1 Introduction

Given that some practical systems, such as robotic systems and industrial process systems, etc, are modeled as interconnected nonlinear systems. Therefore, the investigation of interconnected nonlinear systems has attracted widespread interest from scholars. Initially, the centralized control approach is impractical and suffers from the computational burden of redundancy due to the utilization of overall state information. Notably, the decentralized-based adaptive backstepping control schemes [1,2,3] in each subsystem have capable of independently handling the control assignment of the interconnected system via its local information. Meanwhile, the majority of existing decentralized control results require that the interconnection term of the interconnected system contains all states of the entire interconnected system, which is referred to as strong interconnections. Hence, an adaptive decentralized control method was investigated for interconnected nonlinear systems with strong interconnections in [4]. Due to the existence of unknown nonlinearities in interconnected nonlinear systems, with the aid of radial basis function neural networks (RBFNNs) and fuzzy logic systems, the adaptive decentralized control schemes were presented in [2, 3]. Generally, the backstepping technique entails the issue of complexity in analytical computation stemming from the iterative calculation of the virtual control function. To this end, the dynamic surface control (DSC) schemes [5,6,7] were proposed to address the explosive calculation by introducing a first-order filter. Although the above DSC methods solve the above issues, filter errors between the filter output and the virtual control function were not eliminated. Thus, the command filter backstepping control (CFBC) methods [8, 9] were constructed by means of an error compensation signal. Meanwhile, Song et al. [10] developed an adaptive fuzzy secure control solution for nonlinear systems, where a fractional-order filter (FOF) was introduced to eradicate the effect of filter errors. Noteworthy, the above CFBC-based control results neglect the impact of the communication burden.

Until today, event-triggered control (ETC) plays an important role in the realm of transmission resource constraints, where the measurement error satisfies the trigger criterion, the control signal will be updated and applied to the system. Targeting the aircraft wing rock motion [11], an adaptive event-triggered control was presented to economize communication resources. [12] discussed the prescribed-time synchronization issue for nonlinear systems via ETC strategy. Unfortunately, the event-triggered mechanism is imposed to continuously monitor the trigger criteria, which requires relatively unavailable hardware situations for industrial applications. With this in mind, an adaptive self-triggered control solution was investigated in [13], which can calculate the next trigger instant following the current information. Meanwhile, [14] reported a neural adaptive self-triggered control method for nonlinear systems with unmeasurable states.

To further handle the limitation of bandwidth in networked control, quantized control was used to degrade the communication rate to satisfy that the system can operate normally within the specified bandwidth. Initially, a logarithmic quantizer was investigated in [15], where the bandwidth of the constrained transmission was alleviated. However, the chattering phenomenon will inevitably occur in the quantization control of continuous-time systems. Soon afterward, the hysteresis quantizer was developed in [16], which can decrease the risk of chattering. In [17], an adaptive fuzzy quantized control was proposed for nonlinear systems. In addition, networked control may result in discontinuous control signals at two trigger instants, which will drastically affect the control system’s performance. Hence, it is imperative to concentrate on the trade-off between communication cost and tracking precision.

It is widely acknowledged that prescribed performance control (PPC) enables tracking errors to converge to a predefined range. Particularly, Sun et al. [18] and Song et al. [19] investigated the adaptive CFBC methods for the different nonlinear systems, where transient and steady performances were maintained. In [20], an adaptive PPC scheme was analyzed for nonlinear systems. Soon afterward, an improved performance function in comparison to PPC is proposed, called the finite-time prescribed performance (FTPP) function, which guarantees that the tracking error is confined to a small origin in finite time. In particular, [21] developed a fuzzy adaptive FTPP control strategy for nonlinear systems with dynamic uncertainty. Nevertheless, the above results do not both consider the time delay and unmodeled dynamics that are ubiquitous in modern industrial applications.

Naturally speaking, if this obstacle is not overcome, the system performance will deteriorate and even resulting in the instability of the closed-loop system. Thus, many remarkable results have been reported to ensure the stability of the systems (see [22,23,24,25,26] and reference therein). Among them, by introducing a dynamic signal, an adaptive decentralized tracking control strategy was addressed for nonlinear systems with dynamical uncertainties in [23]. In [24], with the help of Lyapunov–Krasovskii functional, an adaptive neural control scheme was discussed for interconnected nonlinear systems with time delay. Meanwhile, Li et al. [26] developed an adaptive CFBC method for nonlinear time-delay systems.

Guided by the foregoing analysis, this paper developed the neural adaptive FTPP quantized control strategy by utilizing the FOF for interconnected nonlinear time-delay systems with unmodeled dynamics and self-triggered input. The highlights of this article are enumerated below:

  1. 1.

    Different from the event-triggered mechanism [10, 11, 26], the investigated self-triggered control solution was developed, where the next trigger instant was determined by the current information. In addition, the effect of bandwidth limitation was synthesized in the interconnected nonlinear time-delay system as a challenging issue in comparison to [7, 16, 17]. However, the networked control mentioned above may affect the performance of the system. Tactfully, the FTPP function was considered to optimize the transient and steady-state performance of the interconnected nonlinear system.

  2. 2.

    In contrast to the integer-order filter in [9, 18, 26, 27], an improved FOF was introduced such that the tremendous “ amount of calculation ” was avoided and the filter performance was skillfully enhanced. Unlike [7, 21, 28], which only considered the PPC issue or error compensation signal, the proposed self-triggered quantized control scheme in this paper takes FTPP and error compensation signal into account to eliminate the filter errors simultaneously.

  3. 3.

    To address the potential singularity problem that may exist in interconnected nonlinear systems, hyperbolic tangent functions, and RBFNNs were introduced in [16, 26]. Furthermore, this article extends both time delay and unmodeled dynamics to interconnected nonlinear systems, which makes it applicable to more general situations [3, 7, 23].

2 System formulation

Consider the following interconnected nonlinear time-delay plant

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{\psi }_m=&\,q_m(\psi _m,\check{x}_m),\\ \dot{x}_{m,i}=&\,x_{m,i + 1}+f_{m,i}(\bar{x}_{m,i})+g_{m,i}(\bar{y})+\Delta _{m,i}(\check{x}_m,\psi _i)\\ {}&+h_{m,i}(\bar{x}_{m,i}(t-\tau _{m,i}(t))),\\ \dot{x}_{m,{n_m}}=&\,q(u_m)+f_{m,n_m}(\bar{x}_{m,n_m})+g_{m,n_m}(\bar{y})+\Delta _{m,n_m}(\check{x}_m,\psi _i)\\ {}&+h_{m,n_m}(\bar{x}_{m,n_m}(t-\tau _{m,n_m}(t))),\\ y_m=&\,x_{m,1}, \end{array} \right. \end{aligned}$$
(1)

where \(\bar{x}_{m,i} = [x_{m,1}, \ldots , x_{m,i}]^ T \in R^i\) and \(y_m \in R\) are the state vector and output vector of the mth subsystem, respectively. \(\check{x}_m=[x_{m,1}, \ldots , x_{m,n_m}]^ T \in R^{n_m}\), \(\bar{y}=[y_1, \ldots , y_M]^ T \in R^M\). \(f_{m,i}(\cdot )\) and \(h_{m,i}(\cdot )\) with \(1\le m \le M\) and \(1 \le i \le n_m\) are the unknown smooth nonlinear functions, \(g_{m,i}(\bar{y})\) denotes the unknown smooth interconnected term between the mth subsystem and other subsystems. \(\tau _{m,i}(t)\) represents the time-varying delay satisfying \(\mid \tau _{m,i}(t)\mid \le \bar{\tau }_{m,i}<\infty\) and \(\mid \dot{\tau }_{m,i}(t)\mid \le \check{\tau }_{m,i}<1\), where \(\bar{\tau }_{m,i}>0\) and \(\check{\tau }_{m,i}>0\) are constants. The \(\psi _m\)-dynamics and \(\Delta _{m,i}(\cdot )\) denote the unmodeled dynamics and dynamic disturbances, respectively. \(q_m\) and \(\Delta _{m,i}(\cdot )\) indicate the Lipschitz continuous functions. \(q(u_m)\) and \(u_m\) indicate the quantized control signal and quantized input signal. Meanwhile, the hysteresis quantizer is described as follow:

$$\begin{aligned} q(u_m)= \left\{ \begin{array}{ll}u_{m}^w{\text{sgn}}(u_{m}), &\quad \frac{u_{m}^w}{1+\hslash _{m}}<\mid u_{m}\mid \le u_{m}^w, \dot{u}_{m}<0, \mathrm \ or\\ &\quad u_{m}^w<\mid u_{m}\mid \le \frac{u_{m}^w}{1-\hslash _{m}}, \dot{u}_{m}>0 \\u_{m}^w{\vartheta }_m, &\quad u_{m}^w<\mid u_{m}\mid \le \frac{u_{m}^w}{1-\hslash _{m}}, \dot{u}_{m}<0, \mathrm \ or\\&\quad \frac{u_{m}^w}{1-\hslash _{m}}<\mid u_{m}\mid \le \frac{u_{m}^w(1+\hslash _{m})}{1-\hslash _{m}}, \dot{u}_{m}>0 \\ 0, &\quad 0\le \mid u_{m}\mid<\frac{u_{m}^{\rm{min}}}{1+\hslash _{m}}, \dot{u}_{m}<0, \mathrm \ or\\ &\quad \frac{u_{m}^{\rm{min}}}{1+\hslash _{m}}\le \mid u_{m}\mid \le u_{m}^{\rm{min}}, \dot{u}_{m}>0 \\ q(u_m(t^{-})), &\quad \mathrm other\ case, \end{array} \right. \end{aligned}$$

where \({\vartheta }_m=(1+\hslash _{m}){\text{sgn}}(u_m)\), \(u_{m}^w=\delta _{m}^{1-w}u_{m}^{\rm{min}}\ (w=1,2,\ldots )\) with \(0<\delta _{m}<1\) stands for a measure of quantization density and \(\hslash _{m}=\frac{1-\delta _{m}}{1+\delta _{m}}\), \(u_{m}^{\rm{min}}\) indicates the scope of the dead-zone for \(q(u_m)\) and \(q(u_m)\) is in the set \(\bigcup _m=(0,\pm u_{m}^w,\pm u_{m}^w(1+\hslash _{m}))\).

To support the controller design, we need the following assumptions, lemmas, and definitions.

Definition 1

[29] Assume \(F(t):[t_0,+\infty )\rightarrow R\) is a continuous function together with its fractional derivative of order \(P_m\) under Caputo’s definition is expressed as:

$$\begin{aligned} D^{P_m}F(t)=\frac{1}{\Gamma ({n_m} - {P_m})}\int _{0}^t \frac{F^{n_m}(\tau )}{\left( t - \tau \right) ^{{P_m} + 1-{n_m}}}\text {d}\tau , \end{aligned}$$

where \(n_m\) denotes an integer such that \({n_m-1}\le {P_m}\le n_m\). \(\Gamma (x)=\int _{0}^{+\infty }\tau ^{x-1}e^{-\tau }\text {d}\tau (x>0)\) denotes Euler’s Gamma function with \(\Gamma (1)=1\).

Definition 2

[28] A smooth finite-time performance function (FTPF) \(\ell _m(t)\) satisfies the following characteristics: \((1)\ \ell _m(t)>0\), \((2)\ \dot{\ell }_m(t)\le 0,\) \((3)\ \lim \nolimits _{t \rightarrow T_f} \ell _m(t)=\ell _{mT_f}>0\) and \(\ell _m(t)=\ell _{mT_f}>0\) for any \(t>T_f\) with \(T_f\) and \(\ell _{mT_f}\) are the settling time and the arbitrarily small constant, respectively.

Lemma 1

[30] To further analyze the quantization impact, the hysteretic quantizer is reconstructed as \(q(u_m(t))=(1-\bar{\kappa }_{m})u_m(t)+\bar{\kappa }_{m} \bar{\varsigma }_m(t)\) such that the nonlinearity function \(\bar{\varsigma }_m(t)\) satisfies the following inequalities:

$$\begin{aligned} \left( \bar{\varsigma }_m(t)\right) ^{2}\le&\left( \frac{\bar{\kappa }_{m}+\hslash _{m}}{\bar{\kappa }_{m}}u_m(t)\right) ^{2}, \ \ \forall \mid u_m(t)\mid \ge u_{m}^{\rm{min}},\\ \left( \bar{\varsigma }_m(t)\right) ^{2}\le&\left( \frac{1-\bar{\kappa }_{m}}{\bar{\kappa }_{m}}u_{m}^{\rm{min}}\right) ^{2}, \ \ \ \ \ \ \ \forall \mid u_m(t)\mid \le u_{m}^{\rm{min}}, \end{aligned}$$

where \(0<\bar{\kappa }_{m}<1\) is an adjustable constant to be designed.

Assumption 1

[31] There exist unknown non-negative smooth functions \(\varphi _{m,i1}(\Vert \bar{x}_{m,i}\Vert )\) and \(\varphi _{m,i2}(\Vert \psi _i\Vert )\), the dynamic disturbance \(\Delta _{m,i}(\psi _i,\check{x}_m)\) satisfying

$$\begin{aligned} \Delta _{m,i}(\psi _i,\check{x}_m)\le \varphi _{m,i1}(\Vert \bar{x}_{m,i}\Vert )+\varphi _{m,i2}(\Vert \psi _i\Vert ). \end{aligned}$$

Assumption 2

[32] Let us consider the unmodeled dynamics \(\dot{\psi }_m=q_m(\psi _m,\check{x}_m)\), which is exponentially input-to-state practically stable (exp-ISpS). Meanwhile, \(V_m(\psi _m)\) is an exp-ISpS Lyapunov function (LF) satisfying

$$\begin{aligned}&\Upsilon _{m,1}(\mid \psi _m\mid )\le V_m(\psi _m) \le \Upsilon _{m,2}(\mid \psi _m\mid ),\nonumber \\&\frac{\partial V_m(\psi _m)}{\partial \psi _m}q_m(\psi _m,\check{x}_m)\le -r_{m,1}V_m(\psi _m)+\pi _{m0}(\Vert x_{m,1}\Vert )+r_{m,2}, \end{aligned}$$
(2)

where \(\Upsilon _{m,1}, \Upsilon _{m,2}\), and \(\pi _{m0}\) are class \(K_\infty\)-functions and \(r_{m,1}>0\) and \(r_{m,2}>0\) denote known constants.

Assumption 3

[4] The interconnected term \(g_{m,i}(\bar{y})\) has the form of \(\mid g_{m,i}(\bar{y})\mid \le \sum \nolimits _{l= 1}^{M} j_{m,l}\aleph _i(\bar{x}_{l,i})\), with \(\aleph _i(\bar{x}_{l,i})\ (i=1,\ldots ,n_m)\) and \(j_{m,l}\) denote the unknown continuous function and unknown constant, respectively.

Lemma 2

[31] Assume exp-ISpS LF \(V_m(\psi _m)\) satisfies the equation condition (2), then, for \(\forall \bar{r}_{m,1} \in (0,r_{m,1})\), the initial condition \(\psi _{m0}=\psi _m(t_0)\), \(c_0>0\), and the function \(\bar{\pi }_{m0}(\Vert y_m\Vert )\ge \pi _{m0}(\Vert y_m\Vert )\), there exist a finite time \(T_{m0}=T_{m0}(\bar{r}_{m,1},c_0,\psi _{m0})\), a non-negative function \(Q(t_0,t)(t\ge t_0)\), and a dynamic signal expressed as:

$$\begin{aligned} \dot{\lambda }_m=-\bar{r}_{m,1}\lambda _m+\bar{\pi }_{m0}(\Vert x_{m,1}(t)\Vert )+r_{m,2},\ \lambda _m(t_0)=\lambda _{m0}, \end{aligned}$$

satisfying \(Q(t_0,t)=0\) for \(\forall t\ge t_0+T_{m0}\)

$$\begin{aligned} V_m(\psi _m(t))\le \lambda _m(t)+Q(t_0,t),\ \forall t \ge t_0, \end{aligned}$$

where \(t_0\) denotes the initial time and assume that \(\bar{\pi }_{m0}(\Vert x_{m,1}(t)\Vert )=\pi _{m0}(\Vert x_{m,1}\Vert )\).

Lemma 3

[33] For variable \(\varrho \in R\) and \(\forall \jmath >0\), one has

$$\begin{aligned} -\varrho {\text{tanh}}\left( \frac{\varrho }{\jmath }\right) \le 0,\ \ 0\le \mid \varrho \mid -\varrho {\text{tanh}}\left( \frac{\varrho }{\jmath }\right) \le 0.2785\jmath . \end{aligned}$$

Lemma 4

[34] The compact set \(\Omega _{m_k}\) is defined in \(\Omega _{m_k}=\left\{ s_{m,k}\mid \mid s_{m,k}\mid <0.2554m_k\right\}\). Next, for the case of \(\forall s_{m,k} \notin \Omega _{{m,k}}\), the term \(1-16{\text{tanh}}^2(s_{m,k}/{m_k})\le 0\) holds.

Lemma 5

[35] Consider the unknown smooth nonlinearities \(f_{m,i}(\bar{z}_{m,i})\) on the compact set \(\Omega _X\), which can be approximated by RBFNNs

$$\begin{aligned} f_{m,i}(\bar{z}_{m,i})=W_{m,i}^{\rm{T}}S_{m,i}(\bar{z}_{m,i})+\varepsilon _{m,i}, \end{aligned}$$

where \(W_{m,i}\) and \(S_{m,i}(\bar{z}_{m,i})=[\psi _{m,i1}(\bar{z}_{m,i}),\ldots ,\psi _{m,iK}(\bar{z}_{m,i})]^{\rm{T}}\) are, respectively, the weight vector and basic function vector, with \(\bar{z}_{m,i}\) is the input of the RBFNNs, \(K\ge 1\) is the number of neuron, \(\varepsilon _{m,i}\) is the approximation error. There exists a positive constant \(\varepsilon _{m,i}^*>0\) such that \(\Vert \varepsilon _{m,i}\Vert \le \varepsilon _{m,i}^*\). Meanwhile, the Gaussian function is expressed as:

$$\begin{aligned} \psi _{m,i}^\iota (\bar{z}_{m,i})=\exp \left[ \frac{-(\bar{z}_{m,i} -h_{m,i})^{\rm{T}}(\bar{z}_{m,i} -h_{m,i})}{b_{m,i}^2}\right] , \end{aligned}$$

where \(h_{m,i}=\left[ h_{m,i1},\ldots ,h_{m,iM}\right] ^{\rm{T}}\) and \(b_{m,i}\) denote the center and width of the basis function, respectively.

This paper aims at synthesizing a decentralized-based adaptive neural quantized control algorithm for the interconnected nonlinear time-delay system in (1) such that all signals of the resulting closed-loop system (CLS) are semi-globally uniformly ultimately bounded (SGUUB), the system output is confined to a small adjustable region in a finite time interval, and as well as the Zeno phenomenon is ruled out.

3 Main results

In this section, a decentralized-based adaptive self-triggered neural control scheme will be put forward for interconnected nonlinear time-delay systems.

3.1 State transformation

To confine the system output \(x_{m,1}\) to the range \((-\ell _m,\ell _m)\), the FTPF can be chosen from Definition 2.

$$\begin{aligned} \ell _m(t)=\left\{ \begin{array}{ll} \left( \ell _{m0}-\frac{t}{T_f}\right) e^{\left( 1-\frac{T_f}{T_f-t}\right) }+\ell _{mT_f},&\quad t\in [0,T_f),\\ \ell _{mT_f}, &\quad t\in [T_f,+\infty ), \end{array} \right. \end{aligned}$$
(3)

where \(\ell _{m0}>0\) and \(\ell _{mT_f}>0\) are design parameters.

Furthermore, it can be obtained that \(\ell _m(0)=\ell _{m0}+\ell _{mT_f}\) from Definition 2 and (3). The error transformation function with the transformed error \(\partial _m\) is selected as:

$$\begin{aligned} T(\partial _m)=\frac{e^{\partial _m}-e^{-\partial _m}}{e^{\partial _m}+e^{-\partial _m}}, \end{aligned}$$
(4)

The transformation is defined as:

$$\begin{aligned}{} & {} x_{m,1}=\ell _m(t)T(\partial _m), \end{aligned}$$
(5)
$$\begin{aligned}{} & {} \partial _m=T^{-1}\left( \frac{x_{m,1}}{\ell _m}\right) = \frac{1}{2}\ln \left( \frac{x_{m,1}/\ell _m+1}{1-x_{m,1}/\ell _m}\right) , \end{aligned}$$
(6)

and

$$\begin{aligned} \dot{\partial }_m=p_{m,1}\left( \dot{x}_{m,1}-\frac{\dot{\ell }_m x_{m,1}}{\ell _m}\right) , \end{aligned}$$
(7)

where \(p_{m,1}=\frac{1}{2\ell _m}\left( \frac{1}{x_{m,1}/\ell _m+1}-\frac{1}{x_{m,1}/\ell _m-1}\right)\).

3.2 Controller design

In contrast to the advancement of integer-order calculus [9, 18, 26, 27], fractional-order calculus [36,37,38,39] possesses the property of the favorable filter and enhances the freedom of control design attributed to its distinctive historical memory characteristics. In a nutshell, we propose a FOF-based adaptive self-triggered control algorithm for interconnected nonlinear time-delay systems in (1), which can not only overcome the complicated “ amount of calculation ” but also effectively upgrade the filter performance of the existing results in [9, 18, 26, 27].

Now, the fractional-order filter (FOF) are constructed:

$$\begin{aligned} \left\{ \begin{aligned} D^{P_m}\beta _{m,1}=&\,I_{m,1},\\ I_{m,1}=&\,-k_{m,l1} {\lceil \beta _{m,1}-\alpha _{m,i - 1} \rfloor }^{\frac{1}{2}}\\&-k_{m,l2}{\lceil \beta _{m,1}-\alpha _{m,i - 1} \rfloor }^{\frac{3}{2}}+\beta _{m,2},\\ D^{P_m}\beta _{m,2}=&\,-k_{m,l3}{\lceil \beta _{m,1}-\alpha _{m,i - 1} \rfloor }^{\frac{1}{2}}, \end{aligned} \right. \end{aligned}$$
(8)

where \(D^{P_m}\) indicates the fractional operator with \(0<P_m<1\) and \(k_{m,li}\ (i=1,2,3)\) is the design parameter. The virtual control function \(\alpha _{m,{i - 1}}\) indicates the filter input, \(\zeta _{m,i}=\beta _{m,1}\) and \(D^{P_m}\zeta _{m,i}=I_{m,1}\) are the filter output.

Define the change of coordinate as follows:

$$\begin{aligned} \left\{ \begin{aligned}&z_{m,1}=\partial _m,\\&z_{m,i}=x_{m,i}-\zeta _{m,i},\ \ \ \ \ \ 1\le m \le N, \\&\xi _{m,i} = \zeta _{m,i} - \alpha _{m,i - 1},\ \ i=2,\ldots ,n_m, \end{aligned} \right. \end{aligned}$$
(9)

where \(z_{m,i}\) denotes the error surface and \(\xi _{m,i}\) denotes the filter error. Additionally, the compensated error signal \(s _{m,i}\) with \(i=1,\ldots ,n_m\) is designed as follows:

$$\begin{aligned} s_{m,i}=z_{m,i}-\nu _{m,i}, \end{aligned}$$
(10)

where \(\nu _{m,i}\) is the error compensation signal, which can be specifically designed as:

$$\begin{aligned} \left\{ \begin{aligned}&\dot{\nu }_{m,1}=-c_{m,1}\nu _{m,1}+p_{m,1}\nu _{m,2}+p_{m,1}\xi _{m,2},\\&\dot{\nu }_{m,i}=-c_{m,i}\nu _{m,i}-p_{m,{i-1}}\nu _{m,{i-1}}+\nu _{m,{i + 1}}+\xi _{m,{i+1}},\\&\dot{\nu }_{m,n_m}=-c_{m,n_m}\nu _{m,n_m}-p_{m,{n_m-1}}\nu _{m,{n_m-1}}, \end{aligned} \right. \end{aligned}$$
(11)

where \(\nu _{m,i}(0)=0,\ p_{m,i}=1\) and \(c_{m,i}>0 \ (i=2,\ldots ,n_m)\) denotes the positive design parameters.

The virtual control functions and the adaptive laws are designed for each subsystem \((i=2,\ldots ,{n_m-1})\) as follows:

$$\begin{aligned} \alpha _{m,1} =&\,-\frac{c_{m,1}}{p_{m,1}}z_{m,1}- \frac{p_{m,1}s_{m,1}}{2a_{m,1}^2}\hat{\Theta }_{m,1}S_{m,1}^{\rm{T}} (\cdot )S_{m,1}(\cdot )\nonumber \\&+\frac{\dot{\ell }_m x_{m,1}}{\ell _m}, \end{aligned}$$
(12)
$$\begin{aligned} \alpha _{m,i}=&\,-c_{m,i}z_{m,i}-\frac{s_{m,i}}{2a_{m,i}^2}\hat{\Theta }_{m,i}S_{m,i}^{\rm{T}} (\cdot )S_{m,i}(\cdot )\nonumber \\&-p_{m,{i-1}}z_{m,{i-1}}, \end{aligned}$$
(13)
$$\begin{aligned} \alpha _{m,n_m}=&\,\frac{1}{(1-\bar{\kappa }_{m})\underline{\zeta }_{m0}}\left[ -c_{m,n_m}z_{m,n_m}\right. \nonumber \\&-(1-\bar{\kappa }_{m}){\text{sgn}}(s_{m,n_m})u_{m}^{\rm{min}}\nonumber \\&-\frac{s_{m,n_m}}{2a_{m,n_m}^2}\hat{\Theta }_{m,n_m}S_{m,n_m}^{\rm{T}}(\cdot )S_{m,n_m}(\cdot )\nonumber \\&\left. -p_{m,{n_m-1}}z_{m,{n_m-1}}\right] , \end{aligned}$$
(14)
$$\begin{aligned} \dot{\hat{\Theta }}_{m,1}=&\,\mu _{m,1}\left( \frac{p_{m,1}^2s_{m,1}^2}{2a_{m,1}^2}S_{m,1}^{\rm{T}}(\cdot )S_{m,1}(\cdot )-\rho _{m,1}\hat{\Theta }_{m,1}\right) , \end{aligned}$$
(15)
$$\begin{aligned} \dot{\hat{\Theta }}_{m,i}=&\,\mu _{m,i}\left( \frac{s_{m,i}^2}{2a_{m,i}^2}S_{m,i}^{\rm{T}} (\cdot )S_{m,i}(\cdot )- \rho _{m,i}\hat{\Theta }_{m,i}\right) , \end{aligned}$$
(16)

where \(c_{m,i}\) and \(\rho _{m,i}\) are positive design parameters. Let us define that \(\hat{\Theta }_{m,i}\) denotes the estimates of \(\Theta _{m,i}\). \(\tilde{\Theta }_{m,i}= \Theta _{m,i}-\hat{\Theta }_{m,i}\) being the parameter estimation error with \(\Theta _{m,i} = \max \nolimits _{ 1 \le i \le n_m} \left\{ {\left\| W_{m,i}^*\right\| }^2 \right\}\). Analogous to (16), \(\dot{\hat{\Theta }}_{m,n_m}\) can be expressed by replacing i with \(n_m\).

For simplicity, the corresponding abbreviations are considered: \(f_{m,i}\) indicates \(f_{m,i}(\bar{x}_{m,i})\), \(\Delta _{m,i}\) indicates \(\Delta _{m,i}(\check{x}_m,\psi _i)\), \(g_{m,i}\) indicates \(g_{m,i}(\bar{y})\), \(h_{m,i}(x_{m,i\tau })\) indicates \(h_{m,i}(\bar{x}_{m,i}(t-\tau _{m,i}(t)))\), and \(S_{m,i}^{\rm{T}} (\cdot )S_{m,i}(\cdot )\) indicates \(S_{m,i}^{\rm{T}}(\bar{z}_{m,i})S_{m,i}(\bar{z}_{m,i})\).

Proof

The specific control design procedures of this paper are expressed in detail as follows.

Step m, 1: From (7) and (9), the time derivative of \(z_{m,1}\) yields

$$\begin{aligned} \dot{z}_{m,1} =\,p_{m,1}\left( z_{m,2} +\xi _{m,2}+ \alpha _{m,1}+f_{m,1}+\Delta _{m,1}+g_{m,1} +h_{m,1}(x_{m,1\tau })-\frac{\dot{\ell }_mx_{m,1}}{\ell _m}\right) . \end{aligned}$$
(17)

It follows from (10)–(11) that

$$\begin{aligned} \dot{s}_{m,1} =&\,p_{m,1}\left( s_{m,2}+\nu _{m,2}+\xi _{m,2}+\alpha _{m,1}+f_{m,1}+\Delta _{m,1}\right. \nonumber \\&\left. +g_{m,1}+h_{m,1}(x_{m,1\tau })-\frac{\dot{\ell }_m x_{m,1}}{\ell _m}\right) -\dot{\nu }_{m,1}. \end{aligned}$$
(18)

Now, we consider the following Lyapunov function:

$$\begin{aligned} V_1 = \sum \limits _{m = 1}^M \left[ \frac{1}{2}s_{m,1}^2 + \frac{1}{2\mu _{m,1}}\tilde{\Theta }_{m,1}^2+\frac{\lambda _m}{\varrho _{m0}}+\bar{V}_{m,1}\right] , \end{aligned}$$

where \(\mu _{m,1}\) and \(\varrho _{m0}\) are positive constants. The Lyapunov–Krasovskii functional is formulated as \(\bar{V}_{m,1}=[e^{\kappa _{m,1}\bar{\tau }_{m,1}}/2(1-\check{\tau }_{m,1})]\int _{t - \tau _{m,1}(t)}^t e^{-\kappa _{m,1}(t-s)}h_{m,1}^2(x_{m,1}(s))ds\) to tackle the time delay challenge, where \(\kappa _{m,1}>0\) denotes a constant.


Through calculation, one has

$$\begin{aligned}&\dot{\bar{V}}_{m,1}\le -\kappa _{m,1}\bar{V}_{m,1}+\Psi _{m,1}-\frac{1}{2}h_{m,1}^2(x_{m,1\tau }),\nonumber \\&p_{m,1}s_{m,1}h_{m,1}(x_{m,1\tau })\le \frac{1}{2}p_{m,1}^2s_{m,1}^2+\frac{1}{2}h_{m,1}^2(x_{m,1\tau }), \end{aligned}$$
(19)

where \(\Psi _{m,1}=[(e^{\kappa _{m,1}\bar{\tau }_{m,1}})/2(1-\check{\tau }_{m,1})]h_{m,1}^2(x_{m,1}).\)

By utilizing Assumption 3 and Young’s inequality yields

$$\begin{aligned} p_{m,1}s_{m,1}g_{m,1}\le&\frac{1}{2}p_{m,1}^2s_{m,1}^2+\sum \limits _{l= 1}^{M} \bar{\varpi }_{m,l}\aleph _1^2(\bar{x}_{l,1})\nonumber \\ \le&\frac{p_{m,1}^2}{2}s_{m,1}^2+16\tanh ^2\left( \frac{s_{m,1}}{\sigma _{m,1}}\right) \sum \limits _{l= 1}^M\bar{\varpi }_{m,l}\aleph _1^2(\bar{x}_{m,1})\nonumber \\&+\left( 1-16\tanh ^2\left( \frac{s_{m,1}}{\sigma _{m,1}}\right) \right) \sum \limits _{l= 1}^{M}\bar{\varpi }_{m,l}\aleph _1^2(\bar{x}_{m,1})\nonumber \\&+\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _1^2(\bar{x}_{l,1})-\aleph _1^2(\bar{x}_{m,1})), \end{aligned}$$
(20)

where \(\bar{\varpi }_{m,l}=(1/2)Mj_{m,l}^2\).

In light of Assumption 1 and Lemma 3 holds

$$\begin{aligned} p_{m,1}s_{m,1}\Delta _{m,1}\le&p_{m,1}\mid s_{m,1}\mid \bar{\varphi }_{m,11}(\Vert x_{m,1}\Vert )+0.2785\jmath _{m,11}\nonumber \\&+p_{m,1}\mid s_{m,1}\mid \varphi _{m,12}(\Vert \psi _i\Vert ), \end{aligned}$$
(21)

where \(\jmath _{m,11}>0\) denotes a constant and \(\bar{\varphi }_{m,11}(\Vert x_{m,1}\Vert )=\varphi _{m,11}(\Vert x_{m,1}\Vert )\tanh \left( (p_{m,1}\mid s_{m,1}\mid \varphi _{m,11}(x_{m,1}))/\jmath _{m,11}\right)\).

In view of Assumption 2 and Lemma 3, one obtains

$$\begin{aligned} p_{m,1}\mid s_{m,1}\mid \varphi _{m,12}(\Vert \psi _i\Vert )\le&\frac{1}{2}\varphi _{m,12}^2(\vartheta _{m,1}^{-1}(2\bar{D}_m(t_0,t)))\nonumber \\&+\frac{p_{m,1}^2s_{m,1}^2}{2}\nonumber \\&+p_{m,1}s_{m,1}\bar{\varphi }_{m,12}(x_{m,1},\lambda _m)\nonumber \\&+0.2785\jmath _{m,12}, \end{aligned}$$
(22)

where \(\jmath _{m,12}>0\) denotes a constant and \(\bar{\varphi }_{m,12}(x_{m,1},\lambda _m)=\varphi _{m,12}(\vartheta _{m,1}^{-1}(2\lambda _m))\tanh ((p_{m,1}s_{m,1}\varphi _{m,12}\left( \vartheta _{m,1}^{-1}(2\lambda _m)))/\jmath _{m,12}\right)\).

Calculating \(\dot{V}_{1}\), one obtains

$$\begin{aligned} \dot{V}_1 \le&\sum \limits _{m = 1}^M \left[ p_{m,1}s_{m,1}\left( s_{m,2}+\nu _{m,2}+\xi _{m,2}+\alpha _{m,1}\right. \right. \nonumber \\&+\frac{3}{2}p_{m,1}s_{m,1}+f_{m,1}-\frac{\dot{\ell }_m x_{m,1}}{\ell _m}\nonumber \\&\left. -\frac{\dot{\nu }_{m,1}}{p_{m,1}}\right) -\mu _{m,1}^{-1}\tilde{\Theta }_{m,1}\dot{\hat{\Theta } }_{m,1}-\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\nonumber \\&-\kappa _{m,1}\bar{V}_{m,1}+p_{m,1}s_{m,1}\varpi _{m,1}\nonumber \\& \left.+16\tanh ^2\left( \frac{s_{m,1}}{\sigma _{m,1}}\right) \sum \limits _{l= 1}^M \bar{\varpi }_{m,l}\aleph _1^2(\bar{x}_{m,1})\right)+\frac{\bar{\pi }_{m0}(\Vert x_{m,1}(t)\Vert )}{\varrho _{m0}} \nonumber \\&+\left( 1-16\tanh ^2\left( \frac{s_{m,1}}{\sigma _{m,1}}\right) \right) \sum \limits _{l= 1}^M\bar{\varpi }_{m,l}\aleph _1^2(\bar{x}_{m,1})+\Psi _{m,1}\nonumber \\&\left. +\sum \limits _{l=1}^M \bar{\varpi }_{m,l}(\aleph _1^2(\bar{x}_{l,1})-\aleph _1^2(\bar{x}_{m,1}))+\chi _{m,1}\right] , \end{aligned}$$
(23)

where \(\varpi _{m,1}=\bar{\varphi }_{m,11}(\Vert x_{m,1}\Vert )+\bar{\varphi }_{m,12}(x_{m,1},\lambda _m)\) and \(\chi _{m,1}=0.2785\omega _{m,11}+0.2785\omega _{m,12}+r_{m,2}/\varrho _{m0}+\frac{1}{2}\varphi _{m,12}^2(\vartheta _{m,1}^{-1}(2\bar{D}_m(t_0,t)))\).

The hyperbolic tangent function \(16\tanh ^2(s_{m,1}/\sigma _{m,1})\) is utilized to solve the \(\Psi _{m,1},\ \sum \nolimits _{l= 1}^M \bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{l,1})\), and \((\bar{\pi }_{m0}(\Vert x_{m,1}(t)\Vert )/\varrho _{m0})\).

$$\begin{aligned} \dot{V}_1 \le&\sum \limits _{m = 1}^M \left[ p_{m,1}s_{m,1}\left( s_{m,2}+\nu _{m,2}+\xi _{m,2}+\alpha _{m,1}\right. \right. \nonumber \\&+\frac{3}{2}p_{m,1}s_{m,1}+f_{m,1}-\frac{\dot{\ell }_m x_{m,1}}{\ell _m}\nonumber \\&\left. -\frac{\dot{\nu }_{m,1}}{p_{m,1}}+\frac{16}{p_{m,1}s_{m,1}}\tanh ^2 \left( \frac{s_{m,1}}{\sigma _{m,1}}\right) G_{m,1}\right) \nonumber \\&-\mu _{m,1}^{-1}\tilde{\Theta }_{m,1} \dot{\hat{\Theta } }_{m,1}-\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\nonumber \\&+p_{m,1}s_{m,1}\varpi _{m,1}+\left( 1-16\tanh ^2\left( \frac{s_{m,1}}{\sigma _{m,1}}\right) \right) G_{m,1} -\kappa _{m,1}\bar{V}_{m,1}\nonumber \\&\left. +\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _1^2(\bar{x}_{l,1})-\aleph _1^2(\bar{x}_{m,1}))+\chi _{m,1}\right] , \end{aligned}$$
(24)

where \(G_{m,1}=\Psi _{m,1}+(\bar{\pi }_{m0}(\Vert x_{m,1}(t)\Vert ))/\varrho _{m0}+\sum \nolimits _{l= 1}^M \bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{m,1})\) and \(\sigma _{m,1}>0\) denotes a constant. \(\square\)

Remark 1

Since \(\lim \nolimits _{s_{m,1} \rightarrow 0} [G_{m,1}/(p_{m,1}s_{m,1})]=\infty\), the lumped terms \(G_{m,1}/(p_{m,1}s_{m,1})\) cannot be directly approximated by the RBFNNs. To tackle this obstacle, a hyperbolic tangent item \((16/p_ms_{m,1})\tanh ^2(s_{m,1}/\sigma _{m,1}))G_{m,1}\) is used to prevent the singularity of the potential challenge. Meanwhile, two different cases will be further elaborated to handle the term \((1-16\tanh ^2(s_{m,1}/\sigma _{m,1}))G_{m,1}\) in Sect. 3.3.

To further simplify the \(\dot{V}_1\) in (24), the RBFNNs is employed to approximate the unknown item \(F_{m,1}(\bar{z}_{m,1})=f_{m,1}+(16/p_ms_{m,1})\tanh ^2(s_{m,1}/\sigma _{m,1})G_{m,1}+\varpi _{m,1}+2p_{m,1}s_{m,1}\), where the RBFNNs input vector \(\bar{z}_{m,1}=[x_{m,1},s_{m,1},p_{m,1},\lambda _1(t)]^{\rm{T}}\).

Thus, one has

$$\begin{aligned} F_{m,1}(\bar{z}_{m,1})=W_{m,1}^{*T}&S_{m,1}(\bar{z}_{m,1})+\varepsilon _{m,1},\nonumber \\ p_{m,1}s_{m,1}F_{m,1}(\bar{z}_{m,1})& \le \frac{p_{m,1}^2s_{m,1}^2}{2a_{m,1}^2}\Theta _{m,1}S_{m,1}^{\rm{T}}(\cdot )S_{m,1}(\cdot )+\frac{a_{m,1}^2}{2}\nonumber \\&\quad+\frac{1}{2}p_{m,1}^2s_{m,1}^2+\frac{\varepsilon _{m,1}^{*2}}{2}, \end{aligned}$$
(25)

where \(\varepsilon _{m,1}\) is the approximation error with \(\mid \varepsilon _{m,1}\mid \le \varepsilon _{m,1}^*\).

Substituting (11), (12), (15), and (25) into (24) becomes

$$\begin{aligned} \dot{V}_1 \le&\sum \limits _{m = 1}^M \left[ -c_{m,1}s_{m,1}^2+p_{m,1}s_{m,1}s_{m,2}+\rho _{m,1}\tilde{\Theta }_{m,1} \hat{\Theta }_{m,1}-\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\right. \nonumber \\&\left. +\left( 1-16\tanh ^2\left( \frac{s_{m,1}}{\sigma _{m,1}}\right) \right) G_{m,1} -\kappa _{m,1}\bar{V}_{m,1}+\chi _{m,1}\right. \nonumber \\&\left. +\frac{\varepsilon _{m,1}^{*2}}{2}+\frac{a_{m,1}^2}{2}+\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _1^2(\bar{x}_{l,1})-\aleph _1^2(\bar{x}_{m,1}))\right] . \end{aligned}$$
(26)

Step m, i. From (10) and (11), it deduces that

$$\begin{aligned} \dot{s}_{m,i} =&\, s_{m,i+1} +\nu _{m,i+1}+\xi _{m,i+1}+ \alpha _{m,i}+f_{m,i}+\Delta _{m,i}\nonumber \\&+g_{m,i}+h_{m,i}(x_{m,i\tau })- \dot{\zeta } _{m,i}-\dot{\nu }_{m,i}. \end{aligned}$$
(27)

By utilizing Assumption 3 and Young’s inequality holds

$$\begin{aligned} s_{m,i}g_{m,i}\le&\frac{1}{2}s_{m,i}^2+16\tanh ^2\left( \frac{s_{m,i}}{\sigma _{m,i}}\right) \sum \limits _{l= 1}^M\bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{m,i})\nonumber \\&+\left( 1-16\tanh ^2\left( \frac{s_{m,i}}{\sigma _{m,i}}\right) \right) \sum \limits _{l= 1}^M\bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{m,i})\nonumber \\&+\sum \limits _{l= 1}^M\bar{\varpi }_{m,l}(\aleph _i^2(\bar{x}_{l,i})-\aleph _i^2(\bar{x}_{m,i})). \end{aligned}$$
(28)

Select the following Lyapunov function candidate:

$$\begin{aligned} V_i = V_{i-1}+ \sum \limits _{m = 1}^M \left[ \frac{1}{2}s_{m,i}^2 + \frac{1}{2\mu _{m,i}}\tilde{\Theta }_{m,i}^2+\bar{V}_{m,i}\right] , \end{aligned}$$

where \(\kappa _{m,i}>0\) denotes a constant and \(\bar{V}_{m,i}=(e^{\kappa _{m,i}\bar{\tau }_{m,i}}/2(1-\check{\tau }_{m,i}))\int _{t - \tau _{m,i}(t)}^t e^{-\kappa _{m,i}(t-s)}h_{m,i}^2(x_{m,i}(s))ds.\)

Through calculation, one has

$$\begin{aligned}&\dot{\bar{V}}_{m,i}\le -\kappa _{m,i}\bar{V}_{m,i}+\Psi _{m,i}-\frac{1}{2}h_{m,i}^2(x_{m,i\tau }),\nonumber \\&s_{m,i}h_{m,i}(x_{m,i\tau })\le \frac{1}{2}s_{m,i}^2+\frac{1}{2}h_{m,i}^2(x_{m,i\tau }), \end{aligned}$$
(29)

where \(\Psi _{m,i}=(e^{\kappa _{m,i}\bar{\tau }_{m,i}}/2(1-\check{\tau }_{m,i}))h_{m,i}^2(x_{m,i})\).

Similar to (21) and (22) yields

$$\begin{aligned} s_{m,i}\Delta _{m,i}\le&\mid s_{m,i}\mid \varphi _{m,i1}(\Vert \bar{x}_{m,i}\Vert )\tanh \left( \frac{\mid s_{m,i}\mid \varphi _{m,i1}(\bar{x}_{m,i})}{\jmath _{m,i1}}\right) \nonumber \\&+0.2785\jmath _{m,i1}+\mid s_{m,i}\mid \varphi _{m,i2}(\Vert \psi _i\Vert ), \end{aligned}$$
(30)

where \(\jmath _{m,i1}>0\) denotes a constant.

$$\begin{aligned} \mid s_{m,i}\mid \varphi _{m,i2}(\Vert \psi _i\Vert )\le&s_{m,i}\bar{\varphi }_{m,i2}(\bar{x}_{m,i},\lambda _m)\nonumber \\&+0.2785\jmath _{m,i2}+\frac{s_{m,i}^2}{2}\nonumber \\&+\frac{1}{2}\varphi _{m,i2}^2(\vartheta _{m,i}^{-1}(2\bar{D}_m(t_0,t))), \end{aligned}$$
(31)

where \(\jmath _{m,i2}>0\) denotes a constant and \(\bar{\varphi }_{m,i2}(\bar{x}_{m,i},\lambda _m)=\varphi _{m,i2}(\vartheta _{m,i}^{-1}(2\lambda _m))\tanh ((s_{m,i}\varphi _{m,i2}(\vartheta _{m,i}^{-1}(2\lambda _m)))/\jmath _{m,i2})\).

The time derivative of \(V_i\) can be obtained

$$\begin{aligned} \dot{V}_i \le&\dot{V}_{i-1}+\sum \limits _{m = 1}^M \left[ s_{m,i}\left( s_{m,i+1}+\nu _{m,i+1}+\frac{3}{2}s_{m,i}\right. \right. \nonumber \\&+\xi _{m,i+1}+\alpha _{m,i}-\dot{\zeta }_{m,i}\nonumber \\&+\frac{16}{s_{m,i}}\tanh ^2\left( \frac{s_{m,i}}{\sigma _{m,i}}\right) \sum \limits _{l= 1}^M\bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{m,i})+f_{m,i}\nonumber \\&\left. -\dot{\nu }_{m,i}\right) -\kappa _{m,i}\bar{V}_{m,i}\nonumber \\&- \mu _{m,i}^{-1}\tilde{\Theta }_{m,i} \dot{\hat{\Theta } }_{m,i}+\left( 1-16\tanh ^2\left( \frac{s_{m,i}}{\sigma _{m,i}}\right) \right) \nonumber \\&\sum \limits _{l=1}^M\bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{m,i})\nonumber \\&+s_{m,i}\varpi _{m,i}+\Psi _{m,i}+\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _i^2(\bar{x}_{l,i})\nonumber \\&\left. -\aleph _i^2(\bar{x}_{m,i}))+\chi _{m,i}\right] , \end{aligned}$$
(32)

where \(\varpi _{m,i}=\varphi _{m,i1}(\Vert \bar{x}_{m,i}\Vert )\tanh \left( (\mid s_{m,i}\mid \varphi _{m,i1}(\bar{x}_{m,i}))/\jmath _{m,i1}\right) +\bar{\varphi }_{m,i2}(\bar{x}_{m,i},\lambda _m)\) and \(\chi _{m,i}=\frac{1}{2}\varphi _{m,i2}^2(\vartheta _{m,i}^{-1}(2\bar{D}_m(t_0,t)))+0.2785\jmath _{m,i1}+0.2785\jmath _{m,i2}.\)

By applying the hyperbolic tangent function, \(\dot{V}_i\) becomes

$$\begin{aligned} \dot{V}_i \le&\dot{V}_{i-1}+\sum \limits _{m = 1}^M \left[ s_{m,i}\left( s_{m,i+1}+\nu _{m,i+1}+\xi _{m,i+1}+\frac{3}{2}s_{m,i}\right. \right. \nonumber \\&- \dot{\zeta } _{m,i}-\dot{\nu }_{m,i}\nonumber \\&\left. +f_{m,i}+\alpha _{m,i}+\frac{16}{s_{m,i}}\tanh ^2\left( \frac{s_{m,i}}{\sigma _{m,i}}\right) G_{m,i}\right) - \mu _{m,i}^{-1}\tilde{\Theta }_{m,i} \dot{\hat{\Theta } }_{m,i}\nonumber \\&+s_{m,i}\varpi _{m,i}-\kappa _{m,i}\bar{V}_{m,1} +\left( 1-16\tanh ^2\left( \frac{s_{m,i}}{\sigma _{m,i}}\right) \right) G_{m,i}\nonumber \\&\left. +\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _i^2(\bar{x}_{l,i})-\aleph _i^2(\bar{x}_{m,i}))+\chi _{m,i}\right] , \end{aligned}$$
(33)

where \(G_{m,i}=\Psi _{m,i}+\sum \nolimits _{l= 1}^M \bar{\varpi }_{m,l}\aleph _i^2(\bar{x}_{m,i}).\)

The RBFNNs is utilized to approximate the unknown item \(F_{m,i}(\bar{z}_{m,i})=f_{m,i}+(16/s_{m,i})\tanh ^2(s_{m,i}/\sigma _{m,i})G_{m,i}+\varpi _{m,i}- \dot{\zeta } _{m,i}+2s_{m,i}\) in (33), where the RBFNNs input vector \(\bar{z}_{m,i}=[x_{m,i},s_{m,i},\lambda _i(t),\dot{\zeta } _{m,i}]^{\rm{T}}\). Therefore, one obtains

$$\begin{aligned} F_{m,i}(\bar{z}_{m,i})=W_{m,i}^{*T}&S_{m,i}(\bar{z}_{m,i})+ \varepsilon _{m,i},\nonumber \\ s_{m,i}F_{m,i}(\bar{z}_{m,i})\le&\frac{s_{m,i}^2}{2a_{m,i}^2}\Theta _{m,i}S_{m,i}^{\rm{T}}(\cdot )S_{m,i}(\cdot )+\frac{a_{m,i}^2}{2}\nonumber \\&+\frac{1}{2}s_{m,i}^2+\frac{\varepsilon _{m,i}^{*2}}{2}, \end{aligned}$$
(34)

where \(\varepsilon _{m,i}\) is the approximation error with \(\mid \varepsilon _{m,i}\mid \le \varepsilon _{m,i}^*\).

Substituting (11), (13), (16), and (34) into (33) yields

$$\begin{aligned} \dot{V}_i \le&\sum \limits _{m = 1}^M \left[ -\sum \limits _{k = 1}^{i}c_{m,k}s_{m,k}^2+s_{m,i}s_{m,i+1}+\sum \limits _{k = 1}^{i}\rho _{m,k}\tilde{\Theta }_{m,k} \hat{\Theta }_{m,k}\right. \nonumber \\&-\sum \limits _{k =1}^{i}\kappa _{m,k}\bar{V}_{m,k}\nonumber \\&+\sum \limits _{k = 1}^{i}\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k} -\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\nonumber \\&+\sum \limits _{k = 1}^{i}\frac{\varepsilon _{m,k}^{*2}}{2} +\sum \limits _{k =1}^{i}\frac{a_{m,k}^2}{2}\nonumber \\&\left. +\sum \limits _{k = 1}^{i}\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _k^2(\bar{x}_{l,k})-\aleph _k^2(\bar{x}_{m,k}))+\sum \limits _{k = 1}^{i}\chi _{m,k}\right] . \end{aligned}$$
(35)

Step \(m,{n_m}\). In industrial process, the existence of actuator saturation nonlinearity will affect the system stability. Therefore, the following saturation function is considered:

$$\begin{aligned} u_m={\text{sat}}(\bar{u}_m)= \left\{ \begin{array}{ll} {\text{sign}}(\bar{u}_m)u_m^{\rm{Max}}, &\quad \ \mid \bar{u}_m\mid \ge u_m^{\rm{Max}}, \\ \bar{u}_m,&\quad \ \mid \bar{u}_m\mid <u_m^{\rm{Max}}, \end{array} \right. \end{aligned}$$

where \(u_m^{\rm{Max}}\) is the bound of \(u_m\) and \(\bar{u}_m\) is the input of the following saturation nonlinearity:

$$\begin{aligned} \bar{b}_m(\bar{u}_m)=u_m^{\rm{Max}}*\frac{e^{\frac{\bar{u}_m}{u_m^{\rm{Max}}}}-e^{-\frac{\bar{u}_m}{u_m^{\rm{Max}}}}}{ e^{\frac{\bar{u}_m}{u_m^{\rm{Max}}}}+e^{-\frac{\bar{u}_m}{u_m^{\rm{Max}}}}}. \end{aligned}$$

Meanwhile the quantized parameters satisfy \((1-\bar{\kappa }_{m})u_{m}^{\rm{min}}/(\bar{\kappa }_{m}+\hslash _{m})\ge u_m^{\rm{Max}}\). Referring to [40], \({\text{sat}}(\bar{u}_m)\) becomes

$$\begin{aligned} u_m={\text{sat}}(\bar{u}_m)=\bar{b}_m(\bar{u}_m)+\bar{\varrho }_m(\bar{u}_m), \end{aligned}$$
(36)

where \(\mid \bar{\varrho }(\bar{u}_m)\mid =\mid {\text{sat}}(\bar{u}_m))-\bar{b}_m(\bar{u}_m)\mid \le u_m^{\rm{Max}}(1-\tanh (1))=\bar{\underline{\varrho }}_{m}.\)


According to the mean-value theorem, \(\bar{b}_m(\bar{u}_m)\) can be expressed as:

$$\begin{aligned} \bar{b}_m(\bar{u}_m)=\bar{b}_m(\bar{u}_{m0})+\frac{\partial \bar{b}_{m}(\cdot )}{\partial \bar{u}_m} \mid _{\bar{u}_m=\bar{u}_{m}^{c_m}}(\bar{u}_{m}-\bar{u}_{m0}), \end{aligned}$$
(37)

where \(\bar{u}_{m}^{c_m}=c_m\bar{u}_m+(1-c_m)\bar{u}_{m0}\) and \(0<c_m<1\).

Let \(\bar{u}_{m0}=0\), (37) becomes

$$\begin{aligned} \bar{b}_m(\bar{u}_m)=\frac{\partial \bar{b}_{m}(\cdot )}{\partial \bar{u}_m}\mid _{\bar{u}_m =\bar{u}_{m}^{c_m}}\bar{u}_{m}=\zeta _{m0}(\bar{u}_{m}^{c_m})\bar{u}_{m}. \end{aligned}$$
(38)

For the positive constants \(\underline{\zeta }_{m0}\) and \(\bar{\zeta }_{m0}\), the expression \(\underline{\zeta }_{m0}<\zeta _{m0}(\bar{u}_{m}^{c_m})<\bar{\zeta }_{m0}\) holds. Thus, (36) becomes

$$\begin{aligned} u_m={\text{sat}}(\bar{u}_m)=\zeta _{m0}(\bar{u}_{m}^{c_m})\bar{u}_{m}+\bar{\varrho }_m(\bar{u}_m). \end{aligned}$$
(39)

According to (9)–(11) and (39), \(\dot{s}_{m,n_m}\) can be expressed as:

$$\begin{aligned} \dot{s}_{m,n_m} =&\,(1-\bar{\kappa }_{m})\zeta _{m0}\bar{u}_{m}+(1-\bar{\kappa }_{m})\bar{\varrho }_m+\bar{\kappa }_{m} \bar{\varsigma }_m\nonumber \\&+f_{m,n_m}+\Delta _{m,n_m}\nonumber \\&+h_{m,n_m}(x_{m,n_m\tau })+g_{m,n_m}-\dot{\zeta } _{m,n_m}-\dot{\nu }_{m,n_m}. \end{aligned}$$
(40)

Let us construct the Lyapunov function candidate as \(V_{n_m} = V_{n_m-1}+ \sum \limits _{m = 1}^M \left[ \frac{1}{2}s_{m,n_m}^2 + \frac{1}{2\mu _{m,n_m}}\tilde{\Theta }_{m,n_m}^2+\bar{V}_{m,n_m}\right].\)

By utilizing Young’s inequality, one has

$$\begin{aligned} s_{m,n_m}(1-\bar{\kappa }_{m})\bar{\varrho }_m\le \frac{1}{2}s_{m,n_m}^2+\frac{1}{2}(1-\bar{\kappa }_{m})^2\bar{\underline{\varrho }}_{m}^2. \end{aligned}$$
(41)

Then, it can be deduced that

$$\begin{aligned} \dot{V}_{n_m} \le&\dot{V}_{n_m-1}+\sum \limits _{m = 1}^M \left[ s_{m,n_m}((1-\bar{\kappa }_{m})\zeta _{m0}\bar{u}_{m}+2s_{m,n_m}+f_{m,n_m}- \dot{\zeta } _{m,n_m}\right. \nonumber \\&\left. +\frac{16}{s_{m,n_m}}\tanh ^2\left( \frac{s_{m,n_m}}{\sigma _{m,n_m}}\right) \sum \limits _{l= 1}^M\bar{\varpi }_{m,l}\aleph _{n_m}^2(\bar{x}_{m,n_m})-\dot{\nu }_{m,n_m}\right. \nonumber \\&\left. +\left( 1-16\tanh ^2\left( \frac{s_{m,n_m}}{\sigma _{m,n_m}}\right) \right) \sum \limits _{l= 1}^M \bar{\varpi }_{m,l}\aleph _{n_m}^2(\bar{x}_{m,n_m}))+\bar{\kappa }_{m} \bar{\varsigma }_m\right. \nonumber \\&\left. -\kappa _{m,n_m}\bar{V}_{m,n_m}+s_{m,n_m}\varpi _{m,n_m}-\mu _{m,n_m}^{-1}\tilde{\Theta }_{m,n_m} \dot{\hat{\Theta } }_{m,n_m}\right. \nonumber \\&\left. +\Psi _{m,n_m}+\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _{n_m}^2(\bar{x}_{l,n_m})-\aleph _{n_m}^2(\bar{x}_{m,n_m}))+\chi _{m,n_m}\right] , \end{aligned}$$
(42)

where

$$\begin{aligned} \Psi _{m,n_m}=&\,[e^{\kappa _{m,n_m}\bar{\tau }_{m,n_m}}h_{m,n_m}^2(x_{m,n_m})/2 (1-\check{\tau }_{m,n_m})],\\ \varpi _{m,n_m}=&\,\varphi _{m,n_m1}(\Vert \bar{x}_{m,n_m}\Vert )\tanh ((\mid s_{m,n_m}\mid \varphi _{m,n_m1}(\bar{x}_{m,n_m}))/\jmath _{m,{n_m1}})\\&+\bar{\varphi }_{m,n_m2}(\bar{x}_{m,n_m},\lambda _m),\\ \chi _{m,n_m}=&\,0.2785\jmath _{m,{n_m1}}+0.5\varphi _{m,n_m2}^2(\vartheta _{m,n_m}^{-1} (2\bar{D}_m(t_0,t)))\\&+0.2785\jmath _{m,{n_m2}}+0.5(1-\bar{\kappa }_{m})^2\bar{\underline{\varrho }}_{m}^2. \end{aligned}$$

Define \(G_{m,n_m}=\Psi _{m,n_m}+\sum \nolimits _{l= 1}^M \bar{\varpi }_{m,l}\aleph _{n_m}^2(\bar{x}_{m,n_m})\) yields

$$\begin{aligned} \dot{V}_{n_m}\le&\dot{V}_{n_m-1}+\sum \limits _{m = 1}^M \left[ s_{m,n_m}\left( (1-\bar{\kappa }_{m})\zeta _{m0}\bar{u}_{m}+\bar{\kappa }_{m}\bar{\varsigma }_m+f_{m,n_m}-\dot{\zeta } _{m,n_m}\right. \right. \nonumber \\&\left. \left. +\frac{16}{s_{m,n_m}}\tanh ^2\left( \frac{s_{m,n_m}}{\sigma _{m,n_m}}\right) G_{m,n_m} +\varpi _{m,n_m}-\dot{\nu }_{m,n_m}+2s_{m,n_m}\right) \right. \nonumber \\&- \mu _{m,n_m}^{-1}\tilde{\Theta }_{m,n_m} \dot{\hat{\Theta } }_{m,n_m}+\left( 1-16\tanh ^2\left( \frac{s_{m,n_m}}{\sigma _{m,n_m}}\right) \right) G_{m,n_m}\nonumber \\&+\chi _{m,n_m} -\kappa _{m,n_m}\bar{V}_{m,n_m}+\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _{n_m}^2(\bar{x}_{l,n_m})\nonumber \\&\left.-\aleph _{n_m}^2(\bar{x}_{m,n_m}))\right] . \end{aligned}$$
(43)

The RBFNNs is employed to approximate the unknown function \(F_{m,n_m}(\bar{z}_{m,n_m})=f_{m,n_m}+(16/s_{m,n_m})\tanh ^2(s_{m,n_m}/\sigma _{m,n_m})G_{m,n_m}+\varpi _{m,n_m}- \dot{\zeta } _{m,n_m}+2.5s_{m,n_m}\) in (43), where the RBFNNs input vector \(\bar{z}_{m,n_m}=[x_{m,n_m},s_{m,n_m},\lambda _{n_m}(t),\dot{\zeta } _{m,n_m}]^{\rm{T}}\). Therefore, one obtains

$$\begin{aligned} F_{m,n_m}(\bar{z}_{m,n_m})=&\,W_{m,n_m}^{*T}S_{m,i}(\bar{z}_{m,n_m})+ \varepsilon _{m,n_m},\nonumber \\ s_{m,n_m}F_{m,n_m}(\bar{z}_{m,n_m})&\le \frac{s_{m,n_m}^2}{2a_{m,n_m}^2}\Theta _{m,n_m}S_{m,n_m}^{\rm{T}}(\cdot )S_{m,n_m}(\cdot )\nonumber \\&+\frac{a_{m,n_m}^2}{2}+\frac{1}{2}s_{m,n_m}^2+\frac{\varepsilon _{m,n_m}^{*2}}{2}. \end{aligned}$$
(44)

where \(\varepsilon _{m,n_m}\) is the approximation error with \(\mid \varepsilon _{m,n_m}\mid \le \varepsilon _{m,n_m}^*\).

Meanwhile, the self-triggered scheme is developed as follows:

$$\begin{aligned}&\bar{u}_m(t)=v_m(t_k ), \ \forall t \in [ t_k , t_{k + 1} ),\nonumber \\&t_{k + 1}=t_k+\frac{\Lambda _m\mid \bar{u}_m(t)\mid +E_m}{\max \left\{ L_m,\mid \dot{v}_m(t)\mid \right\} }, \end{aligned}$$
(45)

where \(t_k,t_{k+1}\in Z^+; 0<\Lambda _m<1\), \(E_m\) and \(L_m\) are positive constant. \(\Lambda _m\mid \bar{u}_m(t)\mid +E_m\) denotes control signal interval between two successive triggered instants; \(L_m\) and \(\mid \dot{v}_m(t)\mid\) are change rates of control signal interval.

When the conditions of (45) are satisfied, \(\bar{u}_m(t)=v_m(t_k)\) will be used to the control plant in (1). The next trigger point \(t_{k+1}\) will be obtained, and control signal \(\bar{u}_m(t)\) holds as a constant \(v_m(t_k )\) in the period interval \([t_k,t_{k+1})\).

\(v_m(t_k )\) is constructed as follows:

$$\begin{aligned} v_m =&\,-(1+\Lambda _m )\left( \alpha _{m,n_m}\tanh \left( \frac{s_{m,n_m}\alpha _{m,n_m}}{\jmath _m} \right) \right. \nonumber \\&\left. +\,\bar{E}_m\tanh \left( \frac{s_{m,n_m}\bar{E}_m}{\jmath _m}\right) \right) , \end{aligned}$$
(46)

where \(\jmath _m> 0\) and \(\bar{E}_m>E_m/(1-\Lambda _m)\).

From (45), it has \(\mid \bar{u}_m(t_{k+1})-u_m(t)\mid \le \Lambda _m\mid \bar{u}_m(t)\mid +E_m\). There exist the continuous time-varying parameters \(\theta _{m,1}, \theta _{m,2}\) satisfying \(\theta _{m,1}(t_k)=\theta _{m,2}(t_k)=0, \theta _{m,1}(t_{k+1})=\theta _{m,2}(t_{k+1})=\pm 1\) and \(\mid \theta _{m,1}\mid \le 1\), \(\mid \theta _{m,2}\mid \le 1, \forall t\in [t_k,t_{k+1})\),

$$\begin{aligned} \bar{u}_m= \frac{v_m - \theta _{m,2}E_m}{1 + \theta _{m,1}\Lambda _m}. \end{aligned}$$
(47)

Therefore, \(\dot{V}_{n_m}\) can be expressed as:

$$\begin{aligned} \dot{V}_{n_m}\le&\dot{V}_{n_m-1}+\sum \limits _{m = 1}^M \left[ s_{m,n_m}\left( (1-\bar{\kappa }_{m})\zeta _{m0}\frac{v_m - \theta _{m,2}E_m}{1 + \theta _{m,1}\Lambda _m}\right. \right. \nonumber \\&+\bar{\kappa }_{m}\bar{\varsigma }_m-\dot{\nu }_{m,n_m}\nonumber \\&\left. +\frac{s_{m,n_m}}{2a_{m,n_m}^2}\Theta _{m,n_m}S_{m,n_m}^{\rm{T}}S_{m,n_m}\right) \nonumber \\&- \mu _{m,n_m}^{-1}\tilde{\Theta }_{m,n_m} \dot{\hat{\Theta } }_{m,n_m}+\frac{a_{m,n_m}^2}{2}\nonumber \\&-\kappa _{m,n_m}\bar{V}_{m,n_m}+\left( 1-16\tanh ^2\left( \frac{s_{m,n_m}}{\sigma _{m,n_m}}\right) \right) G_{m,n_m}\nonumber \\&+\frac{\varepsilon _{m,n_m}^{*2}}{2}\nonumber \\&\left. +\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _{n_m}^2(\bar{x}_{l,n_m})-\aleph _{n_m}^2(\bar{x}_{m,n_m}))+\chi _{m,n_m}\right] . \end{aligned}$$
(48)

On the basis of (46) and Lemma 3, it has \(s_{m,n_m}v_m<0\). In light of \(0< [(s_{m,n_m}v_m)/(1 + \theta _{m,1}\Lambda _m)]< [(s_{m,n_m}v_m)/(1 +\Lambda _m)]\) and \(- [(s_{m,n_m}\theta _{m,2}E_m)/(1 + \theta _{m,1}\Lambda _m)] \le \mid s_{m,n_m}E_m/1 -\Lambda _m\mid \le \mid {s_{m,n_m}}{\bar{E}_m} \mid,\) the following inequality holds

$$\begin{aligned} s_{m,n_m}&(1-\bar{\kappa }_{m})\zeta _{m0}\left( \frac{v_m - \theta _{m,2}E_m}{1 + \theta _{m,1}\Lambda _m}-\alpha _{m,n_m}\right) \nonumber \\ \le&\frac{s_{m,n_m}(1-\bar{\kappa }_{m})\zeta _{m0}v_m}{1 +\Lambda _m}+\mid {s_{m,n_m}}(1-\bar{\kappa }_{m})\zeta _{m0}{\bar{E}_m}\mid \nonumber \\&-s_{m,n_m}(1-\bar{\kappa }_{m})\zeta _{m0}\alpha _{m,n_m}\nonumber \\ \le&0.557\jmath _m(1-\bar{\kappa }_{m})\zeta _{m0}. \end{aligned}$$
(49)

Since \(\tilde{\Theta }_{m,i}\hat{\Theta }_{m,i} \le - (\tilde{\Theta } _{m,i}^2/2) + (\Theta _{m,i}^2/2)\) and invoking (11), (14), (48) and (49) results in

$$\begin{aligned} \dot{V}_{n_m}\le&\sum \limits _{m = 1}^M \left[ -\sum \limits _{k = 1}^{n_m}c_{m,k}s_{m,k}^2-\sum \limits _{k = 1}^{n_m}\frac{\rho _{m,k}}{2}\tilde{\Theta }_{m,k}^2-\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\right. \nonumber \\&-\sum \limits _{k = 1}^{n_m}\kappa _{m,k}\bar{V}_{m,k}\nonumber \\&+\sum \limits _{k = 1}^{n_m}\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}-(1-\bar{\kappa }_{m})s_{m,n_m}u_{m}^{\rm{min}}\nonumber \\&\left. +s_{m,n_m}\bar{\kappa }_{m} \bar{\varsigma }_m+\bar{\chi }_m+\sum \limits _{k = 1}^{n_m}\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _{k}^2(\bar{x}_{l,k})-\aleph _{k}^2(\bar{x}_{m,k}))\right] , \end{aligned}$$
(50)

where \(\bar{\chi }_m=\sum \nolimits _{k = 1}^{n_m}\chi _{m,k}+\sum \nolimits _{k = 1}^{n_m}(\rho _{m,k}/2)\Theta _{m,k}^2 +\sum \nolimits _{k = 1}^{n_m}(a_{m,k}^2/2)+\sum \nolimits _{k = 1}^{n_m}(\varepsilon _{m,k}^{*2}/2)+0.557\jmath _m(1-\bar{\kappa }_{m})\zeta _{m0}\).

From the analysis in [16, 41], the Lyapunov function is designed as \(V=\bar{\delta }_mV_{n_m}\), where \(\bar{\delta }_m>0\) refers to the cofactor of the mth diagonal element by instead of \(r_{m,l}\) with \(\bar{\varpi }_{m,l}\). Thus, (50) becomes

$$\begin{aligned} \dot{V}_{n_m}\le&\sum \limits _{m = 1}^M \bar{\delta }_m\left[ -\sum \limits _{k = 1}^{n_m}c_{m,k}s_{m,k}^2-\sum \limits _{k = 1}^{n_m}\frac{\rho _{m,k}}{2}\tilde{\Theta }_{m,k}^2-\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\right. \nonumber \\&-\sum \limits _{k = 1}^{n_m}\kappa _{m,k}\bar{V}_{m,k} +\sum \limits _{k =1}^{n_m}\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}\nonumber \\& -(1-\bar{\kappa }_{m})s_{m,n_m}u_{m}^{\rm{min}}\nonumber \\&\left. +s_{m,n_m}\bar{\kappa }_{m} \bar{\varsigma }_m+\bar{\chi }_m+\sum \limits _{k = 1}^{n_m}\sum \limits _{l= 1}^M \bar{\varpi }_{m,l}(\aleph _{k}^2(\bar{x}_{l,k})-\aleph _{k}^2(\bar{x}_{m,k}))\right] . \end{aligned}$$
(51)

In light of [4], \(\sum \nolimits _{m = 1}^M\sum \nolimits _{k = 1}^{n_m}\sum \nolimits _{l= 1}^M\bar{\delta }_m\bar{\varpi }_{m,l}(\aleph _{k}^2(\bar{x}_{l,k})-\aleph _{k}^2(\bar{x}_{m,k}))=0\) holds. Therefore, (51) arrives at

$$\begin{aligned} \dot{V}\le&\sum \limits _{m = 1}^M \bar{\delta }_m\left[ -\sum \limits _{k = 1}^{n_m}c_{m,k}s_{m,k}^2-\sum \limits _{k = 1}^{n_m}\frac{\rho _{m,k}}{2}\tilde{\Theta }_{m,k}^2 -\frac{\bar{r}_{m,1}\lambda _m}{\varrho _{m0}}\right. \nonumber \\&\left. +\sum \limits _{k = 1}^{n_m}\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}-(1-\bar{\kappa }_{m})s_{m,n_m}u_{m}^{\rm{min}}\right. \nonumber \\&\left. +s_{m,n_m}\bar{\kappa }_{m} \bar{\varsigma }_m-\sum \limits _{k = 1}^{n_m}\kappa _{m,k}\bar{V}_{m,k}+\bar{\chi }_{m}\right] . \end{aligned}$$
(52)

To this end, the adaptive self-triggered neural control scheme was preliminary accomplished. The results are summarized in the following Theorem 1.

3.3 Stability analysis

Theorem 1

Let us consider the interconnected nonlinear time-delay system in (1), under Assumptions 1–3, the error compensation signals in (11), the virtual control functions in (12)–(14), the adaptive updating law in (15)–(16), then the decentralized-based adaptive quantized control algorithm guarantees the following features:

  • The system output was confined to a small adjustable region in finite time.

  • All the signals of the CLS remain SGUUB.

  • The Zeno phenomenon was ruled out.

Proof

Let us select the Lyapunov function \(\bar{V}_{\nu }=\sum \nolimits _{m = 1}^M\sum \nolimits _{i=1}^{n_m}\frac{1}{2}\nu _{m,i}^2\) to prove the boundedness of \(\nu _{m,i}\).

$$\begin{aligned} \dot{\bar{V}}_\nu =&\,\sum \limits _{m = 1}^M\left[ - c_{m,1}\nu _{m,1}^2+p_{m,1}\nu _{m,1}\nu _{m,2}+p_{m,1}\nu _{m,1}\xi _{m,2}\right. \nonumber \\&-c_{m,2}\nu _{m,2}^2- p_{m,{1}}\nu _{m,{1}}\nu _{m,2}+\nu _{m,2}\nu _{m,{3}}+\nu _{m,2}\xi _{m,{3}}\nonumber \\&\left. +\cdots - c_{m,n_m}\nu _{m,n_m}^2- p_{m,{n_m-1}}\nu _{m,{n_m-1}}\nu _{m,n_m}\right] \nonumber \\ =&\,\sum \limits _{m = 1}^M\left[ -\sum _{l=1}^{n_m} c_{m,l}\nu _{m,l}^2+\sum _{l=1}^{n_m-1}p_l\nu _{m,l}\xi _{m,l+1}\right] , \end{aligned}$$
(53)

where \(p_l=\max \left\{ {p_{m,1},1}\right\}\).

From [27], one has \(\xi _{m,l+1}=\mid \zeta _{m,{l+1}}-\alpha _{m,l}\mid \le \Delta _{m,l}\ (l=1,\cdots ,n_m-1)\) and define \(D_1=\min \{(c_{m,1}-p_{m,1}^2/2),\ (c_{m,2}-1/2), \ldots ,\)

\((c_{m,n_m}-1/2)\}\) and \(D_2=\frac{1}{2}\Delta _{m,l}^2\). Thus, (53) yields

$$\begin{aligned} \dot{\bar{V}}_\nu&\le \sum \limits _{m = 1}^M\left[ -\sum _{l=1}^{n_m} c_{m,l}\nu _{m,l}^2+\sum _{l=1}^{n_m}p_l\Delta _{m,l}\mid \nu _{m,l}\mid \right] \nonumber \\&\le -D_1\bar{V}_\nu +D_2. \end{aligned}$$
(54)

Therefore, the boundness of \(\nu _{m,i}\) can be obtained. This completes the proof. \(\square\)

Let \(A=\min \bar{\delta }_m\left\{ 2c_{m,k}, \rho _{m,k}\mu _{m,k}, \bar{r}_{m,1}, \kappa _{m,k}\right\}\) and \(\chi _{n_m}= \sum \nolimits _{m = 1}^M \bar{\delta }_m\bar{\chi }_{m}\), (52) is further simplified as:

$$\begin{aligned} \dot{V}\le&-A{V}+ \chi _{n_m}+\sum \limits _{m = 1}^M\sum \limits _{k = 1}^{n_m}\bar{\delta }_m\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}\nonumber \\&+\sum \limits _{m = 1}^M(s_{m,n_m}\bar{\kappa }_{m} \bar{\varsigma }_m-(1-\bar{\kappa }_{m})s_{m,n_m}u_{m}^{\rm{min}}). \end{aligned}$$
(55)

Firstly, by using Lemma 1, let us discuss the following two cases about \(\sum \nolimits _{m = 1}^M(s_{m,n_m}\bar{\kappa }_{m} \bar{\varsigma }_m-(1-\bar{\kappa }_{m})s_{m,n_m}u_{m}^{\rm{min}})\).


Case I: \(u_{m}^{\rm{min}}\le u_m^{\rm{Max}}\), there exist two probabilities, i.e.,

(a): \(\mid u_m\mid \le u_{m}^{\rm{min}}\),based on Lemma 1 with \(\mid \bar{\varsigma }_m(t)\mid \le (1-\bar{\kappa }_{m})u_{m}^{\rm{min}}/\bar{\kappa }_{m}\), it can be obtained

$$\begin{aligned} \dot{V}\le -A{V}+\sum \limits _{m = 1}^M\sum \limits _{k = 1}^{n_m}\bar{\delta }_m\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}+ \chi _{n_m}. \end{aligned}$$
(56)

(b): \(\mid u_m\mid \ge u_{m}^{\rm{min}}\), based on Lemma 1 and \(\mid \bar{\varsigma }_m(t)\mid \le (\bar{\kappa }_{m}+\hslash _{m})\mid u_m\mid /\bar{\kappa }_{m}\), one has

$$\begin{aligned} \dot{V}\le&-A{V}+\chi _{n_m}+\sum \limits _{m = 1}^M\sum \limits _{k = 1}^{n_m}\bar{\delta }_m\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}\nonumber \\&+\sum \limits _{m = 1}^M\left( s_{m,n_m}(\bar{\kappa }_{m}+\hslash _{m})\mid u_m\mid -(1-\bar{\kappa }_{m})s_{m,n_m}u_{m}^{\rm{min}}\right) . \end{aligned}$$
(57)

Since \((1-\bar{\kappa }_{m})u_{m}^{\rm{min}}/(\bar{\kappa }_{m}+\hslash _{m})\ge u_m^{\rm{Max}}\), then we can obtain \(\mid u_m\mid \le u_m^{\rm{Max}}\le (1-\bar{\kappa }_{m})u_{m}^{\rm{min}}/(\bar{\kappa }_{m}+\hslash _{m})\). By invoking (57), the similar results with the form in (56) can be derived.

Case II


\(u_{m}^{\rm{min}}\ge u_m^{\rm{Max}}\), then \(\mid u_m\mid \le u_{m}^{\rm{min}}\). The similar result can be derived by referring to (a) in Case I.

In a nutshell, by combining with the Case I and II, the inequality (56) can be obtained accordingly.

Next, let us discuss the sign of \(\sum \nolimits _{m = 1}^M\sum \nolimits _{k = 1}^{n_m}\bar{\delta }_m(1-16\tanh ^2(s_{m,k}/\sigma _{m,k}))G_{m,k}\), two cases exist as follows:

Case A


\(s_{m,k}\in \Omega _{m_k}=\left\{ s_{m,k}\mid \mid s_{m,k}\mid <0.2554m_k\right\}\) for \(1\le m \le M\) and \(1\le k\le n_m\). Thus, \(s_{m,k}\) is bounded. According to \(z_{m,k}=s_{m,k}+\nu _{m,k}\) and the boundedness of \(\nu _{m,k}\), \(z_{m,k}\) is bounded. Because of (16), \(\hat{\Theta }_{m,k}\) is bounded. Due to \(\Theta _{m,k}\) being constant, the estimation error \(\tilde{\Theta }_{m,k}\) is also bounded. Thus, the boundedness of \(\alpha _{m,k}\) and \(u_m\) are bounded, and \(\zeta _{m,k}\) is also bounded. According to (9), \(x_{m,k}\) and \(\partial _m\) are bounded. Since \(G_{m,k}\) is a non-negative function, \(\bar{\delta }_m\left( 1-16\tanh ^2\left( \frac{s_{m,k}}{\sigma _{m,k}}\right) \right) G_{m,k}\) is bounded and suppose \(G_{m0}\) is its upper bound. On the basis of (56), one obtains

$$\begin{aligned} \dot{V} \le -A{V}+ \hat{\chi }_{n_m}, \end{aligned}$$
(58)

where \(\hat{\chi }_{n_m}=\chi _{n_m}+G_{m0}\). Furthermore, it can be obtained that \(0\le V \le [V(0)-(\hat{\chi }_{n_m}/A)]e^{-At}+ (\hat{\chi }_{n_m}/A)\). Consequently, all signals of the CLS are regulated to a compact set \(\Xi _1\).

$$\begin{aligned} \Xi _1= \left\{ (x_{m,k}, z_{m,k}, s_{m,k}, \tilde{\Theta }_{m,k})\mid \mid V\le \frac{\hat{\chi }_{n_m}}{A} \right\} . \end{aligned}$$

Case B


\(s_{m,k}\notin \Omega _{m_k}\) for \(1\le m \le M\) and \(1 \le k \le n_m\). In light of Lemma 4, one has \(\bar{\delta }_m(1-16{\text{tanh}}^2(s_{m,k}/{m_k}))G_{m,k}\le 0\). Thus, (56) yields

$$\begin{aligned} \dot{V}\le -A{V}+\chi _{n_m}. \end{aligned}$$
(59)

The same as Case A, all signals of the CLS are regulated to a compact set \(\Xi _2\).

$$\begin{aligned} \Xi _2= \left\{ (x_{m,k}, z_{m,k}, s_{m,k}, \tilde{\Theta }_{m,k})\mid \mid V\le \frac{\chi _{n_m}}{A} \right\} . \end{aligned}$$

To sum up the above, it can be concluded that all signals in the CLS remain SGUUB. According to the above discussions, the parameters selection guideline of the proposed control solution is given in Algorithm 1.

Remark 2

In contrast to the classical ETC utilized in References [10, 11, 26], which requires continuous monitoring of the measurement error to determine whether it needs to be triggered, the developed self-triggered control scheme avoids the use of an additional observer by obtaining the next trigger moment from the current one. In addition, although the self-triggered control approach can reduce the transmission cost, the stabilization performance of the control system may be weakened based on the control signal being discontinuous between two consecutive trigger moments. In particular, the interconnected nonlinear time-delay system has a relatively complicated structure. To this end, the FTPF is introduced to achieve the stabilization control goal, which also increases the challenges and difficulties of control design to a certain extent.

Remark 3

In this brief, the trial-and-error-based tactic for choosing the design parameters is utilized to achieve satisfactory control performance. By increasing the design parameters \(c_{m,i}, \mu _{m,k}, \bar{r}_{m,1}\), and \(\kappa _{m,k}\) or decreasing the design parameters \(\Lambda _m, E_m\), and \(\rho _{m,i}\) such that the stabilization error is as small as possible. Theoretically speaking, provided that the design parameters are larger, it may result in a larger amplitude of the control input. Therefore, it is essential to make a trade-off between control performance and control energy by choosing the design parameters appropriately.

figure a

Remark 4

Following the above discussion, we can derive that the control input \(\bar{u}_m\) is bounded. Hence, \((\Lambda _m\mid \bar{u}_m(t)\mid +E_m)/(\max \left\{ L_m,\mid \dot{v}_m(t)\mid \right\} )\) is also bounded. Given the circumstances of (45), it is available that the lower bound for inter-execution intervals satisfies \(\mathop t\nolimits ^ * =t_{k+1}-t_k> 0\), that is, excluding the Zeno-behavior.

4 Simulation example

In this section, two simulation examples are used to validate the effectiveness of the proposed adaptive self-triggered quantized control scheme.

Example 1

In the background of the chemical industry, two-stage chemical reactors are universally prevalent. However, it is well known that interconnected nonlinear systems consisting of two two-stage chemical reactors have significant application potential, which not only models more complex chemical reaction systems but also improves the overall conversion rate. To sum up, the nonlinear systems composed of two two-stage chemical systems have certain practical implications to some extent.

Let us consider two-stage chemical reactors with dynamic uncertainly and delayed recycle streams, as exhibited in Fig. 1. As in [42], the mathematical model of the reactors can be expressed as:

$$\begin{aligned} \left\{ \begin{aligned} \dot{\psi }_1=&\,-\psi _1+0.5x_{1,1}^2+0.5,\\ \dot{x}_{1,1}=&\,-\frac{1}{Y_{1,1}}x_{1,1}-\frac{1}{Y_{1,1}}x_{1,1}(t-{\tau _{1,1}})+\frac{1-T_{1,2}}{M_{1,1}}x_{1,2}\\ {}&-H_{1,1}x_{1,1}+0.1{\text{sin}}(x_{1,1}+x_{2,1})+\psi _1x_{1,1}{\text{sin}}(x_{1,1}),\\ \dot{x}_{1,2}=&\,-\frac{1}{Y_{1,2}}x_{1,2}^2+\frac{T_{1,1}}{M_{1,2}}x_{1,1}(t-{\tau _{1,1}})-\frac{2Q_{1,2}^*}{Y_{1,2}}x_{1,2}\\ {}&+\frac{E_1}{M_{1,2}}q(u_1)-H_{1,2}x_{1,2}+\frac{T_{1,2}}{M_{1,2}}x_{1,2}(t-{\tau _{1,2}})\\ {}&+0.2{\text{sin}}(x_{2,1})+\psi _1x_{1,1}x_{1,2},\\ \dot{\psi }_2=&\,-\psi _2+0.5x_{2,1}^2+0.5,\\ \dot{x}_{2,1}=&\,-\frac{1}{Y_{2,1}}x_{2,1}-\frac{1}{Y_{2,1}}x_{2,1}(t-{\tau _{2,1}})+\frac{1-T_{2,2}}{M_{2,1}}x_{2,2}\\ {}&-H_{2,1}x_{2,1}+0.1{\text{sin}}(x_{1,1})+\psi _2x_{2,1}{\text{sin}}(x_{2,1}),\\ \dot{x}_{2,2}=&\,-\frac{1}{Y_{2,2}}x_{2,2}^2+\frac{T_{2,1}}{M_{2,2}}x_{2,1}(t-{\tau _{2,1}})-\frac{2Q_{2,2}^*}{Y_{2,2}}x_{2,2}\\ {}&+\frac{E_2}{M_{2,2}}q(u_2)-H_{2,2}x_{2,2}+\frac{T_{2,2}}{M_{2,2}}x_{2,2}(t-{\tau _{2,2}})\\ {}&+0.2{\text{sin}}(x_{1,1}-x_{2,1})+\psi _2x_{2,1}{\text{sin}}(x_{2,2}),\\ \end{aligned} \right. \end{aligned}$$

where \(x_{m,1}\) and \(x_{m,2}\ (m=1,2)\) are the compositions to be controlled; \(q(u_m)\) is the quantized signal; the representation and values of the system parameters are shown in Table 1.

Fig. 1
figure 1

Structure chart of a two-stage chemical reactors

Full size image
Table 1 The model parameters
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To confirm the validity of Assumption 2 for \(\psi _m\)-dynamic, let us choose \(\check{V}_m(\psi _m)=\psi _m^2\), then

$$\begin{aligned} \dot{\check{V}}_m(\psi _m)=&\,2\psi _m(-\psi _m+0.5x_{m,1}^2+0.5)\\&\le -2\psi _m^2+\frac{1}{4l_m}\psi _m^2+\frac{1}{l_m}\psi _m^2+l_mx_{m,1}^4+\frac{1}{4}l_m. \end{aligned}$$

By selecting \(l_m=2.5\), results in

$$\begin{aligned} \dot{\check{V}}_m(\psi _m)\le -1.5\psi _m^2+2.5x_{m,1}^4+0.625. \end{aligned}$$

Define \(\Upsilon _{m,1}(\mid \psi _m\mid )=0.6\psi _m^2, \Upsilon _{m,2}(\mid \psi _m\mid )=0.5\psi _m^2, r_{m,1}=1.5, r_{m,2}=0.625\), and \(\pi _{m0}(\Vert x_{m,1}\Vert )=2.5x_{m,1}^4\), Assumption 2 is certified successful. On the basis of Lemma 1 and taking \(\bar{r}_{m,1}=1.2 \in (0,r_{m,1})\), a dynamic signal \(\lambda _m\) is represented as:

$$\begin{aligned} \dot{\lambda }_m=-1.2\lambda _m+2.5x_{m,1}^4+0.625. \end{aligned}$$
Table 2 The design parameters
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Following the control design in Sect. 3, the corresponding design parameters are identified in Table 2. To intuitively show the superiority of the proposed methodology, a comparison simulation between the proposed approach and the algorithm in [16] is exhibited in Figs. 2, 3, 4, 5, and 6. Specifically, the states \(x_{m,1}\) and \(x_{m,2}\ (m=1,2)\) are shown for each subsystem in Fig. 2. Figure 3 plots the profiles of system output \(x_{m,1}\) for each subsystem. From Figs. 2 and 3, it is evidence obtained that the system states are SGUUB and system output \(x_{m,1}\) can converge better to a predetermined range \((-\ell _m,\ell _m)\) than the method in [16] in finite time. Furthermore, the response curves of the adaptive parameters \(\hat{\Theta }_{m,1}\) and \(\hat{\Theta }_{m,2}\) are depicted for each subsystem in Fig. 4. Figure 5 illustrates the graphs of the control input \(v_m\) and quantized input \(q(u_m)\) for each subsystem. It can be seen that the quantization amplitude of the proposed control method is relatively small. Figure 6 expresses the trigger interval for each subsystem. To sum up, the proposed FTPP quantized control approach is superior to that investigated in [16] for achieving a better trade-off between the networked control and stability performance, and also excluding the Zeno phenomenon.

Fig. 2
figure 2

System states \(x_{m,1}\) and \(x_{m,2}\)

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Fig. 3
figure 3

System output \(x_{m,1}\)

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Fig. 4
figure 4

Adaptive parameters \(\hat{\Theta }_{m,1}\) and \(\hat{\Theta }_{m,2}\)

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Fig. 5
figure 5

Control signal \(v_m\) and quantized signal \(q(u_m)\)

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Fig. 6
figure 6

Inter-event intervals

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Example 2

Consider the following interconnected nonlinear time-delay plants:

$$\begin{aligned} \left\{ \begin{aligned} \dot{\psi }_1=&\,-\psi _1+0.5x_{1,1}^2+0.5,\\ \dot{x}_{1,1} =&\, x_{1,2}+\frac{x_{1,1}-x_{1,1}^3}{1+x_{1,1}^2}+0.1{\text{sin}}(x_{1,1}+x_{2,1})\\&-0.5x_{1,1}(t-{\tau _{1,1}})+\psi _1x_{1,1}{\text{sin}}(x_{1,1}),\\ \dot{x}_{1,2} =&\, q(u_1)-(x_{1,1}^2+2x_{1,2}){\text{sin}}(x_{1,1})+x_{1,2}(t-{\tau _{1,2}})\\&+0.2{\text{sin}}(x_{2,1})+\psi _1x_{1,1}x_{1,2},\\ \dot{\psi }_2=&\,-\psi _2+0.5x_{2,1}^2+0.5,\\ \dot{x}_{2,1} =&\, x_{2,2}+\frac{x_{2,1}-x_{2,1}^3}{1+x_{2,1}^2}-0.5x_{2,1}(t-{\tau _{2,1}})+0.1{\text{sin}}(x_{1,1})\\&+\psi _2x_{2,1}{\text{sin}}(x_{2,1}),\\ \dot{x}_{2,2} =&\,q(u_2) -(x_{2,1}^2+2x_{2,2}){\text{sin}}(x_{2,1})+x_{2,2}(t-{\tau _{2,2}})\\&+0.2{\text{sin}}(x_{1,1}-x_{2,1})+\psi _2x_{2,1}{\text{sin}}(x_{2,2}),\\ \dot{\psi }_3=&-\psi _3+0.5x_{3,1}^2+0.5,\\ \dot{x}_{3,1} =&\,x_{3,2}+\frac{x_{3,1}-x_{3,1}^3}{1+x_{3,1}^2}-0.5x_{3,1}(t-{\tau _{3,1}})\\&+0.1{\text{sin}}(x_{1,1}+x_{3,1})+\psi _3x_{3,1}{\text{sin}}(x_{3,1}),\\ \dot{x}_{3,2} =&\,q(u_3) -(x_{3,1}^2+2x_{3,2}){\text{sin}}(x_{3,1})+x_{3,2}(t-{\tau _{3,2}})\\&+0.2{\text{sin}}(x_{2,1}-x_{3,1})+\psi _3x_{3,1}{\text{sin}}(x_{3,2}),\\ \end{aligned} \right. \end{aligned}$$

where \(f_{m,1}=(x_{m,1}-x_{m,1}^3)/(1+x_{m,1}^2),\ f_{m,2}=-(x_{m,1}^2+2x_{m,2}){\text{sin}}(x_{m,1}).\ g_{1,1}=0.1{\text{sin}}(x_{1,1}+x_{2,1}),\ g_{1,2}=0.2{\text{sin}}(x_{2,1}),\ g_{2,1}=0.1{\text{sin}}(x_{1,1}),\ g_{2,2}=0.2{\text{sin}}(x_{1,1}-x_{2,1}),\ g_{3,1}=0.1{\text{sin}}(x_{1,1}+x_{3,1}),\ g_{3,2}=0.2{\text{sin}}(x_{2,1}-x_{3,1}).\ \Delta _{1,1}=\psi _1x_{1,1}{\text{sin}}(x_{1,1}),\ \Delta _{1,2}=\psi _1x_{1,1}x_{1,2},\ \Delta _{2,1}=\psi _2x_{2,1}{\text{sin}}(x_{2,1}),\ \Delta _{2,2}=\psi _2x_{2,1}{\text{sin}}(x_{2,2}),\ \Delta _{3,1}=\psi _3x_{3,1}{\text{sin}}(x_{3,1}),\ \Delta _{3,2}=\psi _3x_{3,1}{\text{sin}}(x_{3,2}).\ \tau _{m,1}=0.2+0.08{\text{sin}}(2t),\ \tau _{m,2}=0.3+0.12{\text{sin}}(2t)\).

We select the \(\psi _m\)-dynamic, the RBFNN basis function, virtual control signals, and adaptive laws of the numerical simulations like Example 1. The different parameters related to Example 1 are selected as follows:\(x_{m,1}(0)=0.2,\ x_{m,2}(0) =1.2,\ c_{m,1}= c_{m,2}=5\ (m=1,2,3)\). To further illustrate the stabilization performance investigated in this paper, the responses of the states \(x_{m,1}\) and \(x_{m,2}\ (m=1,2,3)\) are exhibited for each subsystem in Fig. 7. Figure 8 depicts the graphs of system output \(x_{m,1}\) for each subsystem. The investigated self-triggered control scheme can guarantee that the system states in the resulting CLS are SGUUB and system output \(x_{m,1}\) can regulated to a predetermined range \((-\ell _m,\ell _m)\) in finite time. In addition, the response curves of the adaptive parameters \(\hat{\Theta }_{m,1}\) and \(\hat{\Theta }_{m,2}\) are shown for each subsystem in Fig. 9. Figure 10 plots the graphs of the control input \(v_m\) and quantized input \(q(u_m)\) for each subsystem. Figure 11 exhibits the trigger interval for each subsystem. To sum up, the presented self-triggered control approach can achieve a better trade-off between networked control and stability performance, and the Zeno phenomenon is also ruled out.

Fig. 7
figure 7

System states \(x_{m,1}\) and \(x_{m,2}\)

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Fig. 8
figure 8

System output \(x_{m,1}\)

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Fig. 9
figure 9

Adaptive parameters \(\hat{\Theta }_{m,1}\) and \(\hat{\Theta }_{m,2}\)

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Fig. 10
figure 10

Control signal \(v_m\) and quantized signal \(q(u_m)\)

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Fig. 11
figure 11

Inter-event intervals

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5 Conclusion

Self-triggered-based adaptive neural decentralized control strategy for interconnected nonlinear time-delay systems involving finite-time prescribed performance has been developed. By utilizing the hyperbolic tangent function and RBFNNs, the potential singularity issue has been eliminated. Meanwhile, an improved FOF was incorporated into the controller design, which can avoid the tremendous “ amount of calculation ” and the adverse impact of filter error has been excluded. Besides, the effect of bandwidth limitation was considered in the self-triggered adaptive FTPP control scheme, achieving a trade-off between stability performance and networked control. The presented controller ensured all signals in CLS are SGUUB, and the system output fluctuated to a small adjustable range within a finite time interval. Last but not least, the effectiveness and superiority of the proposed scheme have been verified by two simulation examples. In the future, the author will work to consider the self-triggered adaptive secure control issue for interconnected nonlinear time-delay systems with cyberattacks. In addition, inspired by [43] in dealing with the tracking of small-amplitude signals, a straightforward and effective nonlinearity is implemented in the controller, resulting in better signal-to-noise ratio performance, which will be another topic for our future research.