Quantized practical fixed-time consensus tracking for networked Euler–Lagrange systems under the predetermined workspace

Abstract

This paper tries to solve the quantized practical fixed-time consensus tracking problem for networked Euler–Lagrange systems under the predetermined workspace. To realize the information interaction under the limited bandwidth, a set of encoder, decoder, and average quantizers are constructed to process the interaction data. On this basis, a fixed-time observer is proposed so that each follower can estimate the leader’s information within the quantized communication environment. Afterward, the local tracking control algorithm is designed by using backstepping strategy and adaptive technology, and the state constraint function is introduced to cope with the asymmetric time-varying constraint problem. With the Lyapunov stability criterion, all error signals are guaranteed to remain in the compact sets near the origin within the fixed time. Ultimately, a numerical example is carried out to testify the validity of the proposed scheme.

Keywords

Fixed-time tracking control networked Euler–Lagrange systems quantized communication adaptive technology asymmetric time-varying constraint

Introduction

In recent dozen years, cooperative control of multi-agent systems has emerged as a prominent research topic, driven by the rapid growth of digital communication and computer network technology. This field has expanded to include related areas such as consensus (Liu et al., 2020; Sun et al., 2022), containment (Xiao et al., 2019; Yang and Hu, 2022), and formation (Chen et al., 2022; Yan et al., 2022). Among these, consensus tracking has been a top priority. In military and industrial fields, a variety of mechatronic systems are usually characterized as Euler–Lagrange systems, for example, aircraft, underwater robots, and robot manipulators. With the increase in control task complexity and the improvement of industrial automation, the consensus tracking control for networked Euler–Lagrange systems (NELSs) has been gradually applied in various sophisticated technology fields. However, NELSs are coupling and strong nonlinearity in nature, so it is valuable to derive the corresponding dynamic model accurately. To deal with the inadequacies of NELSs, adaptive technique integration of fuzzy logic systems (Li et al., 2022b; Wang and Cao, 2021) or neural networks (He et al., 2016; Kong et al., 2021; Wang et al., 2022) is the current appropriate methodology, which has increased significantly in prominence.

As is common knowledge, convergence rate is one of the performance indexes in modern control theory. To pursue fast-response performance and high precision, research on finite-time control approaches for NELSs has attracted a great deal of interest and attention (He et al., 2018; Hu et al., 2019). It is worth pointing out the estimation of convergence time in above results subject to the initial conditions. Fortunately, the research on fixed-time consensus tracking (FTCT) has started to develop since the fixed-time control concept was originally investigated by Polyakov (2012), in which the convergence time is regardless of initial conditions. Resulting from this trait, some recent works on FTCT protocol subject to the undirected graph (Li et al., 2022d) or digraph (Mi et al., 2023; Ni et al., 2021b; Zuo et al., 2020) have been displayed. More recently, Li et al. (2022a) established an observer-based local control protocol for NELSs to solve actuator fault problems. Hong et al. (2022) investigated a fixed-time attitude consensus control algorithm for NELSs by combining adaptive strategies, and the parameters of the controller have been chosen independently.

In the aforementioned consensus tracking scheme, it is assumed that the local communication channel between agents is ideal and has unlimited bandwidth, thus enabling an agent to receive actual information from its neighbors. However, in a realistic communication network, information transmission between agents typically occurs through a digital channel with finite bandwidth and limited storage capacity. As a result, actual information should be quantized into finite symbols before transmission. Therefore, since quantization consensus has been first introduced by Kashyap et al. (2007), numerous scholars have made great achievements on this basis. For real-value multi-agent systems with quantized communication, switching topologies, and sampled information, Lu et al. (2022) presented a minimum consensus control strategy to achieve accurate consensus under the limited bandwidth. To lessen the communication load of multi-agent systems, the quantized communication event-triggered protocol has been designed by Wu et al. (2018) to achieve practical consensus. Yao et al. (2021) explored a quantized fault-tolerant control strategy for NELSs subject to sensor and actuator faults. Li et al. (2022c) solved the consensus problem for NELSs with quantized sampling data and external disturbances. Unfortunately, due to the difficulties caused by the asymmetry of digraphs and inaccurate information exchange, the problem of quantized FTCT for NELSs remains to be further addressed.

In the majority of real engineering applications, system motions are generally required to operate within a predetermined workspace. If system motions exceed the specified task space, it may lead to inaccurate control, performance degradation, and even accidents. Therefore, it is particularly important to design an appropriate controller to perform tasks with workspace constraints. Note that the barrier Lyapunov function is generally regarded as a valid tool to handle state constraints. For this reason, Cai et al. (2020) proposed the leader-following tracking control strategy for position-constrained NELSs by adopting the barrier Lyapunov function. With the help of the barrier Lyapunov function and backstepping technique, Liu et al. (2022) studied the adaptive tracking control for state-constrained NELSs, which only considered the special case of symmetric constraints. For further improvement, He et al. (2017) solved the asymmetric state constraint issue for Euler–Lagrange systems by employing an asymmetric barrier Lyapunov function. Nevertheless, such a form only applies to constrained cases, and expanding to unconstrained cases is challenging. To address this restriction, Cao et al. (2022) introduced a comprehensive barrier Lyapunov function to constrain the tracking errors, which can also handle unconstrained case. Nevertheless, this strategy is only applicable if the upper bound of the reference signal can be known. Consequently, it is more significant and realistic to investigate the unified form with respect to state constraints, which can be flexibly transformed among unconstrained case and asymmetric/symmetric constrained case.

In light of the results reported above, there are still certain limitations in existing achievements of FTCT under workspace constraint, system uncertainties, and quantized communication, which are worth further improved. Therefore, the objective of this paper is to address the quantized practical FTCT problem for NELSs under the predetermined workspace. The challenges faced in this work include the difficulty of constructing fixed-time observers for each follower, which is due to the asymmetry of the directed network topology and the presence of quantized communication. Furthermore, fixed-time stability analysis under state constraints adds to the complexity of the task at hand. The primary contributions are listed below:

Under the quantized communication environment, a novel fixed-time observer is designed to drive all followers to estimate the states of the leader. The quantification scheme developed in this paper effectively reduces the number of bits to be transmitted in the communication channel and enhances quantification precision. Different from Ni et al. (2021b), Zuo et al. (2020), and Mi et al. (2023), the observer of this paper has the following advantages: fewer adjustable parameters, relatively simple stability analysis process, and less conservativeness of convergence time upper bound.

This paper introduces a novel adaptive fixed-time local tracking control strategy to ensure that each follower tracks the leader’s trajectory without violating the workspace constraint. Adaptive parameters are adapted to compensate for disturbances resulting from estimation errors. Compared with Cai et al. (2020), Fan and Zhao (2021), Liu et al. (2022), He et al. (2017), and Cao et al. (2022), the unified state-constrained function of this paper can flexibly switch between unconstrained and time-varying constrained forms without changing the adaptive structure and has the ability to impose constraints directly on the states to circumvent additional transformation from errors to states.

For NELSs, workspace constraints, system uncertainties, and quantized communication are simultaneously considered in the research of the FTCT issue, which tremendously broadens the applicability of this work. Moreover, the presented distributed estimation local control structure effectively attenuates the difficulty of stability analysis and control algorithm design.

This paper is organized as follows. The problems and necessary concepts are briefly reviewed in section “Preparation and problem formation.” Section “Main results” exhibits the main results. Subsequently, the simulation results are drawn in section “Simulation results.” Finally, some conclusions are given in section “Conclusion.”

Notations: $R^{n}$ and $R^{n \times m}$ indicate the $n$ -dimensional Euclidean space and $n \times m$ real matrix, respectively. $R^{+}$ refers to the set of positive constants, 0 refers to a zero scalar, $1_{n} = [1, \dots, 1]^{T} \in R^{n}$ , and $I$ indicates an identity matrix. For matrix $Q$ , $Q^{- 1}$ and $Q^{T}$ refer to the inverse matrix and the transpose matrix, respectively. $diag {\dots}$ refers to a diagonal matrix. $| | • | | = \sqrt{•^{T} •}$ refers to the Euclidean norm of a vector, |•| indicates an absolute value, and the sign function is represented as $sign (*)$ . For the $n$ -dimensional vector $v$ , $sig (v)^{α} = {[| v^{1} |^{α} sign (v^{1}), \dots, | v^{n} |^{α} sign (v^{n})]}^{T}$ . ⊗ is the Kronecker product. $λ_{max} (Q)$ and $λ_{min} (Q)$ denote the maximum and minimum eigenvalues of the matrix $Q$ , respectively.

Preparation and problem formation

Graph theory

The communication network of $N$ followers is modeled by a digraph $G = {V, E}$ , consisting of a set of vertices $V = {1, \dots, N}$ and a set of edges $E \subseteq V \times V$ . An edge from the starting vertex $i$ to the terminating vertex $j$ is represented by $(i, j) \in E$ . $A = [a_{ij}] \in R^{N \times N}$ is called adjacency matrix, where the weighted elements $a_{ij} > 0$ if $(j, i) \in E$ ; otherwise, $a_{ij} = 0$ . The set of neighbors for vertex $i$ is given by $N_{i} = {j \in V : (j, i) \in E}$ . The Laplacian matrix $L$ associated with the $G$ is written as $L = D - A = [l_{ij}] \in R^{N \times N}$ , where $l_{ii} = \sum_{j = 1}^{N} a_{ij}$ , $l_{ij} = - a_{ij}$ for $j \neq i$ , and $D = diag {\sum_{j = 1}^{N} a_{ij}, i \in V}$ . $G$ is called to have a directed spanning tree, if there exists a root vertex such that this vertex has a directed path to all remaining vertices. Let $\bar{G}$ denote the leader-following topology between a leader and $N$ followers. Define $H = L + A_{0}$ for the leader-following topology, where $A_{0} = diag {a_{10}, \dots, a_{N 0}}$ refers to the leader’s adjacency matrix, and $a_{i 0} > 0$ indicates that vertex $i$ can receive information from leader, and $a_{i 0} = 0$ otherwise.

System description

Consider the NELSs composed of one leader and $N$ followers. For $i \in V$ , the ith follower dynamics is described by

M_{i} (q_{i}) {\overset{\cdot\cdot}{q}}_{i} + C_{i} (q_{i}, {\dot{q}}_{i}) {\dot{q}}_{i} + O_{i} (q_{i}) = U_{i}

(1)

where ${\overset{\cdot\cdot}{q}}_{i} \in R^{n}$ , ${\overset{\cdot}{q}}_{i} \in R^{n}$ , and $q_{i} \in R^{n}$ refer to the state vectors of the ith follower, $M_{i} (q_{i}) \in R^{n \times n}$ represents the symmetric positive-definite inertia matrix, $C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) \in R^{n \times n}$ refers to the centrifugal-coriolis matrix, $O_{i} (q_{i}) \in R^{n}$ indicates the gravity force vector, and $U_{i} \in R^{n}$ refers to the vector of control torque.

Hereafter, we define $x_{1 i} = q_{i}$ and $x_{2 i} = {\overset{\cdot}{q}}_{i}$ , thus the state-space form of equation (1) is rewritten as

\begin{matrix} {\overset{\cdot}{x}}_{1 i} = x_{2 i} \\ {\overset{\cdot}{x}}_{2 i} = M_{i}^{- 1} (x_{1 i}) [U_{i} - C_{i} (x_{1 i}, x_{2 i}) x_{2 i} - O_{i} (x_{1 i})] \end{matrix}

(2)

for $i \in V$ , where $x_{1 i} \in R^{n}$ and $x_{2 i} \in R^{n}$ represent the position variable and the velocity variable of the ith follower, respectively. Then, the predetermined workspace is expressed as ${\underline{k}}_{i}^{r} (t) < x_{1 i}^{r} (t) < {\bar{k}}_{i}^{r} (t)$ , where $r = 1, \dots, n$ , ${\underline{k}}_{i}^{r} (t)$ and ${\bar{k}}_{i}^{r} (t)$ represent positive time-varying functions, and its second-order derivatives are bounded and continuous. Moreover, the initial state satisfies ${\underline{k}}_{i}^{r} (0) < x_{1 i}^{r} (0) < {\bar{k}}_{i}^{r} (0)$ . In the subsequent design, $M_{i}$ , $C_{i}$ , and $O_{i}$ denote $M_{i} (x_{1 i})$ , $C_{i} (x_{1 i}, x_{2 i})$ , and $O (x_{1 i})$ , respectively.

The dynamics of the leader is described by

{\overset{\cdot}{x}}_{10} = x_{20}, {\overset{\cdot}{x}}_{20} = u_{0}

(3)

where $x_{10} \in R^{n}$ , $x_{20} \in R^{n}$ , and $u_{0} \in R^{n}$ represent the position state vector, the velocity state vector, and the control input vector of the leader, respectively. Similarly, the state of the leader also needs to satisfy ${\underline{k}}_{i}^{r} (t) < x_{10}^{r} (t) < {\bar{k}}_{i}^{r} (t)$ , where $r = 1, \dots, n$ .

As most results on consensus tracking for NELSs, the following assumptions are indispensable.

Assumption 1. The graph $\bar{G}$ contains a directed spanning tree, and the leader is seen as the root.

Assumption 2. The leader’s states are bounded due to performance considerations, security, and environment constraints, and the leader’s control input $u_{0}$ is limited by ${\bar{u}}_{0} \in R^{+}$ , that is, $∥ u_{0} ∥ \leq {\bar{u}}_{0}$ .

Remark 1. Assumption 1 is a standard assumption widely employed in the existing results to achieve the leader–follower consensus, as demonstrated in previous studies such as Mi et al. (2023)and Ni et al. (2021a). Assumption 2 is justifiable as the upper bound of the leader’s input can be acquired through prior knowledge of the physical systems (Zuo et al., 2020).

Definition 1. (Ni et al., 2021a) For $i \in V$ and $m = 1, 2$ , if $lim_{t \to T} ∥ x_{mi} - x_{m 0} ∥ \leq δ$ , then NELSs is called to be able to achieve practical fixed-time consensus tracking, where $T$ indicates the consensus time function and $δ$ refers to the desired accuracy.

Uniform quantizer, encoder, and decoder design

In an actual digital network, the communication bandwidth is limited. Generally, the actual information needs to be quantified as symbolic data, encoded at the sending port, then transmitted as a digital signal, and decoded at the receiving port. In this paper, it is assumed that the actual information of the follower’s neighbor is not available to be directly acquired, but can only be transmitted to each other via symbolic data. At the same time, the communication channel is modeled as a noiseless digital channel, and each channel has a pair of encoder and decoder.

The vector uniform quantizer $q_{c} :$ $R^{n} \to Γ^{n}$ is given by

q_{c} (x_{mi}) = σ ⌊ \frac{x_{mi}}{σ} + \frac{1}{2} ⌋

(4)

where $i \in V$ , $m = 1, 2$ , $σ \in R^{+}$ refers to the quantizer gain. Let $Δ_{ci}$ be the quantization error and $Δ_{ci} = ‖ q_{c} (x_{mi}) - x_{mi} ‖ \leq \frac{1}{2} n σ$ .

To transmit information from vertex $j$ to vertex $i$ under the limited bandwidth communication environment, the encoder $Φ_{j}^{e}$ of the jth agent is given by

Φ_{j}^{e} : {\begin{matrix} η_{j} (0) = 0 \\ η_{j} (t) = g_{j} (t - 1) λ_{j} (t) + η_{j} (t - 1) \\ λ_{j} (t) = q_{c} (\frac{1}{g_{j} (t - 1)} (x_{mj} (t) - η_{j} (t - 1))) \end{matrix}

(5)

where $η_{j}$ and $λ_{j} (t)$ refer to the internal state and output of the encoder $Φ_{j}^{e}$ , respectively. $q_{c}$ is a uniform quantizer defined in equation (4), and $g_{j} (t)$ represents the scaling function.

Note from equation (4) that the $λ_{j} (t)$ will be transmitted from vertex $j$ to vertex $i$ when $(j, i) \in E$ . Accordingly, the decoder $Φ_{j}^{d}$ is given as follows

Φ_{j}^{d} : {\begin{matrix} q_{j} (0) = 0 \\ ψ_{j} (t) = g_{j} (t - 1) λ_{j} (t) + q_{j} (t - 1) \end{matrix}

(6)

where $q_{j} (t)$ represents the output signal of $Φ_{j}^{d}$ .

Remark 2. It is observed from equation (5) that the quantification information is based on the prediction error $x_{mj} (t) - η_{j} (t - 1)$ rather than the state $x_{mj} (t)$ . Since the amplitude of the prediction error is smaller than the amplitude of the state itself, the number of bits transmitted by the prediction error signal in the communication channel is smaller than that of the traditional average quantizer. Therefore, this scheme can reduce quantization error.

Nonsmooth analysis

Consider the differential function given below (Wu et al., 2018)

\overset{\cdot}{x} (t) = f (x (t), t), x (t_{0}) = x_{0}

(7)

where $f (x (t), t), R \times R^{n} \to R^{n}$ represents a measurable discontinuous function. Here, the presence of a solution to a continuous differential function cannot be guaranteed. Consequently, the Krasovskii solution is shown below to analyze the differential function’s solution.

If $x (t)$ is continuous within $[t_{0}, t_{1})$ and satisfies

\overset{\cdot}{x} (t) \in K [f (x (t), t)] = ⋂_{δ > 0} \bar{co} {f (B (x, δ), t)}

(8)

then a vector function $x (t)$ is called the Krasovskii solution of (7), where $x (t)$ represents the Krasovskii solution of (7) on $[t_{0}, t_{1})$ , $\bar{co}$ refers to the convex closure, and $B (x, δ) = {y : ∥ y - x ∥ \leq δ}$ .

For the locally Lipschitz function $V (x (t)) : R^{r} \to R$ , the set-valued Lie-derivative is given as

\overset{\cdot}{V} (x (t)) = ⋂_{θ \in \partial V (x (t))} θ^{T} K [f (x (t), t)]

where $\partial V (x (t)) = \bar{co} {lim_{i \to \infty} \nabla V (x_{i}) : x_{i} \to x, x_{i} \notin Ω_{V} \cup S}$ refers to Clarke’s generalized gradient, $Ω_{V}$ stands for zero measure set, and $S$ represents any zero measure set in $R^{r}$ .

Uniform state-constraint function

To guarantee the workspace constraint requirement for NELSs, a uniform state-constraint function is constructed as follows

ξ_{i}^{r} (t) = \frac{{\underline{k}}_{i}^{r} (t) {\bar{k}}_{i}^{r} (t) x_{1 i}^{r} (t)}{({\underline{k}}_{i}^{r} (t) + x_{1 i}^{r} (t)) ({\bar{k}}_{i}^{r} (t) - x_{1 i}^{r} (t))}

(9)

where the initial state $x_{1 i}^{r} (0) \in (- {\underline{k}}_{i}^{r} (0), {\bar{k}}_{i}^{r} (0))$ , $i \in V$ , $r = 1, \dots, n$ . Clearly, if $x_{1 i}^{r} (t) \to {\bar{k}}_{i}^{r} (t)$ or $x_{1 i}^{r} (t) \to - {\underline{k}}_{i}^{r} (t)$ , then $ξ_{i}^{r} (t) \to \pm \infty$ , meanwhile, $ξ_{i}^{r} = 0$ if and only if $x_{1 i}^{r} = 0$ . Thus, the function (9) has the ability to make $x_{1 i}^{r} (t)$ not violate the constraint, that is, $x_{1 i}^{r} (t) \in (- {\underline{k}}_{i}^{r} (t), {\bar{k}}_{i}^{r} (t))$ if and only if $ξ_{i}^{r} (t) \in L_{\infty}$ . Taking time derivative of $ξ_{i}^{r} (t)$ , we have

{\overset{\cdot}{ξ}}_{i}^{r} (t) = c_{1 i}^{r} (t) {\overset{\cdot}{x}}_{1 i}^{r} (t) + v_{1 i}^{r} (t)

(10)

where $c_{1 i}^{r} (t) = \frac{{\underline{k}}_{i}^{r} (t) {\bar{k}}_{i}^{r} (t) ({\underline{k}}_{i}^{r} (t) {\bar{k}}_{i}^{r} (t) + {(x_{1 i}^{r} (t))}^{2})}{{({\underline{k}}_{i}^{r} (t) + x_{1 i}^{r} (t))}^{2} {({\bar{k}}_{i}^{r} (t) - x_{1 i}^{r} (t))}^{2}}$ , $v_{1 i}^{r} (t) = V_{11 i}^{r} (t) {\underline{\overset{\cdot}{k}}}_{i}^{r} (t) + V_{21 i}^{r} (t) {\overset{\cdot}{\bar{k}}}_{i}^{r} (t)$ , $V_{11 i}^{r} (t) = \frac{\partial ξ_{i}^{r} (t)}{\partial {\underline{k}}_{i}^{r} (t)} = \frac{{\bar{k}}_{i}^{r} (t) {(x_{1 i}^{r} (t))}^{2}}{{({\underline{k}}_{i}^{r} (t) + x_{1 i}^{r} (t))}^{2} ({\bar{k}}_{i}^{r} (t) - x_{1 i}^{r} (t))}$ , $V_{21 i}^{r} (t) = \frac{\partial ξ_{i}^{r} (t)}{\partial {\bar{k}}_{i}^{r} (t)} = - \frac{{\underline{k}}_{i}^{r} (t) {(x_{1 i}^{r} (t))}^{2}}{({\underline{k}}_{i}^{r} (t) + x_{1 i}^{r} (t)) {({\bar{k}}_{i}^{r} (t) - x_{1 i}^{r} (t))}^{2}}$

Remark 3. It is worth mentioning that if the constraints are symmetric, that is, ${\hat{k}}_{i}^{r} (t) = {\underline{k}}_{i}^{r} (t) = {\bar{k}}_{i}^{r} (t)$ , then equation (9) will be transformed into $ξ_{i}^{r} (t) = \frac{{({\hat{k}}_{i}^{r} (t))}^{2} x_{1 i}^{r} (t)}{({\hat{k}}_{i}^{r} (t) + x_{1 i}^{r} (t)) ({\hat{k}}_{i}^{r} (t) - x_{1 i}^{r} (t))}$ . Furthermore, if there is no constraint requirement imposed on $x_{1 i}^{r} (t)$ , which is equivalently seen as ${\hat{k}}_{i}^{r} (t) \to + \infty$ , that is, $lim_{{\hat{k}}_{i}^{r} (t) \to + \infty} ξ_{i}^{r} (t) = x_{1 i}^{r} (t)$ , it indicates that an unconstrained system can be viewed as a particular form of the constrained system. Hence, by introducing the function (9), the consensus tracking problem of asymmetric/symmetric constrained and unconstrained systems can be handled uniformly without changing adaptive structures.

Remark 4. For most state constraints-related results of Cai et al. (2020), Fan and Zhao (2021), Liu et al. (2022), He et al. (2017), and Cao et al. (2022), the barrier Lyapunov function method is usually used to constrain the error signal, and then, the additional transition from the error to the state is used to achieve the state constraint under the assumption that the reference signal is known bounded continuous. In the paper, the function (9) constrains position states directly, which avoids the additional transformation and also applies to situations in which the reference signal is not directly accessible.

Distributed estimation local control structure

For FTCT with practical scenarios, only a few followers have access to the leader’s information. To keep all followers informed of the leader’s state, a feasible approach is to construct an observer for each follower to estimate the state of the leader and then employ the estimated information for subsequent local tracking control. Based on this approach, we propose a distributed estimation local control structure to handle the two major challenges under the fixed-time convergence framework: directed network topology and complex dynamics. This structure is depicted in Figure 1. Benefitting from the flexible structure design, we can pay attention to solving the complexity of asymmetric topology and quantized communication in the network communication part and focus on the complexity of system dynamics in the local control part.

Figure 1.

Distributed estimation local control structure.

The primary task of this paper is to achieve the following objective: for NELSs with quantized communication, an observer-based local tracking control algorithm is designed to achieve practical FTCT, and at the same time, the predetermined constraints are not violated.

Preliminary lemmas

Lemma 1. (Ni et al., 2021b) For any $h_{r} > 0$ , $r = 1, \dots, a$ , $0 < b \leq 1$ , we have

{(\sum_{r = 1}^{a} h_{r})}^{b} \leq \sum_{r = 1}^{a} h_{r}^{b}

Lemma 2. (Ni et al., 2021b) For any $h_{r} > 0$ , $r = 1, \dots, a$ , and $b > 1$ , then

a^{1 - b} \cdot {(\sum_{r = 1}^{a} h_{r})}^{b} \leq \sum_{r = 1}^{a} h_{r}^{b} \leq {(\sum_{r = 1}^{a} h_{r})}^{b}

Lemma 3. (Wang et al., 2014) For any $p, q \in R,$ and $κ > 0, m > 1, n > 1, (m - 1) (n - 1) = 1$ , one has

p \cdot q \leq \frac{κ^{m}}{m} | p |^{m} + \frac{1}{n κ^{n}} | q |^{n}

Lemma 4. (He et al., 2016) Suppose $M \in R^{n \times n}$ is a real positive-definite symmetric matrix. For $\forall θ \in R^{n}$ , we get

λ_{min} (M) ∥ θ ∥^{2} \leq θ^{T} M θ \leq λ_{max} (M) ∥ θ ∥^{2}

Lemma 5. (Aldana-López et al., 2019) Consider a nonlinear system

\overset{\cdot}{x} (t) = f (t, x), x (0) = x_{0}

(11)

where $x \in R^{n}$ , the mapping $f : R^{+} \times R^{n} \to R^{n}$ . If there exists a radially unbounded and positive-definite function $L (x) : R^{n} \to R$ for equation (11) satisfying $\overset{\cdot}{L} (x) \leq - (α L^{μ} (x) + β L^{ν} (x))^{w}$ , where $α, β, μ, ν, w > 0$ and $μ w < 1, ν w > 1$ , then the solution of equation (11) is fixed-time stable, and the setting time $T_{f}$ satisfies

\overset{\cdot}{L} (x) \leq - \frac{F (z)}{T_{f}} {(α L^{μ} (x) + β L^{ν} (x))}^{w}, \forall x \in R^{n} \ {0}

where $F (z) = \frac{Γ (\frac{1 - w μ}{ν - μ}) Γ (\frac{w ν - 1}{ν - μ})}{α^{w} Γ (w) (ν - μ)} {(\frac{α}{β})}^{\frac{1 - w μ}{ν - μ}},$ and $z = [α, β, μ, ν, w]^{T} \in R^{5}, Γ (\cdot)$ is defined as $Γ (c) = \int_{0}^{\infty} e^{- t} t^{c - 1} dt$ .

Main results

In this section, the quantized FTCT problem for NELSs is treated in two processes based on the distributed estimation local control structure.

Process 1: With the quantized communication environment, a fixed-time observer is established to estimate the states of the leader.

Process 2: For NELSs under the predetermined workspace, a novel adaptive local tracking control strategy is presented to ensure that all errors converge to neighborhood sets of the origin in fixed time.

Fixed-time observer design

In this paper, the following fixed-time observers are designed for each follower

{\overset{\cdot}{φ}}_{1 i} (t) = w_{i} (Λ_{1 i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\overset{\cdot}{φ}}_{1 j} (t | t - t_{s}) - a_{i 0} {\overset{\cdot}{x}}_{10})

(12)

{\overset{\cdot}{φ}}_{2 i} (t) = w_{i} (Λ_{2 i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\overset{\cdot}{φ}}_{2 j} (t | t - t_{s}) - a_{i 0} {\overset{\cdot}{x}}_{20})

(13)

where $w_{i} = - \frac{1}{l_{i i} + a_{i 0}}$ , $Λ_{1 i} = \frac{1}{2} sig (ϕ_{1 i}^{*}) + w_{11} sig {(ϕ_{1 i}^{*})}^{s_{1}} + w_{12}^{*} sig {(ϕ_{1 i}^{*})}^{s_{2}}$ , $ϕ_{1 i}^{*} = \sum_{j = 0}^{N} a_{ij} (φ_{1 i} - q_{j} (φ_{1 j}))$ , $Λ_{2 i} = \frac{1}{2} sig (ϕ_{2 i}^{*}) + w_{21} sig (ϕ_{2 i}^{*})^{s_{1}} + w_{22}^{*} sig (ϕ_{2 i}^{*})^{s_{2}}$ , $ϕ_{2 i}^{*} = \sum_{j = 0}^{N} a_{ij} (φ_{2 i} - q_{j} (φ_{2 j}))$ , $φ_{1 i}$ and $φ_{2 i}$ stand for the estimation of leader’s position $x_{10}$ and velocity $x_{20}$ , respectively, $φ_{10} = x_{10}$ , $φ_{20} = x_{20}$ , $t_{s}$ is the sampling period, and ${\overset{\cdot}{φ}}_{1 j} (t | t - t_{s})$ and ${\overset{\cdot}{φ}}_{2 j} (t | t - t_{s})$ represent the previous derivative values at $t$ , which are derived from $t - t_{s}$ . $w_{11}$ , $w_{12}^{*}$ , $w_{21}$ and $w_{22}^{*}$ are observer gains, where $w_{m 2}^{*} = w_{m 2} (\frac{2^{s_{2} - 1} - 2^{1 - s_{2}}}{2} sign (ϕ_{mi}^{*}) + \frac{2^{s_{2} - 1} + 2^{1 - s_{2}}}{2})$ , $m = 1, 2$ . $s_{1}$ and $s_{2}$ satisfy $0 < s_{1} < 1 < s_{2}$ . After encoding, quantization, and decoding by equations (4)–(6), the neighbor states obtained by each follower are represented as $q_{j} (φ_{mj})$ .

Remark 5. If the design methodology of the observer presented in Liu and Yang (2021) is adopted, that is, the derivatives ${\overset{\cdot}{φ}}_{1 j} (t)$ and ${\overset{\cdot}{φ}}_{2 j} (t)$ of the jth neighbor at time $t$ are directly substituted into equations (12) and (13), respectively, then we have

\begin{matrix} {\overset{\cdot}{φ}}_{1 i} (t) = w_{i} (Λ_{1 i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\overset{\cdot}{φ}}_{1 j} (t) - a_{i 0} {\overset{\cdot}{x}}_{10}) \\ {\overset{\cdot}{φ}}_{2 i} (t) = w_{i} (Λ_{2 i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\overset{\cdot}{φ}}_{2 j} (t) - a_{i 0} {\overset{\cdot}{x}}_{20}) \end{matrix}

then it can be found that the logic loop phenomenon occurs in the observers, which makes the leader’s states impossible to estimate. To address this issue, we use ${\overset{\cdot}{φ}}_{1 j} (t | t - t_{s})$ and ${\overset{\cdot}{φ}}_{2 j} (t | t - t_{s})$ at time $t - t_{s}$ to replace the neighbors’ derivatives at current time $t$ , which avoids the logic loop phenomenon in an asynchronous manner.

The quantification error and derivative error are expressed as

ψ_{mj} = φ_{mj} - q_{j} (φ_{mj})

(14)

{\tilde{ψ}}_{mj} = {\overset{\cdot}{φ}}_{mj} - {\overset{\cdot}{φ}}_{mj} (t | t - t_{s})

(15)

where $m = 1, 2$ . According to quantizer (4), encoder (5), and decoder (6), it is easy to get $‖ ψ_{mj} ‖ \leq \frac{1}{2} g_{mj} (t - 1) σ_{mj}$ . If the quantizer is not saturated, the quantization error bound is determined by the dynamic quantization coefficient $g_{mj} (t - 1)$ and quantization interval $σ_{mj}$ . Reducing $g_{mj} (t - 1)$ can improve quantization accuracy. Moreover, since the sampling period $t_{s}$ is relatively short and negligible, it is reasonable to assume that the errors are bounded, that is, there exists ${\tilde{ϵ}}_{m j} \in R^{+}$ such that $‖ {\tilde{ψ}}_{m j} ‖ \leq {\tilde{ϵ}}_{m j}$ .

Theorem 1. Under Assumption 1, the presented observers (12) and (13) can drive all followers to estimate the leader’s position $x_{10}$ and velocity $x_{20}$ in fixed time for NELSs (2) and (3), the estimation errors can achieve practical fixed-time convergence, and the settling time $T_{ob}$ satisfies

T_{ob} = T_{1} + T_{2}

where $T_{m} = \frac{Γ (\frac{1 - s_{1}}{s_{2} - s_{1}}) Γ (\frac{s_{2} - 1}{s_{2} - s_{1}})}{2^{\frac{s_{1} + 1}{2}} w_{m 1} Γ (1) (s_{2} - s_{1})} {(\frac{2^{\frac{s_{1} + 1}{2}} w_{m 1}}{(1 - ν_{1}) 2^{\frac{s_{2} + 1}{2}} w_{m 2} N^{\frac{1 - s_{2}}{2}}})}^{\frac{1 - s_{1}}{s_{2} - s_{1}}}$ , $m = 1, 2$ .

Proof: For $i \in V$ , the estimation errors of the ith follower are given as ${\tilde{φ}}_{1 i} = φ_{1 i} - x_{10},$ ${\tilde{φ}}_{2 i} = φ_{2 i} - x_{20}$ , and then we have

\begin{matrix} ϕ_{1 i} = \sum_{j = 1}^{N} a_{ij} (φ_{1 i} - φ_{1 j}) + a_{i 0} (φ_{1 i} - x_{10}) \\ = \sum_{j = 1}^{N} a_{ij} ({\tilde{φ}}_{1 i} - {\tilde{φ}}_{1 j}) + a_{i 0} {\tilde{φ}}_{1 i} \\ ϕ_{2 i} = \sum_{j = 1}^{N} a_{ij} (φ_{2 i} - φ_{2 j}) + a_{i 0} (φ_{2 i} - x_{20}) \\ = \sum_{j = 1}^{N} a_{ij} ({\tilde{φ}}_{2 i} - {\tilde{φ}}_{2 j}) + a_{i 0} {\tilde{φ}}_{2 i} \end{matrix}

Accordingly, the compact forms of $ϕ_{1 i}$ and $ϕ_{2 i}$ are given below

ϕ_{1} = (H \otimes I_{n}) {\tilde{φ}}_{1}

(16)

ϕ_{2} = (H \otimes I_{n}) {\tilde{φ}}_{2}

(17)

where $ϕ_{1} = [ϕ_{11}^{T}, \dots, ϕ_{1 N}^{T}]^{T}$ , ${\tilde{φ}}_{1} = [{\tilde{φ}}_{11}^{T}, \dots, {\tilde{φ}}_{1 N}^{T}]^{T}$ , $ϕ_{2} = [ϕ_{21}^{T}, \dots, ϕ_{2 N}^{T}]^{T}$ , and ${\tilde{φ}}_{1} = [{\tilde{φ}}_{21}^{T}, \dots, {\tilde{φ}}_{2 N}^{T}]^{T}$ . Then, the time derivatives of $ϕ_{1}$ and $ϕ_{2}$ are computed as

\begin{matrix} {\overset{\cdot}{ϕ}}_{1} = (H \otimes I_{n}) {\overset{\cdot}{\tilde{φ}}}_{1} = ((L + A_{0}) \otimes I_{n}) {\overset{\cdot}{φ}}_{1} - A_{0} 1_{N} \otimes x_{20} \\ {\overset{\cdot}{ϕ}}_{2} = (H \otimes I_{n}) {\overset{\cdot}{\tilde{φ}}}_{2} = ((L + A_{0}) \otimes I_{n}) {\overset{\cdot}{φ}}_{2} - A_{0} 1_{N} \otimes u_{0} \end{matrix}

which means that the elements of ${\overset{\cdot}{ϕ}}_{1}$ and ${\overset{\cdot}{ϕ}}_{2}$ can be written

{\overset{\cdot}{ϕ}}_{1 i} = (l_{ii} + a_{i 0}) {\overset{\cdot}{φ}}_{1 i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\overset{\cdot}{φ}}_{1 j} - a_{i 0} x_{20}

(18)

{\overset{\cdot}{ϕ}}_{2 i} = (l_{ii} + a_{i 0}) {\overset{\cdot}{φ}}_{2 i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\overset{\cdot}{φ}}_{2 j} - a_{i 0} u_{0}

(19)

Substituting equation (12) into equation (18) leads to

{\overset{\cdot}{ϕ}}_{1 i} = - K [Λ_{1 i}] + \sum_{j = 1, j \neq i}^{N} l_{ij} {\tilde{ψ}}_{1 ij}

(20)

Define a Lyapunov function $L_{ob}$ as

L_{ob} = \frac{1}{2} \sum_{i = 1}^{N} ‖ ϕ_{1 i} ‖^{2}

then the time derivative of $L_{ob}$ can be derived as

{\overset{\cdot}{L}}_{ob} = ⋂_{θ_{1} \in \partial L_{ob}} θ_{1}^{T} K [{\overset{\cdot}{ϕ}}_{1 i}] = \sum_{i = 1}^{N} sig (ϕ_{1 i}^{T}) {P_{1 i} + P_{2 i}}

(21)

where $P_{1 i} = - Λ_{1 i} + P_{3 i}$ , $P_{2 i} = \sum_{j = 1, j \neq i}^{N} l_{ij} {\tilde{ψ}}_{1 ij} - P_{3 i}$ , and $P_{3 i} = \frac{1}{2} sig (ϕ_{1 i}) + w_{11} sig (ϕ_{1 i})^{s_{1}} + w_{12} sig (ϕ_{1 i})^{s_{2}}$ .

For term $P_{1 i}$ , the classification considers the following four cases:

Case 1: When $ϕ_{1 i}^{*} \geq 0$ , $ϕ_{1 i} \geq 0$ , let the observer gain be $w_{12}^{*} = 2^{s_{2} - 1} w_{12}$ , and combine with Lemmas 1 and 2. Then, we have

\begin{matrix} ‖ P_{1 i} ‖ = ‖ - \frac{1}{2} ‖ ϕ_{1 i}^{*} ‖ - w_{11} {‖ ϕ_{1 i}^{*} ‖}^{s_{1}} - w_{12}^{*} {‖ ϕ_{1 i}^{*} ‖}^{s_{2}} \\ + \frac{1}{2} ‖ \sum_{j = 0}^{N} a_{ij} (φ_{1 i} - q ({\hat{φ}}_{1 j}) - ψ_{1 j}) ‖ \\ + w_{11} {‖ \sum_{j = 0}^{N} a_{ij} (φ_{1 i} - q ({\hat{φ}}_{1 j}) - ψ_{1 j}) ‖}^{s_{1}} \\ + w_{12} {‖ \sum_{j = 0}^{N} a_{ij} (φ_{1 i} - q ({\hat{φ}}_{1 j}) - ψ_{1 j}) ‖}^{s_{2}} ‖ \\ \leq ‖ - \frac{1}{2} ‖ ϕ_{1 i}^{*} ‖ - w_{11} {‖ ϕ_{1 i}^{*} ‖}^{s_{1}} - w_{12}^{*} {‖ ϕ_{1 i}^{*} ‖}^{s_{2}} + P_{4 i} ‖ \\ \leq \frac{1}{2} P_{5 i} + w_{11} P_{5 i}^{s_{1}} + 2^{s_{2} - 1} w_{12} P_{5 i}^{s_{2}} \\ \leq \frac{1}{2} P_{6 i} + w_{11} P_{6 i}^{s_{1}} + 2^{s_{2} - 1} w_{12} P_{6 i}^{s_{2}} \end{matrix}

where $P_{4 i} = \frac{1}{2} [‖ ϕ_{1 i}^{*} ‖ + P_{5 i}] + w_{11} {[‖ ϕ_{1 i}^{*} ‖ + P_{5 i}]}^{s_{1}} + w_{12} {[‖ ϕ_{1 i}^{*} ‖ + P_{5 i}]}^{s_{2}}$ , $P_{5 i} = ‖ \sum_{j = 0}^{N} a_{ij} ψ_{1 j} ‖$ , and $P_{6 i} = | \sum_{j = 0}^{N} a_{ij} \frac{1}{2} n g_{1 j} (t - 1) σ_{1 j} |$ .

With a similar analysis in Case 1, the following results are deduced:

Case 2: When $ϕ_{1 i}^{*} \geq 0$ , $ϕ_{1 i} \leq 0$ , let the observer gain be $w_{12}^{*} = 2^{s_{2} - 1} w_{12}$ . Then, we have $‖ P_{1 i} ‖ \leq \frac{3}{2} P_{6 i} + 3 w_{11} P_{6 i}^{s_{1}} + (2^{2 s_{2} - 1} + 1) w_{12} P_{6 i}^{s_{2}}$ .

Case 3: When $ϕ_{1 i}^{*} \leq 0$ , $ϕ_{1 i} \leq 0$ , let the observer gain be $w_{12}^{*} = 2^{1 - s_{2}} w_{12}$ . Then, we can validate that $‖ P_{1 i} ‖ \leq \frac{1}{2} P_{6 i} + w_{11} P_{6 i}^{s_{1}} + w_{12} P_{6 i}^{s_{2}}$ .

Case 4: When $ϕ_{1 i}^{*} \leq 0$ , $ϕ_{1 i} \geq 0$ , let the observer gain be $w_{12}^{*} = 2^{1 - s_{2}} w_{12}$ . Then, one has $‖ P_{1 i} ‖ \leq \frac{3}{2} P_{6 i} + 3 w_{11} P_{6 i}^{s_{1}} + 3 w_{12} P_{6 i}^{s_{2}}$ .

Combining the above four cases gives

‖ P_{1 i} ‖ \leq Γ_{i}

(22)

where $Γ_{i} = \frac{3}{2} P_{6 i} + 3 w_{11} P_{6 i}^{s_{1}} + (2^{2 s_{2} - 1} + 1) w_{12} P_{6 i}^{s_{2}}$ .

Therefore, according to equations (21) and (22), we can further get

\begin{matrix} {\overset{\cdot}{L}}_{ob} = \sum_{i = 1}^{N} sig (ϕ_{1 i}^{T}) {P_{1 i} + P_{2 i}} \\ \leq \sum_{i = 1}^{N} {‖ ϕ_{1 i} ‖ [Γ_{i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\tilde{ψ}}_{1 ij}] - sig (ϕ_{1 i}^{T}) P_{3 i}} \\ \leq \sum_{i = 1}^{N} {\frac{1}{2} {‖ ϕ_{1 i} ‖}^{2} + \frac{1}{2} {\bar{Γ}}_{i}^{2} - P_{7 i}} \\ \leq - \sum_{i = 1}^{N} {w_{11} {‖ ϕ_{1 i} ‖}^{s_{1} + 1} + w_{12} {‖ ϕ_{1 i} ‖}^{s_{2} + 1}} + \frac{1}{2} N {\bar{Γ}}_{max}^{2} \\ \leq \frac{1}{2} N {\bar{Γ}}_{max}^{2} - 2^{\frac{s_{1} + 1}{2}} w_{11} L_{ob}^{\frac{s_{1} + 1}{2}} - 2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}} L_{ob}^{\frac{s_{2} + 1}{2}} \end{matrix}

(23)

where $P_{7 i} = \frac{1}{2} ‖ ϕ_{1 i} ‖^{2} + w_{11} ‖ ϕ_{1 i} ‖^{s_{1} + 1} + w_{12} ‖ ϕ_{1 i} ‖^{s_{2} + 1}$ , ${\bar{Γ}}_{i} = Γ_{i} + \sum_{j = 1, j \neq i}^{N} l_{ij} {\tilde{ψ}}_{1 ij}$ , and ${\bar{Γ}}_{max} = max {{\bar{Γ}}_{i}, i \in V}$ .

From equation (23), we know that $L_{ob}^{\frac{s_{2} + 1}{2}} \geq \frac{\frac{1}{2} N {\bar{Γ}}_{max}^{2}}{2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}}}$ holds due to ${\overset{\cdot}{L}}_{ob} < - 2^{\frac{s_{1} + 1}{2}} w_{11} L_{ob}^{\frac{s_{1} + 1}{2}} \leq 0$ , thereby $L_{ob}$ is bounded. When $L_{o b}^{\frac{s_{2} + 1}{2}} \geq \frac{\frac{1}{2} N {\bar{Γ}}_{\max^{2}}}{ν_{1} 2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}}}$ with $0 < ν_{1} < 1$ , one has $\frac{1}{2} N {\bar{Γ}}_{max}^{2} \leq ν_{1} 2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}} L_{ob}^{\frac{s_{2} + 1}{2}}$ . Then, it follows from equation (23) that

{\overset{\cdot}{L}}_{ob} \leq - 2^{\frac{s_{1} + 1}{2}} w_{11} L_{ob}^{\frac{s_{1} + 1}{2}} - (1 - ν_{1}) 2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}} L_{ob}^{\frac{s_{2} + 1}{2}}

According to Lemma 5, $L_{ob}$ will converge to the set ${L_{ob} | L_{ob} \leq {(\frac{\frac{1}{2} N {\bar{Γ}}_{max}^{2}}{2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}}})}^{\frac{2}{s_{2} + 1}}}$ in fixed time bounded by $T_{1}$ . Accordingly, it is proved that $ϕ_{1}$ will converge to neighborhood sets of the origin. By a similar procedure to equations (20)–(23), we can also verify that $ϕ_{2}$ will converge to neighborhood sets of the origin within $T_{2}$ , which is omitted here.

From Assumption 1, it is easy to get that $H$ is invertible. Therefore, it follows from equations (16) and (17) that all followers will estimate the states of the leader when $t > T_{ob}$ . The proof of Theorem 1 is completed.□

Remark 6. It is easy to get $‖ ϕ_{1 i} ‖ \leq \sqrt{{((\frac{1}{2} N {\bar{Γ}}_{max}^{2}) / (2^{\frac{s_{2} + 1}{2}} w_{12} N^{\frac{1 - s_{2}}{2}}))}^{\frac{2}{s_{2} + 1}} \frac{2}{N}}$ and $‖ ϕ_{2 i} ‖ \leq \sqrt{{((\frac{1}{2} N {\bar{Γ}}_{max}^{2}) / (2^{\frac{s_{2} + 1}{2}} w_{22} N^{\frac{1 - s_{2}}{2}}))}^{\frac{2}{s_{2} + 1}} \frac{2}{N}}$ . By appropriately adjusting the observer parameters, the size of the estimation error bounds can be reduced. For example, increasing the values of parameters $w_{12}$ and $w_{22}$ while decreasing the values of parameters $g_{1 j} (t - 1)$ and $g_{2 j} (t - 1)$ will bring smaller estimation error bounds.

Remark 7. It is worth pointing out that the standard fixed-time observer (Mi et al., 2023 ; Ni et al., 2021b) usually contains a fractional exponential term, a power exponent term, and a symbolic function term, which means that more adjustable parameters need to be introduced, and conservativeness of the convergence time bound is increased accordingly. Subsequently, the observer proposed in Zuo et al. (2020) only adopted one power exponent term and one sign function term, and thus it is simpler in design and parameter tuning. However, the observer estimation error only achieves the fixed-time set attractivity by this observer, which brings difficulties for the subsequent local control design. Compared with the aforementioned observers, the observers (12) and (13) of this paper not only has less conservativeness of convergence time bound but also has a relatively simple analysis process.

Design of adaptive local tracking control algorithm

According to Theorem 1, it is proved that each follower can estimate the leader’s state under the observers (16) and (17) when $t > T_{ob}$ . Hereafter, a novel adaptive local tracking control strategy is proposed for NELSs (2) by the backstepping design procedure. The local control part is presented in Figure 1.

Step 1 : Let $e_{1 i} = x_{1 i} - φ_{1 i}$ be the position error vector, where $e_{1 i} = [e_{1 i}^{1}, \dots, e_{1 i}^{n}]^{T}$ , $i \in V$ . And its time derivative is given as

{\overset{\cdot}{e}}_{1 i} = x_{1 i} - {\overset{\cdot}{φ}}_{1 i}

(24)

where ${\overset{\cdot}{e}}_{1 i} = [{\overset{\cdot}{e}}_{1 i}^{1}, \dots, {\overset{\cdot}{e}}_{1 i}^{n}]^{T}$ . Based on equation (10), equation (24) can be rewritten as

{\overset{\cdot}{ξ}}_{i} = c_{1 i} x_{2 i} + v_{1 i} - c_{di} (φ_{2 i} + d_{i}) - v_{di}

(25)

where $v_{1 i} = [v_{1 i}^{1}, \dots, v_{1 i}^{n}]^{T}$ , $c_{1 i} = diag {c_{1 i}^{1}, \dots, c_{1 i}^{n}}$ , $c_{di} = diag {c_{di}^{1}, \dots, c_{di}^{n}}$ , $c_{di}^{r} = \frac{{\underline{k}}_{i}^{r} {\bar{k}}_{i}^{r} ({\underline{k}}_{i}^{r} {\bar{k}}_{i}^{r} + {(φ_{1 i}^{r})}^{2})}{{({\underline{k}}_{i}^{r} + φ_{1 i}^{r})}^{2} {({\bar{k}}_{i}^{r} - φ_{1 i}^{r})}^{2}}$ , $d_{i} = {\overset{\cdot}{φ}}_{1 i} - φ_{2 i}$ , $v_{di} = [v_{di}^{1}, \dots, v_{di}^{n}]^{T}$ , $v_{di}^{r} = V_{1 di}^{r} {\underline{\overset{\cdot}{k}}}_{i}^{r} + V_{2 di}^{r} {\overset{\cdot}{\bar{k}}}_{i}^{r}$ , $V_{1 di}^{r} = \frac{{\bar{k}}_{i}^{r} {(φ_{1 i}^{r})}^{2}}{{({\underline{k}}_{i}^{r} + φ_{1 i}^{r})}^{2} ({\bar{k}}_{i}^{r} - φ_{1 i}^{r})}$ , $V_{2 di}^{r} = - \frac{{\underline{k}}_{i}^{r} {(φ_{1 i}^{r})}^{2}}{({\underline{k}}_{i}^{r} + φ_{1 i}^{r}) {({\bar{k}}_{i}^{r} - φ_{1 i}^{r})}^{2}}$ , $i \in V$ , and $r = 1, \dots, n$ .

Remark 8. Note that $d_{i}$ in equation (25) caused by quantization error can be considered disturbances. According to Assumption 2, $φ_{2 i}$ is bounded. In addition, we know that ${\overset{\cdot}{φ}}_{1 i}$ is also bounded based on the proof of Theorem 1. Thus, we have that $d_{i}$ is bounded, that is, a positive constant ${\bar{d}}_{i}^{r}$ can be found such that $| d_{i}^{r} | \leq {\bar{d}}_{i}^{r}$ , $r = 1, \dots, n$ .

Define velocity error vector as

e_{2 i} = x_{2 i} - β_{i}

(26)

where $e_{2 i} = [e_{2 i}^{1}, \dots, e_{2 i}^{n}]^{T}$ and $β_{i} = [β_{i}^{1}, \dots, β_{i}^{n}]^{T}$ is the virtual control. Thus, equation (25) is rewritten as

{\overset{\cdot}{ξ}}_{i} = c_{1 i} (e_{2 i} + β_{i}) + v_{1 i} - c_{di} (φ_{2 i} + d_{i}) - v_{di}

(27)

To further improve the accurate control performance, the disturbance $d_{i}$ should be compensated. We define $W_{i} = {\bar{d}}_{i}$ , ${\tilde{W}}_{i} = {\hat{W}}_{i} - W_{i}$ , where ${\bar{d}}_{i} = [{\bar{d}}_{i}^{1}, \dots, {\bar{d}}_{i}^{n}]^{T}$ , ${\hat{W}}_{i} \in R^{n}$ refers to the estimation of $W_{1 i}$ , ${\tilde{W}}_{i} \in R^{n}$ indicates the estimation error, and the adaptive law is governed by

\dot{{\hat{W}}_{i}} = F_{1 i} [c_{d i} {\hat{ξ}}_{i} - {\hat{W}}_{i} {\hat{W}}_{i}^{T} {\hat{W}}_{i} - ρ_{1 i} {\hat{W}}_{i}]

(28)

where $F_{1 i}$ is a positive-definite symmetric matrix, ${\hat{ξ}}_{i} = {[| ξ_{i}^{1} |, \dots, | ξ_{i}^{n} |]}^{T}$ , $ρ_{1 i} \in R^{+}$ .

Remark 9. It is worth noting that the additional term ${\hat{W}}_{i} {\hat{W}}_{i}^{T} {\hat{W}}_{i}$ in adaptive update law (28) is added to be consistent with the requirement of fixed-time convergence presented in Lemma 5. $ρ_{1 i}$ -modification term is added such that the strong robustness can still be achieved even though there exist approximation errors.

Then, the virtual controller $β_{i}$ is selected as

\begin{matrix} β_{i} = c_{1 i}^{- 1} [- K_{11 i} {(\frac{1}{2})}^{\frac{3}{4}} Ξ_{1 i} - K_{12 i} {(\frac{1}{2})}^{2} ξ_{i} ξ_{i}^{T} ξ_{i} \\ - v_{1 i} + c_{di} φ_{2 i} + v_{di} + {\bar{ξ}}_{i} c_{di} {\hat{W}}_{i}] \end{matrix}

(29)

where $K_{11 i} \in R^{+}$ and $K_{12 i} \in R^{+}$ are the gain parameters and ${\bar{ξ}}_{i} = diag {sign (ξ_{i}^{1}), \dots, sign (ξ_{i}^{n})}$ . Motivated by Jin (2019), to circumvent a singularity issue, the function $Ξ_{1 i}$ is chosen as

Ξ_{1 i} = {\begin{matrix} ξ_{i} {(ξ_{i}^{T} ξ_{i})}^{- \frac{1}{4}} & if ‖ ξ_{i} ‖ \geq ϱ_{1} \\ \frac{5}{4} ξ_{i} ϱ_{1}^{- \frac{1}{2}} - \frac{1}{4} ξ_{i} ξ_{i}^{T} ξ_{i} ϱ_{1}^{- \frac{5}{2}} & otherwise \end{matrix}

where $ϱ_{1} > 0$ . Therefore, we have $lim_{ξ_{i} \to 0} Ξ_{1 i} = lim_{ξ_{i} \to 0} \frac{5}{4} ξ_{i} ϱ_{1}^{- \frac{1}{2}} - \frac{1}{4} ξ_{i} ξ_{i}^{T} ξ_{i} ϱ_{1}^{- \frac{5}{2}} = 0$ .

Select the Lyapunov function candidate as

L_{c 1 i} = \frac{1}{2} ξ_{i}^{T} ξ_{i} + \frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i}

Taking the time derivative of $L_{c 1 i}$ yields

\begin{matrix} {\overset{\cdot}{L}}_{c 1 i} \leq - K_{11 i} {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{\frac{3}{4}} - K_{12 i} {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{2} + ξ_{i}^{T} c_{1 i} e_{2 i} \\ + {\hat{ξ}}_{i}^{T} c_{di} {\tilde{W}}_{i} + {\tilde{W}}_{i}^{T} {\overset{\cdot}{\hat{W}}}_{i} + ς_{1 i} \\ = - K_{11 i} {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{\frac{3}{4}} - K_{12 i} {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{2} \\ + ξ_{i}^{T} c_{1 i} e_{2 i} - {\tilde{W}}_{i}^{T} {\hat{W}}_{i} {\hat{W}}_{i}^{T} {\hat{W}}_{i} \\ - ρ_{1 i} {\tilde{W}}_{i}^{T} {\hat{W}}_{i} + ς_{1 i} \end{matrix}

(30)

where

\begin{matrix} ς_{1 i} = {\begin{matrix} 0 & ‖ ξ_{i} ‖ \geq ϱ_{1} \\ K_{11 i} {(\frac{1}{2})}^{\frac{3}{4}} Θ_{1 i} & otherwise \end{matrix} \\ Θ_{1 i} = {(ξ_{i}^{T} ξ_{i})}^{\frac{3}{4}} - \frac{5}{4} ξ_{i}^{T} ξ_{i} ϱ_{1}^{- \frac{1}{2}} + \frac{1}{4} {(ξ_{i}^{T} ξ_{i})}^{2} ϱ_{1}^{- \frac{5}{2}} \end{matrix}

For the term $- ρ_{1 i} {\tilde{W}}_{i}^{T} {\hat{W}}_{i}$ in equation (30), it is obtained that

- ρ_{1 i} {\tilde{W}}_{i}^{T} {\hat{W}}_{i} \leq - \frac{ρ_{1 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + \frac{ρ_{1 i}}{2} W_{i}^{T} W_{i}

(31)

For the term $- \frac{ρ_{1 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i}$ in equation (31), we have

\begin{matrix} - \frac{ρ_{1 i}}{2} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} = - \frac{ρ_{1 i}}{4} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} - ρ_{1 i} {({\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{3}{4}} \\ - \frac{ρ_{1 i}}{4} {[{({\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2}} - 2 {({\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{4}}]}^{2} \\ + ρ_{1 i} {({\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2}} \end{matrix}

(32)

From Lemmas 3 and 4, it follows that

- ρ_{1 i} {({\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{3}{4}} \leq - ρ_{1 i} {(\frac{2}{λ_{max} (F_{1 i}^{- 1})})}^{\frac{3}{4}} {(\frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i})}^{\frac{3}{4}}

(33)

ρ_{1 i} {({\tilde{W}}_{i}^{T} {\tilde{W}}_{i})}^{\frac{1}{2}} \leq \frac{ρ_{1 i}}{8} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} + 2 ρ_{1 i}

(34)

Applying equations (32)–(34) to equation (31), it yields that

\begin{matrix} - ρ_{1 i} {\tilde{W}}_{i}^{T} {\tilde{W}}_{i} \leq - ρ_{1 i} {(\frac{2}{λ_{max} (F_{1 i}^{- 1})})}^{\frac{3}{4}} {(\frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i})}^{\frac{3}{4}} \\ + \frac{ρ_{1 i}}{2} W_{i}^{T} W_{i} + 2 ρ_{1 i} \end{matrix}

(35)

Then, the term ${\tilde{W}}_{i}^{T} {\hat{W}}_{i} {\hat{W}}_{i}^{T} {\hat{W}}_{i}$ in equation (30) is given by

\begin{matrix} - {\tilde{W}}_{i}^{T} {\hat{W}}_{i} {\hat{W}}_{i}^{T} {\hat{W}}_{i} = - ‖ {\tilde{W}}_{i} ‖^{4} - 3 ‖ W_{i} ‖^{2} ‖ {\tilde{W}}_{i} ‖^{2} \\ - ‖ W_{i} ‖^{2} {\tilde{W}}_{i}^{T} W_{i} - 3 ‖ {\tilde{W}}_{i} ‖^{2} {\tilde{W}}_{i}^{T} W_{i} \end{matrix}

(36)

Using Lemma 3, we get

- 3 ‖ {\tilde{W}}_{i} ‖^{2} {\tilde{W}}_{i}^{T} W_{i} \leq 3 ‖ {\tilde{W}}_{i} ‖^{3} ‖ W_{i} ‖ \leq \frac{9}{4} κ_{1 i}^{\frac{4}{3}} ‖ {\tilde{W}}_{i} ‖^{4} + \frac{3}{4 κ_{i}^{4}} ‖ W_{i} ‖^{4}

(37)

- ‖ W_{i} ‖^{2} {\tilde{W}}_{i}^{T} W_{i} \leq 3 ‖ W_{i} ‖^{2} ‖ {\tilde{W}}_{i} ‖^{2} + \frac{1}{12} ‖ W_{i} ‖^{4}

(38)

where $0 < κ_{1 i} < {(\frac{2}{3})}^{\frac{3}{2}}$ . Based on Lemma 4, one obtains

- ‖ {\tilde{W}}_{i} ‖^{4} \leq - \frac{4}{λ_{max}^{2} (F_{1 i}^{- 1})} {(\frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i})}^{2}

(39)

Taking into account equations (37)–(39), equation (36) is rewritten as

\begin{matrix} - {\tilde{W}}_{i}^{T} {\hat{W}}_{i} {\hat{W}}_{i}^{T} {\hat{W}}_{i} \leq - \frac{4 - 9 κ_{1 i}^{\frac{4}{3}}}{λ_{max}^{2} (F_{1 i}^{- 1})} {(\frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i})}^{2} \\ + (\frac{3}{4 κ_{1 i}^{4}} + \frac{1}{12}) ‖ W_{i} ‖^{4} \end{matrix}

(40)

Applying equations (35) and (40) to equation (30), one has

\begin{matrix} {\overset{\cdot}{L}}_{c 1 i} \leq - K_{11 i} {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{\frac{3}{4}} - K_{12 i} {(\frac{1}{2} ξ_{i}^{T} ξ_{i})}^{2} + ξ_{i}^{T} c_{1 i} e_{2 i} \\ - (ρ_{1 i} {(\frac{2}{λ_{max} (F_{1 i}^{- 1})})}^{\frac{3}{4}} {(\frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i})}^{\frac{3}{4}}) \\ - (\frac{4 - 9 κ_{1 i}^{\frac{4}{3}}}{λ_{max}^{2} (F_{1 i}^{- 1})} {(\frac{1}{2} {\tilde{W}}_{i}^{T} F_{1 i}^{- 1} {\tilde{W}}_{i})}^{2}) + ς_{1 i} + B_{1 i} \end{matrix}

(41)

where $B_{1 i} = [(\frac{3}{4 κ_{1 i}^{4}} + \frac{1}{12}) {‖ W_{i} ‖}^{4} + \frac{ρ_{1 i}}{2} {‖ W_{i} ‖}^{2} + 2 ρ_{1 i}]$

Step 2 : From equations (2) and (26), the time derivative of $e_{2 i}$ satisfies

{\overset{\cdot}{e}}_{2 i} = M_{i}^{- 1} [U_{i} - C_{i} x_{2 i} - O_{i}] - {\overset{\cdot}{β}}_{i}

Consider the Lyapunov function candidate at this step as

L_{c 2 i} = \frac{1}{2} e_{2 i}^{T} M_{i} e_{2 i}

and the time derivative of $L_{c 2 i}$ is given by

{\overset{\cdot}{L}}_{c 2 i} = e_{2 i}^{T} [U_{i} - C_{i} x_{2 i} - O_{i}] - e_{2 i}^{T} M_{i} {\overset{\cdot}{β}}_{i} + \frac{1}{2} e_{2 i}^{T} {\overset{\cdot}{M}}_{i} e_{2 i}

(42)

From He et al. (2016), we know that $\frac{1}{2} ({\overset{\cdot}{M}}_{i} - 2 C_{i})$ is a skew-symmetric matrix, so we obtain $\frac{1}{2} e_{2 i}^{T} ({\overset{\cdot}{M}}_{i} - 2 C_{i}) e_{2 i} = 0$ .

Hence, equation (42) turns to be

{\overset{\cdot}{L}}_{c 2 i} = e_{2 i}^{T} [U_{i} - M_{i} {\overset{\cdot}{β}}_{i} - C_{i} β_{i} - O_{i}]

(43)

Then, the adaptive neural networks technique (He et al., 2016; Kong et al., 2021) is employed to compensate for the uncertain term $M_{i}$ , $C_{i}$ , and $O_{i}$ . To this end, the radial basis function neural network approximation can be formulated as

f_{i} = Ψ_{i}^{T} S_{i} (Z_{i}) + ε_{i} (Z_{i}) = M_{i} {\overset{\cdot}{β}}_{i} + C_{i} β_{i} + O_{i}

(44)

where $Ψ_{i} = [Ψ_{i}^{1}, \dots, Ψ_{i}^{n}] \in R^{l \times n}$ represents the vector of optimal weight, $l$ indicates the number of neural nodes, $Z_{i} = [x_{1 i}^{T}, x_{2 i}^{T}, β_{i}^{T}, {\overset{\cdot}{β}}_{i}^{T}]^{T} \in R^{n \times 4}$ refers to the input vector, and $S_{i} (Z_{i})$ indicates the basis function. $ε_{i} (Z_{i}) \in R^{n}$ refers to the approximation error, which satisfies $∥ ε_{i} (Z_{i}) ∥ \leq {\bar{ε}}_{i}$ with ${\bar{ε}}_{i} \in R^{+}$ . Let ${\tilde{Ψ}}_{i} = {\hat{Ψ}}_{i} - Ψ_{i}$ , where ${\hat{Ψ}}_{i}$ indicates the estimation of optimal weight $Ψ_{i}$ and ${\tilde{Ψ}}_{i}$ stands for the weight error.

Substituting equation (44) into equation (43) yields

{\overset{\cdot}{L}}_{c 2 i} = e_{2 i}^{T} [U_{i} - Ψ_{i}^{T} S_{i} (Z_{i}) - ε_{i} (Z_{i})]

(45)

The control law is established as

U_{i} = - K_{21 i} {(\frac{1}{2})}^{\frac{3}{4}} Ξ_{2 i} - K_{22 i} {(\frac{1}{2})}^{2} e_{2 i} e_{2 i}^{T} e_{2 i} + {\hat{Ψ}}_{i}^{T} S_{i} (Z_{i}) - c_{1 i} ξ_{i} - \frac{1}{2} e_{2 i}

(46)

where the gain parameters $K_{21 i} \in R^{+}$ and $K_{22 i} \in R^{+}$ . Moreover, the function $Ξ_{2 i}$ in $U_{i}$ is selected as

Ξ_{2 i} = {\begin{matrix} e_{2 i} {(e_{2 i}^{T} e_{2 i})}^{- \frac{1}{4}} & if ‖ e_{2 i} ‖ \geq ϱ_{2} \\ \frac{5}{4} e_{2 i} ϱ_{2}^{- \frac{1}{2}} - \frac{1}{4} e_{2 i} e_{2 i}^{T} e_{2 i} ϱ_{2}^{- \frac{5}{2}} & otherwise \end{matrix}

where $ϱ_{2} \in R^{+}$ .

Then, the adaptive laws are designed as

\overset{\cdot}{{\hat{Ψ}}_{i}^{r}} = - F_{2 i}^{r} [S_{i}^{r} (Z_{i}) e_{2 i}^{r} + {\hat{Ψ}}_{i}^{r} {({\hat{Ψ}}_{i}^{r})}^{T} {\hat{Ψ}}_{i}^{r} + ρ_{2 i}^{r} {\hat{Ψ}}_{i}^{r}]

(47)

where $F_{2 i}^{r}$ is a symmetric positive-definite matrix, $r = 1, \dots, n$ , $ρ_{2 i}^{r} \in R^{+}$ .

Substituting equation (46) into equation (45), we obtain

\begin{matrix} {\overset{\cdot}{L}}_{c 2 i} = - K_{21 i} {(\frac{1}{2} e_{2 i}^{T} e_{2 i})}^{\frac{3}{4}} - K_{22 i} {(\frac{1}{2} e_{2 i}^{T} e_{2 i})}^{2} \\ - e_{2 i}^{T} c_{1 i} ξ_{i} - \frac{1}{2} e_{2 i}^{T} e_{2 i} + e_{2 i}^{T} {\tilde{Ψ}}_{i}^{T} S_{i} (Z_{i}) \\ - e_{2 i}^{T} ε_{i} (Z_{i}) + ς_{2 i} \end{matrix}

(48)

where

\begin{matrix} ς_{2 i} = {\begin{matrix} 0 & ‖ e_{2 i} ‖ \geq ϱ_{2} \\ K_{21 i} {(\frac{1}{2})}^{\frac{3}{4}} Θ_{2 i} & otherwise \end{matrix} \\ Θ_{2 i} = {(e_{2 i}^{T} e_{2 i})}^{\frac{3}{4}} - \frac{5}{4} e_{2 i}^{T} e_{2 i} ϱ_{2}^{- \frac{1}{2}} + \frac{1}{4} {(e_{2 i}^{T} e_{2 i})}^{2} ϱ_{2}^{- \frac{5}{2}} \end{matrix}

For the term $- e_{2 i}^{T} ε_{i} (Z_{i})$ in equation (48), it is easy to get

- e_{2 i}^{T} ε_{i} (Z_{i}) \leq \frac{1}{2} {\bar{ε}}_{i}^{2} + \frac{1}{2} e_{2 i}^{T} e_{2 i}

(49)

Then, by using Lemma 4, we have

- K_{21 i} {(\frac{1}{2} e_{2 i}^{T} e_{2 i})}^{\frac{3}{4}} \leq - \frac{K_{21 i}}{λ_{max}^{\frac{3}{4}} (M_{i})} {(L_{c 2 i})}^{\frac{3}{4}}

(50)

- K_{22 i} {(\frac{1}{2} e_{2 i}^{T} e_{2 i})}^{2} \leq - \frac{K_{22 i}}{λ_{max}^{2} (M_{i})} {(L_{c 2 i})}^{2}

(51)

With equations (49)–(51) applied to equation (48), we get

\begin{matrix} {\overset{\cdot}{L}}_{c 2 i} \leq - \frac{K_{21 i} {(L_{c 2 i})}^{\frac{3}{4}}}{λ_{max}^{\frac{3}{4}} (M_{i})} - \frac{K_{22 i} {(L_{c 2 i})}^{2}}{λ_{max}^{2} (M_{i})} - e_{2 i}^{T} c_{1 i} ξ_{i} + e_{2 i}^{T} {\tilde{Ψ}}_{i}^{T} S_{i} (Z_{i}) \\ + \frac{1}{2} {\bar{ε}}_{i}^{2} + ς_{2 i} \end{matrix}

(52)

Theorem 2. For the NELSs (2) and (3), when the virtual controller, controller, and adaptive law are selected as equations (29), (46), and (47), the following results can be obtained

(i) The settling time function $T_{c}$ satisfies

T_{c} = \frac{4 Γ (\frac{1}{5}) Γ (\frac{4}{5})}{5 χ_{1} Γ (1)} {(\frac{(2 + n) χ_{1}}{(1 - ν_{2}) χ_{2}})}^{\frac{1}{5}}

where $0 < ν_{2} < 1$ , $χ_{1}, χ_{2} \in R^{+}$ .

(ii) If the initial values $q_{i} (0)$ , ${\overset{\cdot}{q}}_{i} (0)$ , and ${\hat{Ψ}}_{i}^{r} (0)$ ( $i \in V$ , $r = 1, \dots, n$ ) are bounded, then the errors $ξ_{i}$ , $e_{2 i}$ , and ${\tilde{Ψ}}_{i}$ will converge into compact sets $Ω_{ξ_{i}}$ , $Ω_{e_{2 i}}$ , and $Ω_{{\tilde{Ψ}}_{i}}$ within fixed time, where $Ω_{ξ_{i}} : = {ξ_{i} | ‖ ξ_{i} ‖ \leq \sqrt{A_{i}}}$ , $Ω_{e_{2 i}} : = {e_{2 i} | ‖ e_{2 i} ‖ \leq \sqrt{\frac{A_{i}}{λ_{min} (M_{i})}}}$ , $Ω_{{\tilde{Ψ}}_{i}} : = {{\tilde{Ψ}}_{i} \in R^{l \times n} | ‖ {\tilde{Ψ}}_{i}^{r} ‖ \leq \sqrt{\frac{A_{i}}{λ_{min} ({(F_{2 i}^{r})}^{- 1})}}}$ , and $A_{i} : = 2 (\sqrt{\frac{B_{i} (2 + n)}{ν_{2} χ_{2 i}}})$ .

Proof. See Appendix A. □

Remark 10. By properly adjusting parameters, it is possible to make the sizes of the compact sets $Ω_{ξ_{i}}$ and $Ω_{e_{2 i}}$ as small as possible. For instance, increasing parameters $χ_{2}$ , $κ_{1 i}$ , $κ_{2 i}^{r}$ , and $ν_{2}$ and decreasing parameters $ρ_{1 i}$ and $ρ_{2 i}^{r}$ will reduce the sizes of the compact sets.

Simulation results

To exhibit the viability of the proposed methods, simulation is conducted based on a manipulator cooperation platform established with a virtual leader and six two-link manipulators as depicted in Figure 2, where the network topology satisfies Assumption 1. The system model for the ith two-link manipulator is offered below

M_{i} (q_{i}) {\overset{\cdot\cdot}{q}}_{i} + C_{i} (q_{i}, {\overset{\cdot}{q}}_{i}) {\overset{\cdot}{q}}_{i} + O_{i} (q_{i}) = U_{i}, i = 1, \dots, 6

where physical parameters are provided in He et al. (2016).

Figure 2.

Network topology for multiple two-link manipulators.

In the simulation, the control input of the leader is presented as $u_{10} = [\begin{matrix} - 0.1 \sin (t) - \cos (t) & - 0.1 \sin (t) - \cos (t) \end{matrix}]^{T}$ .

Next, we choose the initial values as $x_{10} (0) = [\begin{matrix} 1.0 & 1.0 \end{matrix}]^{T} rad$ , $q_{1} (0) = [\begin{matrix} 2.5 & 0.5 \end{matrix}]^{T} rad$ , $q_{2} (0) = [\begin{matrix} 2.1 & 0.9 \end{matrix}]^{T} rad$ , $q_{3} (0) = [\begin{matrix} 1.7 & 1.3 \end{matrix}]^{T} rad$ , $q_{4} (0) = [\begin{matrix} 1.3 & 1.7 \end{matrix}]^{T} rad$ , $q_{5} (0) = [\begin{matrix} 0.9 & 2.1 \end{matrix}]^{T} rad$ , and $q_{6} (0) = [\begin{matrix} 0.5 & 2.5 \end{matrix}]^{T} rad$ . The workspace constraints are defined as ${\bar{k}}_{i}^{m} = 1.65 + 0.2 \sin (t) + \cos (t)$ , ${\underline{k}}_{i}^{m} = 1 - 0.2 \sin (t) - \cos (t)$ , and $m = 1, 2$ .

The simulation parameters are summarized in Table 1, so it can be inferred that $T_{ob} = 0.6024 (s)$ and $T_{c} = 1.1644 (s)$ . Then, the Gaussian function is chosen as the basis function (He et al., 2016), $2^{8}$ nodes are employed for each $S_{i, k}^{r} (Z_{i})$ , $k = 1, \dots, 2^{8}$ , and the centers are chosen in the area of $[- 1, 1] \times [- 1, 1] \times [- 1, 1] \times [- 1, 1] \times [- 1, 1] \times [- 1, 1] \times [- 1, 1] \times [- 1, 1]$ .

Table 1.

Simulation parameters.

Notation	Value	Notation	Value
$w_{11}$	10	$w_{12}$	10
$w_{21}$	10	$w_{22}$	10
$s_{1}$	0.6	$s_{2}$	1.6
$ν_{1}$	0.01	$ν_{2}$	0.01
$K_{11 i}$	8	$K_{12 i}$	8
$K_{21 i}$	10	$K_{22 i}$	10
$F_{1 i}$	100	$F_{2 i}$	$100 I_{2^{8} \times 2^{8}}$
$κ_{1 i}$	0.54	$κ_{2 i}$	0.54
$ρ_{1 i}$	0.01	$ρ_{2 i}$	0.01
$λ_{max} (M_{i})$	0.0314	Sampling time	$1 \times 10^{- 4} (s)$
$σ_{i}$	0.07	$g_{i} (t)$	$0.07 + 0.2 \cdot 0 . 8^{t}$
$m$	1,2	$i$	$1, \dots, 6$

The simulation results are manifested in Figures 3 –8. The responses of position estimation and velocity estimation for each manipulator are depicted in Figures 3 and 4. Under the quantized communication environment, the states of the leader can be captured by each follower within $0.3 (s)$ through the observers (12) and (13), and the convergence time $t$ satisfies $t < T_{ob} = 0.6024 (s)$ . Moreover, the comparisons among $T_{ob}$ are listed in Table 2. For the sake of fairness, the simulation parameters involved in Ni et al. (2021b), Zuo et al. (2020), and Mi et al. (2023) are uniformly chosen as $p = 13$ , $q = 17$ , $λ_{min} (Q) = 0.6696$ , $γ = 2.3$ $N = 6$ , and $p_{m} = 6$ . From Table 2, it is revealed that the $T_{ob}$ of this paper is less conservativeness.

Figure 3.

Angular estimation for each manipulator.

Figure 4.

Angular velocity estimation for each manipulator.

Figure 5.

Top: angular tracking for each manipulator under constraints. Bottom: errors $ξ_{1}$ .

Figure 6.

Top: angular velocity tracking for each manipulator. Bottom: errors $e_{2}$ .

Figure 7.

Control input $U$ .

Figure 8.

Adaptive parameters $∥ \hat{Ψ} ∥$ .

Table 2.

Comparison of $T_{ob}$ .

Schemes	$T_{ob} (s)$
Theorem 1 of Ni et al. (2021b)	4.2106
Theorem 2 of Zuo et al. (2020)	2.6001
Theorem 1 of Mi et al. (2023)	7.1834
Theorem 1 in this article	0.6024

Subsequently, based on the adaptive local tracking control strategy, it is seen that the trajectory of the leader can be tracked by each follower within $0.7 (s)$ from the top of Figures 5 and 6, and the convergence time $t$ satisfies $t \leq T_{co} = 1.1644 (s)$ . Apparently, the asymmetric state constraints are not violated, that is, $- {\underline{k}}_{i}^{r} (t) < q_{i}^{r} (t) < {\bar{k}}_{i}^{r}$ is ensured. The bottom of Figures 5 and 6 depicts that the error signals converge rapidly to the compact sets near the origin. The response of the control signal for each manipulator is portrayed in Figure 7. Furthermore, Figure 8 exhibits the trajectories of adaptive parameters during the tracking process.

Conclusion

In this paper, the problem of quantized FTCT has been investigated for NELSs under the predetermined workspace. Note that the proposed distributed estimation local control structure, which establishes a bridge between the consensus of multi-agent systems and the tracking of single Euler-Lagrange systems, has effectively reduced the difficulty of solving the FTCT issue. Furthermore, the quantization scheme based on encoder, quantizer, and decoder has reduced the number of bits transmitted in communication links and improved the quantization accuracy. Unlike existing barrier Lyapunov function methods, a uniform state-constrained function has been introduced to cope with the problem of asymmetric time-varying state constraints, while an unconstrained form is also applicable. In the follow-up study, we will focus on the event-based predefined-time consensus tracking problem for NELSs.

Footnotes

Appendix A

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grants 61973139 and 61991402 and the Fundamental Research Funds for the Central Universities under Grant JUSRP22014.

ORCID iDs

He Li

Cheng-Lin Liu

Ya Zhang

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