)\right)★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-\overline{\tilde{\Xi }}{\tau }^{\alpha })\right)+{\tilde{\Xi }}_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\overline{\tilde{\Xi }}}_1{\tau }^{\alpha }).\end{array}$$

(17)

According to (6) and (17), we have

$$\begin{array}{c}\hfill \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)=\mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })-\mathrm{z}\left(0\right){\mathcal{E}}_{\alpha }(-\overline{{\tilde{\Xi }}_1}{\tau }^{\alpha })+\mathrm{z}\left(\tau \right)+\overline{{\rm Y}}\left(\tau \right),\end{array}$$

(18)

where $\overline{M}\left(\tau \right)=\left({\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel )-{\rm Y}\left(\tau \right)\right)★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })\right)$ $+{\tilde{\Xi }}_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\tilde{\Xi }}_1{\tau }^{\alpha })-{\overline{\tilde{\Xi }}}_5\left(\mathrm{s}\left(\tau \right)\right)$ $★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\overline{\tilde{\Xi }}}_1{\tau }^{\alpha })\right)-{\tilde{\Xi }}_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\overline{\tilde{\Xi }}}_1{\tau }^{\alpha }).$ Thus, we deduce that $\overline{{\rm Y}}\left(\tau \right)<0$ for $\tau >0$. Because ${\rm Y}\left(\tau \right)\ge 0$ and ${\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })>0$, it follows that ${\rm Y}\left(\tau \right)★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })\right)\ge 0$. Furthermore, ${\mathcal{E}}_{\alpha ,\alpha }(-\varrho )$ and ${\mathcal{E}}_{\alpha ,\alpha +1}(-\varrho )$ are monotonically non-increasing mappings for $\varrho >0$ (as seen in Lemma 3), which indicates ${\mathcal{E}}_{\alpha ,\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })\le {\mathcal{E}}_{\alpha ,\alpha }(-{\overline{\tilde{\Xi }}}_1{\tau }^{\alpha })$ and ${\mathcal{E}}_{\alpha ,\alpha +!}(-{\tilde{\Xi }}_1{\tau }^{\alpha })\le {\mathcal{E}}_{\alpha ,\alpha +1}(-{\overline{\tilde{\Xi }}}_1{\tau }^{\alpha })$ combined by ${\overline{\tilde{\Xi }}}_5\left(\mathrm{s}\left(\tau \right)\right)\ge {\tilde{\Xi }}_5(\parallel \mathrm{s}\left(\tau \right)\parallel )\Rightarrow \overline{{\rm Y}}\left(\tau \right)\le 0.$ So, (18) yields

$$\begin{array}{c}\hfill \mathrm{U}\left(\tau ,{\mathrm{y}}_1\left(\tau \right)\right)\le \mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1{\tau }^{\alpha })-\mathrm{z}\left(0\right){\mathcal{E}}_{\alpha }(-\overline{{\tilde{\Xi }}_1}{\tau }^{\alpha })+\mathrm{z}\left(\tau \right).\end{array}$$

(19)

Assume that $U_1(0,\tau )=max\left\{0,\mathrm{U}\left(0\right){\mathcal{E}}_{\alpha }(-{\tilde{\Xi }}_1){\tau }^{\alpha }-\mathrm{z}\left(0\right){\mathcal{E}}_{\alpha }(-\overline{{\tilde{\Xi }}_1}{\tau }^{\alpha })\right\}.$ For $\overline{{\tilde{\Xi }}_1}\in (0,{\tilde{\Xi }}_1)$ and $\mathrm{z}\left(0\right)>0$, there must be ${\mathbf{T}}_0={\mathbf{T}}_0({\overline{\tilde{\Xi }}}_1,{\mathrm{z}}_0,{\mathrm{y}}_0)\ge 0$ such that $U_1(0,\tau )=0\forall \tau \ge {\mathbf{T}}_0$, resulting in (16). This concludes the proof. □

2.1. Problem Formulation

The FON SF framework with uncertain controlling orientations is examined as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\sigma }={\wp }_{\sigma }{\mathrm{y}}_{\sigma +1}+{\rho }_{\sigma }\left({\overline{\mathrm{y}}}_{\sigma }\right),\sigma =1,\dots ,\mathrm{n}-1,\hfill \\ {}^c\mathrm{D}^{\alpha }{\mathrm{y}}_j={\wp }_j\mathrm{s}\left(\tau \right)+{\rho }_j\left({\overline{\mathrm{y}}}_{\sigma }\right),\hfill \\ \mathrm{w}={\mathrm{y}}_1,\hfill \end{array}\right.\end{array}$$

(20)

where $\alpha \in (0,1),$ ${\overline{\mathrm{y}}}_{\ell }={\left[{\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\ell }\right]}^{\mathbf{T}}\in {\mathbb{R}}^{\ell },(\ell =1,\dots ,j)$, where ${\mathrm{y}}_{\ell }$ represents the structure state, ${\rho }_{\ell }\left({\overline{\mathrm{y}}}_{\ell }\right)$ represents an independent smooth function, $\mathrm{s}$ comprises the control data, and $\mathrm{w}$ represents the system output. The control orientation, ${\wp }_{\ell }$, represents an undetermined constant.

Control mechanism: In SF nonlinear structures involving uncertain control prescriptions, an optimized controller $\mathrm{s}$ is implemented for guaranteeing every signal in scheme (20) is confined. The results of the system ${\mathrm{y}}_1$ can monitor the source signal ${\mathrm{y}}_{\mathrm{d}}$ while allowing the tracking error $|{\mathrm{y}}_1-{\mathrm{y}}_{\mathrm{d}}|$ to get closer to a predetermined precision over time.

Remark 1. 

Numerous control algorithms were proposed for nonlinear networks involving uncertain control orientation and predetermined effectiveness [27,28]. However, these oversight strategies do not allow you to configure the monitoring duration in advance. This study proposes a control approach where the duration characteristic ${\mathbf{T}}_{\tilde{\Xi }}$ can be freely chosen to ensure that the tracking error tends to a predefined interval in a predetermined time.

To simplify the argument, this study makes a specific hypothesis:

Hypothesis 1 

([26]). The reference signal ${\mathrm{y}}_{\mathrm{d}}\in {\mathrm{l}}_{\infty }$ and its differentiation ${\mathrm{y}}_{\mathrm{d}}^{\left(\sigma \right)}\in {\mathrm{l}}_{\infty }(\sigma =1,\dots ,\mathrm{n}).$

2.2. Nussbaum-Type Criteria

Definition 5 

([66]). Let $\Omega \left(\eta \right)$ define a mapping that, based on the identity, has both the upper and lower integrals approach ∞ and can be represented as follows:

$$\begin{array}{c}\hfill \underset{\gamma \mapsto \infty }{lim}sup{\displaystyle \frac{1}{\gamma }}\underset{0}{\overset{\gamma }{\int }}\Omega \left(\eta \right)d\eta =+\infty ,\end{array}$$

(21)

$$\begin{array}{c}\hfill \underset{\gamma \mapsto \infty }{lim}inf{\displaystyle \frac{1}{\gamma }}\underset{0}{\overset{\gamma }{\int }}\Omega \left(\eta \right)d\eta =-\infty .\end{array}$$

(22)

Thus, Definition 5 makes clear that the Nussbaum function provides an unlimited transforming frequency and intensity. The most frequently encountered Nussbaum functions for controlling include $exp\left({\eta }^2\right)cos(\eta \pi /2),{\eta }^{2}cos\left(\eta \right)$, and ${\eta }^{2}sin\left(\eta \right)$. This study uses $exp\left({\eta }^2\right)cos(\eta \pi /2)$ for establishing a control system.

Lemma 5 

([66]). For the SF nonlinear framework (2), the subsequent variant satisfies:

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\mathcal{L}(\mathrm{y},\tau )\le -{\mathcal{P}}_1\mathcal{L}(\mathrm{y},\tau )+\sum _{\sigma =1}^{\mathrm{n}}\left[{\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right]{}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\mathcal{P}}_2,\end{array}$$

(23)

where $\mathcal{L}(\mathrm{y},\tau )>0,\Omega \left({\eta }_{\sigma }\right)=exp\left({\eta }_{\sigma }^2\right)cos({\eta }_{\sigma }\pi /2),$ ${\eta }_{\sigma }\left(\tau \right)$ represents the smooth function with $\tau \in [0.\infty ),{\mathcal{P}}_1>0,{\mathcal{P}}_2>0$ and ${\wp }_{\sigma }$, which are unknown constants. Consequently, $\mathcal{L}(\mathrm{y},\tau )\in {\mathrm{l}}_{\infty }$ and ${\displaystyle \sum _{\sigma =1}^{\mathrm{n}}}\left[{\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right]{}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }\in {\mathrm{l}}_{\infty }.$

2.3. Radial Basis Function NN (RBFNN)

It is widely acknowledged that RBFNNs are highly effective at approximating arbitrary nonlinear mappings [67]. For any continuous mapping $\hslash \left(Z_1\right)$ specified within the compact set ${\nabla }_{\mathrm{Z}}\subset {\mathbb{R}}^{\mathrm{n}},$ the subsequent NN can be expressed as

$$\begin{array}{c}\hfill \hslash \left(Z_1\right)={\mho }_{\hslash }^{★\mathbf{T}}{\Lambda }_{\hslash }\left(Z_1\right)+{\epsilon }_{\hslash }\left(Z_1\right),\left|{\epsilon }_{\hslash }\left(Z_1\right)\right|\le {\overline{\epsilon }}_{\hslash },\end{array}$$

(24)

where ${\mho }_{\hslash }^{★}$ represents the optimum density vector, $Z_1\in {\nabla }_{\mathrm{Z}}$ is the supplied vector, ${\overline{\epsilon }}_{\hslash }$ is an undetermined fixed number, and ${\Lambda }_{\hslash }\left(Z_1\right)={\left[{\psi }_{{\hslash }_1}\left(Z_1\right),\dots ,{\psi }_{{\hslash }_{\ell }}\left(Z_1\right)\right]}^{\mathbf{T}}$ represents the basis function vector as

$$\begin{array}{c}\hfill {\psi }_{\hslash }=exp\left(-{\displaystyle \frac{\parallel Z_1-{\omega }_{1\sigma }{\parallel }^2}{2{\omega }_{2\sigma }^2}}\right),\sigma =1,\dots ,\ell ,\end{array}$$

(25)

where ${\omega }_{\sigma 1}={\left[{\omega }_{1\sigma }^{\left(1\right)},\dots ,{\omega }_{\sigma 1}^{\left(\mathrm{n}\right)}\right]}^{\mathbf{T}}$ stands for center, ${\omega }_{2\sigma }$ for increment, and ℓ for node number.

2.4. Transform Operator

In order to solve the predetermined efficiency monitoring issue, the transformation function that follows is shown as

$$\begin{array}{c}\hfill \Xi \left(\tau \right)=\left\{\begin{array}{c}{\displaystyle \frac{{\tau }^2}{{({\mathbf{T}}_{\tilde{\Xi }}-\tau )}^2+\aleph {\tau }^2}},0\le \tau \le {\mathbf{T}}_{\tilde{\Xi }},\hfill \\ {\displaystyle \frac{1}{\aleph }},\tau \ge {\mathbf{T}}_{\tilde{\Xi }},\hfill \end{array}\right.\end{array}$$

(26)

where ${\mathbf{T}}_{\tilde{\Xi }}$ represents the predetermined time, and ℵ is the layout factor.

Lemma 6 

([26]). The function (26) exhibits the subsequent qualities:

(1) 

$\Xi \left(0\right)=0.$

(2) 

$\Xi \left(\tau \right)$ increases monotonically from 0 to ${\mathbf{T}}_{\tilde{\Xi }}$ and remains constant at $1/\aleph$ in $[{\mathbf{T}}_{\epsilon },+\infty ).$

(3) 

$\Xi \left(\tau \right)$ is a differentiable mapping and ${}^c\mathrm{D}^{\alpha }\Xi \left(\tau \right)\in {\mathrm{l}}_{\infty .}$

Lemma 7 

([68,69]). For all $({\beta }_1,{\beta }_2)\in {\mathbb{R}}^2,$ the subsequent variant satisfies

$$\begin{array}{c}\hfill {\beta }_1{\beta }_2\le {\displaystyle \frac{{\xi }^{{\beta }_3}}{{\beta }_3}}|{\beta }_1{|}^{{\beta }_3}+{\displaystyle \frac{1}{{\beta }_4{\xi }^{{\beta }_3}}}{\left|{\beta }_2\right|}^{{\beta }_4},\end{array}$$

(27)

where $\xi >0,{\beta }_3,{\beta }_4>1$ and $({\beta }_3-1)({\beta }_4-1)=1.$

3. Control Scheme Design and Stability Analysis

The following section introduces the adaptive NN controllers for the FON system (25). Depending upon the suggested controllers, a new adaptive NN controller is developed to lessen interface strain.

Optimized Controller Design

This work employs ${\widehat{\Psi }}_{\sigma }(\sigma =1,\dots ,\mathrm{n}-1)$ to create the optimized artificial control ${\Psi }_{\sigma }^{★}$ and performs the subsequent coordinate inversion as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\epsilon ={\mathrm{y}}_1-{\mathrm{y}}_{\mathrm{d}},\hfill \\ {\phi }_1=\tilde{\Xi }({\mathrm{y}}_1-{\mathrm{y}}_{\mathrm{d}}),\hfill \\ {\phi }_{\sigma }={\mathrm{y}}_{\sigma }-{\widehat{\Psi }}_{\sigma -1},\hfill \end{array}\right.\end{array}$$

(28)

where ${\phi }_{\ell },(\ell =1,\dots ,\mathrm{n})$ represents the error factor.

Case (1): In this study, the effectiveness index function is constructed to guarantee that tracking errors gradually tend to a predetermined reliability as

$$\begin{array}{c}\hfill {\mathcal{J}}_1\left(\tau \right)=\underset{\tau }{\overset{\infty }{\int }}\left(ln{\tilde{\Xi }}_1^4-ln({\tilde{\Xi }}_1^4-{\phi }^4\left(\vartheta \right))+{\Psi }_1^2\left(\phi \left(\vartheta \right)\right)\right)d\vartheta ,\end{array}$$

(29)

where ${\Psi }_1$ represents the initial artificial controller.

Consider ${\beth }_1=\left\{{\phi }_1:-{\tilde{\Xi }}_1<{\phi }_1<{\tilde{\Xi }}_1\right\}$ being a compact set and ${\nabla }_{{\beth }_1}$ as the accepted controlling set of ${\Psi }_1$. Furthermore, the optimized effectiveness index function is defined as follows:

$$\begin{array}{ccc}\hfill {\mathcal{J}}_1\left(\left(\tau \right)\right)& & =\underset{{\Psi }_1\in {\nabla }_{{\beth }_1}}{min}\left\{\underset{\tau }{\overset{\infty }{\int }}\left(ln{\tilde{\Xi }}_1^4-ln({\tilde{\Xi }}_1^4-{\phi }^4\left(\vartheta \right))+{\Psi }_1^2\left(\phi \left(\vartheta \right)\right)\right)d\vartheta \right\}\hfill \\ & & =\underset{\tau }{\overset{\infty }{\int }}\left(ln{\tilde{\Xi }}_1^4-ln({\tilde{\Xi }}_1^4-{\phi }^4\left(\vartheta \right))+{\Psi }_1^{★2}\left(\phi \left(\vartheta \right)\right)\right)d\vartheta ,\hfill \end{array}$$

(30)

The computation of ${}^c\mathrm{D}^{\alpha }\epsilon$ using (28) produces

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }\epsilon ={\wp }_1{\phi }_2+{\wp }_1\widehat{{\Psi }_1}+{\rho }_1\left({\overline{\mathrm{y}}}_1\right)-{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}.\end{array}$$

(31)

Additionally, we acquire

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\phi }_1={}^c\mathrm{D}^{\alpha }\Xi \epsilon +\Xi \left({\wp }_1{\phi }_2+{\wp }_1\widehat{{\Psi }_1}+{\rho }_1\left({\overline{\mathrm{y}}}_1\right)-{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}\right).\end{array}$$

(32)

Based on (30)–(32), the H–J–B equation connected to (29) can be determined and obtained as follows:

$$\begin{array}{ccc}\hfill {\mathcal{H}}_1\left({}^c\mathrm{D}^{\alpha }{\mathcal{J}}_1^{★}\left({\phi }_1\right),{\phi }_1,{\Psi }_1^{★}\right)& & ={}^c\mathrm{D}^{\alpha }{\mathcal{J}}_1^{★}\left({\phi }_1\right)\left({}^c\mathrm{D}^{\alpha }\Xi \epsilon +\Xi \left({\wp }_1{\phi }_2+{\wp }_1{\Psi }_1^{★}+{\rho }_1\left({\overline{\mathrm{y}}}_1\right)-{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}\right)\right)+ln{\tilde{\Xi }}_1^4\hfill \\ & & -ln({\tilde{\Xi }}_1^4-{\phi }^4\left(\vartheta \right))+{\left({\Psi }_1^{★}\right)}^2=0.\hfill \end{array}$$

(33)

Figuring out the formula for ${\mathcal{D}}_1^{\alpha }{\mathcal{H}}_1\left({\Psi }_1^{★}\right)=0$, we obtain

$$\begin{array}{c}\hfill {\Psi }_1^{★}=-{\displaystyle \frac{\Xi {\wp }_1}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_1^{★}\left({\phi }_1\right).\end{array}$$

(34)

Assume that

$${\mathcal{F}}_1\left({\mathrm{Z}}_{\mathcal{F}}\right)={\displaystyle \frac{{\rho }_1\left({\overline{\mathrm{y}}}_1\right)-{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}+({\wp }_1^2+1/2){\phi }^3\Xi }{({\tilde{\Xi }}_1^4-{\phi }^4)+\frac{{\phi }^2{\epsilon }^3}{({\tilde{\Xi }}^4-{\phi }^4)}}}$$

with ${\mathrm{Z}}_{\mathcal{F}}={\left[\overline{{\mathrm{y}}_1},{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}},\Xi \right]}^{\mathbf{T}}\in {\mathbb{R}}^3.$ Also, ${\mathcal{G}}_1\left({\mathrm{Z}}_{\mathcal{G}}\right)=\Omega \left({\eta }_1\right)\left[{\theta }_1\epsilon +\mathcal{F}\left({\mathrm{Z}}_{\mathcal{F}}\right)\right]+\left(\Xi {\wp }_1/2\right)$ ${}^c\mathrm{D}^{\alpha }{\mathcal{J}}_1^{★}\left({\phi }_1\right)$ with ${\mathrm{Z}}_{\mathcal{G}}={\left[{\mathrm{Z}}_{\mathcal{F}}^{\mathbf{T}},{\eta }_1\right]}^{\mathbf{T}}\in {\mathbb{R}}^4.$ The expression $-{\displaystyle \frac{\Xi {\wp }_1}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_1^{★}\left({\phi }_1\right)$ is disassembled as follows:

$$\begin{array}{c}\hfill -{\displaystyle \frac{\Xi {\wp }_1}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_1^{★}\left({\phi }_1\right)=\Omega \left({\eta }_1\right)\left({\theta }_1\epsilon +\mathcal{F}\left({\mathrm{Z}}_{\mathcal{F}}\right)\right)-{\mathcal{G}}_1\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right),\end{array}$$

(35)

where ${\theta }_1$ denotes the layout factor.

As ${\mathcal{F}}_1\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)$ and ${\mathcal{G}}_1\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)$ represent undefined processes, the present research employs NNs to determine them in the manner outlined below:

$$\begin{array}{c}\hfill {\mathcal{F}}_1\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)={\mho }_{{\mathcal{F}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)+{\epsilon }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right),\left|{\epsilon }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right|<{\overline{\epsilon }}_{{\mathcal{F}}_1},\end{array}$$

(36)

$$\begin{array}{c}\hfill {\mathcal{G}}_1\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)={\mho }_{{\mathcal{G}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)+{\epsilon }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right),\left|{\epsilon }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)\right|<{\overline{\epsilon }}_{{\mathcal{G}}_1},\end{array}$$

(37)

where ${\mho }_{\mathcal{F}}^{★}$ and ${\mho }_{\mathcal{G}}^{★}$ are optimal strengths; ${\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)$ and ${\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)$ are basis vectors; ${\epsilon }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)$ and ${\epsilon }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)$ represent estimate inaccuracies; ${\overline{\epsilon }}_{{\mathcal{F}}_1}$ and ${\overline{\epsilon }}_{{\mathcal{G}}_1}$ are undetermined positive values, respectively.

After substituting the unidentified functions in (35) using (36) and (37), we obtain

$$\begin{array}{c}\hfill {\Psi }_1^{★}=\Omega \left({\eta }_1\right)\left({\theta }_1\epsilon +{\mho }_{{\mathcal{F}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)+{\epsilon }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right)-{\mho }_{{\mathcal{G}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)-{\epsilon }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right).\end{array}$$

(38)

Obviously, the initial artificial controller can be created as

$$\begin{array}{c}\hfill {\widehat{\Psi }}_1=\Omega \left({\eta }_1\right)\left({\theta }_1\epsilon +{\widehat{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)\end{array}$$

(39)

involving the conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_1}={\Gamma }_{{\mathcal{F}}_1}\left({\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{\phi }_1^4}}{\Lambda }_{\mathcal{F}}\left({\mathrm{Z}}_{\mathcal{F}}\right)-{\upsilon }_{{\mathcal{F}}_1}{\widehat{\mho }}_{{\mathcal{F}}_1}\right),\hfill \\ {}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_1}=-{\upsilon }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{\mathcal{G}}\right){\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1},\hfill \\ {}^c\mathrm{D}^{\alpha }{\eta }_1={\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{\phi }_1^4}}\left({\theta }_1\epsilon +{\widehat{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right),\hfill \end{array}\right.\end{array}$$

(40)

where ${\upsilon }_{{\mathcal{F}}_1}>0$ and ${\upsilon }_{\mathcal{G}}\ge 1$ constitute architectural values and ${\Gamma }_{{\mathcal{F}}_1}$ represents a positive-definite matrix.

Case (i) $(\sigma =2,\dots ,\mathrm{n}-1)$ As a result of no limitation regarding the $\sigma$th error factor, we construct the $\sigma$th efficiency index formula below:

$$\begin{array}{c}\hfill {\mathcal{J}}_{\sigma }\left(\tau \right)=\underset{\tau }{\overset{\infty }{\int }}\left({\phi }_{\sigma }^2\left(\vartheta \right)+{\Psi }_{\sigma }^2\left({\phi }_{\sigma }\left(\vartheta \right)\right)\right)d\vartheta ,\end{array}$$

(41)

where ${\alpha }_{\sigma }$ constitutes the $\sigma$th artificial control.

Suppose ${\nabla }_{\sigma }$ is considered the feasible control collection for ${\alpha }_{\sigma }$. According to the previous one, we specify ${\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\left(\tau \right)\right)$ as

$$\begin{array}{c}\hfill {\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\left(\tau \right)\right)=\underset{{\Psi }_{\sigma }\in {\nabla }_{\sigma }}{min}\left\{\underset{\tau }{\overset{\infty }{\int }}\left({\phi }_{\sigma }^2\left(\vartheta \right)+{\Psi }_{\sigma }^2\left({\phi }_{\sigma }\left(\vartheta \right)\right)\right)d\vartheta \right\}=\underset{\tau }{\overset{\infty }{\int }}\left({\phi }_{\sigma }^2\left(\vartheta \right)+{\Psi }_{\sigma }^{★2}\left({\phi }_{\sigma }\left(\vartheta \right)\right)\right)d\vartheta .\end{array}$$

(42)

Determining ${}^c\mathrm{D}^{\alpha }{\phi }_{\sigma }$ results in

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\phi }_{\sigma }={\wp }_{\sigma }{\phi }_{\sigma +1}+{\wp }_{\sigma }{\widehat{\Psi }}_{\sigma }+{\rho }_{\sigma }\left({\overline{\mathrm{y}}}_{\sigma }\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1},\end{array}$$

(43)

where ${}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}={\displaystyle \sum _{\ell =1}^{\sigma -1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\mathrm{y}}_{\ell }\right)\right)\left({\mathrm{y}}_{\ell +1}+{\mathcal{F}}_{\ell }\left({\overline{\mathrm{y}}}_{\ell }\right)\right)+{\kappa }_{\sigma -1}$, including ${\kappa }_{\sigma -1}=\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\mathrm{y}}_{\mathrm{d}}\right)\right){}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}$ $+{\displaystyle \sum _{\ell =1}^{\sigma -1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\mathrm{y}}_{\mathrm{d}}^{(\ell )}\right)\right){\mathrm{y}}_{\mathrm{d}}^{(\ell +1)}$ $+{\displaystyle \sum _{\ell =1}^{\sigma -1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\eta }_{\ell }\right)\right){}^c\mathrm{D}^{\alpha }{\eta }_{\ell }$ $+{\displaystyle \sum _{\ell =1}^{\sigma -1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\widehat{\mho }}_{{\mathcal{F}}_{\ell }}\right)\right){}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\ell }}$ $+{\displaystyle \sum _{\ell =1}^{\sigma -1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\widehat{\mho }}_{{\mathcal{G}}_{\ell }}\right)\right){}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\ell }}.$

Using (38)–(43), the H–J–B formula corresponding to (41) is generated in this way:

$$\begin{array}{c}\hfill {\mathcal{H}}_{\sigma }\left({}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\right),{\phi }_{\sigma },{\Psi }_{\sigma }^{★}\right)={}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\right)\left({\wp }_{\sigma }{\Psi }_{\sigma }^{★}+{\wp }_{\sigma }{\widehat{\Psi }}_{\sigma }^{★}+{\rho }_{\sigma }\left({\overline{\mathrm{y}}}_{\sigma }\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\right)+{\phi }_{\sigma }^2\left(\vartheta \right)+{\left({\Psi }_{\sigma }^{★}\right)}^2=0.\end{array}$$

(44)

Analogous to Case (1), we can determine ${\Psi }_{\sigma }^{★}$ by using the formula ${}^c\mathrm{D}^{\alpha }{\mathcal{H}}_{\sigma }\left({\Psi }_{\sigma }^{★}\right)=0$ as

$$\begin{array}{c}\hfill {\Psi }_{\sigma }^{★}=-{\displaystyle \frac{{\wp }_{\sigma }}{2}}{}^c\mathrm{D}{\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\right).\end{array}$$

(45)

Assume that ${\mathcal{F}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)={\rho }_{\sigma }\left({\overline{\mathrm{y}}}_{\sigma }\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}+({\wp }_{\sigma }^2-1){\phi }_{\sigma }$ involving ${\mathrm{Z}}_{\mathcal{F}}={\left[{\overline{\mathrm{y}}}_{\sigma },{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\mathrm{y}}_1\right),\dots ,{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\mathrm{y}}_{\sigma -1}\right),{\kappa }_{\sigma -1}\right]}^{\mathbf{T}}\in {\mathbb{R}}^{2\sigma +1}.$ Therefore, the factor $-{\displaystyle \frac{{\wp }_{\sigma }}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\right)$ can be articulated as

$$\begin{array}{c}\hfill -{\displaystyle \frac{{\wp }_{\sigma }}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\sigma }^{★}\left({\phi }_{\sigma }\right)=\Omega \left({\eta }_{\sigma }\right)\left({\theta }_{\sigma }{\phi }_{\sigma }+{\mathcal{F}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)\right)-{\mathcal{G}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right),\end{array}$$

(46)

where ${\theta }_{\sigma }>0$ is the layout factor.

In an analogous way, we estimate unidentified functions ${\mathcal{F}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)$ and ${\mathcal{G}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)$ using NNs as follows:

$$\begin{array}{c}\hfill {\mathcal{F}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)={\mho }_{{\mathcal{F}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)+{\epsilon }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right),\left|{\epsilon }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)\right|<{\overline{\epsilon }}_{{\mathcal{F}}_{\sigma }},\end{array}$$

(47)

$$\begin{array}{c}\hfill {\mathcal{G}}_{\sigma }\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)={\mho }_{{\mathcal{G}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)+{\epsilon }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right),\left|{\epsilon }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)\right|<{\overline{\epsilon }}_{{\mathcal{G}}_{\sigma }},\end{array}$$

(48)

where ${\mho }_{{\mathcal{F}}_{\sigma }}^{★}$ and ${\mho }_{{\mathcal{G}}_{\sigma }}^{★}$ represent appropriate strengths. The variables ${\Lambda }_{\mathcal{F}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)$ and ${\Lambda }_{\mathcal{G}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)$ indicate basis function vectors, ${\epsilon }_{\mathcal{F}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)$ and ${\epsilon }_{\mathcal{G}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)$ represent estimation errors, and ${\overline{\epsilon }}_{\mathcal{F}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)$ and ${\overline{\epsilon }}_{\mathcal{G}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)$ comprise undetermined non-negative factors.

By substituting the unidentified functions in (46) alongside (47) and (48), we obtain

$$\begin{array}{c}\hfill {\Psi }_{\sigma }^{★}=\Omega \left({\eta }_{\sigma }\right)\left({\theta }_{\sigma }{\phi }_{\sigma }+{\mho }_{{\mathcal{F}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)+{\epsilon }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)\right)-{\mho }_{{\mathcal{G}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)-{\epsilon }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right).\end{array}$$

(49)

Next, we create the $\sigma$th artificial control as

$$\begin{array}{c}\hfill {\widehat{\Psi }}_{\sigma }=\Omega \left({\eta }_{\sigma }\right)\left({\theta }_{\sigma }{\phi }_{\sigma }+{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)\end{array}$$

(50)

containing

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}={\Gamma }_{{\mathcal{F}}_{\sigma }}\left({\phi }_{\sigma }{\Lambda }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)-{\upsilon }_{{\mathcal{F}}_{\sigma }}{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}\right),\hfill \\ {}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}=-{\upsilon }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)-{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right){\widehat{\mho }}_{{\mathcal{G}}_{\sigma }},\hfill \\ {}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }={\phi }_{\sigma }\left({\theta }_{\sigma }{\phi }_{\sigma }+{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right)\right),\hfill \end{array}\right.\end{array}$$

(51)

where ${\upsilon }_{{\mathcal{F}}_{\sigma }}>0$ and ${\upsilon }_{{\mathcal{G}}_{\sigma }}\ge 1$ act as design values, and ${\Gamma }_{{\mathcal{F}}_{\sigma }}$ represents a positive-definite matrix.

Case ($\mathrm{n}$): The efficiency index mapping within the control is outlined below:

$$\begin{array}{c}\hfill {\mathcal{J}}_{\mathrm{n}}\left(\tau \right)=\underset{\tau }{\overset{\infty }{\int }}\left({\phi }_{\mathrm{n}}^2\left(\vartheta \right)+{\mathrm{s}}^2\left({\phi }_{\mathrm{n}}\left(\vartheta \right)\right)\right)d\vartheta ,\end{array}$$

(52)

where $\mathrm{s}$ denotes the real controller.

Suppose ${\nabla }_{\mathrm{n}}$ is considered the feasible control collection for $\mathrm{s}$. Analogously, we define ${\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\left(\tau \right)\right)$ as

$$\begin{array}{c}\hfill {\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\left(\tau \right)\right)=\underset{\mathrm{s}\in {\nabla }_{\mathrm{n}}}{min}\left\{\underset{\tau }{\overset{\infty }{\int }}\left({\phi }_{\mathrm{n}}^2\left(\vartheta \right)+{\mathrm{s}}^2\left({\phi }_{\mathrm{n}}\left(\vartheta \right)\right)\right)d\vartheta \right\}=\underset{\tau }{\overset{\infty }{\int }}\left({\phi }_{\mathrm{n}}^2\left(\vartheta \right)+{\Psi }_{\mathrm{n}}^{★2}\left({\phi }_{\mathrm{n}}\left(\vartheta \right)\right)\right)d\vartheta .\end{array}$$

(53)

Computing ${}^c\mathrm{D}^{\alpha }{\phi }_{\mathrm{n}}$ results in

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\phi }_{\mathrm{n}}={\wp }_{\sigma }{\widehat{u}}_1+{\rho }_{\mathrm{n}}\left({\overline{\mathrm{y}}}_{\mathrm{n}}\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1},\end{array}$$

(54)

where ${}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}={\displaystyle \sum _{\ell =1}^{\mathrm{n}-1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\mathrm{y}}_{\ell }\right)\right)\left({\mathrm{y}}_{\ell +1}+{\mathcal{F}}_{\ell }\left({\overline{\mathrm{y}}}_{\ell }\right)\right)+{\kappa }_{\mathrm{n}-1}$, including ${\kappa }_{\mathrm{n}-1}=\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\left({\mathrm{y}}_{\mathrm{d}}\right)\right){}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}$ $+{\displaystyle \sum _{\ell =1}^{\mathrm{n}-1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\mathrm{y}}_{\mathrm{d}}^{(\ell )}\right)\right){\mathrm{y}}_{\mathrm{d}}^{(\ell +1)}$ $+{\displaystyle \sum _{\ell =1}^{\mathrm{n}-1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\eta }_{\ell }\right)\right){}^c\mathrm{D}^{\alpha }{\eta }_{\ell }$ $+{\displaystyle \sum _{\ell =1}^{\mathrm{n}-1}}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\widehat{\mho }}_{{\mathcal{F}}_{\ell }}\right)\right){}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\ell }}$ $+\sum _{\ell =1}^{\mathrm{n}-1}\left({}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\widehat{\mho }}_{{\mathcal{G}}_{\ell }}\right)\right){}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\ell }}.$

In view of (53) and (54), the H–J–B expression for (52) is produced as follows:

$$\begin{array}{c}\hfill {\mathcal{H}}_{\mathrm{n}}\left({}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\right),{\phi }_{\mathrm{n}},{\mathrm{s}}^{★}\right)={}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\right)\left({\wp }_{\mathrm{n}}{\widehat{u}}_1+{\rho }_{\mathrm{n}}\left({\overline{\mathrm{y}}}_{\mathrm{n}}\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\right)+{\phi }_{\mathrm{n}}^2\left(\vartheta \right)+{\left({\mathrm{s}}^{★}\right)}^2=0.\end{array}$$

(55)

After calculating the formula ${}^c\mathrm{D}^{\alpha }{\mathcal{H}}_{\mathrm{n}}\left({\mathrm{s}}^{★}\right)=0,$ we get

$$\begin{array}{c}\hfill {\mathrm{s}}^{★}=-{\displaystyle \frac{-\Xi {\wp }_{\mathrm{n}}}{2}}{}^c\mathrm{D}{\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\right).\end{array}$$

(56)

Assume that ${\mathcal{F}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)={\rho }_{\mathrm{n}}\left({\overline{\mathrm{y}}}_{\mathrm{n}}\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}+\left({\displaystyle \frac{1}{2}}\right){\wp }_{\mathrm{n}}^2+1){\phi }_{\mathrm{n}}$ involving ${\mathrm{Z}}_{\mathcal{F}}={\left[{\overline{\mathrm{y}}}_{\mathrm{n}},{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\mathrm{y}}_1\right),\dots ,{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\left({\mathrm{y}}_{\mathrm{n}-1}\right),{\kappa }_{\mathrm{n}-1}\right]}^{\mathbf{T}}\in {\mathbb{R}}^{2\mathrm{n}}.$ Also, ${\mathcal{G}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)=\Omega \left({\eta }_{\mathrm{n}}\right)\left({\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}+{\mathcal{F}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)\right)+\left({\displaystyle \frac{{\wp }_{\mathrm{n}}}{2}}\right){}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\right)$ alongside ${\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}={\left[{\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}},{\eta }_{\mathrm{n}}\right]}^{\mathbf{T}}\in {\mathbb{R}}^{2\mathrm{n}+1}.$

The expression $-{\displaystyle \frac{{\wp }_{\mathrm{n}}}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\right)$ can be reduced as follows.

$$\begin{array}{c}\hfill -{\displaystyle \frac{{\wp }_{\mathrm{n}}}{2}}{}^c\mathrm{D}^{\alpha }{\mathcal{J}}_{\mathrm{n}}^{★}\left({\phi }_{\mathrm{n}}\right)=\Omega \left({\eta }_{\mathrm{n}}\right)\left({\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}+{\mathcal{F}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)\right)-{\mathcal{G}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right),\end{array}$$

(57)

where ${\theta }_{\mathrm{n}}>0$ is the layout factor.

The NNs estimate ${\mathcal{F}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)$ and ${\mathcal{G}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)$ in the following way:

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)={\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)+{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right),\left|{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)\right|<{\overline{\epsilon }}_{{\mathcal{F}}_{\mathrm{n}}},\end{array}$$

(58)

$$\begin{array}{c}\hfill {\mathcal{G}}_{\mathrm{n}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)={\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)+{\epsilon }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right),\left|{\epsilon }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)\right|<{\overline{\epsilon }}_{{\mathcal{G}}_{\mathrm{n}}},\end{array}$$

(59)

where ${\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★}$ and ${\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★}$ represent appropriate strengths. The variables ${\Lambda }_{\mathcal{F}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)$ and ${\Lambda }_{\mathcal{G}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)$ indicate basis function vectors, ${\epsilon }_{\mathcal{F}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)$ and ${\epsilon }_{\mathcal{G}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)$ represent estimation errors, and ${\overline{\epsilon }}_{\mathcal{F}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)$ and ${\overline{\epsilon }}_{\mathcal{G}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)$ comprise undetermined non-negative factors.

By substituting the unidentified functions in (57) alongside (58) and (59), we obtain

$$\begin{array}{c}\hfill {\mathrm{s}}^{★}=\Omega \left({\eta }_{\mathrm{n}}\right)\left({\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}+{\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)+{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)\right)-{\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)-{\epsilon }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right).\end{array}$$

(60)

Next, we create the $\mathrm{n}$th artificial control as

$$\begin{array}{c}\hfill {\mathrm{s}}^{★}=\Omega \left({\eta }_{\mathrm{n}}\right)\left({\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}+{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right)\end{array}$$

(61)

containing

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}={\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}\left({\phi }_{\mathrm{n}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)-{\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}\right),\hfill \\ {}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}=-{\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right){\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}\left({\mathrm{Z}}_{{\mathcal{G}}_{\mathrm{n}}}\right){\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}},\hfill \\ {}^c\mathrm{D}^{\alpha }{\eta }_{\mathrm{n}}={\phi }_{\mathrm{n}}\left({\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}+{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}\left({\mathrm{Z}}_{{\mathcal{F}}_{\mathrm{n}}}\right)\right),\hfill \end{array}\right.\end{array}$$

(62)

where ${\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}>0$ and ${\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}\ge 1$ act as design values, and ${\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}$ represents a positive definite matrix.

4. Stability Analysis

Theorem 1. 

SF dynamical structures (20), optimized virtual controls (39) and (50), optimized real control (61), and adaptation principles (40), (51), and (62) ensure that the subsequent findings are generally true. (i)The signals in structure (20) remain bounded. (ii)The tracking deviation ε merges to a specified precision ${\tilde{\Xi }}_1\aleph$ during a predetermined time ${\mathbf{T}}_{\tilde{\Xi }}$.

Proof. 

In the following documentation, ${\mho }_{{\mathcal{F}}_{\sigma }}$ indicates ${\mho }_{{\mathcal{F}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{F}}_{\sigma }}\right),$ ${\mho }_{{\mathcal{G}}_{\sigma }}$ represents ${\mho }_{{\mathcal{G}}_{\sigma }}\left({\mathrm{Z}}_{{\mathcal{G}}_{\sigma }}\right)$, and ${\epsilon }_{\mathcal{F}}$ is employed for ${\epsilon }_{\mathcal{F}}\left({\mathrm{Z}}_{\mathcal{F}}\right)$, respectively.

Case I: Assuming the predefined tracking preciseness, we construct the Lyapunov function choice ${\mathcal{L}}_1$ in the following way:

$$\begin{array}{c}\hfill {\mathcal{L}}_1={\displaystyle \frac{1}{4}}\left(ln{\tilde{\Xi }}_1^4-ln({\tilde{\Xi }}_1^4-{\phi }_1^4)\right)+{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{\mathcal{F}}^{\mathbf{T}}{\Gamma }_{\mathcal{F}}^{-1}{\tilde{\mho }}_{\mathcal{F}}+{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{\mathcal{G}}^{\mathbf{T}}{\tilde{\mho }}_{\mathcal{G}}.\end{array}$$

(63)

Taking into account (32), we have

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_1& & ={\displaystyle \frac{{\phi }^2{}^c\mathrm{D}^{\alpha }{\phi }_1}{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}-{\tilde{\mho }}_{\mathcal{F}}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_1}^{-1}{}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{F}}_1}-{\mho }_{{\mathcal{G}}_1}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{G}}_1}\hfill \\ & & ={\displaystyle \frac{{\phi }_1^3\epsilon {}^c\mathrm{D}^{\alpha }\Xi }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}+{\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}\left(\wp {\phi }_2+{\wp }_1{\widehat{\Psi }}_1+{\rho }_1\left({\overline{\mathrm{y}}}_1\right)-{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}\right)-{\tilde{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_1}^{-1}{}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{F}}_1}-{\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{G}}_1}\hfill \\ & & ={\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}({\wp }_1\Omega \left({\eta }_1\right)\left({\theta }_1\epsilon +{\widehat{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\right)-\left({\wp }_1^2+{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{{{\phi }_1}^3\Xi }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}\hfill \\ & & -{\displaystyle \frac{{{\phi }_1}^2{\epsilon }^3}{2({\tilde{\Xi }}_1^4-{{\phi }_1}^4)}}+{\wp }_1{\phi }_2-{\wp }_1{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}+{\widehat{\mho }}_{{\mathcal{F}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}+{\epsilon }_{{\mathcal{F}}_1})\hfill \\ & & +{\displaystyle \frac{{\phi }_1^3\epsilon {}^c\mathrm{D}^{\alpha }\Xi }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}-{\tilde{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_1}^{-1}{}^c\mathrm{D}^{\alpha }{\tilde{\mho }}_{{\mathcal{F}}_1}-{\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_1}.\hfill \end{array}$$

(64)

Employing Lemma 4 to address specific factors in (64), we find

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{{\phi }_1}^3\Xi {\wp }_1{\phi }_2}{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}\le {\displaystyle \frac{{{\phi }_1}^6{\Xi }^2{\wp }_1^2}{2{({\tilde{\Xi }}_1^4-{{\phi }_1}^4)}^2}}+{\displaystyle \frac{1}{2}}{\phi }_2^2,\hfill \\ & & {\displaystyle \frac{{{\phi }_1}^3\Xi {\epsilon }_{{\mathcal{F}}_1}}{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}\le {\displaystyle \frac{{{\phi }_1}^6{\Xi }^2}{2{({\tilde{\Xi }}_1^4-{{\phi }_1}^4)}^2}}+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_1}^2,\hfill \\ & & {\displaystyle \frac{{{\phi }_1}^3{}^c\mathrm{D}^{\alpha }\Xi \epsilon }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}\le {\displaystyle \frac{{{\phi }_6}^3{\epsilon }^2}{2{({\tilde{\Xi }}_1^4-{{\phi }_1}^4)}^2}}+{\displaystyle \frac{1}{2}}{}^c\mathrm{D}^{\alpha }{\Xi }^2,\hfill \\ & & -{\displaystyle \frac{{{\phi }_1}^3\Xi {\wp }_1}{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}\le {\displaystyle \frac{{{\phi }_1}^6{\Xi }^2{\wp }_1^2}{2{({\tilde{\Xi }}_1^4-{{\phi }_1}^4)}^2}}+{\displaystyle \frac{1}{2}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{\mathcal{G}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}.\hfill \end{array}$$

(65)

Plugging (65) into (64) produces

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_1& & \le -{\displaystyle \frac{{\theta }_1{\phi }_1^4}{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}+{\tilde{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}\left({\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}{\Lambda }_{{\mathcal{F}}_1}-{\Gamma }_{\mathcal{F}}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_1}\right)+{\displaystyle \frac{1}{2}}{}^c\mathrm{D}^{\alpha }{\Xi }^2+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_1}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}-{\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_1}+\left({\wp }_1\Omega \left({\eta }_1\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_1+{\displaystyle \frac{1}{2}}{\phi }_2^2\hfill \\ & & =-{\displaystyle \frac{{\theta }_1{\phi }_1^4}{{\tilde{\Xi }}_1^4-{{\phi }_1}^4}}+{\upsilon }_{{\mathcal{F}}_1}{\tilde{\mho }}_{{\mathcal{F}}_1}{\widehat{\mho }}_{{\mathcal{F}}_1}+{\upsilon }_{{\mathcal{G}}_1}{\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}+{\displaystyle \frac{1}{2}}{}^c\mathrm{D}^{\alpha }{\Xi }^2+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_1}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}+\left({\wp }_1\Omega ({\eta }_1+1)\right){}^c\mathrm{D}^{\alpha }{\eta }_1+{\displaystyle \frac{1}{2}}{\phi }_2^2.\hfill \end{array}$$

(66)

Using the descriptions of ${\tilde{\mho }}_{{\mathcal{F}}_1}$ and ${\tilde{\mho }}_{{\mathcal{G}}_1}$, we can derive

$$\begin{array}{c}\hfill {\tilde{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{F}}_1}\le -{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{F}}_1}+{\displaystyle \frac{1}{2}}{\mho }_{{\mathcal{F}}_1}^{★\mathbf{T}}{\mho }_{{\mathcal{F}}_1}^{★},\end{array}$$

(67)

$$\begin{array}{c}\hfill {\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}=-{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_1}-{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}+{\displaystyle \frac{1}{2}}{\mho }_{{\mathcal{G}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\mho }_{{\mathcal{G}}_1}^{★}.\end{array}$$

(68)

Putting (67) and (68) into (66) produces

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_1& & \le -{\displaystyle \frac{{\theta }_1{\phi }_1^4}{{\tilde{\Xi }}_1^4-{\phi }_1^4}}-{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_1}}{2}}{\tilde{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{F}}_1}-{\displaystyle \frac{{\upsilon }_{{\mathcal{G}}_1}}{2}}{\tilde{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_1}\hfill \\ & & +{\displaystyle \frac{1-{\upsilon }_{{\mathcal{G}}_1}}{2}}{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_1}+\left[{\wp }_1\Omega \left({\eta }_1\right)+1\right]{}^c\mathrm{D}^{\alpha }{\eta }_1+{\lambda }_1+{\displaystyle \frac{1}{2}}{\phi }_2^2,\hfill \end{array}$$

(69)

where ${\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{2}}\left({\upsilon }_{{\mathcal{G}}_1}{\mho }_{{\mathcal{G}}_1}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}{\mho }_{{\mathcal{G}}_1}^{★}+{\upsilon }_{{\mathcal{F}}_1}{\mho }_{{\mathcal{F}}_1}^{★\mathbf{T}}{\mho }_{{\mathcal{F}}_1}^{★}+{}^c\mathrm{D}^{\alpha }{\Xi }^2\left(\tau \right)+{\epsilon }_{{\mathcal{F}}_1}^2\left(\tau \right)\right)$, and ${P_1}_1$ indicates an upper limit on ${\lambda }_1\left(\tau \right)$, while ${\delta }_{{\mathcal{F}}_1}^{max}$ is the maximum eigenvalues of matrix ${\Gamma }_{{\mathcal{F}}_1}$. Also, ${\delta }_{{\mathcal{G}}_1}^{min}$ is the minimum eigenvalues of matrix ${\Lambda }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}^{\mathbf{T}}.$

Case $\left(\sigma \right)$: ($\sigma =2,\dots ,\mathrm{n}-1$) in contrast to Case (1), we specify the Lyapunov function choice for Case ($\sigma$) simply written as

$$\begin{array}{c}\hfill {\mathcal{L}}_{\sigma }={\displaystyle \frac{1}{2}}{\phi }_{\sigma }^2+{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}+{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}.\end{array}$$

(70)

Evaluating ${}^c\mathrm{D}^{\alpha }\mathcal{L}$ produces

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }\mathcal{L}& & ={\phi }_{\sigma }{}^c\mathrm{D}^{\alpha }{\phi }_{\sigma }-{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}-{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & ={\phi }_{\sigma }\left({\wp }_{\sigma }{\phi }_{\sigma +1}+{\wp }_{\sigma }{\widehat{\Psi }}_{\sigma }+{\rho }_{\sigma }\left({\overline{\mathrm{y}}}_{\sigma }\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\sigma -1}\right)-{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}-{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & ={\phi }_{\sigma }\left\{{\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)\left({\theta }_{\sigma }{\phi }_{\sigma }+{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\sigma }}\right)-({\wp }_{\sigma }^2+1){\phi }_{\sigma }+{\wp }_{\sigma }{\phi }_{\sigma +1}-{\wp }_{\sigma }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}+{\mho }_{{\mathcal{F}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\sigma }}+{\epsilon }_{{\mathcal{F}}_{\sigma }}\right\}\hfill \\ & & -{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}-{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}.\hfill \end{array}$$

(71)

Similar to (65) and using Lemma 4 provides

$$\begin{array}{ccc}\hfill 2{\phi }_{\sigma }{\wp }_{\sigma }{\phi }_{\sigma +1}& \le & {\wp }_{\sigma }^2{\phi }_{\sigma }^2+{\phi }_{\sigma +1}^2\hfill \\ \hfill 2{\phi }_{\sigma }{\epsilon }_{{\mathcal{F}}_{\sigma }}& \le & {\phi }_{\sigma }^2+{\epsilon }_{{\mathcal{F}}_1}^2\hfill \\ \hfill -2{\wp }_{\sigma }{\phi }_{\sigma }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}& \le & {\wp }_{\sigma }^2{\phi }_{\sigma }^2+{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}.\hfill \end{array}$$

(72)

Plugging (72) into (71) yields

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_{\sigma }& & \le -{\theta }_{\sigma }{\phi }_{\sigma }^2+{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}\left({\phi }_{\sigma }{\Lambda }_{{\mathcal{F}}_{\sigma }}-{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}\right)+\left({\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & -{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_{\sigma }}^2+{\displaystyle \frac{1}{2}}{\phi }_{\sigma +1}^2-{\displaystyle \frac{1}{2}}{\phi }_{\sigma }^2\hfill \\ & & =-{\theta }_{\sigma }{\phi }_{\sigma }^2+{\upsilon }_{{\mathcal{F}}_{\sigma }}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}+{\upsilon }_{{\mathcal{G}}_{\sigma }}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_{\sigma }}^2+{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & +\left({\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\displaystyle \frac{1}{2}}{\phi }_{\sigma +1}^2-{\displaystyle \frac{1}{2}}{\phi }_{\sigma }^2.\hfill \end{array}$$

(73)

Implementing the techniques outlined in (67) and (68) produces

$$\begin{array}{c}\hfill 2{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}\le -{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}+{\mho }_{{\mathcal{F}}_{\sigma }}^{★\mathbf{T}}{\mho }_{{\mathcal{F}}_{\sigma }}^{★}.\end{array}$$

(74)

It follows that

$$\begin{array}{c}\hfill {\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}=-{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}-{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}+{\displaystyle \frac{1}{2}}{\mho }_{{\mathcal{G}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\mho }_{{\mathcal{G}}_{\sigma }}^{★}.\end{array}$$

(75)

By inserting (74) and (75) into (73), one gets

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_{\sigma }& & \le -{\theta }_{\sigma }{\phi }_{\sigma }^2-{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_{\sigma }}}{2}}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}-{\displaystyle \frac{{\upsilon }_{{\mathcal{G}}_{\sigma }}}{2}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}+{\displaystyle \frac{1-{\upsilon }_{{\mathcal{G}}_{\sigma }}}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & +\left({\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\lambda }_{\sigma }+{\displaystyle \frac{1}{2}}{\phi }_{\sigma +1}^2-{\displaystyle \frac{1}{2}}{\phi }_{\sigma }^2\hfill \\ & & \le -{\theta }_{\sigma }{\phi }_{\sigma }-{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_{\sigma }}}{2{\delta }_{{\mathcal{F}}_{\sigma }}^{max}}}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}-{\displaystyle \frac{{\upsilon }_{{\mathcal{G}}_{\sigma }}}{2}}{\delta }_{{\mathcal{F}}_{\sigma }}^{min}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & +\left({\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\mathcal{P}}_{\sigma }+{\displaystyle \frac{1}{2}}{\phi }_{\sigma +1}^2-{\displaystyle \frac{1}{2}}{\phi }_{\sigma }^2,\hfill \end{array}$$

(76)

where ${\lambda }_{\sigma }\left(\tau \right)={\displaystyle \frac{1}{2}}\left({\upsilon }_{{\mathcal{G}}_{\sigma }}{\mho }_{{\mathcal{G}}_{\sigma }}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\mho }_{{\mathcal{G}}_{\sigma }}^{★}+{\upsilon }_{{\mathcal{F}}_{\sigma }}{\mho }_{{\mathcal{F}}_{\sigma }}^{★\mathbf{T}}{\mho }_{{\mathcal{F}}_{\sigma }}^{★}+{\epsilon }_{{\mathcal{F}}_{\sigma }}^2\left(\tau \right)\right)$ and ${\mathcal{P}}_{\sigma }$ represents the upper bound on ${\lambda }_{\sigma }\left(\tau \right)$, ${\delta }_{{\mathcal{F}}_{\sigma }}^{max}$ is the highest eigenvalue of matrix ${\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}$, and ${\delta }_{{\mathcal{G}}_{\sigma }}^{min}$ is the lowest eigenvalue of matrix ${\Lambda }_{{\mathcal{G}}_{\sigma }}{\Lambda }_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}$.

Case $\left(\mathrm{n}\right)$: Parallel to Case (1), we construct the Lyapunov function choice as follows:

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{n}}={\displaystyle \frac{1}{2}}{\phi }_{\mathrm{n}}^2+{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}+{\displaystyle \frac{1}{2}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\end{array}$$

(77)

The Caputo form of ${\mathcal{L}}_{\mathrm{n}}$ is

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_{\mathrm{n}}& & ={\phi }_{\mathrm{n}}{}^c\mathrm{D}^{\alpha }{\phi }_{\mathrm{n}}-{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}-{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & ={\phi }_{\mathrm{n}}\left({\wp }_{\mathrm{n}}\widehat{\mathrm{s}}+{\rho }_{\mathrm{n}}\left({\overline{\mathrm{y}}}_{\mathrm{n}}\right)-{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\mathrm{n}-1}\right)-{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}-{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & ={\phi }_{\mathrm{n}}\{{\wp }_{\mathrm{n}}\Omega \left({\eta }_{\mathrm{n}}\right)\left({\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}+{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}\right)-\left({\displaystyle \frac{{\wp }_{\mathrm{n}}^2}{2}}+1\right){\phi }_{\mathrm{n}}-{\wp }_{\mathrm{n}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & +{\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★\mathbf{T}}{\psi }_{{\mathcal{F}}_{\mathrm{n}}}+{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}\}-{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}{}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{F}}_{\mathrm{n}}}-{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}.\hfill \end{array}$$

(78)

Employing Lemma 4 provides

$$\begin{array}{ccc}\hfill 2{\phi }_{\mathrm{n}}{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}& \le & {\phi }_{\mathrm{n}}^2+{\epsilon }_{{\mathcal{F}}_1}^2\hfill \\ \hfill -2{\wp }_{\mathrm{n}}{\phi }_{\mathrm{n}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}& \le & {\wp }_{\mathrm{n}}^2{\phi }_{\mathrm{n}}^2+{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}.\hfill \end{array}$$

(79)

Plugging (79) into (78) yields

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_{\mathrm{n}}& & \le -{\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}^2+{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}\left({\phi }_{\mathrm{n}}{\Lambda }_{{\mathcal{F}}_{\mathrm{n}}}-{\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}\right)+\left({\wp }_{\mathrm{n}}\Omega \left({\eta }_{\mathrm{n}}\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\mathrm{n}}+{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & -{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}^2-{\displaystyle \frac{1}{2}}{\phi }_{\mathrm{n}}^2\hfill \\ & & =-{\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}^2+{\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}+{\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}+{\displaystyle \frac{1}{2}}{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}^2+{\displaystyle \frac{1}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & +\left({\wp }_{\mathrm{n}}\Omega \left({\eta }_{\mathrm{n}}\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\mathrm{n}}-{\displaystyle \frac{1}{2}}{\phi }_{\mathrm{n}}^2.\hfill \end{array}$$

(80)

As mentioned before, the subsequent inequalities exist as follows:

$$\begin{array}{ccc}\hfill 2{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}& & \le -{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}+{\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★\mathbf{T}}{\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★},\hfill \\ \hfill 2{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}& & =-{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}-{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & +{\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★}.\hfill \end{array}$$

(81)

By inserting (81) into (80), we get

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }{\mathcal{L}}_{\mathrm{n}}& & \le -{\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}^2-{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}}{2}}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}-{\displaystyle \frac{{\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}}{2}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}+{\displaystyle \frac{1-{\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}}{2}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\widehat{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & +\left({\wp }_{\mathrm{n}}\Omega \left({\eta }_{\mathrm{n}}\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\mathrm{n}}+{\lambda }_{\mathrm{n}}-{\displaystyle \frac{1}{2}}{\phi }_{\mathrm{n}}^2\hfill \\ & & \le -{\theta }_{\mathrm{n}}{\phi }_{\mathrm{n}}-{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}}{2{\delta }_{{\mathcal{F}}_{\mathrm{n}}}^{max}}}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}{\tilde{\mho }}_{{\mathcal{F}}_{\mathrm{n}}}-{\displaystyle \frac{{\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}}{2}}{\delta }_{{\mathcal{F}}_{\mathrm{n}}}^{min}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\mathrm{n}}}\hfill \\ & & +\left({\wp }_{\mathrm{n}}\Omega \left({\eta }_{\mathrm{n}}\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\mathrm{n}}+{\mathcal{P}}_{\mathrm{n}}-{\displaystyle \frac{1}{2}}{\phi }_{\mathrm{n}}^2,\hfill \end{array}$$

(82)

where ${\lambda }_{\mathrm{n}}\left(\tau \right)={\displaystyle \frac{1}{2}}\left({\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}{\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★\mathbf{T}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}{\mho }_{{\mathcal{G}}_{\mathrm{n}}}^{★}+{\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}{\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★\mathbf{T}}{\mho }_{{\mathcal{F}}_{\mathrm{n}}}^{★}+{\epsilon }_{{\mathcal{F}}_{\mathrm{n}}}^2\left(\tau \right)\right)$, and ${\mathcal{P}}_{\mathrm{n}}$ represents the upper bound on ${\lambda }_{\mathrm{n}}\left(\tau \right)$; ${\delta }_{{\mathcal{F}}_{\mathrm{n}}}^{max}$ is the highest eigenvalue of matrix ${\Gamma }_{{\mathcal{F}}_{\mathrm{n}}}^{-1}$, and ${\delta }_{{\mathcal{G}}_{\mathrm{n}}}^{min}$ is the lowest eigenvalue of matrix ${\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}{\Lambda }_{{\mathcal{G}}_{\mathrm{n}}}^{\mathbf{T}}$.

Following the previous study, we determine $\mathcal{L}$ as

$$\begin{array}{c}\hfill \mathcal{L}=\sum _{\sigma =1}^{\mathrm{n}}{\mathcal{L}}_{\sigma }.\end{array}$$

(83)

According to (69), (76), and (82), the subsequent inequality possesses

$$\begin{array}{ccc}\hfill {}^c\mathrm{D}^{\alpha }& & \le -{\theta }_{1}ln\left({\displaystyle \frac{{\tilde{\Xi }}_1^4}{{\tilde{\Xi }}_1^4-{\phi }_1^4}}\right)-\sum _{\sigma =2}^{\mathrm{n}}{\theta }_{\sigma }{\phi }_{\sigma }^2-\sum _{\sigma =2}^{\mathrm{n}}{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_{\sigma }}}{2{\delta }_{{\mathcal{F}}_{\sigma }}^{max}}}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}^{\mathbf{T}}{\Gamma }_{{\mathcal{F}}_{\sigma }}^{-1}{\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}-\sum _{\sigma =2}^{\mathrm{n}}{\displaystyle \frac{{\upsilon }_{{\mathcal{G}}_{\sigma }}}{2{\delta }_{{\mathcal{F}}_{\sigma }}^{min}}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}^{\mathbf{T}}{\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}\hfill \\ & & +\sum _{\sigma =1}^{\mathrm{n}}\left({\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+\sum _{\sigma =1}^{\mathrm{n}}{\mathcal{P}}_{\sigma }\le -{\Pi }_1\mathcal{L}+\sum _{\sigma =1}^{\mathrm{n}}\left({\wp }_{\sigma }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\Pi }_2,\hfill \end{array}$$

(84)

where ${\Pi }_1=min\left\{4{\theta }_1,2{\theta }_2,\dots ,2{\theta }_{\mathrm{n}},{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_1}}{{\delta }_{{\mathcal{F}}_1}^{max}}},\dots ,{\displaystyle \frac{{\upsilon }_{{\mathcal{F}}_{\mathrm{n}}}}{{\delta }_{{\mathcal{F}}_{\mathrm{n}}}^{max}}},{\upsilon }_{{\mathcal{G}}_1}{\delta }_{{\mathcal{G}}_1}^{min},\dots ,{\upsilon }_{{\mathcal{G}}_{\mathrm{n}}}{\delta }_{{\mathcal{G}}_{\mathrm{n}}}^{min}\right\}$, and ${\Pi }_2={\displaystyle \sum _{\sigma =1}^{\mathrm{n}}}{\mathcal{P}}_{\sigma }$. Furthermore, Definition 5 states that $\mathcal{L}\in {\mathrm{l}}_{\infty },{\eta }_{\sigma }\in {\mathrm{l}}_{\infty }(\sigma =1,\dots ,\mathrm{n})$, and ${}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }\in {\mathrm{l}}_{\infty }.$ This implies that $|{\phi }_1|<{\tilde{\Xi }}_1,{\phi }_{\ell }\in {\mathrm{l}}_{\infty },(\ell =2,..,\mathrm{n})$, ${\tilde{\mho }}_{{\mathcal{F}}_{\sigma }}\in {\mathrm{l}}_{\infty }$, and ${\tilde{\mho }}_{{\mathcal{G}}_{\sigma }}\in {\mathrm{l}}_{\infty }$, implying that ${\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}\in {\mathrm{l}}_{\infty }$ and ${\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\in {\mathrm{l}}_{\infty }.$ Using the equation ${\phi }_1=\Xi \epsilon =\Xi ({\mathrm{y}}_1-{\mathrm{y}}_{\mathrm{d}})$, we may conclude that $\epsilon \in {\mathrm{l}}_{\infty }$, implying that ${\mathrm{y}}_1\in {\mathrm{l}}_{\infty }$. It suggests that ${\widehat{\Psi }}_1\in {\mathrm{l}}_{\infty },{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_1}\in {\mathrm{l}}_{\infty }$ and ${}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{G}}_1}\in {\mathrm{l}}_{\infty }$ since ${\eta }_1,\epsilon ,{\widehat{\mho }}_{{\mathcal{F}}_1},{\widehat{\mho }}_{{\mathcal{G}}_1}$ and ${\Lambda }_{{\mathcal{G}}_1}$ are bounded, which reveals ${}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_1\in {\mathrm{l}}_{\infty }$. Using an analogous inquiry, we can generate ${\mathrm{y}}_{\ell },{\widehat{\Psi }}_{\ell },{}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_{\ell },{}^c\mathrm{D}^{\alpha }{\mho }_{{\mathcal{F}}_{\ell }}$ and ${}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_{\ell }}\in {\mathrm{l}}_{\infty }.$ Thus, the initial component of Theorem 1 is verified.

Furthermore, if $\left|{\phi }_1\right|<{\tilde{\Xi }}_1$, it concludes that

$$\begin{array}{c}\hfill \left|\epsilon \right|<{\displaystyle \frac{{\tilde{\Xi }}_1}{\Xi }}=\left\{\begin{array}{c}{\left({\displaystyle \frac{{\mathbf{T}}_{{\tilde{\Xi }}_1}-\tau }{\tau }}\right)}^2{\displaystyle \frac{{\tilde{\Xi }}_1}{\Gamma (\alpha +1)}}+\aleph {\tilde{\Xi }}_1,0<\tau <{\mathbf{T}}_{{\tilde{\Xi }}_1},\hfill \\ \aleph {\tilde{\Xi }}_1,\tau \ge {\mathbf{T}}_{\tilde{\Xi }}.\hfill \end{array}\right.\end{array}$$

(85)

Ultimately, the tracking error $\epsilon$ can reach the predefined precision $(-\aleph {\tilde{\Xi }}_1,\aleph {\tilde{\Xi }}_1)$ throughout the specified time ${\mathbf{T}}_{\tilde{\Xi }}$. To summarize, the argument in support of Theorem 1 is accomplished. □

Theorem 2. 

Suppose the supposition of Lemma 4 is fulfilled for mechanism (20). The proposed virtualized control (38), ultimate controller (50), and adaptation rules (61) can ensure

(a) Every signal within a closed-loop structure remains bounded.

The signal $\mathrm{x}={({\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{n}})}^{\mathbf{T}}$ tends to a small neighborhood around its starting point.

Proof. 

By means of the formulation of $U_1(0,\tau )$ along with the evidence that $U_1(0,\tau )=0,(\tau \ge {\mathbf{T}}_0)$, it is evident that $U_1(0,\tau )$ is continuous in $[0,{\mathbf{T}}_0]$. Additionally, a positive constant ${\chi }_0$ occurs to ensure that

$$\begin{array}{c}\hfill \sum _{\ell =1}^{\mathrm{n}}{\widetilde{U_1}}_{\ell }\left(\tau \right)\le {\chi }_0,\tau \ge 0.\end{array}$$

(86)

Using (63), (70), and (80), we obtain

$$\begin{array}{c}\hfill {\mathrm{U}}_{\mathrm{n}}=\sum _{\ell =1}^{\mathrm{n}}\left({\displaystyle \frac{{\mathrm{x}}_{\ell }^2{\wp }_{\ell }+{\tilde{\rho }}_{\ell }^2}{2{\wp }_{\ell }}}+{\displaystyle \frac{\mathrm{z}\left(\tau \right)}{{\wp }_0}}+{\displaystyle \frac{1}{2}}{\hslash }^2\left(\tau \right)\right).\end{array}$$

(87)

Choosing ${\chi }_1=\underset{\ell \in (1,\mathrm{n})}{min}\left\{a{\delta }_{\ell },1,{\overline{\tilde{\Xi }}}_1,{\mathcal{G}}_{{\mathrm{n}}_0}\right\}.$

Therefore, (86) and (87) imply that

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\mathrm{U}}_{\mathrm{n}}\le {\chi }_1{\mathrm{U}}_{\mathrm{n}}+{\chi }_0+{\displaystyle \frac{(1-{{\mathrm{x}}_1}^2)}{{\wp }_0}}{\overline{\tilde{\Xi }}}_5\left(\overline{{\mathrm{y}}_1}\right)+{\eta }_{\mathrm{n}}.\end{array}$$

(88)

Considering (88), there are additionally two instances:

Case (1): If $|{\mathrm{x}}_1|\le 1,$ then the result indicates that ${\mathrm{x}}_1$ is bounded, and, since ${\mathrm{x}}_1={\mathrm{y}}_1-{\mathrm{w}}_{\mathrm{d}}$, it follows that ${\mathrm{y}}_1$ is also bounded. So, ∃ a non-negative fixed value ${\chi }_{{\mathrm{x}}_1}$ such that $\left|{\displaystyle \frac{(1-{{\mathrm{x}}_1}^2)}{{\wp }_0}}{\overline{\tilde{\Xi }}}_5\left(\overline{{\mathrm{y}}_1}\right)\right|\le {\chi }_{{\mathrm{x}}_1}.$ Then, (88) reduces to

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\mathrm{U}}_{\mathrm{n}}\le -{\chi }_1{\mathrm{U}}_{\mathrm{n}}+{\chi }_0+{\chi }_{{\mathrm{x}}_1}+{\Xi }_{\mathrm{n}}.\end{array}$$

(89)

Case (2): If $|{\mathrm{x}}_1>1|,$ then this indicates that ${\displaystyle \frac{(1-{{\mathrm{x}}_1}^2)}{{\wp }_0}}{\overline{\tilde{\Xi }}}_5\left(\overline{{\mathrm{y}}_1}\right)\le 0.$ Therefore, (88) can be expressed as

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\mathrm{U}}_{\mathrm{n}}\le -{\chi }_1{\mathrm{U}}_{\mathrm{n}}+{\chi }_0+{\Xi }_{\mathrm{n}}.\end{array}$$

(90)

Given that ${\mathrm{U}}_{\mathrm{n}}={\mathrm{U}}_{\mathrm{n}}(\mho \left(\tau \right))$ alongside $\mho \left(\tau \right)={({}_{\mathrm{x}}\mathbf{T},{\tilde{\rho }}_1,\dots ,{\tilde{\rho }}_{\mathrm{n}},\mathrm{z},\hslash )}^{\mathbf{T}},$ the subsequent inequality applies to both scenarios.

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\mathrm{U}}_{\mathrm{n}}={}^c\mathrm{D}^{\alpha }{\mathrm{U}}_{\mathrm{n}}(\mho \left(\tau \right))\le -{\chi }_1{\mathrm{U}}_{\mathrm{n}}+{\chi }_2,\end{array}$$

(91)

where ${\chi }_2={\chi }_0+{\chi }_{{\mathrm{x}}_1}+{\Xi }_{\mathrm{n}}$ satisfies $|{\mathrm{x}}_1|<1$ and ${\chi }_2={\chi }_0+{\Xi }_{\mathrm{n}}$ satisfies $|{\mathrm{x}}_1|>1.$ In view of (91), we can identify a positive mapping ${\rm Y}\left(\tau \right)(\tau >0)$ such that

$$\begin{array}{c}\hfill {}^c\mathrm{D}^{\alpha }{\mathrm{U}}_{\mathrm{n}}(\mho \left(\tau \right))+{\rm Y}\left(\tau \right)=-{\chi }_1{\mathrm{U}}_{\mathrm{n}}(\mho )+{\chi }_2.\end{array}$$

(92)

Performing the Laplace transform on (92), we get

$$\begin{array}{c}\hfill {\varrho }^{\alpha }{\mathrm{U}}_{\mathrm{n}}\left(\varrho \right)-{\varrho }^{\alpha -1}{\mathrm{U}}_{\mathrm{n}}\left(0\right)+{\rm Y}\left(\phi \right)=-{\chi }_1{\mathrm{U}}_{\mathrm{n}}\left(\varrho \right)+{\displaystyle \frac{{\chi }_2}{\varrho }}.\end{array}$$

It follows that

$$\begin{array}{c}\hfill {\mathrm{U}}_{\mathrm{n}}\left(\varrho \right)={\displaystyle \frac{1}{{\varrho }^{\alpha }+{\chi }_1}}\left({\mathrm{U}}_{\mathrm{n}}\left(0\right){\varrho }^{\alpha -1}-{\rm Y}\left(\mathrm{x}\right)+{\displaystyle \frac{{\chi }_2}{\varrho }}\right),\end{array}$$

(93)

where ${\mathrm{U}}_{\mathrm{n}}\left(0\right)={\mathrm{U}}_{\mathrm{n}}(\mho \left(0\right)),{\mathrm{U}}_{\mathrm{n}}\left(\varrho \right)$ and ${\rm Y}\left(\varrho \right)$ indicate the Laplace transforms of ${\mathrm{U}}_{\mathrm{n}}(\mho \left(\tau \right))$ and ${\rm Y}\left(\tau \right)$, respectively. Applying the inverse Laplace transform to (93), we obtain

$$\begin{array}{c}\hfill {\mathrm{U}}_{\mathrm{n}}(\mho \left(\tau \right))={\mathrm{U}}_{\mathrm{n}}\left(0\right){\mathcal{E}}_{\alpha }(-{\chi }_1{\tau }^{\alpha })-{\rm Y}\left(\tau \right)★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\chi }_1{\tau }^{\alpha })\right)+{\chi }_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\chi }_1{\tau }^{\alpha }),\end{array}$$

where ★ indicates the convolutional process. Lemma 1 leads to the subsequent variants:

$$\begin{array}{ccc}& & {\chi }_2{\tau }^{\alpha }{\mathcal{E}}_{\alpha ,\alpha +1}(-{\chi }_1{\tau }^{\alpha })\le {\displaystyle \frac{{\chi }_2{\chi }_3{\tau }^{\alpha }}{1+{\chi }_1{\tau }^{\alpha }}}\le {\displaystyle \frac{{\chi }_2{\chi }_3}{{\chi }_1}},\hfill \\ & & {\mathcal{E}}_{\alpha }(-{\chi }_1{\tau }^{\alpha })\le {\displaystyle \frac{{\chi }_4}{1+{\chi }_1{\tau }^{\alpha }}},\hfill \end{array}$$

where ${\chi }_3$ and ${\chi }_4$ are fixed constants. It is worth mentioning that ${\rm Y}\left(\tau \right)★\left({\tau }^{\alpha -1}{\mathcal{E}}_{\alpha ,\alpha }(-{\chi }_1{\tau }^{\alpha })\right)$ $\ge 0$, and, employing the aforesaid variants, we obtain

$$\begin{array}{c}\hfill {\mathrm{U}}_{\mathrm{n}}(\mho \left(\tau \right))\le {\displaystyle \frac{{\chi }_4{\mathrm{U}}_{\mathrm{n}}\left(0\right)}{1+{\chi }_1{\tau }^{\alpha }}}+{\displaystyle \frac{{\chi }_2{\chi }_3}{{\chi }_1}}\le {\chi }_4{\mathrm{U}}_{\mathrm{n}}\left(0\right)+{\displaystyle \frac{{\chi }_2{\chi }_3}{{\chi }_1}}.\end{array}$$

(94)

Consequently, variables ${\mathrm{x}}_{\sigma },{\tilde{\rho }}_{\sigma }(\sigma =1,\dots ,\mathrm{n}),\mathrm{z}$, and ℏ are bounded, implying that ${\widehat{\rho }}_{\sigma }={\tilde{\rho }}_{\sigma }+{\rho }_{\sigma }$ is also bounded. Since ${\mathrm{y}}_{\mathrm{d}}$ and ${\mathrm{x}}_1$ are bounded, we can conclude that ${\mathrm{y}}_1$ is also bounded. The boundedness of ${\mathrm{y}}_1$ and ${\widehat{\rho }}_1$ implies that $\Xi$ is bounded. Using ${\mathrm{x}}_2={\mathrm{y}}_2-\Xi$, we can determine the boundedness of ${\mathrm{y}}_2$. We can gradually achieve the boundedness of ${\mathrm{y}}_{\sigma }(\sigma =3,\dots ,\mathrm{n})$ and $v_1$. Furthermore, using (94), one obtains ${\parallel \mathrm{x}\parallel }^2\le 2{\chi }_4{\mathrm{U}}_{\mathrm{n}}\left(0\right)+{\displaystyle \frac{2{\chi }_2{\chi }_3}{{\chi }_1}}.$ By selecting suitable criteria, we may ensure that ${\mathrm{x}}_{\sigma }(\sigma =1,\dots ,\mathrm{n})$ converges to a small neighborhood of the origin. The evidence is finished. □

5. Experimental Examples

In this section, we conduct a few modeling experiments to illustrate the effectiveness of the suggested strategy.

Example 1. 

A collection of modeling exercises are conducted for the subsequent FON systems:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_1\left(\tau \right)={\wp }_1{\mathrm{y}}_2+{\rho }_1\left({\overline{\mathrm{y}}}_1\right),\hfill \\ {}^c\mathrm{D}^{\alpha }{\mathrm{y}}_2\left(\tau \right)={\wp }_2\mathrm{s}+{\rho }_2\left({\overline{\mathrm{y}}}_2\right),\hfill \end{array}\right.\end{array}$$

(95)

where $\alpha =0.95,{\wp }_1=-3.5,{\wp }_2=1.3,$ ${\rho }_1\left({\overline{\mathrm{y}}}_1\right)=cos\left({\mathrm{y}}_1\right)exp(-{\mathrm{y}}_1^2),$ ${\rho }_2\left({\overline{\mathrm{y}}}_2\right)=sin({\mathrm{y}}_1+{\mathrm{y}}_1)exp(-{\mathrm{y}}_1^2-{\mathrm{y}}_2^2)$, and the corresponding signal ${\mathrm{y}}_{\mathrm{d}}\left(\tau \right)=3/4sin(3\tau /2).$ The suggested layout phases can result in the subsequent controls:

$$\begin{array}{ccc}& & {\widehat{\Psi }}_1=\Omega \left({\eta }_1\right)\left({\theta }_1\epsilon +{\widehat{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right),\hfill \\ & & \widehat{\mathrm{s}}=\Omega \left({\eta }_2\right)\left({\theta }_2{\phi }_2+{\widehat{\mho }}_{{\mathcal{F}}_2}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_2}\left({\mathrm{Z}}_{{\mathcal{F}}_2}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_2}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_2}\left({\mathrm{Z}}_{{\mathcal{G}}_2}\right),\hfill \end{array}$$

(96)

containing

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_1}={\Gamma }_{{\mathcal{F}}_1}\left\{{\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{\phi }_1^4}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)-{\upsilon }_{{\mathcal{F}}_1}{\widehat{\mho }}_{{\mathcal{F}}_1}\right\},\hfill \\ {}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{F}}_1}={\Gamma }_{{\mathcal{F}}_1}\left\{{\phi }_2{\Lambda }_{{\mathcal{F}}_2}\left({\mathrm{Z}}_{{\mathcal{F}}_2}\right)-{\upsilon }_{{\mathcal{F}}_2}{\widehat{\mho }}_{{\mathcal{F}}_2}\right\},\hfill \\ {}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_1}=-{\upsilon }_{{\mathcal{G}}_1}{\Lambda }_{{\mathcal{G}}_1}{\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right)}^{\mathbf{T}}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right){\widehat{\mho }}_{{\mathcal{G}}_1}\},\hfill \\ {}^c\mathrm{D}^{\alpha }{\widehat{\mho }}_{{\mathcal{G}}_2}=-{\upsilon }_{{\mathcal{G}}_2}{\Lambda }_{{\mathcal{G}}_2}{\left({\mathrm{Z}}_{{\mathcal{G}}_2}\right)}^{\mathbf{T}}\left({\mathrm{Z}}_{{\mathcal{G}}_2}\right){\widehat{\mho }}_{{\mathcal{G}}_2}\},\hfill \\ {}^c\mathrm{D}^{\alpha }{\eta }_1={\displaystyle \frac{{\phi }_1^3\Xi }{{\tilde{\Xi }}_1^4-{\phi }_1^4}}\left({\theta }_1\epsilon +{\widehat{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right),\hfill \\ {}^c\mathrm{D}^{\alpha }{\eta }_2={\phi }_2\left({\theta }_2{\phi }_2+{\widehat{\mho }}_{{\mathcal{F}}_2}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_2}\left({\mathrm{Z}}_{{\mathcal{F}}_2}\right)\right),\hfill \end{array}\right.\end{array}$$

(97)

where $\epsilon ={\mathrm{y}}_1-{\mathrm{y}}_{\mathrm{d}},{\phi }_1=\eta \epsilon ,$ ${\phi }_2={\mathrm{y}}_2-{\widehat{\Psi }}_1,{\mathrm{Z}}_{{\mathcal{F}}_1}={\left[{\mathrm{y}}_1,{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_1^{\mathrm{d}},\Xi \right]}^{\mathbf{T}},$ ${\mathrm{Z}}_{{\mathcal{G}}_1}={\left[{\mathrm{y}}_1,{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_1^{\mathrm{d}},\Xi ,{\eta }_1\right]}^{\mathbf{T}},{\mathrm{Z}}_{{\mathcal{F}}_2}={\left[{\mathrm{y}}_1,{\mathrm{y}}_2,{}^c\mathrm{D}^{\alpha }\widehat{{\Psi }_1}\left({\mathrm{y}}_1\right),{\kappa }_1\right]}^{\mathbf{T}},$ ${\mathrm{Z}}_{{\mathcal{G}}_2}={\left[{\mathrm{y}}_1,{\mathrm{y}}_2,{}^c\mathrm{D}^{\alpha }\widehat{{\Psi }_1}\left({\mathrm{y}}_1\right),{\kappa }_1,{\eta }_2\right]}^{\mathbf{T}},$ ${\kappa }_1={}^c\mathrm{D}^{\alpha }{\widehat{\Psi }}_1\left({\mathrm{y}}_{\mathrm{d}}^{\left(1\right)}\right){\mathrm{y}}_{\mathrm{d}}^{\left(2\right)}$ $+{}^c\mathrm{D}^{\alpha }\widehat{\Psi }\left(\widehat{{\eta }_1}\right){}^c\mathrm{D}^{\alpha }\left(\widehat{{\eta }_1}\right)$ $+{}^c\mathrm{D}^{\alpha }\widehat{\Psi }\left({\widehat{\mho }}_{{\mathcal{F}}_1}\right){}^c\mathrm{D}^{\alpha }\left({\widehat{\mho }}_{{\mathcal{F}}_1}\right)$ $+{}^c\mathrm{D}^{\alpha }\widehat{\Psi }\left({\widehat{\mho }}_{{\mathcal{G}}_1}\right){}^c\mathrm{D}^{\alpha }\left({\widehat{\mho }}_{{\mathcal{G}}_1}\right).$

In the modeling process, we select the following settings: ${\theta }_1={\theta }_2=10,$ ${\upsilon }_{{\mathcal{F}}_1}=2,$ ${\upsilon }_{{\mathcal{F}}_2}=3,{\upsilon }_{{\mathcal{G}}_1}=1.2,$ ${\upsilon }_{{\mathcal{G}}_2}=2.5,{\Gamma }_{{\mathcal{F}}_1}=0.9,$ ${\Gamma }_{{\mathcal{F}}_2}=2.$ For ${\Lambda }_{{\mathcal{F}}_1},{\Lambda }_{{\mathcal{G}}_1},{\Lambda }_{{\mathcal{F}}_2}$, and ${\Lambda }_{{\mathcal{G}}_2},$ we designed 32 clusters with evenly dispersed centers in the range $[-15,15]$ and lengths of 1. Furthermore, the predetermined precision $\left({\aleph }_{{\tilde{\Xi }}_1}\right)=0.05$, whereas the predefined time ${\mathbf{T}}_{{\tilde{\Xi }}_1}=1,0.9,0.7.$

The model’s initial settings have been determined as ${\mathrm{y}}_1\left(0\right)=0.1,{\mathrm{y}}_2=-0.3,$ ${\widehat{\mho }}_{{\mathcal{F}}_1}\left(0\right)={\left[0.1,\dots ,0.1\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{\mho }}_{{\mathcal{G}}_1}\left(0\right)={\left[0.4,\dots ,0.4\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{\mho }}_{{\mathcal{F}}_2}={\left[0.3,\dots ,0.3\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{\mho }}_{{\mathcal{F}}_2}={\left[0.7,\dots ,0.7\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ $\widehat{{\eta }_1}\left(0\right)=0$, and $\widehat{{\eta }_2}\left(0\right)=0.3.$

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 show the modeling outcomes. Figure 1 shows an analogous path $\left({\mathrm{y}}_{\mathrm{d}}\right)$ and the output results $\left({\mathrm{y}}_1\right)$ in various predetermined times. Figure 2a,b show that the technique’s tracking failure $\left(\epsilon \right)$ continually fluctuates inside the time frame $\left[-{\tilde{\Xi }}_1/\Xi ,{\tilde{\Xi }}_1/\Xi \right].$ Figure 3a,b depict the paths of network failures ${\phi }_1$ and ${\phi }_2$. Figure 4a,b display the paths of the optimized control $\widehat{\mathrm{s}}$ at various predetermined times. Figure 5b and Figure 6a show the cost function dynamics for ${\hslash }_1=ln\left({{\tilde{\Xi }}_1}^4/({{\tilde{\Xi }}_1}^4-{{\phi }_1}^4)\right)+{\widehat{\Psi }}_1^2$ and ${\hslash }_2={{\phi }_1}^2+{\widehat{\mathrm{s}}}^2$, respectively. Figure 6a,b, Figure 7a,b, and Figure 8a show the trends of $\parallel {\widehat{\mho }}_{{\mathcal{F}}_{\sigma }}\parallel$, $\parallel {\widehat{\mho }}_{{\mathcal{G}}_{\sigma }}\parallel$ and ${\eta }_{\sigma }$ and $\Omega \left({\eta }_{\sigma }\right)(\sigma =1.3)$ at various predefined times.

Example 2. 

Simulations are undertaken to compare the suggested method’s performance to the monitoring technique in [70] regarding an SF mechanism featuring unspecified control orientations. The structure of the framework is presented as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_1\left(\tau \right)={\wp }_1{\mathrm{y}}_2+{\rho }_1\left({\overline{\mathrm{y}}}_1\right),\hfill \\ {}^c\mathrm{D}^{\alpha }{\mathrm{y}}_2\left(\tau \right)={\wp }_2\mathrm{s}+{\rho }_2\left({\overline{\mathrm{y}}}_2\right),\hfill \end{array}\right.\end{array}$$

(98)

where $\alpha =0.95,{\wp }_1=-3,$ ${\wp }_2=1.3,{\rho }_1\left({\overline{\mathrm{y}}}_1\right)=sin({\mathrm{y}}_1/2)cos({\mathrm{y}}_1/4),$ ${\rho }_2\left(\overline{{\mathrm{y}}_2}\right)=cos({\mathrm{y}}_2/2+{\mathrm{y}}_1/2)sin\left({\mathrm{y}}_1{\mathrm{y}}_2\right)$, and corresponding signal ${\mathrm{y}}_{\mathrm{d}}\left(\tau \right)=3/4sin(3\tau /2).$ For this research test, the predefined precision is $0.1$, and the predetermined runtime is one second. We employ an analogue controller approach and change settings to attain equivalent controlling efficacy in the proposed study and the strategy adopted in [70]. Using the control approach suggested in this work, the controller’s structure is constructed in the following manner:

$$\begin{array}{ccc}& & {\widehat{\Psi }}_1=\Omega \left({\eta }_1\right)\left({\theta }_1\epsilon +{\widehat{\mho }}_{{\mathcal{F}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_1}\left({\mathrm{Z}}_{{\mathcal{F}}_1}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_1}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_1}\left({\mathrm{Z}}_{{\mathcal{G}}_1}\right),\hfill \\ & & \widehat{\mathrm{s}}=\Omega \left({\eta }_2\right)\left({\theta }_2{\phi }_2+{\widehat{\mho }}_{{\mathcal{F}}_2}^{\mathbf{T}}{\Lambda }_{{\mathcal{F}}_2}\left({\mathrm{Z}}_{{\mathcal{F}}_2}\right)\right)-{\widehat{\mho }}_{{\mathcal{G}}_2}^{\mathbf{T}}{\Lambda }_{{\mathcal{G}}_2}\left({\mathrm{Z}}_{{\mathcal{G}}_2}\right)\hfill \end{array}$$

(99)

with ${\theta }_1=15,{\theta }_2=10,$ ${\upsilon }_{{\mathcal{F}}_1}=2,{\upsilon }_{{\mathcal{F}}_2}=3,$ ${\upsilon }_{{\mathcal{G}}_1}=1.2,$ ${\upsilon }_{{\mathcal{G}}_2}=2.5,{\Gamma }_{{\mathcal{F}}_1}=0.91,$ ${\Gamma }_{{\mathcal{F}}_2}=2.1,{\mathrm{y}}_1\left(0\right)=0.2,$ ${\mathrm{y}}_2\left(0\right)=-0.3,{\widehat{\mho }}_{{\mathcal{F}}_1}\left(0\right)={\left[0.2,\dots ,0.2\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{\mho }}_{{\mathcal{G}}_1}\left(0\right)={\left[0.4,\dots ,0.4\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{\mho }}_{{\mathcal{F}}_2}\left(0\right)={\left[0.4,\dots ,0.4\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{\mho }}_{{\mathcal{G}}_2}\left(0\right)={\left[0.8,\dots ,0.8\right]}^{\mathbf{T}}\in {\mathcal{R}}^{31},{\widehat{\eta }}_1\left(0\right)=0$, and ${\widehat{\eta }}_2\left(0\right)=1.$

Using the control approach given in [70], the control system is constructed as follows:

$$\begin{array}{ccc}& & {\Psi }_1=\Omega \left({\xi }_1\right){\overline{\Psi }}_1,{}^c\mathrm{D}^{\alpha }{\xi }_1={\wp }_1{\left({\mathrm{x}}_1\right)}^3{\overline{\Psi }}_1,{\overline{\Psi }}_1={\displaystyle \frac{{\tilde{\Xi }}_1{\mathrm{x}}_1}{{\wp }_1}}+\widehat{G_1}{\Phi }_1\left({\mathrm{Y}}_1\right)-{\displaystyle \frac{{\mathrm{y}}_{\mathrm{d}}{}^c\mathrm{D}^{\alpha }\theta }{\theta }}-{}^c\mathrm{D}^{\alpha }{\mathrm{y}}_{\mathrm{d}}+\widehat{Q_1}tanh\left({\displaystyle \frac{{\mathrm{x}}_1^3{\wp }_1}{\gamma }}\right),\hfill \\ & & \mathrm{s}=\Omega \left({\xi }_2\right){\overline{\Psi }}_2,{}^c\mathrm{D}^{\alpha }{\xi }_2={\left({\mathrm{x}}_2\right)}^3{\overline{\Psi }}_2,{\overline{\Psi }}_2={\tilde{\Xi }}_2{\mathrm{x}}_2+\widehat{G_2}{\Phi }_2\left({\mathrm{Y}}_2\right)+\widehat{Q_2}tanh\left({\displaystyle \frac{{\mathrm{x}}_2^3{\lambda }_2}{\gamma }}\right)\hfill \end{array}$$

(100)

with

$$\begin{array}{ccc}& & {}^c\mathrm{D}^{\alpha }{\widehat{W}}_1={\Gamma }_1\left({\left({\mathrm{x}}_1\right)}^3{\wp }_1{\Phi }_1-{\kappa }_1{\widehat{W}}_1\right),\hfill \\ & & {}^c\mathrm{D}^{\alpha }{\widehat{W}}_2={\Gamma }_2\left({\left({\mathrm{x}}_2\right)}^3{\wp }_2{\Phi }_2-{\kappa }_2{\widehat{W}}_2\right),\hfill \\ & & {}^c\mathrm{D}^{\alpha }{\widehat{Q}}_1={\delta }_1{\left({\mathrm{x}}_1\right)}^3{\wp }_{1}tanh\left({\displaystyle \frac{{\left({\mathrm{x}}_1\right)}^3{\wp }_1}{\gamma }}\right)-{\kappa }_1{\widehat{Q}}_1,\hfill \\ & & {}^c\mathrm{D}^{\alpha }{\widehat{Q}}_2={\delta }_2{\left({\mathrm{x}}_2\right)}^3{\wp }_{2}tanh\left({\displaystyle \frac{{\left({\mathrm{x}}_2\right)}^3{\wp }_2}{\gamma }}\right)-{\kappa }_2{\widehat{Q}}_2,\hfill \end{array}$$

(101)

where ${\tilde{\Xi }}_1={\tilde{\Xi }}_2=25,{\kappa }_1=0.7,$ ${\kappa }_2=0.1,{\delta }_1=0.2,{\delta }_2=0.6,{\kappa }_1=0.4,$ ${\kappa }_2=0.8,{\Gamma }_1=0.31,{\Gamma }_2=0.5,$ ${\mathrm{y}}_1\left(0\right)=0.1,{\mathrm{y}}_2\left(0\right)=-0.3,$ ${\widehat{G}}_1={[0,\dots ,0]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ $\widehat{W_2}={[1.3,\dots ,1.3]}^{\mathbf{T}}\in {\mathcal{R}}^{31},$ ${\widehat{Q}}_1\left(0\right)=\left[0\right],\widehat{D}\left(0\right)={[0.5,0.5]}^{\mathbf{T}}\in {\mathcal{R}}^2,$ ${\widehat{\xi }}_1\left(0\right)=0$, and ${\widehat{\xi }}_2\left(0\right)=-1.$ Considering ${\Phi }_1$ and ${\Phi }_2$, we generated 31 networks with evenly dispersed centers in the range $[-15,15]$ and width of 1. Liu et al. [70] provide detailed descriptions of the expressions ${\mathrm{x}}_1,{\mathrm{x}}_2,{\wp }_1,\theta ,{\mathrm{Y}}_1,{\mathrm{Y}}_2,\gamma ,{\psi }_2$.

Figure 9 and Figure 10 show that the developed control technique is considerably less costly than typical fixed-functionality controllers with comparable controlling efficacy. We use the integrated absolute cost and integrated temporal absolute cost to assess cost effectiveness.

Table 1. Comparison analysis of proposed scheme.

	Control in FON System	Control in [57]	Control in [70]
Integrated absolute cost	27.0023	33.0699	47.9955
Integrated temporal absolute cost	3.9823	11.7391	52.8275 3

summarizes the empirical data, demonstrating the optimized controller’s cost reductions.

6. Stochastic Analysis

Assume a large-scale nonlinear framework with $\mathcal{N}$ subsystems associated via outcomes. The $\sigma$th component ${\displaystyle \sum _{\sigma }}(\sigma =1,2,\dots ,\mathcal{N})$ is represented as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{d}^{\alpha }{\mathrm{y}}_{\sigma ,1}=\left({\mathrm{y}}_{\sigma ,2}+{\mathcal{F}}_{\sigma ,1}\left({\underline{\mathrm{y}}}_{\sigma ,1}\right)+{\Delta }_{\sigma ,1}\left(\overline{\mathrm{w}}\right)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{\sigma ,1}\left(\overline{\mathrm{w}}\right)d{\mathrm{v}}_{\sigma },\hfill \\ d^{\alpha }{\mathrm{y}}_{\sigma ,2}=\left({\mathrm{y}}_{\sigma ,3}+{\mathcal{F}}_{\sigma ,2}\left({\underline{\mathrm{y}}}_{\sigma ,2}\right)+{\Delta }_{\sigma ,2}\left(\overline{\mathrm{w}}\right)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{\sigma ,2}\left(\overline{\mathrm{w}}\right)d{\mathrm{v}}_{\sigma },\hfill \\ ⋮\hfill \\ d^{\alpha }{\mathrm{y}}_{\sigma ,{\mathrm{n}}_{\sigma }-1}=\left({\mathrm{y}}_{\sigma ,{\mathrm{n}}_{\sigma }}+{\mathcal{F}}_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left({\underline{\mathrm{y}}}_{\sigma ,{\mathrm{n}}_{\sigma }-1}\right)+{\Delta }_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left(\overline{\mathrm{w}}\right)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left(\overline{\mathrm{w}}\right)d{\mathrm{v}}_{\sigma },\hfill \\ d^{\alpha }{\mathrm{y}}_{\sigma ,{\mathrm{n}}_{\sigma }}=\left({\mathcal{F}}_{\sigma ,{\mathrm{n}}_{\sigma }}\left({\underline{\mathrm{y}}}_{\sigma ,{\mathrm{n}}_{\sigma }}\right)+{\Delta }_{\sigma ,{\mathrm{n}}_{\sigma }}\left(\overline{\mathrm{w}}\right)+{{\beta }_2}_{\sigma ,0}{\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{\sigma ,{\mathrm{n}}_{\sigma }}\left(\overline{\mathrm{w}}\right)d{\mathrm{v}}_{\sigma },\hfill \\ {\mathrm{w}}_{\sigma }={\mathrm{y}}_{\sigma ,1},\hfill \end{array}\right.\end{array}$$

(102)

where ${\underline{\mathrm{y}}}_{\sigma ,\ell }={[{\mathrm{y}}_{\sigma ,1},\dots ,{\mathrm{y}}_{\sigma ,\ell }]}^{\mathbf{T}}\in {\mathbb{R}}^{\sigma },(\sigma =1,\dots ,\mathcal{N},\ell =1,\dots ,{\mathrm{n}}_{\sigma })$ represents the state vector, whereas ${\mathrm{s}}_{\sigma }\in \mathbb{R}$ and ${\mathrm{w}}_{\sigma }\in \mathbb{R}$ represent the system’s controller input as well as output, respectively. ${\mathcal{F}}_{\sigma ,\ell }(.)$ represent undefined continuous nonlinear functions. The unidentified continuous nonlinear function ${\mathcal{F}}_{\sigma ,\ell }\left(0\right)=0$ and ${\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right)\ne 0.$ The unidentified smooth functions ${\Delta }_{\sigma ,\ell }\left(\overline{\mathrm{w}}\right)$ and ${\mathcal{G}}_{\sigma ,\ell }\left(\overline{\mathrm{w}}\right),\left(\overline{\mathrm{w}}={[{\mathrm{w}}_1,\dots ,{\mathrm{w}}_{\mathcal{N}}]}^{\mathbf{T}}\right)$ reflect the interrelated impacts that occur among $\sigma$th and additional components. ${{\beta }_2}_{\sigma ,0}\ne 0$ corresponds to an undetermined consistent factor ($|{{\beta }_2}_{\sigma ,0}|\le {\overline{{\beta }_2}}_{\sigma ,0},$ where ${\overline{{\beta }_2}}_{\sigma ,0}$ constitutes a definite non-negative integer), and the symbol of ${\overline{{\beta }_2}}_{\sigma ,0}$ is undetermined. ${\mathrm{v}}_{\sigma }\in \mathbb{R}$ denotes an independent standard Brownian motion specified for a complete probability space. This study assumes that just the output ${\mathrm{w}}_{\sigma }$ is accessible for analysis.

Hypothesis 2 

([48]). The nonlinear mappings ${\Delta }_{\sigma ,\ell }\left(\overline{\mathrm{w}}\right)$ and ${\mathcal{G}}_{\sigma ,\ell }\left(\overline{\mathrm{w}}\right)$, fulfill

$$\begin{array}{c}\hfill \left|{\Delta }_{\sigma ,\ell }\left(\overline{\mathrm{w}}\right)\right|\le \sum _{\sigma =1}^{\mathcal{N}}{q_1}_{\sigma ,j}{\Xi }_j\left(\right|{\mathrm{w}}_j\left|\right),|{\mathcal{G}}_{\sigma ,\ell }\left(\overline{\mathrm{w}}\right)|\le \sum _{j=1}^{\mathcal{N}}{\lambda }_{\sigma ,j}{\overline{\varsigma }}_{\sigma ,j}\left(\right|{\mathrm{w}}_j\left|\right),\end{array}$$

(103)

where ${q_1}_{\sigma ,j}$ and ${\lambda }_{\sigma ,j}$ represent undefined values representing the power of connections: $\sigma =1,2,\dots ,\mathcal{N},\ell =1,\dots ,{\mathrm{n}}_{\sigma },{\Xi }_j\left(\right|{\mathrm{w}}_j\left|\right)\ge 0.$ The nonlinear smooth mappings ${\overline{\varsigma }}_j\left(\right|{\mathrm{w}}_j\left|\right)\ge 0$ with ${\Xi }_j\left(0\right)=0$ and ${\overline{\varsigma }}_j\left(0\right)=0.$

Hypothesis 3. 

The control-acquired ${\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right),(\sigma =1,\dots ,\mathcal{N})$ is defined as ${\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right)={\overline{\gamma }}_{\sigma }\left({\mathrm{w}}_{\sigma }\right)+\Delta {\overline{\gamma }}_{\sigma }({\mathrm{w}}_{\sigma },\tau ),$ where ${\overline{\gamma }}_{\sigma }\left({\mathrm{w}}_{\sigma }\right)$ constitutes a well-defined mapping $({\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right)\ne 0):|\Delta {\overline{\gamma }}_{\sigma }({\mathrm{w}}_{\sigma },\tau )|<{\varsigma }_{\sigma }<\left|{\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right)\right|$, and ${\varsigma }_{\sigma }$ denotes a non-negative factor.

In the literature, researchers [40] demonstrated the Nussbaum function condition for single-input and single-output uncertain nonlinear frameworks with undetermined control orientations. This has been employed to construct and stabilize closed-loop networks. However, the results from this study show stochastic large-scale nonlinear frameworks fail to satisfy the resulting Nussbaum function feature.

Furthermore, a novel Nussbaum function feature can be created as follows.

Theorem 3. 

For $\alpha \in (0,1]$ and considering this specific Nussbaum gain mapping $\Omega \left(\eta \right)=cos\left({\eta }^2\right)exp\left({\eta }^2\right)$, assume that there is a smooth mapping $\eta \left(\tau \right)$ specified on $[0,{\tau }_{\mathcal{F}}].$ Assume a large-scale fractional-Brownian-motion complex framework (102) with a non-negative-definite radially unbounded mapping $\mathrm{U}(\tau ,{\mathrm{y}}_1)$ and values $(\sigma =1,2)$ that meet the aforementioned inequality:

$$\begin{array}{c}\hfill \mathcal{L}\mathrm{U}(\tau ,{\mathrm{y}}_1)\le -{\mathcal{W}}_1\mathrm{U}(\tau ,{\mathrm{y}}_1)+\sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left[{\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right]{}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\mathcal{W}}_2,\end{array}$$

(104)

where j is characterized as an infinitesimal operator and ${\mathrm{d}}_{\sigma }$ constitutes an appropriate factor. $\mathbb{E}\mathrm{U}(\tau ,{\mathrm{y}}_1),{\eta }_{\sigma }\left(\tau \right)$ and ${\displaystyle \sum _{\sigma =1}^{\mathcal{N}}}{\mathrm{d}}_{\sigma }\left[{\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right]{}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }$ are all bounded on $[0,{\tau }_{\mathcal{F}}]$.

Proof. 

Initially, we take

$$\begin{array}{c}\hfill \mathcal{Q}(\tau ,{\mathrm{y}}_1)=\mathcal{Q}(\tau ,{\mathrm{y}}_1)exp\left({\mathcal{W}}_1\tau \right)\end{array}$$

(105)

It is straightforward to find

$$\begin{array}{c}\hfill {\mathbb{E}\mathcal{Q}(\phi ,\mathrm{y}\left(\mathrm{y}\right))|}_0^{{\tau }^{\alpha }}=\mathbb{E}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}\mathfrak{L}\mathcal{Q}(\phi ,\mathrm{y}\left(\phi \right))d{\phi }^{\alpha },\forall \tau \in [0,{\tau }_{\mathcal{F}}],\end{array}$$

(106)

which is articulated as

$$\begin{array}{c}\hfill \mathbb{E}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}\left[{\mathcal{W}}_1\mathrm{U}(\phi ,\mathrm{y}\left(\phi \right))exp\left({\mathcal{W}}_1\phi \right)+\mathfrak{L}\mathrm{U}(\phi ,\mathrm{y}\left(\phi \right))exp\left({\mathcal{W}}_1\phi \right)\right]d{\phi }^{\alpha }.\end{array}$$

(107)

Utilizing the fact of (105) and (107) as

$$\begin{array}{ccc}& & \mathbb{E}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}\sum _{\sigma =1}^{\mathcal{N}}\left({\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }+{\mathcal{W}}_2\right)exp\left({\mathcal{W}}_1\phi \right)d{\phi }^{\alpha }\hfill \\ & & \le {\displaystyle \frac{{\mathcal{W}}_2}{{\mathcal{W}}_1}}\left(exp\left({\mathcal{W}}_1\tau \right)-1\right)+\sum _{\sigma =1}^{\mathcal{N}}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}\left({\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }\right)exp\left({\mathcal{W}}_1\phi \right)d{\phi }^{\alpha },\forall \tau \in [0,{\tau }_{\mathcal{F}}].\hfill \end{array}$$

(108)

In view of (104), we have

$$\begin{array}{ccc}\hfill & & \mathbb{E}\mathcal{Q}(\tau ,\mathrm{y}(\tau \left)\right)\hfill \\ & & \le \mathbb{E}\mathcal{Q}(0,\mathrm{y}\left(0\right))+\sum _{\sigma =1}^{\mathcal{N}}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}({\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }exp\left({\mathcal{W}}_1\phi \right)d{\phi }^{\alpha },\forall \tau \in [0,{\tau }_{\mathcal{F}}].\hfill \end{array}$$

(109)

Take advantage of $\mathcal{Q}(\tau ,\mathrm{y}\left(\tau \right))=\mathrm{U}(\tau ,\mathrm{y}\left(\tau \right))exp\left({\mathcal{W}}_1\tau \right)$ for every $\tau \in [0,{\tau }_{\mathcal{F}}]$ and obtain

$$\begin{array}{ccc}\hfill 0\le \mathbb{E}\mathrm{U}\left(\tau ,\mathrm{y}\left(\tau \right)\right)& & \le \mathbb{E}\mathrm{U}\left(0,\mathrm{y}\left(0\right)\right)exp\left(-{\mathcal{W}}_1\tau \right)+{\displaystyle \frac{{\mathcal{W}}_2}{{\mathcal{W}}_1}}+exp\left(-{\mathcal{W}}_1\tau \right)\hfill \\ & & \times \sum _{\sigma =1}^{\mathcal{N}}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }exp\left(-{\mathcal{W}}_1\phi \right)d{\phi }^{\alpha }.\hfill \end{array}$$

(110)

According to (110), we have

$$\begin{array}{ccc}\hfill & & exp\left(-{\mathcal{W}}_1\tau \right)\sum _{\sigma =1}^{\mathcal{N}}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }exp\left(-{\mathcal{W}}_1\phi \right)d{\phi }^{\alpha }\hfill \\ & & =exp\left(-{\mathcal{W}}_1\tau \right)\underset{0}{\overset{{\tau }^{\alpha }}{\int }}\sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }exp\left(-{\mathcal{W}}_1\phi \right)d{\phi }^{\alpha }\hfill \\ & & \le exp\left(-{\mathcal{W}}_1\mathcal{B}\left(\tau \right)-{\mathcal{W}}_1\tau \right)\sum _{\sigma =1}^{\mathcal{N}}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }exp\left(-{\mathcal{W}}_1\phi \right)d{\phi }^{\alpha },\hfill \end{array}$$

(111)

where

$$\begin{array}{c}\hfill \mathcal{B}\left(\tau \right)=\left\{\begin{array}{c}{\tau }^{\alpha }{\displaystyle \sum _{\sigma =1}^{\mathcal{N}}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }\ge 0,\hfill \\ 0{\displaystyle \sum _{\sigma =1}^{\mathcal{N}}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }<0.\hfill \end{array}\right.\end{array}$$

(112)

Selecting

$$\begin{array}{c}\hfill \Omega \left({\eta }_{\sigma }\right)=exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right).\end{array}$$

(113)

The Caputo derivative of $\Omega \left({\eta }_{\sigma }\right)$ concerning ${\eta }_{\sigma }$ is ${}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)={\displaystyle \frac{1}{\Gamma (\mathrm{q}-\alpha )}}\underset{0}{\overset{{\eta }_{\sigma }^{\alpha }}{\int }}{({\eta }_{\sigma }-\tau )}^{\mathrm{q}-1-\alpha }\left(2{\eta }_{\sigma }exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right)-2{\eta }_{\sigma }exp\left({\eta }_{\sigma }^2\right)sin\left({\eta }_{\sigma }^2\right)\right)d{\eta }_{\sigma }^{\alpha }.$ Therefore, we have

$$\begin{array}{ccc}\hfill & & \sum _{\sigma =1}^{\mathcal{N}}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }d{\phi }^{\alpha }\hfill \\ & & \le \sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right)+{\eta }_{\sigma }\right)-\sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\left(0\right)\right)cos\left({\eta }_{\sigma }^2\left(0\right)\right)+{\eta }_{\sigma }\left(0\right)\right).\hfill \end{array}$$

(114)

Plugging (111)–(113) into (114) produces

$$\begin{array}{ccc}\hfill & 0& \le \mathbb{E}\mathrm{U}\left(\tau ,\mathrm{y}\left(\tau \right)\right)\hfill \\ & & \le \mathbb{E}\mathrm{U}\left(0,\mathrm{y}\left(0\right)\right)exp\left(-{\mathcal{W}}_1\tau \right)+{\displaystyle \frac{{\mathcal{W}}_2}{{\mathcal{W}}_1}}+exp\left({\mathcal{W}}_1\mathcal{B}\left(\tau \right)-{\mathcal{W}}_1\tau \right)\sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)+1\right){}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }\hfill \\ & & \le \sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right)+{\eta }_{\sigma }\right)-\sum _{\sigma =1}^{\mathcal{N}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\left(0\right)\right)cos\left({\eta }_{\sigma }^2\left(0\right)\right)+{\eta }_{\sigma }\left(0\right)\right).\hfill \end{array}$$

(115)

To demonstrate that ${\eta }_{\sigma }$ is bounded on $[0,{\tau }_{\mathcal{F}})$, we look for a contradiction. Two distinct scenarios are under consideration.

Step (1): Suppose ${\wp }_{\sigma ,0}>0,(\sigma =1,\dots ,\Omega )$ and suppose that ${\eta }_{\sigma }$ and $exp\left({\eta }_{\sigma }^2\right)$ hold $\underset{\tau \mapsto {\tau }_{\mathcal{F}}}{lim}{\eta }_{\sigma }\left(\tau \right)=\infty$ and $\underset{\tau \mapsto {\tau }_{\mathcal{F}}}{lim}sup{\displaystyle \frac{{\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\left(\tau \right)\right)}{{\wp }_{\sigma +1,0}exp\left({\eta }_{\sigma +1}^2\left(\tau \right)\right)}}={a_1}_{\sigma ,(\sigma =1,\dots ,\Omega -1)},$ where ${a_1}_{\sigma }\ge 0$ is fixed. As

$$\begin{array}{ccc}\hfill & & {\mathrm{d}}_1\left({\wp }_{1,0}\Omega \left({\eta }_1\left(\tau \right)\right)+{\eta }_1\left(\tau \right)\right)+{\mathrm{d}}_2\left({\wp }_{2,0}\Omega \left({\eta }_2\left(\tau \right)\right)+{\eta }_2\left(\tau \right)\right)+\dots +{\mathrm{d}}_{\Omega }\left({\wp }_{\Omega ,0}\Omega \left({\eta }_{\Omega }\left(\tau \right)\right)+{\eta }_{\Omega }\left(\tau \right)\right)\hfill \\ & & ={\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)({\mathrm{d}}_1{\displaystyle \frac{{\wp }_{1,0}exp\left({\eta }_1^2\left(\tau \right)\right)}{{\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}}cos\left({\eta }_1^2\left(\tau \right)\right)\hfill \\ & & +\dots +{\mathrm{d}}_{\Omega -1}{\displaystyle \frac{{\wp }_{\Omega -1,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}{{\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}}cos\left({\eta }_{\Omega }^2\left(\tau \right)\right)+\dots +{\mathrm{d}}_{\Omega }cos\left({\eta }_{\Omega }^2\left(\tau \right)\right)\hfill \\ & & +{\displaystyle \frac{{\wp }_1{\eta }_1\left(\tau \right)+\dots +{\mathrm{d}}_{\Omega }{\eta }_{\Omega }\left(\tau \right)}{{\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}}).\hfill \end{array}$$

(116)

When we select ${\mathrm{d}}_{\Omega }\ge {a_1}_1{a_1}_2\dots {a_1}_{\Omega -1}+{a_1}_2\dots {a_1}_{\Omega -1}+\dots +{a_1}_{\Omega -1}+1$ and ${\tau }_{\mathrm{n}}$ fulfills $\underset{\mathrm{n}\mapsto \infty }{lim}{\tau }_{\mathrm{n}}={\tau }_0$, $(\forall {\tau }_0\in [0,{\tau }_{\mathcal{F}}))cos\left({\eta }_{\Omega }^2\left({\tau }_{\mathrm{n}}\right)\right)=-1,$ it concludes that

$$\begin{array}{ccc}\hfill & & \underset{\mathrm{n}\mapsto \infty }{lim}sup({\mathrm{d}}_1{\displaystyle \frac{{\wp }_{1,0}exp\left({\eta }_1^2\left(\tau \right)\right)}{{\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}}cos\left({\eta }_1^2\left(\tau \right)\right)\hfill \\ & & +\dots +{\mathrm{d}}_{\Omega -1}{\displaystyle \frac{{\wp }_{\Omega -1,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}{{\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}}cos\left({\eta }_{\Omega }^2\left(\tau \right)\right)+\dots +{\mathrm{d}}_{\Omega }cos\left({\eta }_{\Omega }^2\left(\tau \right)\right)\hfill \\ & & +{\displaystyle \frac{{\wp }_1{\eta }_1\left(\tau \right)+\dots +{\mathrm{d}}_{\Omega }{\eta }_{\Omega }\left(\tau \right)}{{\wp }_{\Omega ,0}exp\left({\eta }_{\Omega }^2\left(\tau \right)\right)}})\le -1.\hfill \end{array}$$

(117)

In view of (116) and (117), as $\mathrm{n}\mapsto \infty ,$ ${\mathrm{d}}_1\left({\wp }_{1,0}\Omega \left({\eta }_1\left(\tau \right)\right)+{\eta }_1\left(\tau \right)\right)$ $+{\mathrm{d}}_2\left({\wp }_{2,0}\Omega \left({\eta }_2\left(\tau \right)\right)+{\eta }_2\left(\tau \right)\right)+\dots$ $+{\mathrm{d}}_{\Omega }\left({\wp }_{\Omega ,0}\Omega \left({\eta }_{\Omega }\left(\tau \right)\right)+{\eta }_{\Omega }\left(\tau \right)\right)\mapsto -\infty ,$ contradicting the fact that $\mathbb{E}\mathrm{U}\left(\tau ,\mathrm{y}\left(\tau \right)\right)\ge 0$ (by (115)). Consequently, ${\eta }_{\sigma }$ has an upper bound on $[0,{\tau }_{\mathcal{F}})$, and ${\displaystyle \sum _{\sigma =1}^{\mathcal{N}}}{\mathrm{d}}_{\sigma }\left({\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right)+{\eta }_{\sigma }\right)$, implying that $\mathbb{E}\mathrm{U}\left(\tau ,\mathrm{y}\left(\tau \right)\right)\ge 0$ additionally has to be bounded on $[0,{\tau }_{\mathcal{F}})$.

Step (2): If ∃ certain ${\wp }_{\sigma ,0}<0$, then ${\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right)=-{\wp }_{\sigma ,0}exp\left({\eta }_{\sigma }^2\right)cos({\eta }_{\sigma }^2+\pi /2).$ Applying identical interpretations utilized in Step (1), we can argue that ${\eta }_{\sigma }$ is upper-bounded on $[0,{\tau }_{\mathcal{F}})$. Consequently, it implies that ${\eta }_{\sigma }$ is bounded on $[0,{\tau }_{\mathcal{F}}).$ Finally, ${\displaystyle \sum _{\sigma =1}^{\Omega }}\underset{0}{\overset{{\tau }^{\alpha }}{\int }}{\wp }_{\sigma ,0}{}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right){\eta }_{\sigma }exp\left({\mathcal{W}}_1\phi \right)d{\phi }^{\alpha },{\eta }_{\sigma },\mathrm{U}$, and $\mathbb{E}\mathrm{U}\left(\tau ,\mathrm{y}\left(\tau \right)\right)$ are also bounded on $[0,{\tau }_{\mathcal{F}}).$ Hence, Theorem 3 has been verified. □

6.1. Uncertain Filter Technique

Consider that uncertain-logic systems possess the forms

$$\begin{array}{c}\hfill {\widehat{\mathcal{F}}}_{\sigma ,\ell }\left({\widehat{\underline{\mathrm{y}}}}_{\sigma ,\ell }\right|{\mathcal{Q}}_{\sigma ,\ell })={\mathcal{Q}}_{\sigma ,\ell }^{\mathbf{T}}{\theta }_{\sigma ,\ell }\left({\widehat{\underline{\mathrm{y}}}}_{\sigma ,\ell }\right),\end{array}$$

(118)

where ${\widehat{\underline{\mathrm{y}}}}_{\sigma ,\ell }={\left[{\widehat{\underline{\mathrm{y}}}}_{\sigma ,1},\dots ,{\widehat{\underline{\mathrm{y}}}}_{\sigma ,\ell }\right]}^{\mathbf{T}}$ represents an accurate estimation of the state ${\underline{\mathrm{y}}}_{\sigma ,\ell }$ and

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\sigma ,\ell }^{★}=arg\underset{{\mathcal{Q}}_{\sigma ,\ell }\in {\nabla }_{\sigma }}{min}\left\{\underset{({\underline{\mathrm{y}}}_{\sigma ,\ell },\widehat{{\underline{\mathrm{y}}}_{\sigma ,\ell }})\in {U_1}_{\sigma ,1}\times {U_1}_{\sigma ,2}}{sup}|{\widehat{\mathcal{F}}}_{\sigma }(.)-{\mathcal{F}}_{\sigma }(.)|\right\},\end{array}$$

(119)

where ${\nabla }_{\sigma },{U_1}_{\sigma ,1}$, and ${U_1}_{\sigma ,2}$ represent bound compact domains for ${\mathcal{Q}}_{\sigma ,\ell },{\underline{\mathrm{y}}}_{\sigma ,\ell }$ and ${\widehat{\underline{\mathrm{y}}}}_{\sigma ,\ell }$, respectively. Furthermore, the fuzzy minimal estimation error ${\epsilon }_{\sigma ,\ell }$ is specified as

$$\begin{array}{c}\hfill {\mathcal{F}}_{\sigma ,\ell }\left({\underline{\mathrm{y}}}_{\sigma ,\ell }\right)={\widehat{\mathcal{F}}}_{\sigma ,\ell }\left(\widehat{{\underline{\mathrm{y}}}_{\sigma ,\ell }}|{\mathcal{Q}}_{\sigma ,\ell }^{★}\right)+{\epsilon }_{\sigma ,\ell }.\end{array}$$

(120)

Hypothesis 4. 

∃ an unidentified fixed number ${\epsilon }_{\sigma ,\ell }^{★}>0$ fulfilling that $|{\epsilon }_{\sigma ,\ell }|\le {\epsilon }_{\sigma ,\ell }^{★}>0.$ By inserting (120) into (102), then system (102) can be articulated in terms of fractional Brownian motion as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}d{\mathrm{y}}_{\sigma }^{\alpha }=\left({\mathcal{A}}_{\sigma ,0}{\mathrm{y}}_{\sigma }+{\mathbb{k}}_{\sigma }{\mathrm{w}}_{\sigma }+{{F_1}_{\sigma }}^{\mathbf{T}}{\vartheta }_{\sigma }+{\Delta }_{\sigma }\left(\overline{\mathrm{w}}\right)+{\epsilon }_{\sigma }\right)d{\tau }^{\alpha }+{\mathbb{G}}_{\sigma }\left(\overline{\mathrm{w}}\right)d{\mathrm{v}}_{\sigma },\hfill \\ {\mathrm{w}}_{\sigma }={\mathcal{W}}_{\sigma }^{\mathbf{T}}{\mathrm{y}}_{\sigma },\hfill \end{array}\right.\end{array}$$

(121)

where ${\mathcal{A}}_{\sigma }=\left(\begin{array}{cc}0& {\mathcal{I}}_{{\mathrm{n}}_{\sigma }-1}\\ 0& 0\end{array}\right),$ ${\Phi }_{\sigma }^{\mathbf{T}}=diag\left\{{\varkappa }_{\sigma ,1}^{\mathbf{T}},\dots ,{\varkappa }_{\sigma ,{\mathrm{n}}_{\sigma }}^{\mathbf{T}}\right\},$ ${\Delta }_{\sigma }^{\mathbf{T}}={\left[{\Delta }_{\sigma ,1}\left(\overline{\mathrm{w}}\right),\dots ,{\Delta }_{\sigma ,{\mathrm{n}}_{\sigma }}\left(\overline{\mathrm{w}}\right)\right]}^{\mathbf{T}},$ ${\mathbb{G}}_{\sigma }\left(\overline{\mathrm{w}}\right)={\left[{\mathcal{G}}_{\sigma ,1}\left(\overline{\mathrm{w}}\right),\dots ,{\mathcal{G}}_{\sigma ,{\mathrm{n}}_{\sigma }}\left(\overline{\mathrm{w}}\right)\right]}^{\mathbf{T}},$ ${\mathcal{Q}}_{\sigma }^{★}={\left[{\mathcal{Q}}_{\sigma ,1}^{★},\dots ,{\mathcal{Q}}_{\sigma ,{\mathrm{n}}_{\sigma }}^{★}\right]}^{\mathbf{T}},$ ${\mathcal{W}}_{\sigma }={\left[1,0,\dots ,\right]}^{\mathbf{T}},$ ${\epsilon }_{\sigma }={\left[{\epsilon }_{\sigma ,1},\dots ,{\epsilon }_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}^{\mathbf{T}},$ ${\vartheta }_{\sigma }={\left[{\wp }_{\sigma ,0}{\mathcal{Q}}_{\sigma }^{★\mathbf{T}}\right]}^{\mathbf{T}},$ ${F_1}_{\sigma }^{\mathbf{T}}=\left[\left(\begin{array}{c}{0}_{({\mathrm{n}}_{\sigma }-1)\times 1}\\ 1\end{array}\right){\gamma }_{\sigma }\left({\mathrm{w}}_{\sigma }\right){\mathrm{s}}_{\sigma },{\Phi }_{\sigma }^{\mathbf{T}}\right],$ ${\mathcal{A}}_{\sigma ,0}={\mathcal{A}}_{\sigma }-{\mathbb{k}}_{\sigma }{\mathcal{W}}_{\sigma }^{\mathbf{T}}$ and ${\mathbb{k}}_{\sigma }={\left[{\eta }_{\sigma ,1},\dots ,{\eta }_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}^{\mathbf{T}}.$

Assume that there is a vector ${\mathbb{k}}_{\sigma }$, and ${\mathcal{A}}_{\sigma ,0}$ represent a strict Hurwitz matrix. Then, for each positive-definite matrix ${S_1}_{\sigma }={S_1}_{\sigma }^{\mathbf{T}}>0,\exists$ a positive-definite matrix ${\mathrm{V}}_{\sigma }={\mathrm{V}}_{\sigma }^{\mathbf{T}}>0$ such that

$$\begin{array}{c}\hfill {\mathcal{A}}_{\sigma ,0}^{\mathbf{T}}{\mathrm{V}}_{\sigma }+{\mathrm{V}}_{\sigma }{\mathcal{A}}_{\sigma ,0}=-{S_1}_{\sigma }.\end{array}$$

(122)

To determine the unrestricted phases ${\mathrm{y}}_{\sigma ,2},{\mathrm{y}}_{\sigma ,3},\dots ,{\mathrm{y}}_{\sigma ,{\mathrm{n}}_{\sigma }}$ in systems (121) and (102), use the subsequent filtering techniques. Design the artificial-state approximation and control as

$$\begin{array}{ccc}\hfill & & {\widehat{\chi }}_{\sigma }={\xi }_{\sigma }+{\nabla }_{\sigma }^{\mathbf{T}}{\vartheta }_{\sigma }+{\hslash }_{\sigma },\hfill \\ & & {\mathrm{s}}_{\sigma }={\mathrm{s}}_{\sigma 1}+{\mathrm{s}}_{\sigma 2},\hfill \end{array}$$

(123)

where the principal controller, ${\mathrm{s}}_{\sigma 1}$ is created by the OB architecture approach. The compensating controller, ${\mathrm{s}}_{\sigma 2}$, is utilized to manage variability $\Delta {\overline{\gamma }}_{\sigma }({\mathrm{w}}_{\sigma },\tau ){\nabla }_{\sigma }^{\mathbf{T}}=[{\eta }_{\sigma },{\Theta }_{\sigma }].$ Determine the uncertainty filters as follows:

$$\begin{array}{ccc}\hfill & & d^{\alpha }{\xi }_{\sigma }=\left({\mathcal{A}}_{\sigma ,0}{\xi }_{\sigma }+{\mathbb{k}}_{\sigma }{\mathrm{w}}_{\sigma }\right)d{\tau }^{\alpha },\hfill \\ & & d^{\alpha }{\Theta }_{\sigma }=\left({\mathcal{A}}_{\sigma ,0}{\Theta }_{\sigma }+{\Phi }_{\sigma }^{\mathbf{T}}\right)d{\tau }^{\alpha },\hfill \\ & & d^{\alpha }{\eta }_{\sigma }=\left({\mathcal{A}}_{\sigma ,0}{\eta }_{\sigma }+{\mathbb{B}}_{\sigma }{\overline{\gamma }}_{\sigma }{\mathrm{s}}_{\sigma 1}\right)d{\tau }^{\alpha },\hfill \\ & & {\hslash }_{\sigma }={\mathcal{A}}_{\sigma ,0}{\hslash }_{\sigma }+{\mathbb{B}}_{\sigma }{\mathrm{s}}_{\sigma 2},\hfill \end{array}$$

(124)

where ${\xi }_{\sigma }={\left[{\xi }_{\sigma ,1},\dots ,{\xi }_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}^{\mathbf{T}},$ ${\eta }_{\sigma }={\left[{\eta }_{\sigma ,1},\dots ,{\eta }_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}^{\mathbf{T}},$ ${\hslash }_{\sigma }={\left[{\hslash }_{\sigma ,1},\dots ,{\hslash }_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}^{\mathbf{T}},$ ${\Theta }_{\sigma }={\left[{\Theta }_{\sigma ,1},\dots ,{\Theta }_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}_{{\mathrm{n}}_{\sigma }\times \sigma }^{\mathbf{T}},$ ${\mathbb{B}}_{\sigma }=\underset{\mathrm{n}}{\overset{\mathbf{T}}{\underbrace{\left[0,\dots ,0.1\right]}}}.$

Create the arbitrary observation error vector ${e_1}_{\sigma }$ as

$$\begin{array}{c}\hfill {e_1}_{\sigma }={\left[{e_1}_{\sigma ,1},\dots ,{e_1}_{\sigma ,{\mathrm{n}}_{\sigma }}\right]}^{\mathbf{T}}={\mathrm{y}}_{\sigma }-{\widehat{\chi }}_{\sigma }.\end{array}$$

(125)

The control system will employ the actual condition approximation ${\widehat{\mathrm{y}}}_{\sigma }$ outlined below:

$$\begin{array}{c}\hfill {\widehat{\mathrm{y}}}_{\sigma }={\xi }_{\sigma }+{\Theta }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }+{\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma }+\hslash \sigma ,\end{array}$$

(126)

where ${\widehat{\mathcal{Q}}}_{\sigma }$ and ${\widehat{\wp }}_{\sigma ,0}$ represent the approximations of ${\mathcal{Q}}_{\sigma }^{★}$ and ${\wp }_{\sigma ,0}$, respectively.

Considering (121), (124), and (125) and after disregarding small concerns, the observer error (${e_1}_{\sigma }$) can be approximated as

$$\begin{array}{c}\hfill d^{\alpha }{e_1}_{\sigma }=\left[{\mathcal{A}}_{\sigma ,0}{e_1}_{\sigma }+{\epsilon }_{\sigma }+{\mathbb{B}}_{\sigma }{\wp }_{\sigma ,0}\Delta {\overline{\gamma }}_{\sigma }{\mathrm{s}}_{\sigma 1}-{\mathbb{B}}_{\sigma }{\mathrm{s}}_{\sigma 2}+{\Delta }_{\sigma }\left(\overline{\mathrm{w}}\right)\right]d{\tau }^{\alpha }+{\mathbb{G}}_{\sigma }\left(\overline{\mathrm{w}}\right)d{\mathrm{v}}_{\sigma },\end{array}$$

(127)

Select the compensatory control ${\mathrm{s}}_{\sigma 2}$ as

$$\begin{array}{c}\hfill {\mathrm{s}}_{\sigma 2}=sgn\left({e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{B}}_{\sigma }\right){\overline{\wp }}_{\sigma ,0}{\delta }_{\sigma }\left|{\mathrm{s}}_{\sigma 1}\right|.\end{array}$$

(128)

In order to assess the characteristic corresponding to the uncertain-level filter (124), consider the subsequent theorem.

Theorem 4. 

For the given Lyapunov function for the observer error system (127),

$$\begin{array}{c}\hfill {\mathrm{U}}_{\sigma ,0}={e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{e_1}_{\sigma }.\end{array}$$

(129)

Ultimately, the infinitesimal generator of ${\mathrm{U}}_0$ is

$$\begin{array}{c}\hfill \mathfrak{L}{\mathrm{U}}_0\le \sum _{\sigma =1}^{\mathbb{N}}\left(-{\lambda }_{\sigma ,0}{\parallel {e_1}_{\sigma }\parallel }^2+{\wp }_{\sigma ,0}{\mathrm{w}}_{\sigma }^4+{\upsilon }_{\sigma ,0}{\mathrm{w}}_{\sigma }^4+{Y_1}_{\sigma ,0}\right),\end{array}$$

(130)

where ${\lambda }_{\sigma ,0}={\eta }_{min}\left({O_1}_{\sigma }\right)-2{\parallel \mathrm{V}\parallel }^2$ and ${Y_1}_{\sigma ,0}=4{\mathrm{n}}_{\sigma }^2\left({\sum }_{\mathrm{l}=1}^{\mathcal{N}}{\left({\Xi }_{\mathrm{l}}\left(0\right)\right)}^4\right)+4{\mathrm{n}}_{\sigma }^2\left({\sum }_{\mathrm{l}=1}^{\mathcal{N}}{\left(\overline{{\delta }_{\mathrm{l}}}\left(0\right)\right)}^4\right)+\parallel {\epsilon }_{\sigma }^{★}{\parallel }^2+{\displaystyle \frac{1}{2}}{\parallel \mathrm{V}\parallel }^2+{\displaystyle \frac{1}{2}}.$

Proof. 

By means of (122), (126), (127), and (130), we obtain

$$\begin{array}{c}\hfill \mathfrak{L}{\mathrm{U}}_0\le \sum _{\sigma =1}^{\mathcal{N}}\left\{-{\eta }_{min}\left({O_1}_{\sigma }\right){\parallel {e_1}_{\sigma }\parallel }^2+2{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }\left({\epsilon }_{\sigma }+{\mathbb{B}}_{\sigma }{\wp }_{\sigma ,0}\Delta {\overline{\gamma }}_{\sigma }{\mathrm{s}}_{\sigma 1}-{\mathbb{B}}_{\sigma }{\mathrm{s}}_{\sigma 2}+{\Delta }_{\sigma }\left(\overline{\mathrm{w}}\right)\right)+trac\left({\mathbb{G}}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{G}}_{\sigma }\right)\right\}.\end{array}$$

(131)

Making use of Hypothesis (1), we have

$$\begin{array}{ccc}\hfill & & \sum _{\sigma =1}^{\mathcal{N}}2{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\Delta }_{\sigma }\left(\overline{\mathrm{w}}\right)\le \sum _{\sigma =1}^{\mathcal{N}}\left(\parallel {\mathrm{V}}_{\sigma }{\parallel }^2{\parallel {e_1}_{\sigma }\parallel }^2+{\displaystyle \frac{{\mathrm{n}}_{\sigma }}{2}}\sum _{\ell =1}^{{\mathrm{n}}_{\sigma }}{\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{q_1}_{\sigma ,\mathrm{l}}{\Xi }_{\mathrm{l}}\left(\right|{\mathrm{w}}_{\mathrm{l}}\left|\right)\right)}^4+{\displaystyle \frac{1}{2}}\right),\hfill \\ & & \sum _{\sigma =1}^{\mathcal{N}}trac\left({\mathbb{G}}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{G}}_{\sigma }\right)\le \sum _{\sigma =1}^{\mathcal{N}}\left({\displaystyle \frac{1}{2}}\parallel {\mathrm{V}}_{\sigma }{\parallel }^2{\parallel {e_1}_{\sigma }\parallel }^2+{\displaystyle \frac{{\mathrm{n}}_{\sigma }}{2}}\sum _{\ell =1}^{{\mathrm{n}}_{\sigma }}{\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{\lambda }_{\sigma ,\mathrm{l}}\left|{\overline{\delta }}_{\mathrm{l}}{\mathrm{w}}_{\mathrm{l}}\right|\right)}^4\right),\hfill \\ & & \sum _{\sigma =1}^{\mathcal{N}}2{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\epsilon }_{\sigma }\le \sum _{\sigma =1}^{\mathcal{N}}\parallel {\mathrm{V}}_{\sigma }{\parallel }^2\parallel {e_1}_{\sigma }{\parallel }^2+\sum _{\sigma =1}^{\mathcal{N}}{\parallel {\epsilon }_{\sigma }^{★}\parallel }^2,\hfill \end{array}$$

(132)

where ${\epsilon }_{\sigma }^{★}={\left[{\epsilon }_{\sigma ,1}^{★},\dots ,{\epsilon }_{\sigma ,{\mathrm{n}}_{\sigma }}^{★}\right]}^{\mathbf{T}}.$ Under Hypothesis (2) and (128), we have

$$\begin{array}{ccc}\hfill & & {e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{B}}_{\sigma }\left({\wp }_{\sigma ,0}\Delta {\overline{\gamma }}_{\sigma }{\mathrm{s}}_{\sigma 1}-{\mathrm{s}}_{\sigma 2}\right)\hfill \\ & & \le |{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{B}}_{\sigma }\left|\right|{\wp }_{\sigma ,0}\left|\right|\Delta {\overline{\gamma }}_{\sigma }\left|\right|{\mathrm{s}}_{\sigma 1}|-{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{B}}_{\sigma }{\mathrm{s}}_{\sigma 2}\hfill \\ & & \le \left|{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{B}}_{\sigma }\right|\left(|{\wp }_{\sigma ,0}\left|\right|\Delta {\overline{\gamma }}_{\sigma }\left|\right|{\mathrm{s}}_{\sigma 1}|-{\overline{\wp }}_{\sigma ,0}{\delta }_{\sigma }\left|{\mathrm{s}}_{\sigma 1}\right|\right)\le 0.\hfill \end{array}$$

(133)

As stated by [38], if ${\Xi }_{\sigma }\left(\right|{\mathrm{w}}_{\mathrm{l}}\left|\right)\ge 0$ and ${\overline{\delta }}_{\mathrm{l}}\left(\right|{\mathrm{w}}_{\mathrm{l}}| \ge 0)$, for $\mathrm{l}=1,\dots ,\mathcal{N}$, ∃ smooth positive mappings ${\eta }_{\sigma ,\mathrm{l}}^{\left(1\right)}\left({\mathrm{w}}_{\mathrm{l}}\right)$ and ${\eta }_{\sigma ,\mathrm{l}}^{\left(2\right)}\left({\mathrm{w}}_{\mathrm{l}}\right)$ satisfy

$$\begin{array}{ccc}\hfill & & {\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{q_1}_{\sigma ,\mathrm{l}}{\Xi }_{\mathrm{l}}\left(\right|{\mathrm{w}}_{\mathrm{l}}\left|\right)\right)}^4\le \sum _{\mathrm{l}=1}^{\mathcal{N}}{\eta }_{\sigma ,\mathrm{l}}^{\left(1\right)}\left({\mathrm{w}}_{\mathrm{l}}\right){\mathrm{w}}_{\mathrm{l}}^4+8{\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{\Xi }_{\mathrm{l}}\left(0\right)\right)}^4,\hfill \\ & & {\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{\lambda }_{\sigma ,\mathrm{l}}{\overline{\delta }}_{\mathrm{l}}\left(\right|{\mathrm{w}}_{\mathrm{l}}\left|\right)\right)}^4\le \sum _{\mathrm{l}=1}^{\mathcal{N}}{\eta }_{\sigma ,\mathrm{l}}^{\left(2\right)}\left({\mathrm{w}}_{\mathrm{l}}\right){\mathrm{w}}_{\mathrm{l}}^4+8{\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{\overline{\delta }}_{\mathrm{l}}\left(0\right)\right)}^4.\hfill \end{array}$$

(134)

It follows that

$$\begin{array}{ccc}\hfill & & \sum _{\sigma =1}^{\mathcal{N}}2{e_1}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\Delta }_{\sigma }\left(\overline{\mathrm{w}}\right)\le \sum _{\sigma =1}^{\mathcal{N}}\left(\parallel {\mathrm{V}}_{\sigma }{\parallel }^2{\parallel {e_1}_{\sigma }\parallel }^2+{\wp }_{\sigma ,0}{\mathrm{w}}^4+4{\mathrm{n}}_{\sigma }^2{\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{\Xi }_{\mathrm{l}}\left(0\right)\right)}^4+{\displaystyle \frac{1}{2}}\right),\hfill \\ & & \sum _{\sigma =1}^{\mathcal{N}}trac\left({\mathbb{G}}_{\sigma }^{\mathbf{T}}{\mathrm{V}}_{\sigma }{\mathbb{G}}_{\sigma }\right)\le \sum _{\sigma =1}^{\mathcal{N}}\left({\displaystyle \frac{1}{2}}{\parallel {\mathrm{V}}_{\sigma }\parallel }^2+\sum _{\sigma =1}^{\mathcal{N}}{\upsilon }_{\sigma ,0}{\mathrm{w}}_{\sigma }^4+4{\mathrm{n}}_{\sigma }^2{\left(\sum _{\mathrm{l}=1}^{\mathcal{N}}{\overline{\delta }}_{\mathrm{l}}\left(0\right)\right)}^4\right),\hfill \end{array}$$

(135)

where ${\wp }_{\sigma ,0}={\displaystyle \frac{{\mathrm{n}}_{\sigma }^2}{2}}4{\mathrm{n}}_{\sigma }^2{\displaystyle \sum _{\mathrm{l}=1}^{\mathcal{N}}}{\eta }_{\mathrm{l},\sigma }^{\left(1\right)}$ and ${\upsilon }_{\sigma ,0}={\displaystyle \sum _{\mathrm{l}=1}^{\mathcal{N}}}{\mathrm{n}}_{\sigma }^2\left(\parallel {\mathrm{V}}_{\sigma }{\parallel }^2+{\displaystyle \frac{{\eta }_{max}^2}{2}}\left({\mathrm{V}}_{\sigma }\right){\eta }_{\mathrm{l},\sigma }^{\left(2\right)}\right)$. replacing (132), (133), and (135) in (131) produces the desired outcome (130). □

6.2. Configuration of Fractional Stochastic Adaptive-Uncertain Control

Describe a specific coordinate modification:

$$\begin{array}{ccc}& & {\mathrm{x}}_{\sigma ,1}={\mathrm{w}}_{\sigma },\hfill \\ & & {\mathrm{x}}_{\sigma ,\ell }={\eta }_{\sigma ,\ell }-{\Psi }_{\sigma ,\ell -1},\sigma =1,\dots ,\mathcal{N},\ell =2,\dots ,{\mathrm{n}}_{\sigma }.\hfill \end{array}$$

(136)

Select the tuning and intermediate control mechanisms outlined below:

$$\begin{array}{ccc}\hfill & & {\Psi }_{\sigma ,1}={}^c\mathrm{D}^{\alpha }\Omega \left({\eta }_{\sigma }\right)\left({\tilde{\Xi }}_{\sigma ,1}{\mathrm{x}}_{\sigma ,1}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }+{\displaystyle \frac{9}{4}}{\mathrm{x}}_{\sigma ,1}+{\wp }_{\sigma ,0}{\mathrm{w}}_{\sigma }+{\mathrm{n}}_{\sigma }{\wp }_{\sigma ,1}{\mathrm{w}}_{\sigma }+{\upsilon }_{\sigma ,0}{\mathrm{w}}_{\sigma }+{\mathrm{n}}_{\sigma }{\upsilon }_{\sigma ,1}{\mathrm{w}}_{\sigma }+{\mathrm{x}}_{\sigma ,1}^3\right),\hfill \\ & & {\Psi }_{\sigma ,2}=-{\displaystyle \frac{3}{4}}{\mathrm{x}}_{\sigma ,2}-{\tilde{\Xi }}_{\sigma ,2}{\mathrm{x}}_{\sigma ,2}-{ϝ}_{\sigma ,2}+{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{w}}_{\sigma }\right)\left({\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma ,2}+{\xi }_{\sigma ,2}+{\hslash }_{\sigma ,2}+{\mathrm{v}}_{\sigma }\widehat{{\mathcal{Q}}_{\sigma }}\right)-{\varkappa }_{\sigma ,1,2}{\mathrm{x}}_{\sigma ,2}^3,\hfill \\ & & {\Psi }_{\sigma ,\ell }=-{\displaystyle \frac{3}{4}}{\mathrm{x}}_{\sigma ,\ell }-{\tilde{\Xi }}_{\sigma ,\ell }{\mathrm{x}}_{\sigma ,\ell }-{ϝ}_{\sigma ,\ell }+{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\mathrm{w}}_{\sigma }\right)\left({\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma ,1}+{\xi }_{\sigma ,2}+{\hslash }_{\sigma ,2}+{\mathrm{v}}_{\sigma }\widehat{{\mathcal{Q}}_{\sigma }}\right)\hfill \\ & & -\sum _{\kappa =2}^{\ell }{\varkappa }_{\sigma ,1,\kappa }+{\mathcal{A}}_{\sigma ,1,\kappa }{\mathrm{x}}_{\sigma ,\kappa }^3,\hfill \\ & & {\varpi }_{\sigma ,1}={\mathrm{v}}_{\sigma }^{\mathbf{T}}{\mathrm{x}}_{\sigma ,1}^3,{\varpi }_{\sigma ,\ell }={\varpi }_{\sigma ,\ell }={\varpi }_{\sigma ,\ell -1}-{\mathrm{x}}_{\sigma ,\ell }^3{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\mathrm{w}}_{\sigma }\right){\eta }_{\sigma ,2},(\ell =3,\dots ,{\mathrm{n}}_{\sigma }),\hfill \end{array}$$

(137)

where ${\upsilon }_{\sigma }$ represents a layout factor; $\Omega \left({\eta }_{\sigma }\right)=exp\left({\eta }_{\sigma }^2\right)cos\left({\eta }_{\sigma }^2\right),$ ${\mathrm{v}}_{\sigma }=\left[{\varkappa }_{\sigma ,1}^{\mathbf{T}},0,\dots ,0\right]+{\Theta }_{\sigma ,2},$ ${\wp }_{\sigma ,1}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{\mathrm{l}=1}^{\mathcal{N}}}{\eta }_{\mathrm{l},\sigma }^{\left(1\right)},{\upsilon }_{\sigma ,1}={\displaystyle \frac{3}{4}}{\displaystyle \sum _{\mathrm{l}=1}^{\mathcal{N}}}{\eta }_{\mathrm{l},\sigma }^{\left(2\right)},$ ${\displaystyle \sum _{\ell =2}^{{\mathrm{n}}_{\sigma }}}{\varkappa }_{\sigma ,1,\ell }{\mathrm{x}}_{\sigma ,\ell }^3$ $=-\left({}^c\mathrm{D}^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma }-{\Gamma }_{\sigma }{\varpi }_{\sigma ,1}+{\Gamma }_{\sigma }{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{V}}_{\sigma }\right),{\varkappa }_{\sigma ,1,2}$ $={}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathcal{Q}}_{\sigma }\right)\times {\Gamma }_{\sigma }{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{w}}_{\sigma }\right){\mathrm{v}}_{\sigma }^{\mathbf{T}},$ ${\displaystyle \sum _{\kappa =\ell }^{{\mathrm{n}}_{\sigma }}}{\mathcal{A}}_{\sigma ,1,\kappa }{\mathrm{x}}_{\sigma ,\kappa }^3$ $=-\left({}^c\mathrm{D}^{\alpha }{\widehat{\wp }}_{\sigma ,0}-{\mathrm{z}}_{\sigma ,1}{v_1}_{\sigma ,\ell -1}+{\mathrm{z}}_{\sigma ,1}{\widehat{\wp }}_{\sigma ,0}\right){}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\wp }_{\sigma ,0}\right)$ and ${\mathcal{A}}_{\sigma ,1,\kappa }={}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\kappa -1}\left({\widehat{\wp }}_{\sigma ,0}\right){\mathrm{z}}_{\sigma ,1}{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}{\mathrm{w}}_{\sigma }.$ Moreover, we have

$$\begin{array}{ccc}\hfill & & {}^c\mathrm{D}^{\alpha }{\eta }_{\sigma }={\mathrm{d}}_{\sigma }^{-1}{\mathrm{x}}_{\sigma ,1}^3\left({\tilde{\Xi }}_{\sigma ,1}{\mathrm{x}}_{\sigma ,1}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }+{\displaystyle \frac{9}{4}}{\mathrm{x}}_{\sigma ,1}+{\wp }_{\sigma ,0}{\mathrm{w}}_{\sigma }+{\mathrm{n}}_{\sigma }{\wp }_{\sigma ,1}{\mathrm{w}}_{\sigma }+{\upsilon }_{\sigma ,0}{\mathrm{w}}_{\sigma }+{\mathrm{n}}_{\sigma }{\upsilon }_{\sigma ,1}{\mathrm{w}}_{\sigma }+{\mathrm{x}}_{\sigma ,1}^3\right),\hfill \\ & & F_{\sigma ,2}=-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\widehat{\mathcal{Q}}}_{\sigma }\right){\Gamma }_{\sigma }({\varpi }_{\sigma ,1}-{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma })-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\xi }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\xi }_{\sigma }-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\Theta }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\Theta }_{\sigma }-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\hslash }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\hslash }_{\sigma }\hfill \\ & & -{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\varsigma }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\varsigma }_{\sigma }-{\kappa }_{\sigma ,2}{\eta }_{\sigma ,1}+{\displaystyle \frac{3}{2}}{\left({}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{w}}_{\sigma }\right)\right)}^{4/3}{\mathrm{x}}_{\sigma ,2}+{\displaystyle \frac{1}{4}}{\mathrm{x}}_{\sigma ,2}+{\displaystyle \frac{9}{4}}{\mathrm{x}}_{\sigma ,2}{\left({}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{w}}_{\sigma }\right)\right)}^4\hfill \\ & & +{\displaystyle \frac{3}{4}}{\mathrm{x}}_{\sigma ,2}^3{\left({}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{w}}_{\sigma }\right)\right)}^2-\sum _{\ell =1}^{{\mathrm{n}}_{\sigma }-1}{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\eta }_{\ell }\right)\left({\eta }_{\sigma ,\ell +1}-{\kappa }_{\sigma ,\ell }{\eta }_{\sigma ,1}\right),\hfill \\ & & F_{\sigma ,\ell }=-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\widehat{\mathcal{Q}}}_{\sigma }\right){\Gamma }_{\sigma }({\varpi }_{\sigma ,\ell -1}-{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma })-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\widehat{\wp }}_{\sigma ,0}\right){\mathrm{z}}_{\sigma ,1}\left({v_1}_{\sigma ,\ell -1}-{\upsilon }_{\sigma }{\widehat{\wp }}_{\sigma ,0}\right)\hfill \\ & & -{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -11}\left({\xi }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\xi }_{\sigma }-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\Theta }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\Theta }_{\sigma }\hfill \\ & & -{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\hslash }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\hslash }_{\sigma }-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\varsigma }_{\sigma }\right){}^c\mathrm{D}^{\alpha }{\varsigma }_{\sigma }-{\kappa }_{\sigma ,\ell }{\eta }_{\sigma ,1}+{\displaystyle \frac{3}{2}}{\left({}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\mathrm{w}}_{\sigma }\right)\right)}^{4/3}{\mathrm{x}}_{\sigma ,\ell }\hfill \\ & & +{\displaystyle \frac{1}{4}}{\mathrm{x}}_{\sigma ,\ell }+{\displaystyle \frac{9}{4}}{\mathrm{x}}_{\sigma ,\ell }{\left({}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\mathrm{w}}_{\sigma }\right)\right)}^4+{\displaystyle \frac{3}{4}}{\mathrm{x}}_{\sigma ,\ell }^3{\left({}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\mathrm{w}}_{\sigma }\right)\right)}^2\hfill \\ & & -\sum _{\mathrm{q}=1}^{{\mathrm{n}}_{\sigma }-1}{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\eta }_{\sigma }\right)\left({\eta }_{\sigma ,\mathrm{q}+1}-{\kappa }_{\sigma ,\mathrm{q}}{\eta }_{\sigma ,1}\right).\hfill \end{array}$$

(138)

Select the primary controller $\left({\mathrm{s}}_{\sigma 1}\right)$ and adaptability principles ${\widehat{\mathcal{Q}}}_{\sigma }$ and ${\widehat{\wp }}_{\sigma ,0}$ as follows:

$$\begin{array}{ccc}\hfill {\mathrm{s}}_{\sigma 1}& & ={\displaystyle \frac{1}{{\overline{\gamma }}_{\sigma }\left({\mathrm{w}}_{\sigma }\right)}}\{-{\displaystyle \frac{1}{4}}{\mathrm{x}}_{\sigma ,{\mathrm{n}}_{\sigma }}-{\tilde{\Xi }}_{\sigma ,{\mathrm{n}}_{\sigma }}{\mathrm{x}}_{\sigma ,{\mathrm{n}}_{\sigma }}-{ϝ}_{\sigma ,{\mathrm{n}}_{\sigma }}+{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left({\mathrm{w}}_{\sigma }\right)\left({\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma ,2}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)\hfill \\ & & -\sum _{\kappa =2}^{{\mathrm{n}}_{\sigma }}\left({\varkappa }_{\sigma ,1,\kappa }+{\mathcal{A}}_{\sigma ,1,\kappa }\right){\mathrm{x}}_{\sigma ,\kappa }^3\},\hfill \end{array}$$

(139)

$$\begin{array}{ccc}\hfill & & d^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma }=\left({\Gamma }_{\sigma }({\varpi }_{\sigma ,{\mathrm{n}}_{\sigma }}-{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma })\right)d{\tau }^{\alpha },\hfill \\ & & d^{\alpha }{\widehat{\wp }}_{\sigma ,0}=\left({\mathrm{z}}_{\sigma ,1}({v_1}_{{\sigma ,\mathrm{n}}_{\sigma }}-{\upsilon }_{\sigma }{\widehat{\wp }}_{\sigma ,0})\right)d{\tau }^{\alpha }.\hfill \end{array}$$

(140)

The upcoming findings show that the closed-loop structures are stable and have excellent oversight effectiveness.

Theorem 5. 

In a probabilistic nonlinear large-scale system (102), a controller (123), filtration (124) containing intermediary control procedures (137), and setting adaption laws (140) ensure that each signal within a closed-loop framework is bounded in probability.

Proof. 

Assume that the Lyapunov function can be presented as

$$\begin{array}{c}\hfill \mathrm{U}={\mathrm{U}}_0+\sum _{\sigma =1}^{\mathcal{N}}\sum _{\ell =1}^{{\mathrm{n}}_{\sigma }}\left({\displaystyle \frac{1}{4}}{\mathrm{x}}_{\sigma ,\ell }^4+{\displaystyle \frac{1}{2}}{\tilde{\mathcal{Q}}}_{\sigma }^{\mathbf{T}}{\Gamma }_{\sigma }^{-1}{\tilde{\mathcal{Q}}}_{\sigma }+{\displaystyle \frac{1}{2{\mathrm{z}}_{\sigma ,1}}}{\tilde{\wp }}_{\sigma ,0}^2\right).\end{array}$$

(141)

In view of (130), the infinitesimal generator of $\mathrm{U}$ is

$$\begin{array}{ccc}\hfill \mathfrak{L}\mathrm{U}& & =\mathcal{L}{\mathrm{U}}_0+\sum _{\sigma =1}^{{\mathrm{n}}_{\sigma }}\left(\sum _{\ell =1}^{{\mathrm{n}}_{\sigma }}{\mathrm{x}}_{\sigma ,\ell }^{3}d{\mathrm{x}}_{\sigma ,\ell }-{\tilde{\mathcal{Q}}}_{\sigma }^{\mathbf{T}}{\Gamma }_{\sigma }^{-1}{}^c\mathrm{D}^{\alpha }{\dot{\mathcal{Q}}}_{\sigma }-{\displaystyle \frac{1}{{\mathrm{z}}_{\sigma ,1}}}{\tilde{\wp }}_{\sigma ,0}{}^c\mathrm{D}^{\alpha }{\widehat{\wp }}_{\sigma ,0}\right)\hfill \\ & & \le \mathcal{L}{\mathrm{U}}_0+\sum _{\sigma =1}^{\mathcal{N}}\{{\mathrm{x}}_{\sigma ,1}^3\left({\wp }_{\sigma ,0}{\eta }_{\sigma ,2}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\mathcal{Q}}_{\sigma }^{★}+{\Delta }_{\sigma ,1}+{\epsilon }_{\sigma ,1}+{e_1}_{\sigma ,2}\right)+{\displaystyle \frac{3}{2}}{\mathrm{x}}_{\sigma ,1}^2{\mathcal{G}}_{\sigma ,1}^2-{\tilde{\mathcal{Q}}}_{\sigma }^{\mathbf{T}}{\Gamma }_{\sigma }^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma }\hfill \\ & & +{\tilde{\mathcal{Q}}}_{\sigma }^{\mathbf{T}}({\varpi }_{\sigma ,2}-{\Gamma }_{\sigma }^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma })\}+{\mathrm{x}}_{\sigma ,2}^3\{{\mathrm{x}}_{\sigma ,3}+{\Psi }_{\sigma ,2}+{ϝ}_{\sigma ,2}-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left({\mathrm{w}}_{\sigma }\right)\left({\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma ,2}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)\hfill \\ & & -{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,1}\left(\widehat{{\mathcal{Q}}_{\sigma }}\right)\left({}^c\mathrm{D}^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma }-{\Gamma }_{\sigma }{\varpi }_{\sigma ,1}+{\Gamma }_{\sigma }{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)\}-{\tilde{\wp }}_{\sigma ,0}\left({\mathrm{z}}_{\sigma ,1}^{-1}{}^c\mathrm{D}^{\alpha }{\widehat{\wp }}_{\sigma ,0}-{v_1}_{\sigma ,2}\right)\hfill \\ & & +\sum _{\ell =3}^{{\mathrm{n}}_{\sigma }-1}\left\{{\mathrm{x}}_{\sigma ,\ell }^3\right({\mathrm{x}}_{\sigma ,\ell +1}+{\Psi }_{\sigma ,\ell }+{ϝ}_{\sigma ,\ell }-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\mathrm{w}}_{\sigma }\right)\left({\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma ,1}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)\hfill \\ & & -{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\widehat{\mathcal{Q}}}_{\sigma }\right)\left({}^c\mathrm{D}^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma }-{\Gamma }_{\sigma }{\varpi }_{\sigma ,\ell -1}+{\Gamma }_{\sigma }{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,\ell -1}\left({\widehat{\wp }}_{\sigma ,0}\right)({}^c\mathrm{D}^{\alpha }{\widehat{\wp }}_{\sigma ,0}-{\mathrm{z}}_{\sigma ,1}{v_1}_{\sigma ,\ell -1}\hfill \\ & & +{\mathrm{z}}_{\sigma ,1}{\upsilon }_{\sigma }{\widehat{\wp }}_{\sigma ,0}\left)\right)\}+{\mathrm{x}}_{\sigma ,{\mathrm{n}}_1}\{{\mathrm{s}}_{\sigma }+{ϝ}_{{\sigma ,\mathrm{n}}_{\sigma }}-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left({\mathrm{w}}_{\sigma }\right)\left({\widehat{\wp }}_{\sigma ,0}{\eta }_{\sigma ,1}+{\xi }_{\sigma ,2}+{\mathrm{v}}_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)\hfill \\ & & -{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left({}^c\mathrm{D}^{\alpha }{\widehat{\mathcal{Q}}}_{\sigma }-{\Gamma }_{\sigma }{\varpi }_{\sigma ,{\mathrm{n}}_{\sigma }-1}+{\Gamma }_{\sigma }{\upsilon }_{\sigma }{\widehat{\mathcal{Q}}}_{\sigma }\right)-{}^c\mathrm{D}^{\alpha }{\Psi }_{\sigma ,{\mathrm{n}}_{\sigma }-1}\left({\widehat{\wp }}_{\sigma ,0}\right)({}^c\mathrm{D}^{\alpha }{\widehat{\wp }}_{\sigma ,0}-{\mathrm{z}}_{\sigma ,1}{v_1}_{\sigma ,{\mathrm{n}}_{\sigma }-1}\hfill \\ & & +{\mathrm{z}}_{\sigma ,1}{\upsilon }_{\sigma }{\widehat{\wp }}_{\sigma ,0}\left)\right\},\hfill \end{array}$$

(142)

where ${\lambda }_{\sigma ,{\mathrm{n}}_{\sigma }}={\lambda }_{\sigma ,0}-{\displaystyle \frac{{\mathrm{n}}_{\sigma }}{2}},$ ${Y_1}_{\sigma ,{\mathrm{n}}_{\sigma }}={Y_1}_{\sigma ,0}+{\mathrm{n}}_{\sigma }\left\{{\displaystyle \frac{1}{4}}{\epsilon }_{\sigma ,1}^{★4}+2{\left({\sum }_{\mathrm{l}=1}^{\mathcal{N}}{\Xi }_{\mathrm{l}}\left(0\right)\right)}^4\right\}$ $+8{\left({\sum }_{\mathrm{l}=1}^{\mathcal{N}}{\overline{\delta }}_{\mathrm{l}}\left(0\right)\right)}^4+{\displaystyle \frac{{\wp }_{\sigma ,0}^4}{2}},$ ${\mathcal{W}}_1=\underset{\ell \in [1,{\mathrm{n}}_{\sigma }],\sigma \le \Omega }{min}\left\{4{\tilde{\Xi }}_{\sigma ,\ell },{\upsilon }_{\sigma }/{\eta }_{max}\left({\Gamma }_{\sigma }^{-1}\right),{\upsilon }_{\sigma }{\varpi }_{\sigma ,1}2{\lambda }_{\sigma ,{\mathrm{n}}_{\sigma }}/{\eta }_{min}\left({\mathrm{V}}_{\sigma }\right)\right\}$ and ${\mathcal{W}}_2={\sum }_{\sigma =1}^{\Omega }\left({Y_1}_{\sigma ,{\mathrm{n}}_{\sigma }}+{\displaystyle \frac{1}{2}}{\upsilon }_{\sigma }{\parallel {\mathcal{Q}}_{\sigma }^{★}|}^2+{\displaystyle \frac{{\upsilon }_{\sigma }}{2}}{\wp }_{\sigma ,0}^2\right).$

Theorem 3 and related references [48,52] demonstrate the fact that the closed-loop system’s outputs are bounded in probability. □

6.3. Illustrative Example

In order to verify the suggested solution, we investigate the stabilization problem of tripled reversed pendulums involving fractional Brownian motion as well as unidentified control patterns [71]. The mechanisms can be detailed below:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{d}^{\alpha }{\mathrm{y}}_{1,1}={\mathrm{y}}_{1,2}d{\tau }^{\alpha },\hfill \\ d^{\alpha }{\mathrm{y}}_{1,2}=\left({\mathcal{F}}_{1,2}\left({\underline{\mathrm{y}}}_{1,2}\right)+{\wp }_{1,0}{\gamma }_1\left({\mathrm{w}}_1\right){\mathrm{s}}_1+{\Delta }_{1,2}({\mathrm{w}}_1,{\mathrm{w}}_2,{\mathrm{w}}_3)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{12}d{\mathrm{v}}_1,\hfill \\ {\mathrm{w}}_1={\mathrm{y}}_{1,1},\hfill \end{array}\right.\end{array}$$

(143)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{d}^{\alpha }{\mathrm{y}}_{2,1}={\mathrm{y}}_{2,2}d{\tau }^{\alpha },\hfill \\ d^{\alpha }{\mathrm{y}}_{2,2}=\left({\mathcal{F}}_{2,2}\left({\underline{\mathrm{y}}}_{2,2}\right)+{\wp }_{2,0}{\gamma }_2\left({\mathrm{w}}_2\right){\mathrm{s}}_2+{\Delta }_{2,2}({\mathrm{w}}_1,{\mathrm{w}}_2,{\mathrm{w}}_3)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{22}d{\mathrm{v}}_2,\hfill \\ {\mathrm{w}}_2={\mathrm{y}}_{2,1},\hfill \end{array}\right.\end{array}$$

(144)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{d}^{\alpha }{\mathrm{y}}_{3,1}={\mathrm{y}}_{3,2}d{\tau }^{\alpha },\hfill \\ d^{\alpha }{\mathrm{y}}_{3,2}=\left({\mathcal{F}}_{3,2}\left({\underline{\mathrm{y}}}_{3,2}\right)+{\wp }_{3,0}{\gamma }_3\left({\mathrm{w}}_3\right){\mathrm{s}}_3+{\Delta }_{3,2}({\mathrm{w}}_1,{\mathrm{w}}_2,{\mathrm{w}}_3)\right)d{\tau }^{\alpha }+{\mathcal{G}}_{32}d{\mathrm{v}}_3,\hfill \\ {\mathrm{w}}_3={\mathrm{y}}_{3,1},\hfill \end{array}\right.\end{array}$$

(145)

where ${\mathcal{F}}_{1,2}={\displaystyle \frac{\mathcal{G}}{\mathrm{l}}}sin{\mathrm{y}}_{1,1},$ ${\mathcal{F}}_{2,2}={\displaystyle \frac{\mathcal{G}}{\mathrm{l}}}sin{\mathrm{y}}_{2,1},$ ${\mathcal{F}}_{3,2}={\displaystyle \frac{\mathcal{G}}{\mathrm{l}}}sin{\mathrm{y}}_{3,1},$ ${\Delta }_{1,2}=-{\displaystyle \frac{\left({\kappa }_1{a_1}^2\right)}{\mathrm{q}{\mathrm{l}}^2}}sin{\mathrm{w}}_{1}cos{\mathrm{w}}_1$ $+{\displaystyle \frac{\left({\kappa }_1{a_1}^2\right)}{\mathrm{q}{\mathrm{l}}^2}}\times sin{\mathrm{w}}_{2}cos{\mathrm{w}}_2,$ ${\Delta }_{2,2}=-{\displaystyle \frac{\left({\kappa }_1{a_1}^2\right)}{m_2{\mathrm{l}}^2}}sin{\mathrm{w}}_{1}cos{\mathrm{w}}_1$ $-\left\{{\displaystyle \frac{\left({\kappa }_1{a_1}^2\right)}{\mathrm{q}{\mathrm{l}}^2}}\right\}$ $+{\displaystyle \frac{\left({\kappa }_2{a_1}^2\right)}{m_2{\mathrm{l}}^2}}sin{\mathrm{w}}_{2}cos{\mathrm{w}}_2+{\displaystyle \frac{\left({\kappa }_2{a_1}^2\right)}{m_2{\mathrm{l}}^2}}sin{\mathrm{w}}_{3}cos{\mathrm{w}}_3,$ ${\Delta }_{3,2}=-{\displaystyle \frac{\left({\kappa }_2{a_1}^2\right)}{m_2{\mathrm{l}}^2}}sin{\mathrm{w}}_{2}cos{\mathrm{w}}_2-{\displaystyle \frac{\left({\kappa }_2{a_1}^2\right)}{m_3{\mathrm{l}}^2}}sin{\mathrm{w}}_{3}cos{\mathrm{w}}_3,$ ${\mathcal{G}}_{1,2}\left(\overline{\mathrm{w}}\right)={\mathrm{w}}^{2}cos{\mathrm{w}}_2,$ ${\mathcal{G}}_{2,2}\left(\overline{\mathrm{w}}\right)={\displaystyle \frac{{\mathrm{w}}_1^1{\mathrm{w}}_2^{2}sin\left({\mathrm{w}}_2\right)}{1+{\mathrm{w}}_1^1{\mathrm{w}}_2^2}},$ ${\mathcal{G}}_{3,2}\left(\overline{\mathrm{w}}\right)={\displaystyle \frac{{\mathrm{w}}_{3}exp(-{\mathrm{w}}_1^2)}{1+{\mathrm{w}}_3^2}},$ ${\gamma }_1\left({\mathrm{w}}_1\right)=1-0.2sin\left({\mathrm{w}}_1\right),$ ${\gamma }_2\left({\mathrm{w}}_2\right)=2+0.3cos\left({\mathrm{w}}_2\right),$ ${\gamma }_3\left({\mathrm{w}}_3\right)=1+0.2sin\left({\mathrm{w}}_3\right),$ $\Delta {\overline{\gamma }}_1=-0.1sin\left({\mathrm{w}}_1\right),$ $\Delta {\overline{\gamma }}_2=0.2cos\left({\mathrm{w}}_2\right),$ $\Delta {\overline{\gamma }}_3=-0.2sin\left({\mathrm{w}}_3\right),{\delta }_1=0.11,$ ${\delta }_2=0.22,$ ${\delta }_3=0.3,{\wp }_{1,0}=1,$ ${\wp }_{2,0}=-1$, and ${\wp }_{3,0}=1.$ Here, $d{\mathrm{v}}_{\sigma }\left(\tau \right)$ is considered as a Gaussian noise with a mean of zero and a variance of one. Thus, (143)–(145) have the following criteria of $\mathcal{G}=9.8,\mathrm{l}=9,\mathrm{q}=0.2,m_2=0.4,m_3=0.3,{\kappa }_1=1,{\kappa }_2=1.2$, and $a_1=3.$

We introduce the uncertain membership functions as ${\upsilon }_{{F_1}_{\sigma ,1}^{\mathrm{l}}}=exp\left({\displaystyle \frac{-{({\widehat{\mathrm{y}}}_{\sigma ,1}-62\mathrm{l})}^2}{3}}\right),$ $(\sigma =1,2,3;\mathrm{l}=1,\dots ,5).$ The modeling process employs the following architectural criteria: ${\eta }_{1,1}={\eta }_{2,1}={\eta }_{3,1}=5,$ ${\eta }_{1,2}={\eta }_{2,2}={\eta }_{3,2}=5,{\Gamma }_1={\Gamma }_2={\Gamma }_3=6I,$ ${\mathrm{z}}_{1,1}={\mathrm{z}}_{2,1}={\mathrm{z}}_{3,1}=1,$ ${\upsilon }_1={\upsilon }_1={\upsilon }_3=1,{\mathrm{d}}_1={\mathrm{d}}_2={\mathrm{d}}_3=1,$ ${\tilde{\Xi }}_{1,1}=5,{\tilde{\Xi }}_{1,2}=1,$ ${\tilde{\Xi }}_{2,1}=5,{\tilde{\Xi }}_{2,2}=1,{\tilde{\Xi }}_{3,1}=5$, and ${\tilde{\Xi }}_{3,2}=1.$

The initial settings of the system are provided as ${\mathrm{y}}_{1,1}\left(0\right)=0.3,$ ${\mathrm{y}}_{1,2}\left(0\right)={\mathrm{y}}_{2,1}\left(0\right)$ $={\mathrm{y}}_{2,2}\left(0\right)={\mathrm{y}}_{3,1}\left(0\right)=$ ${\mathrm{y}}_{3,2}\left(0\right)=0,{\xi }_{1,1}\left(0\right)=$ $2,{\xi }_{2,1}\left(0\right)=3,{\xi }_{3,1}\left(0\right)=2,$ ${\kappa }_1\left(0\right)=1.4,{\kappa }_2\left(0\right)=1.2,$ ${\kappa }_3\left(0\right)=1.4,$ ${\widehat{\wp }}_{1,0}\left(0\right)={\widehat{\wp }}_{3,0}\left(0\right)=1,$ ${\widehat{\wp }}_{2,0}\left(0\right)=-1,$ and the remaining starting points are set to zeros. Figure 11a–f illustrate the modeling findings for the trajectories of ${\mathrm{y}}_{\sigma ,1}-{\widehat{\mathrm{y}}}_{\sigma ,1},$ and ${\mathrm{y}}_{\sigma ,2}-{\widehat{\mathrm{y}}}_{\sigma ,2},(\sigma =1,2,3)$. Figure 12a–f illustrate the modeling findings for the trajectories of main controllers ${\mathrm{s}}_{\sigma ,1},(\sigma =1,2,3)$ and Figure 13 illustrates the modeling findings for the trajectories of compensation controllers ${\mathrm{s}}_{\sigma ,2},(\sigma =1,2,3),$ respectively.

In Illustrative Example 6.3, we examine the stabilization dynamics of triple-reversed pendulums influenced by fractional Brownian motion under uncertain control configurations. The governing system is characterized by fractional-order derivatives in the Caputo sense, where $\alpha \in (0,1)$. The baseline fractional order is set at $\alpha =0.95$, and we consider variations by increasing and decreasing this value slightly.

Observations:

• When $\alpha >\mathbf{0}.\mathbf{95}$: The system exhibits faster convergence towards stabilization. The trajectories of the pendulum angles and velocities settle more quickly. This behavior is attributed to the fact that increasing $\alpha$ makes the memory effect less dominant, rendering the system dynamics closer to an integer-order behavior (i.e., less history-dependent), which accelerates the damping of oscillations.
• When $\alpha <\mathbf{0}.\mathbf{95}$: The system response becomes slower and more oscillatory. The convergence to equilibrium takes more time, and the amplitude of fluctuations remains higher for longer durations. This reflects a stronger memory effect, typical in lower-order fractional systems, where the past states heavily influence current dynamics.
• Stochastic influence: The presence of fractional Brownian motion with Gaussian noise maintains boundedness across all values of $\alpha$, but the magnitude and persistence of fluctuations vary. For lower $\alpha$, the stochastic perturbations accumulate more persistently due to long-memory effects, whereas, for higher $\alpha$, they decay more rapidly.

Fractional-order $\alpha$ serves as a tuning parameter that directly influences the rate of stabilization and robustness of the pendulum system. A careful selection of $\alpha$ is crucial to balancing responsiveness and robustness under noisy and uncertain environments.

6.4. Remarks for Better Understanding of the Proposed Methods and Theorems

To achieve reliable and precise tracking under uncertainty, the methodology adopted in this study defines a control structure that integrates Nussbaum functions, adaptive neural–fuzzy approximators, and FON dynamics within a decentralized framework.

(i) Modeling of FON models:

FON dynamics are employed in this study due to their inherent memory and hereditary properties, which make them well-suited for modeling complex physical phenomena characterized by long-term dependencies and filtering effects. Compared to classical integer-order systems, FON models provide a more accurate and flexible representation of nonlinear dynamics.

(ii) The OB technique’s function:

The proposed OB approach, unlike conventional backstepping, introduces a framework that not only ensures system stability but also optimizes control performance. This is achieved by integrating Nussbaum-type functions with adaptive neural networks, allowing the controller to adjust rapidly in response to environmental uncertainties.

(iii) Nussbaum-type function interpretation:

To address the challenge of uncertain control directions, a Nussbaum-type function is introduced. It enhances the system’s robustness by enabling the control law to adapt even when the sign or direction of the control gain is unknown or varying. This eliminates the need for precise prior knowledge of the control input characteristics.

(iv) Objective of the quartic-barrier Lyapunov function:

The quartic-barrier Lyapunov function is employed to ensure that the tracking error remains within a predefined accuracy bound within a finite time. By enforcing a constraint that prevents the error from exceeding this bound, the function guarantees both transient and steady-state performance.

(v) Insight into the theorem proofs:

The proofs are constructed using the fractional Lyapunov direct method, combined with adaptive laws derived from neural networks and fuzzy approximators. By developing Lyapunov functions that incorporate both fractional derivatives and stochastic disturbances, the proofs demonstrate that all closed-loop signals are uniformly ultimately bounded in probability.

(vi) Design of observers and auxiliary signals:

In accordance with the ISpS framework, an auxiliary signal is designed to manage unmeasured states and external disturbances. To estimate these unobservable states, a fuzzy-phase observer is incorporated, which is crucial for practical applications in networks with limited sensors.

(vii) Significance of decentralized control architecture:

A decentralized control architecture plays a vital role in large-scale systems composed of interconnected subsystems. While each subsystem controller is designed independently, adaptive laws are employed to compensate for uncertainties in the interaction dynamics. This approach enhances scalability and eliminates the need for centralized computation.

(viii) Novelty and comparison with existing work:

The proposed approach distinguishes itself from current FON and neuro-adaptive-control methods by integrating several advanced features within a unified decentralized architecture. Unlike previous methods, this work enhances system modeling and control accuracy by combining neural and fuzzy approximators with FON dynamics. Additionally, it employs Nussbaum functions to effectively handle unknown control directions and incorporates predefined performance constraints to guarantee precise tracking. The method also addresses stochastic robustness, ensuring reliable performance in the presence of uncertainties. To the best of our knowledge, the existing literature has not thoroughly explored such a comprehensive integration of these components in a decentralized framework.

7. Conclusions

In this research, we examined fractional-order adaptive NN-optimized control for the predetermined time and precision tracking challenge in SF structures with undetermined controlling orientations. The suggested fractional-order control approach ensures that tracking data tends to a predetermined precision within a specified time frame. Furthermore, the controlling device requires the least controlling capabilities. In order to accomplish the controlling goal, a Nussbaum-type function and a new time-dependent fractional-order functional were proposed to address the control orientation undetermined and predetermined execution shortcomings, respectively. In addition, we addressed the adaptive decentralized uncertain feedback regarding output stabilization issues encountered in unpredictable fractional stochastic complex large-scale systems using stringent control. The probabilistic nonlinear structures under consideration lack specified nonlinear functions, controlling paths, and state observations. The deterministic–stochastic fractional-order nonlinear structures under consideration lack specified nonlinear functions, controlling paths, and state observations. The fuzzy reasoning algorithms revealed unidentified fractional complex processes, while a fuzzy phase filtration observer estimated the unrecorded events. Using the optimized backstepping fractional-order architecture approach and the Nussbaum-type function, an efficient adaptation-based decentralized uncertain-feedback regulation strategy was devised. The closed-loop application system’s robustness was demonstrated. Finally, the suggested approach was validated using multiple sets of scenarios at distinct predetermined times. We plan to apply the findings to various structures, including non-SF and multiple-input, multiple-output systems. Also, the adaptive-control technique solves both state-related unrecorded and unidentified control orientation problems for probabilistic exponential large-scale mechanisms, providing significant improvements.

• Drawbacks:

The proposed approach shows significant progress but has several limitations. Its integration of Nussbaum functions, fractional-order dynamics, and neural–fuzzy approximators increases computational complexity, which may hinder real-time use on limited hardware. The method requires careful tuning of many parameters, and improper choices can reduce accuracy and reliability. Although it addresses stochastic robustness, its performance under varying, especially high-level, noise still requires thorough experimental validation. Lastly, while the decentralized design enhances scalability, complex interconnections and communication delays in large systems may restrict implementation. Addressing these challenges is crucial for broader application.