Nonsingular adaptive finite-time consensus control for uncertain nonlinear multi-agent systems with input quantization

Abstract

This paperinvestigates the distributed consensus control for nonlinear multi-agent systems with input quantization in a directed communication topology. A nonsingular adaptive finite-time control (NAFTC) scheme is proposed by combining the filtered backstepping method with neural network control. Both chattering and singularity problems of the control signals are avoided thanks to its $C^{1}$ continuity, so that the contradiction between discontinuous- and $C^{0}$ continuous finite-time control with input quantization does not exist in our scheme. At the same time, a novel inequality for the input-output relationship of hysteresis quantizer is derived, which can simply handle the input quantization without requiring additional stability analysis. What is more, some common assumptions, such as Lipschitz assumption and Holder-continuity assumption, are not required in our scheme. Finally, stability analysis and numerical simulation are provided.

Keywords

Nonlinear multi-agent systems input quantization consensus adaptive backstepping finite-time control

Introduction

Driven by the application prospect, significant efforts from communities of power electronics, circuit and signal, and control science have been devoted to the coordination control research for multi-agent systems (MASs) and networked interconnected systems over the past decades Olfati-Saber and Murray (2004), Yu et al. (2011), Cao et al. (2013), Mozaffari et al. (2021), Xiang et al. (2022), Dehkordi et al. (2019), and Liang et al. (2023). Consensus is one of the most concerned topics in coordinated control, whose objective is to control all agents/subsystems to track the leader (or reference signal) and then reach an agreement on some state of concern Olfati-Saber and Murray (2004), Yu et al. (2011), Cao et al. (2013), and Mozaffari et al. (2021). The results of consensus control are widely used in sensor networks Mozaffari et al. (2021), distributed microgrids Dehkordi et al. (2019), multiple vehicles Liang et al. (2023), etc.

For the consensus control of MASs, integrator-type MASs and general linear MASs are the first to be concerned, and a wealth of excellent results have been achieved so far Yu et al. (2011), Cao et al. (2013), Ferrari-Trecate et al. (2009), Wen et al. (2013), and Li et al. (2015). In view of the fact that many systems in reality are inherently nonlinear, the cooperative control for nonlinear MASs has been receiving increasing attention. Uncertainties, such as model perturbations and signal noises, are ubiquitous in nonlinear systems, which brings great challenges to control design Sanner and Slotine (1992), Ge and Wang (2002), and Song et al. (2021). With the development of nonlinear control theory, the backstepping method combined with neural network (NN), sliding mode control (SMC), and fuzzy logic (FL) have shown their effectiveness in dealing with the consensus control problems of uncertain nonlinear MASs Chen et al. (2016), Shahvali and Askari (2016), Huang et al. (2015), and Shafiqul et al. (2021). However, the aforementioned control schemes can only obtain the asymptotic consensus or uniformly ultimately bounded (UUB) results, that is, they all belong to infinite-time control, which is not suitable for the control tasks with requirements for settling time.

Finite-time control (FTC) is recognized as an effective technique for many practical systems that have real-time requirements, such as braking system and circuit protection system. Compared with general asymptotic control, FTC has a faster convergence rate and better robustness since it contains fractional power control terms, especially near the equilibrium of the system Bhat and Bernstein (2000) and Yu et al. (2005, 2018). For linear and Lipschitz nonlinear MASs, abundant excellent results of FTC results have been reported yet, such as Yu and Long (2015), Sakthivel et al. (2019), Yin et al. (2020), Zhao et al. (2018), and You et al. (2020). Differently, for uncertain nonlinear systems, it is not easy to achieve finite-time stable results due to unknown residuals that are often generated by adaptive control methods. Hence, the concept of practical finite-time stable/finite-time bounded is proposed by Zhu et al. (2011). For uncertain nonlinear MASs, an NN and SMC-based practical FTC law is proposed by Zamanian et al. (2022), which can drive the consensus error to converge to an arbitrarily small region around zero within a finite time. In Hong et al. (2017), Davila and Pisano (2020), and Zou et al. (2020), three fixed-time consensus control schemes, as a special kind of FTC independent of the initial state, are proposed respectively. However, the aforementioned Yu and Long (2015), Zhao et al. (2018), Zamanian et al. (2022), Hong et al. (2017), and Davila and Pisano (2020); and many other existing finite-/fixed-time control laws are discontinuous, which likely to generate control chattering. To avoid potential control chattering, some continuous and even smooth FTC schemes have been reported successively Wang et al. (2016), Cui et al. (2020), and Sedghi et al. (2022). From the perspective of practical application, the transmission of chattering signals requires a communication channel with unlimited bandwidth, which is impossible for a real networked control system (NCS) and digital/hybrid system.

Input quantization refers to converting the control signal into a set of the piece-wise constant function of time, which is worth considering for practical systems since the control signal from the command unit to the execution unit is transmitted over a bandwidth-limited channel. Some relevant pioneering works were introduced in De Persis (2009), Hayakawa et al. (2009), Liu et al. (2012), and Zhou et al. (2014). For the FTC study of nonlinear individual systems with input quantization, some interesting results have been achieved so far Wang et al. (2018), Sui et al. (2020), Mirzaei et al. (2022), and Jiang et al. (2023). For isomorphic NCSs and heterogeneous MASs with input quantization, two prescribed performance control (PPC) schemes with low computation are proposed by Bikas and Rovithakis (2019) and Kechagias and Rovithakis (2020), respectively. In Liang et al. (2020), an improved PPC-based quantized control scheme with finite-time convergence is proposed for the consensus tracking of unmodeled nonlinear MASs. For stochastic nonlinear MASs, two quantized adaptive FTC schemes by combining the backstepping method and radial basis function (RBF)-NN are proposed by Zhang et al. (2019) and Wu et al. (2021), respectively. In general, the results of combining FTC with input quantization are few, especially for nonlinear MASs and/or NCSs.

Based on our observations, many existing FTC results are not suitable to be combined with quantized control. Specifically, one of the contradictions between FTC and input quantization is that many discontinuous FTC schemes, such as those proposed by Zhao et al. (2018), Zamanian et al. (2022), and Davila and Pisano (2020), tend to generate high-frequency control signals that require bandwidth-infinite channels, which is contrary to the original intention of input quantization. Another is that many existing continuous FTCs only have $C^{0}$ continuity, which often leads to singularity problems. For example, for Sedghi et al. (2022), Wang et al. (2018), Zhang et al. (2019), and Wu et al. (2021), the first derivatives of the proposed virtual control signals tend to infinity as the tracking error approaches zero. The singularity problem is particularly noteworthy for high-order systems, which may cause the control signal to be larger than the finite level of the input quantizer. In our opinion, the research on FTC for uncertain nonlinear MASs with input quantization is challenging and promising.

Motivated by the above observations, this paperstudies the distributed consensus tracking control problem of uncertain strict-feedback nonlinear MASs with input quantization under directed communication topology. Improving on our previous work reported in Jiang et al. (2023), an adaptive nonsingular FTC scheme is designed by using the command-filtered backstepping design and RBF-NN based on the Lyapunov direct method. The contribution of this work can be highlighted from the following aspects:

A $C^{1}$ distributed adaptive finite-time consensus controller is proposed in this paper. Unlike the result of Jiang et al. (2023), which is only applicable to single systems, the proposed scheme is applicable to fully distributed MASs or interconnected systems. In contrast to discontinuous FTC schemes proposed by Yu and Long (2015), Zhao et al. (2018), Zamanian et al. (2022), Hong et al. (2017), Davila and Pisano (2020), Zou et al. (2020), and Mirzaei et al. (2022), the chattering of the control signal is fundamentally avoided in our scheme by using a continuous switching control term. Compared with $C^{0}$ FTC schemes in Sedghi et al. (2022), Wang et al. (2018), Zhang et al. (2019), and Wu et al. (2021), the singularity problem is avoided due to the $C^{1}$ continuity of the designed control law.

The adopted command-filtered backstepping design with adaptive compensation not only avoids the complex differential calculations for the virtual control law but also relaxes/removes several common assumptions/default settings for system dynamics and leader’s input as adopted in Shafiqul et al. (2021), Yin et al. (2020), Zhao et al. (2018), You et al. (2020), and Davila and Pisano (2020). A detailed related discussion is carried out in Remark 7.

The problem of hysteresis input quantization is solved by a new lemma (i.e. Lemma 6). The core idea of the proposed solution is to transform the non-smooth relationship between the control signal and the quantized signal into a simple inequality with an easily satisfied condition, so as to simplify the stability analysis.

The rest of this paper is organized as follows. In Section 2, the concerns and necessary preliminaries are presented. The programmed control design and the complete stability analysis are given in Section 3. In Section 4, the effectiveness of the proposed controller is verified by a numerical simulation. Finally, a brief conclusion of this work and its prospects for improvement are given in Section 5.

Preliminaries and problem statement

Notations

Throughout this paper, $C^{0}$ function refers to a continuous function. Moreover, a function is called $C^{m}$ function if its $m$ -order derivative is continuous. $N$ , $R$ , $R_{+}$ , and $R^{m}$ denote the spaces of positive integers, real numbers, and positive real numbers, respectively.

System description

Consider a networked MAS consisting of $N + 1$ agents, where the leader is marked as 0 and $N$ followers are indexed by a finite set $V = {1, \dots, N}$ . The $i th$ $(i \in V)$ follower’s dynamics is described by the following nonlinear system

{\begin{matrix} {\overset{\cdot}{x}}_{i, m} = f_{i, m} ({\bar{x}}_{i, m}) + x_{i, m + 1}, m = 1, \dots, n - 1 \\ {\overset{\cdot}{x}}_{i, n} = f_{i, n} ({\bar{x}}_{i, n}) + q_{i} (u_{i}) \end{matrix}

(1)

where ${\bar{x}}_{i, k} = [x_{i, 1}, \dots, x_{i, k}]^{T} \in R^{k}$ with $x_{i, k} \in R$ is the measurable state of the system, $k = 1, \dots, n$ . The unknown smooth function $f_{i} ({\bar{x}}_{i, k}) : R^{k} \to R$ represents the uncertain dynamic behavior, $u_{i} \in R$ is the actual control signal, and $q_{i} (u_{i}) \in R$ is the mapping of the control command $u_{i}$ after quantization. Herein, the hysteretic quantizer is considered

q_{i} (u_{i}) = {\begin{matrix} u_{i, ι} sgn (u_{i}), \frac{u_{i, ι}}{1 + δ_{i}} < | u_{i} | \leq u_{i, ι}, {\overset{\cdot}{u}}_{i} < 0, or \\ u_{i, ι} < | u_{i} | \leq \frac{u_{i, ι}}{1 - δ_{i}}, {\overset{\cdot}{u}}_{i} > 0 \\ u_{i, ι} (1 + δ_{i}) sgn (u_{i}), u_{i, ι} < | u_{i} | \leq \frac{u_{i, ι}}{1 - δ_{i}}, {\overset{\cdot}{u}}_{i} < 0, or \\ \frac{u_{i, ι}}{1 - δ_{i}} < | u_{i} | \leq \frac{u_{i, ι} (1 + δ_{i})}{1 - δ_{i}}, {\overset{\cdot}{u}}_{i} > 0 \\ 0, 0 \leq | u_{i} | < \frac{u_{i, ι}}{1 + δ_{i}}, {\overset{\cdot}{u}}_{i} < 0, or \\ \frac{u_{i, min}}{1 + δ_{i}} \leq | u_{i} | < u_{i, min}, {\overset{\cdot}{u}}_{i} > 0 \\ q_{i} (u_{i} (t^{-})), {\overset{\cdot}{u}}_{i} = 0 \end{matrix}

(2)

where $u_{i, ι} = κ_{i}^{1 - ι} u_{i, min}$ with $ι = 1, 2, \dots$ , $δ_{i} = (1 - κ_{i}) / (1 + κ_{i})$ , $u_{i, min} > 0$ , and $0 < κ_{i} < 1$ . Constants $κ_{i}$ and $u_{i, min}$ represent the quantization density and the dead-zone range of the quantizer, respectively. Moreover, the mapping of $q_{i} (u_{i})$ is given in Figure 1.

Figure 1.

The mapping of the quantizer is described by equation (2).

The dynamic system of the leader is given as

\begin{matrix} {\overset{\cdot}{x}}_{0, m} & = f_{0, m} ({\bar{x}}_{0, m}) + x_{i, m + 1}, m = 1, \dots, n - 1 \\ {\overset{\cdot}{x}}_{0, n} & = f_{0, n} ({\bar{x}}_{0, n}) + u_{0} \end{matrix}

(3)

where $x_{0, k} \in R$ $(k = 1, \dots, n)$ , $f_{0, k} ({\bar{x}}_{0, k})$ , and $u_{0}$ are the state, dynamic function and unknown input respectively. Note that the leader acts only as a generator of the reference signal.

Remark 1. System (1) can cover many actual plants in the real world, such as the attitude dynamics models of manipulators and vehicles Wang et al. (2018) and Jiang et al. (2022). Moreover, the systems adopted by Zhao et al. (2018), Zamanian et al. (2022), and Davila and Pisano (2020) can be considered as a special form of system (1). Thus, it can be said that the research results of this paper have good universality.

Herein, the internal communication of the considered MAS is described by a directed graph $\bar{G} = (\bar{V}, \bar{ℰ})$ , which contains a set of nodes $\bar{V} = V \cup {0}$ with $V = {1, \dots, N} \subseteq R^{N}$ and a set of edges $\bar{ℰ} \subseteq \bar{V} \times \bar{V}$ . In $\bar{G}$ , each node denotes an agent and the directed edge $(j, i) \in \bar{ℰ}$ means that information is transmitted one-way from node $j$ to node $i$ . For the subgraph $G = (V, ℰ)$ of followers, $A = [a_{ij}] \subseteq R^{N \times N}$ is defined by $a_{ij} = 1$ if $(j, i) \in ℰ$ , and $a_{ij} = 0$ otherwise. (In this context, $a_{ii} = 0$ since self-loop does not exist). In addition, the set of neighbors of the $i - th$ followers is denoted as $N_{i} = {j \in V | (j, i) \in ℰ}$ . The degree matrix $D = diag {d_{1}, \dots, d_{N}} \in R^{N \times N}$ with $d_{i} = \sum_{j = 1}^{N} a_{ij}$ and the Laplacian matrix is $ℒ = [l_{i j}] = D - A$ . The matrix $ℬ = diag {b_{1}, \dots, b_{N}} \in ℝ^{N}$ represents the interaction between the leader and $N$ followers, there is $b_{i} > 0$ if $(0, i) \in ℰ$ , $b_{i} = 0$ and otherwise.

Control objective

This paper aims to develop a distributed quantized controller for each follower to track the leader so as to achieve the state consensus of the MAS in a finite time. Meanwhile, all signals in the closed-loop system are bounded.

Preliminaries

To promote this work, the following assumption and useful lemmas are necessary.

Assumption 1. In graph $\bar{G}$ , there is a spanning tree with the leader being the root node.

Remark 2 . Assumption 1 is borrowed from Jiang et al. (2022), Li et al. (2015), and Liang et al. (2020), which is necessary for achieving distributed tracking control. Moreover, it indicates that $ℒ + ℬ$ is nonsingular.

Lemma 1. Consider the following dynamic system Yu et al. (2005)

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

(4)

where $x \in R$ . The Lyapunov explanation for finite-time result is as below

\overset{\cdot}{V} (x) \leq - aV (x) - b V^{κ} (x)

(5)

in which $0 < κ < 1$ and $a, b \in R_{+}$ . The settling time of the system is

T_{s} \leq \frac{1}{a (1 - κ)} \ln [\frac{a V^{1 - κ} (x_{0}) + b}{b}]

Lemma 2. For any continuous function $f (x) : R^{n} \to R$ defined on compact set $Ω_{x} \in R^{n}$ , it can be approximated by an RBF-NN, that is, Sanner and Slotine (1992)

f (x) = W^{T} φ (x) + ϵ

(6)

in which

W : = \arg min_{W \in R^{ℓ}} {sup_{x \in Ω_{x}} | f (x) - W^{T} φ (x) |}

where $x \in R^{n}$ is the input vector, $W = [W_{1}, \dots, W_{ℓ}]^{T} \in R^{ℓ}$ and $φ (x) = [φ_{1} (x), \dots, φ_{ℓ} (x)]^{T} \in R^{ℓ}$ denote the ideal weight vector and basis function vector of the NN with $ℓ$ nodes. Moreover, $φ (x)$ is defined as $φ_{m} (x) = \exp [- (x - o_{m})^{T} (x - o_{m}) / ς_{m}^{2}]$ , where $ς_{m} \in R$ and $o_{m} \in R^{n}$ $(m = 1, \dots, ℓ)$ are the width and center of the basis function, respectively. The function approximation error $ε$ satisfies $| ϵ | \leq \bar{ϵ}$ , $\exists \bar{ϵ} \in R_{+}$ .

Lemma 3. Consider the following command filter Ferrari-Trecate et al. (2009)

{\begin{matrix} {\overset{\cdot}{ϑ}}_{1} = μ_{1} ϑ_{2} \\ {\overset{\cdot}{ϑ}}_{2} = - 2 μ_{1} μ_{2} ϑ_{2} - μ_{1} (ϑ_{1} - α) \end{matrix}

(7)

where $α (t)$ is continuous input, $ϑ_{1} (0) = α (0)$ , $ϑ_{2} (0) = 0$ , $μ_{1} > 0$ , and $0 < μ_{2} \leq 1$ . Let $α^{c} (t) = ϑ_{1} (t)$ and ${\overset{\cdot}{α}}^{c} (t) = μ_{1} ϑ_{2}$ , there exists $c_{o} \in R_{+}$ and $c_{d} \in R_{+}$ such that $| α^{c} (t) - α (t) | \leq c_{o}$ and $| {\overset{\cdot}{α}}^{c} (t) - \overset{\cdot}{α} (t) | \leq c_{d}$ .

Lemma 4. For $x, y \in R$ , if odd integers $r_{1}, r_{2} \in R_{+}$ satisfy $r = r_{1} / r_{2} < 1$ , there exists Wang et al. (2016)

x y^{r} \leq - σ_{1} x^{1 + r} + σ_{2} (x + y)^{1 + r}

(8)

in which

σ_{1} = \frac{2^{r - 1} - 2^{(r - 1) (r + 1)}}{1 + r}, σ_{2} = \frac{\frac{2 r + 1}{1 + r} + \frac{2^{- {(r - 1)}^{2} (r + 1)}}{r + 1} - 2^{r - 1}}{1 + r}

Lemma 5. For $x_{i} \in R$ , $i = 1, \dots, n$ , and $0 < r \leq 1$ , there is a fact that ${(\sum_{i = 1}^{n} x_{i})}^{r} \leq \sum_{i = 1}^{n} | x_{i} |^{r} \leq n^{1 - r} {(\sum_{i = 1}^{n} x_{i})}^{r}$ Hardy et al. (1952).

Main results

In this part, the detailed controller design is given step-by-step. First, the tracking errors of the control system under the backstepping design are defined. Then, the design process of the proposed controller is carried out orderly. Finally, a complete stability analysis of the closed-loop system is given, and a comparison discussion with some related existing results is presented.

Error definition

Generally, the first state of each agent is regarded as its output, so that the consensus tracking error of the MAS is defined as

e = [x_{i, 1}, \dots, x_{N, 1}]^{T} - x_{0, 1} \cdot 1_{N}

(9)

If the considered MAS is isomorphic, $x_{i, 1} \to x_{0, 1}$ indicates $x_{i, m} \to x_{0, m}$ , $m = 2, \dots, n$ . Considering that not every follower has access to the leader in a distributed scheme, then the consensus error of each follower based on the local information interaction is constructed as

{\tilde{x}}_{i} = \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1}) + b_{i} (x_{i, 1} - x_{0, 1})

(10)

where $i = 1, \dots, N$ . Let $\tilde{x} = [{\tilde{x}}_{1}, \dots, {\tilde{x}}_{N}]^{T} \in R^{N}$ , there is

\tilde{x} = (L + B) e

(11)

According to Assumption 1 and Remark 2, we know that the boundedness and convergence of $\tilde{x}$ implies those of $e$ .

Based on the backstepping framework, the following coordinate transformation is introduced

{\begin{matrix} v_{i, 1} = {\tilde{x}}_{i} \\ v_{i, m} = x_{i, m} - α_{i, m - 1}^{c} \end{matrix}

(12)

where $m = 2, \dots, n$ , and $α_{i, m - 1}^{c}$ is the output of the following command filter Farrell et al. (2009):

{\begin{matrix} {\overset{\cdot}{ω}}_{i, m} = μ_{1} ϑ_{i, m} \\ {\overset{\cdot}{ϑ}}_{i, m} = - 2 μ_{1} μ_{2} ϑ_{i, m} - μ_{1} (ω_{i, m} - α_{i, m}) \end{matrix}

(13)

where $α_{i, m}^{c} (t) = ω_{i, m} (t)$ and ${\overset{\cdot}{α}}_{i, m}^{c} (t) = μ_{1} ϑ_{i, m}$ are the outputs of the filter, $ω_{i, m} (0) = α_{i, m} (0)$ , $ϑ_{i, m} (0) = 0$ , $μ_{1} > 0$ , and $0 < μ_{2} \leq 1$ are design parameters. According to lemma 3, the filtering errors $(α_{i, m}^{c} - α_{i, m})$ and $({\overset{\cdot}{α}}_{i, m}^{c} - {\overset{\cdot}{α}}_{i, m})$ are bounded. To reduce filtering errors, a $C^{1}$ compensated signal $ϖ_{i, m} (t)$ $(m = 1, \dots, n)$ is constructed as

\begin{matrix} {\overset{\cdot}{ϖ}}_{i, 1} = - λ_{i, 1} ϖ_{i, 1} + h_{i, 1} (α_{i, 1}^{c} - α_{i, 1}) + h_{i, 1} ϖ_{i, 2} - γ_{i} ϖ_{i, 1}^{r} \\ {\overset{\cdot}{ϖ}}_{i, m} = - λ_{i, m} ϖ_{i, m} + (α_{i, m}^{c} - α_{i, m}) - h_{i, m} ϖ_{i, m - 1} \\ + ϖ_{i, m + 1} - γ_{i} ϖ_{i, m}^{r}, m = 2, \dots, n - 1 \\ {\overset{\cdot}{ϖ}}_{i, n} = - λ_{i, n} ϖ_{i, n} - h_{i, n} ϖ_{i, n - 1} - γ_{i} ϖ_{i, n}^{r} \end{matrix}

(14)

where $ϖ_{i, m} (0) = 0$ , $λ_{i, 1}, \dots, λ_{i, n} > 0$ , $h_{i, 1} = h_{i, 2} = d_{i} + b_{i}$ , $h_{i, 3}, \dots, h_{i, n} = 1$ $(n \geq 3)$ , $γ_{i} > 0$ , $0 < r = r_{1} / r_{2} < 1$ with $r_{1}$ and $r_{2}$ are positive odd integers. Further, the compensated tracking error is constructed as

z_{i, m} = v_{i, m} - ϖ_{i, m}, m = 1, \dots, n

(15)

Remark 3. The compensated signals $ϖ_{i, 1}, \dots, ϖ_{i, n}$ are uniformly bounded, that is, there is $| ϖ_{i, k} | \leq \sqrt{2 ξ_{i}}, \exists ξ_{i} \in R_{+}$ . The detailed expression of $ξ_{i}$ and the relevant analysis is given in Appendix A. Different from the dynamic surface control (DSC) adopted by Shahvali and Askari (2016) and Cui et al. (2020), the adaptive compensation mechanism adopted here can well suppress the influence of filtering error, especially for high-order systems.

Controller design

A finite-time distributed consensus controller incorporating NN-based function estimation is proposed in this subsection. Based on the Lyapunov direct method and backstepping idea, a $n$ -steps control design is performed as follows.

Step 1: Design of virtual control signal $α_{i, 1}$ $(i = 1, \dots, N)$ .

Recalling equation (15) and combining equations (1), (10), and (14), the first error dynamics of the closed-loop system is obtained as

\begin{matrix} {\overset{\cdot}{z}}_{i, 1} = {\overset{\cdot}{v}}_{i, 1} - {\overset{\cdot}{ϖ}}_{i, 1} \\ = (d_{i} + b_{i}) x_{i, 2} + {\bar{f}}_{i, 1} (X_{i, 1}) - \sum_{j \in N_{i}} a_{ij} x_{j, 2} - b_{i} x_{0, 2} \\ + λ_{i, 1} ϖ_{i, 1} - h_{i, 1} (α_{i, 1}^{c} - α_{i, 1}) - h_{i, 1} ϖ_{i, 2} + γ_{i} ϖ_{i, 1}^{r} \end{matrix}

(16)

where ${\bar{f}}_{i, 1} (X_{i, 1}) = (d_{i} + b_{i}) f_{i, 1} (x_{i, 1}) - \sum_{j \in N_{i}} a_{ij} f_{j, 1} (x_{j, 1}) - b_{i} f_{0, 1} (x_{0, 1})$ and $X_{i, 1} = [x_{i, 1}, x_{j \in N_{i}, 1}, b_{i} x_{0, 1}]^{T}$ . Based on Lemma 4, ${\bar{f}}_{i, 1}$ can be approximated by a RBF-NN as follows

{\bar{f}}_{i, 1} (X_{i, 1}) = W_{i, 1}^{T} φ_{i, 1} (X_{i, 1}) + ϵ_{i, 1}

(17)

in which $W_{i, 1} \in R^{ℓ_{i, 1}}$ is the ideal weight of the NN with $ℓ_{i, 1}$ nodes, $φ_{i, 1} (X_{i, 1}) \in R^{ℓ_{i, 1}}$ is the Gaussian basis function vector, and $ϵ_{i, 1}$ is the approximation error satisfying $| ϵ_{i, 1} | \leq {\bar{ϵ}}_{i, 1} \in R_{+}$ .

Choose the Lyapunov function candidate as

V_{i, 1} = \frac{1}{2} z_{i, 1}^{2} + \frac{1}{2} {\tilde{θ}}_{i, 1}^{2}

(18)

where ${\tilde{θ}}_{i, 1} = θ_{i, 1} - {\hat{θ}}_{i, 1}$ and ${\hat{θ}}_{i, 1}$ is the estimate of $θ_{i, 1} : = ∥ W_{i, 1} ∥^{2}$ . Then, differentiating $V_{i, 1}$ along with equation (16) yields

\begin{matrix} {\overset{\cdot}{V}}_{i, 1} = z_{i, 1} [(d_{i} + b_{i}) x_{i, 2} + {\bar{f}}_{i, 1} (X_{i, 1}) - \sum_{j \in N_{i}} a_{ij} x_{j, 2} - b_{i} x_{0, 2} + λ_{i, 1} ϖ_{i, 1} \\ - h_{i, 1} (α_{i, 1}^{c} - α_{i, 1}) - h_{i, 1} ϖ_{i, 2} + γ_{i} ϖ_{i, 1}^{r}] + {\tilde{θ}}_{i, 1} ({\overset{\cdot}{θ}}_{i, 1} - {\overset{\cdot}{\hat{θ}}}_{i, 1}) \end{matrix}

(19)

Recalling $z_{i, 2} = x_{i, 2} - α_{i, 1}^{c} - ϖ_{i, 2}$ defined by equations (12) and (15), then equation (19) can be expanded as

\begin{matrix} {\overset{\cdot}{V}}_{i, 1} = z_{i, 1} [(d_{i} + b_{i}) z_{i, 2} + {\bar{f}}_{i, 1} (X_{i, 1}) - \sum_{j \in N_{i}} a_{ij} x_{j, 2} - b_{i} x_{0, 2} \\ + λ_{i, 1} ϖ_{i, 1} + (d_{i} + b_{i}) α_{i, 1} + γ_{i} ϖ_{i, 1}^{r}] - {\tilde{θ}}_{i, 1} {\overset{\cdot}{\hat{θ}}}_{i, 1} \end{matrix}

(20)

Using Young’s inequality, there are

z_{i, 1} {\bar{f}}_{i, 1} (X_{i, 1}) \leq \frac{1}{2 η_{i, 1}^{2}} z_{i, 1}^{2} θ_{i, 1} ∥ φ_{i, 1} (X_{i, 1}) ∥^{2} + \frac{z_{i, 1}^{2}}{2} + \frac{η_{i, 1}^{2}}{2} + \frac{ϵ_{i, 1}^{2}}{2}

(21)

z_{i, 1} γ_{i} ϖ_{i, 1}^{r} \leq \frac{1}{2} z_{i, 1}^{2} + \frac{γ_{i}^{2}}{2} ϖ_{i, 1}^{2 r} \leq \frac{1}{2} z_{i, 1}^{2} + γ_{i}^{2} ξ_{i}^{r}

(22)

where $η_{i, 1} \in R_{+}$ is a control parameter and $ξ_{i} \in R_{+}$ is an adjustable parameter mentioned in Remark 3. Further, substituting equations (21) and (22) into equation (20) yields

\begin{matrix} {\overset{\cdot}{V}}_{i, 1} \leq z_{i, 1} [(d_{i} + b_{i}) α_{i, 1} - \sum_{j \in N_{i}} a_{ij} x_{j, 2} - b_{i} x_{0, 2} + λ_{i, 1} ϖ_{i, 1}] \\ + \frac{z_{i, 1}^{2}}{2 η_{i, 1}^{2}} θ_{i, 1} ∥ φ_{i, 1} (X_{i, 1}) ∥^{2} + \frac{η_{i, 1}^{2}}{2} + \frac{z_{i, 1}^{2}}{2} + \frac{ϵ_{i, 1}^{2}}{2} \\ + (d_{i} + b_{i}) z_{i, 1} z_{i, 2} + \frac{z_{i, 1}^{2}}{2} + γ_{i}^{2} ξ_{i}^{r} - {\tilde{θ}}_{i, 1} {\overset{\cdot}{\hat{θ}}}_{i, 1} \end{matrix}

(23)

Design a $C^{1}$ finite-time virtual control law as

\begin{matrix} α_{i, 1} = \frac{1}{d_{i} + b_{i}} (- λ_{i, 1} v_{i, 1} + \sum_{j \in N_{i}} a_{ij} x_{j, 2} + b_{i} x_{0, 2} - z_{i, 1} \\ - β_{i, 1} ζ_{i, 1} (z_{i, 1}) - \frac{1}{2 η_{i, 1}^{2}} z_{i, 1} {\hat{θ}}_{i, 1} ∥ φ_{i, 1} (X_{i, 1}) ∥^{2}) \end{matrix}

(24)

where $β_{i, 1} \in R_{+}$ and the estimate ${\hat{θ}}_{i, 1}$ is determined by

{\overset{\cdot}{\hat{θ}}}_{i, 1} = \frac{1}{2 η_{i, 1}^{2}} z_{i, 1}^{2} ∥ φ_{i, 1} (X_{i, 1}) ∥^{2} - p_{i, 1} {\hat{θ}}_{i, 1} - q_{i, 1} {\hat{θ}}_{i, 1}^{r}

(25)

where $p_{i, 1}$ , $q_{i, 1} \in R_{+}$ , and $ζ_{i, 1} (z_{i, 1})$ is designed as

ζ_{i, 1} (z_{i, 1}) = {\begin{matrix} z_{i, 1}^{r}, if | z_{i, 1} | \geq τ_{i, 1} \\ ρ_{i, 1} z_{i, 1} + ρ_{i, 1} z_{i, 1}^{3}, if | z_{i, 1} | < τ_{i, 1} \end{matrix}

(26)

where $ρ_{i, 1} = (1 / 2) (3 - r) τ_{i, 1}^{r - 1}$ , $ϱ_{i, 1} = (1 / 2) (r - 1) τ_{i, 1}^{r - 3}$ , and $τ_{i, 1} \in R_{+}$ is a custom parameter.

Substituting equation (24) with equation (25) into equation (23), one has

\begin{matrix} {\overset{\cdot}{V}}_{i, 1} \leq - λ_{i, 1} z_{i, 1}^{2} - β_{i, 1} ζ_{i, 1} (z_{i, 1}) z_{i, 1} + p_{i, 1} {\tilde{θ}}_{i, 1} {\hat{θ}}_{i, 1} \\ + q_{i, 1} {\tilde{θ}}_{i, 1} {\hat{θ}}_{i, 1}^{r} + \frac{η_{i, 1}^{2}}{2} + \frac{ϵ_{i, 1}^{2}}{2} + γ_{i}^{2} ξ_{i}^{r} + h_{i, 1} z_{i, 1} z_{i, 2} \end{matrix}

(27)

where the coupling term $h_{i, 1} z_{i, 1} z_{i, 2}$ will be eliminated in the next step.

Remark 4. It is worth emphasizing that the $C^{1}$ continuity of the designed virtual control signal is guaranteed, that is, both $α_{i, 1}$ and its derivative ${\overset{\cdot}{α}}_{i, 1}$ are continuous. Specifically, it is obvious from equation (25) that $ζ_{i, 1} (τ_{i, 1}^{+}) = lim_{z_{i, 1} \to τ_{i, 1}^{+}} z_{i, 1}^{r} = τ_{i, 1}^{r}$ and $ζ_{i, 1} (τ_{i, 1}^{-}) = lim_{z_{i, 1} \to τ_{i, 1}^{-}} ρ_{i, 1} z_{i, 1} + ρ_{i, 1} z_{i, 1}^{3} = τ_{i, 1}^{r}$ when $z_{i, 1} \geq 0$ . Similarly, $ζ_{i, 1} (- τ_{i, 1}^{+}) = ζ_{i, 1} (- τ_{i, 1}^{-})$ when $z_{i, 1} < 0$ can also be deduced. It can be concluded that $ζ (z_{i, 1})$ is continuous at $τ_{i, 1}$ and $- τ_{i, 1}$ . Similarly, it can be verify that ${\overset{\cdot}{ζ}}_{i, 1} (z_{i, 1})$ is continuous. Thus, the singularity problem is avoided so that the recursive design based on a filtered backstepping framework can proceed.

Step m $(m = 2, \dots, n - 1)$ : Design of virtual control signal $α_{i, m}$ $(i = 1, \dots, N)$

Recalling equation (15) and combining equations (1), (12), and (14), one has

\begin{matrix} {\overset{\cdot}{z}}_{i, m} = {\overset{\cdot}{v}}_{i, m} - {\overset{\cdot}{ϖ}}_{i, m} \\ = x_{i, m + 1} + f_{i, m} ({\bar{x}}_{i, m}) - {\overset{\cdot}{α}}_{i, m - 1}^{c} + λ_{i, m} ϖ_{i, m} - α_{i, m}^{c} \\ + α_{i, m} + h_{i, m} ϖ_{i, m - 1} - ϖ_{i, m + 1} + γ_{i} ϖ_{i, m}^{r} \end{matrix}

(28)

Based on Lemma 4, the unknown function $f_{i, m} ({\bar{x}}_{i, m})$ can be approximated by a RBF-NN

f_{i, m} ({\bar{x}}_{i, m}) = W_{i, m}^{T} φ_{i, m} ({\bar{x}}_{i, m}) + ϵ_{i, m}

(29)

where $W_{i, m} \in R^{ℓ_{i, m}}$ is the ideal weight of the NN with $ℓ_{i, m}$ nodes, $φ_{i, m} ({\bar{x}}_{i, m}) \in R^{ℓ_{i, m}}$ is the basis function vector, and the approximation error $ϵ_{i, m}$ satisfies $| ϵ_{i, m} | \leq {\bar{ϵ}}_{i, m} \in R_{+}$ . Here, we define $θ_{i, m} = ∥ W_{i, m} ∥^{2}$ , $m = 2, \dots, n - 1$ .

Choose the Lyapunov function candidate as

V_{i, m} = V_{i, m - 1} + \frac{1}{2} z_{i, m}^{2} + \frac{1}{2} {\tilde{θ}}_{i, m}^{2}

(30)

where ${\tilde{θ}}_{i, m} = θ_{i, m} - {\hat{θ}}_{i, m}$ and ${\hat{θ}}_{i, m}$ is the estimate of $θ_{i, m}$ . Differentiating $V_{i, m}$ along equation (28) and incorporating $z_{i, m + 1} = x_{i, m + 1} - α_{i, m}^{c} - ϖ_{i, m + 1}$ into it yields

\begin{matrix} {\overset{\cdot}{V}}_{i, m} = {\overset{\cdot}{V}}_{i, m - 1} + z_{i, m} [x_{i, m + 1} + f_{i, m} ({\bar{x}}_{i, m}) - {\overset{\cdot}{α}}_{i, m - 1}^{c} + λ_{i, m} ϖ_{i, m} \\ - α_{i, m}^{c} + α_{i, m} + h_{i, m} ϖ_{i, m - 1} - ϖ_{i, m + 1} + γ_{i} ϖ_{i, m}^{r}] - {\tilde{θ}}_{i, m} {\overset{\cdot}{\hat{θ}}}_{i, m} \\ = {\overset{\cdot}{V}}_{i, m - 1} + z_{i, m} [α_{i, m} + z_{i, m + 1} + f_{i, m} ({\bar{x}}_{i, m}) - {\overset{\cdot}{α}}_{i, m - 1}^{c} \\ + λ_{i, m} ϖ_{i, m} + h_{i, m} ϖ_{i, m - 1} + γ_{i} ϖ_{i, m}^{r}] - {\tilde{θ}}_{i, m} {\overset{\cdot}{\hat{θ}}}_{i, m} \end{matrix}

(31)

Again, using Young’s inequality yields

z_{i, m} f_{i, m} ({\bar{x}}_{i, m}) \leq \frac{1}{2 η_{i, m}^{2}} z_{i, m}^{2} θ_{i, m} ∥ φ_{i, m} ({\bar{x}}_{i, m}) ∥^{2} + \frac{1}{2} z_{i, m}^{2} + \frac{η_{i, m}^{2}}{2} + \frac{ϵ_{i, m}^{2}}{2}

(32)

z_{i, m} γ_{i} ϖ_{i, m}^{r} \leq \frac{1}{2} z_{i, m}^{2} + γ_{i}^{2} ξ_{i}^{r}

(33)

where $η_{i, m} \in R_{+}$ . Substituting equations (32) and (33) into equation (31), we have

\begin{matrix} {\overset{\cdot}{V}}_{i, m} \leq {\overset{\cdot}{V}}_{i, m - 1} + z_{i, m} \\ [α_{i, m} + z_{i, m + 1} - {\overset{\cdot}{α}}_{i, m - 1}^{c} + λ_{i, m} ϖ_{i, m} + h_{i, m} ϖ_{i, m - 1}] \\ + \frac{z_{i, m}^{2}}{2 η_{i, m}^{2}} θ_{i, m} ∥ φ_{i, m} ({\bar{x}}_{i, m}) ∥^{2} + z_{i, m}^{2} + \frac{η_{i, m}^{2}}{2} \\ + \frac{ϵ_{i, m}^{2}}{2} + γ_{i}^{2} ξ_{i}^{r} - {\tilde{θ}}_{i, m} {\overset{\cdot}{\hat{θ}}}_{i, m} . \end{matrix}

(34)

Design the $C^{1}$ finite-time virtual control law $α_{i, m}$ as

\begin{matrix} α_{i, m} = - λ_{i, m} v_{i, m} + {\overset{\cdot}{α}}_{i, m - 1}^{c} - h_{i, m} v_{i, m - 1} - z_{i, m} \\ - β_{i, m} ζ_{i, m} (z_{i, m}) - \frac{1}{2 η_{i, m}^{2}} z_{i, m} {\hat{θ}}_{i, m} ∥ φ_{i, m} ({\bar{x}}_{i, m}) ∥^{2} \end{matrix}

(35)

where $β_{i, m} \in R_{+}$ , the adaptive parameter ${\hat{θ}}_{i, m}$ is updated by

{\overset{\cdot}{\hat{θ}}}_{i, m} = \frac{1}{2 η_{i, m}^{2}} z_{i, m}^{2} ∥ φ_{i, m} ({\bar{x}}_{i, m}) ∥^{2} - p_{i, m} {\hat{θ}}_{i, m} - q_{i, m} {\hat{θ}}_{i, m}^{r}

(36)

where $p_{i, m}, q_{i, m} \in R_{+}$ , and $ζ_{i, m} (z_{i, m})$ is defined as

ζ_{i, m} (z_{i, m}) = {\begin{matrix} z_{i, m}^{r}, if | z_{i, m} | \geq τ_{i, m} \\ ρ_{i, m} z_{i, m} + ϱ_{i, m} z_{i, m}^{3}, if | z_{i, m} | < τ_{i, m} \end{matrix}

(37)

where $ρ_{i, m} = (1 / 2) (3 - r) τ_{i, m}^{r - 1}$ , $ϱ_{i, m} = (1 / 2) (r - 1) τ_{i, m}^{r - 3}$ , and $τ_{i, m} \in R_{+}$ is a custom parameter.

Substituting equation (35) with equation (36) into equation (34), there is

\begin{matrix} {\overset{\cdot}{V}}_{i, m} \leq \sum_{k = 1}^{m} [- λ_{i, k} z_{i, k}^{2} - β_{i, k} ζ_{i, k} (z_{i, k}) z_{i, k} + p_{i, k} {\tilde{θ}}_{i, k} {\hat{θ}}_{i, k} + q_{i, k} {\tilde{θ}}_{i, k} {\hat{θ}}_{i, k}^{r} \\ + \frac{1}{2} (η_{i, k}^{2} + ϵ_{i, k}^{2})] + m γ_{i}^{2} ξ_{i}^{r} + h_{i, m} z_{i, m} z_{i, m + 1} \end{matrix}

(38)

in which the coupling term $h_{i, m} z_{i, m} z_{i, m + 1}$ can be eliminated step-by-step.

Step n: Design of actual control signal $u_{i}$ $(i = 1, \dots, N)$ .

Recalling equation (15) and combining equations (1), (12), and (14), one has

\begin{matrix} {\overset{\cdot}{z}}_{i, n} = {\overset{\cdot}{v}}_{i, n} - {\overset{\cdot}{ϖ}}_{i, n} = q_{i} (u_{i}) + f_{i, n} ({\bar{x}}_{i, n}) \\ - {\overset{\cdot}{α}}_{i, n - 1}^{c} + λ_{i, n} ϖ_{i, n} + h_{i, n} ϖ_{i, n - 1} + γ_{i} ϖ_{i, n}^{r} \end{matrix}

(39)

Similar to the previous design, the unknown function $f_{i, n} ({\bar{x}}_{i, n})$ is approximated by an RB-FNN

f_{i, n} ({\bar{x}}_{i, n}) = W_{i, n}^{T} φ_{i, n} ({\bar{x}}_{i, n}) + ϵ_{i, n}

(40)

where ${\bar{x}}_{i, n}$ is the training input, $W_{i, n} \in R^{ℓ_{i, n}}$ is the ideal weight vector with $ℓ_{i, n}$ nodes, $φ_{i, n} ({\bar{x}}_{i, n}) \in R^{ℓ_{i, n}}$ is the basis function vector, and $ϵ_{i, n}$ is the approximation error satisfying $| ϵ_{i, n} | \leq {\bar{ϵ}}_{i, n} \in R_{+}$ . Define $θ_{i, n} = ∥ W_{i, n} ∥^{2}$ .

Choose the Lyapunov function candidate as

V_{i, n} = V_{i, n - 1} + \frac{1}{2} z_{i, n}^{2} + \frac{1}{2} {\tilde{θ}}_{i, n}^{2}

(41)

where ${\tilde{θ}}_{i, n} = θ_{i, n} - {\hat{θ}}_{i, n}$ and ${\hat{θ}}_{i, n}$ is the estimate of $θ_{i, n}$ . Taking the time derivative of $V_{i, n}$ along equation (39) yields

\begin{matrix} {\overset{\cdot}{V}}_{i, n} = {\overset{\cdot}{V}}_{i, n - 1} + z_{i, n} [q_{i} (u_{i}) + f_{i, n} ({\bar{x}}_{i, n}) - {\overset{\cdot}{α}}_{i, n - 1}^{c} \\ + λ_{i, n} ϖ_{i, n} + h_{i, n} ϖ_{i, n - 1} + γ_{i} ϖ_{i, m}^{r}] - {\tilde{θ}}_{i, n} {\overset{\cdot}{\hat{θ}}}_{i, n} \end{matrix}

(42)

Similar to previous processing, using Young’s inequality, there are

z_{i, n} f_{i, n} ({\bar{x}}_{i, n}) \leq \frac{1}{2 η_{i, n}^{2}} z_{i, n}^{2} θ_{i, n} ∥ φ_{i, n} ({\bar{x}}_{i, n}) ∥^{2} + \frac{η_{i, n}^{2}}{2} + \frac{1}{2} z_{i, n}^{2} + \frac{ϵ_{i, n}^{2}}{2}

(43)

z_{i, n} γ_{i} ϖ_{i, n}^{r} \leq \frac{1}{2} z_{i, n}^{2} + γ_{i}^{2} ξ_{i}^{r}

(44)

where $η_{i, n} \in R_{+}$ . Substituting equations (43) and (44) into equation (42), we can get

\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq {\overset{\cdot}{V}}_{i, n - 1} + z_{i, n} [q_{i} (u_{i}) - {\overset{\cdot}{α}}_{i, n - 1}^{c} + λ_{i, n} ϖ_{i, n} + h_{i, n} ϖ_{i, n - 1}] \\ + \frac{1}{2 η_{i, n}^{2}} z_{i, n}^{2} θ_{i, n} ∥ φ_{i, n} ({\bar{x}}_{i, n}) ∥^{2} + \frac{1}{2} z_{i, n}^{2} + \frac{η_{i, n}^{2}}{2} + \frac{ϵ_{i, n}^{2}}{2} \\ + γ_{i}^{2} ξ_{i}^{r} - {\tilde{θ}}_{i, n} {\overset{\cdot}{\hat{θ}}}_{i, n} \end{matrix}

(45)

A difficulty arises here, that is, the mapping from the actual control signal $u_{i}$ to quantized signal $q_{i} (u_{i})$ is nonlinear. Fortunately, a lemma is then deduced to solve the above difficulty.

Lemma 6. For logarithmic quantizer and hysteresis quantizer, the following relationship holds between the input signal $u$ and the quantized output $q (u)$

z_{i, n} q (u_{i}) \leq (1 - δ_{i}) z_{i, n} (u_{i} + u_{i, \min} sgn (z_{i, n})), \forall z_{i, n} u_{i} \leq 0

(46)

where $δ_{i} = (1 - κ_{i}) / (1 + κ_{i})$ with $0 < κ_{i} < 1$ is the quantization density and $u_{i, \min} > 0$ is the dead-zone range of the quantizer.

Proof. The proof of Lemma 6 is given in Appendix B.

Remark 5. Compared with some related established results, the proposed Lemma 6 is simplified in dealing with input quantization. For example, the quantized input in Zhou et al. (2014), Bikas and Rovithakis (2019), and Kechagias and Rovithakis (2020) is decoupled as

q_{i} (u_{i}) = u_{i} + D_{i} (u_{i})

where

{\begin{matrix} D_{i}^{2} \leq δ_{i} u_{i}^{2}, \forall | u_{i} | \geq u_{i, min} \\ D_{i}^{2} \leq u_{i, min}^{2}, \forall | u_{i} | \leq u_{i, min} \end{matrix}

However, this decoupling method requires discussing the boundedness of the input-related term $D_{i} (u_{i})$ , which leads to the troublesome stability analysis of the closed-loop control system. In contrast, the inequality equation (46) established by Lemma 6 is more convenient for stability analysis.

At this point, the $C^{1}$ FTC signal $u_{i}$ is designed as

u_{i} = - \frac{z_{i, n} {\bar{u}}_{i}^{2}}{(1 - δ_{i}) \sqrt{z_{i, n}^{2} {\bar{u}}_{i}^{2} + χ_{i}^{2}}}

(47)

in which $χ_{i} \in R_{+}$ and the term ${\bar{u}}_{i}$ is constructed as

\begin{matrix} {\bar{u}}_{i} = λ_{i, n} v_{i, n} - {\overset{\cdot}{α}}_{i, n - 1}^{c} + h_{i, n} v_{i, n - 1} + \frac{3}{2} z_{i, n} + β_{i, n} ζ_{i, n} (z_{i, n}) \\ + \frac{1}{2 η_{i, n}^{2}} z_{i, n} {\hat{θ}}_{i, n} ∥ φ_{i, n} ({\bar{x}}_{i, n}) ∥^{2} \end{matrix}

(48)

where $β_{i, n} \in R_{+}$ , the adaptive parameter ${\hat{θ}}_{i, n}$ is updated by

{\overset{\cdot}{\hat{θ}}}_{i, n} = \frac{1}{2 η_{i, n}^{2}} z_{i, n}^{2} ∥ φ_{i, n} ({\bar{x}}_{i, n}) ∥^{2} - p_{i, n} {\hat{θ}}_{i, n} - q_{i, n} {\hat{θ}}_{i, n}^{r}

(49)

where $p_{i, n}$ , $q_{i, n} \in R_{+}$ , and $ζ_{i, n} (z_{i, n})$ is defined as

ζ_{i, n} (z_{i, n}) = {\begin{matrix} z_{i, n}^{r}, if | z_{i, n} | \geq τ_{i, n} \\ ρ_{i, n} z_{i, n} + ϱ_{i, n} z_{i, n}^{3}, if | z_{i, n} | < τ_{i, n} \end{matrix}

(50)

where $ρ_{i, n} = (1 / 2) (3 - r) τ_{i, n}^{r - 1}$ , $ϱ_{i, n} = (1 / 2) (r - 1) τ_{i, n}^{r - 3}$ , and $τ_{i, n} \in R_{+}$ . From equation (47), obviously, $z_{i, n} u_{i} \leq 0$ is always true, so inequality equation (46) is established. Moreover, the proposed control scheme can be vividly illustrated in Figure 2.

Figure 2.

Schematic diagram of the proposed control scheme.

Remark 6. Unlike the fractional power term in the form of $z^{2 σ - 1}$ or $z^{4 σ - 3}$ used in $C^{0}$ FTC schemes designed by Wang et al. (2018), Zhang et al. (2019), Sedghi et al. (2022), and Wu et al. (2021), the switching term $ζ_{i, n} (z_{i, n})$ constructed in our design ensures the $C^{1}$ continuity of virtual control laws, thus avoiding the singularity problem. (Note that $σ = (2 l - 1) / (2 l + 1)$ in Sedghi et al. (2022) and Wu et al. (2021), and $σ = (4 l - 1) / (4 l + 1)$ in Wang et al. (2018) and Zhang et al. (2019) $l \geq 2, l \in N$ . Specifically, for the $C^{0}$ control signal $α_{i} (z_{i})$ proposed by Sedghi et al. (2022), Wang et al. (2018), Zhang et al. (2019), and Wu et al. (2021), there is $lim_{z_{i} \to 0} {\overset{\cdot}{α}}_{i} (z_{i}) = \infty)$ . Further, there are two problems that can result from the singularity of virtual control signals. One problem is that ${\overset{\cdot}{α}}_{i, m} (z_{i, m})$ cannot be approximated by a NN as in Wang et al. (2018), since it is discontinuous at zero when the singularity problem occurs. Another problem is that the filter or differentiator is no longer available because the filtering error $| ϑ_{i, m} - {\overset{\cdot}{α}}_{i, m} (z_{i, m}) |$ is no longer bounded when $lim_{z_{i} \to 0} {\overset{\cdot}{α}}_{i} (z_{i}) = \infty$ .

Remark 7. Thanks to the $C^{1}$ continuity of the proposed control law, several strong assumptions and favorable default settings commonly used in the existing results are not required in our scheme. Specifically, for system dynamics, the Lipchitz assumption in You et al. (2020), the Holder-continuity assumption in Davila and Pisano (2020), and even the assumption in Zhao et al. (2018) and Zou et al. (2020) that the upper bound of the agent dynamics is pre-known, are not required herein. Meanwhile, compared to Yu and Long (2015) in which the leader has no input and Yin et al. (2020) and You et al. (2020) in which the upper bound of the leader’s input is assumed to be available, the leader we considered is driven by a completely unknown input. In general, the removal of the above assumptions increases the practicality of the results.

Stability analysis

Theorem 1. For systems (1) and (3) with Assumption 1 holds, driven by the proposed controller equation (47) with the virtual control laws (24) and (35) and with the parameter adaptive laws (25), (36), and (49), the leader-follower consensus can be achieved with certain precision in a finite time. Moreover, all signals in the close-loop system are bounded.

Proof. Recalling equations (45) and (46), one has

\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq {\overset{\cdot}{V}}_{i, n - 1} + z_{i, n} \\ [(1 - δ_{i}) (u_{i} + u_{i, min} sgn (z_{i, n})) - {\overset{\cdot}{α}}_{i, n - 1}^{c} + λ_{i, n} ϖ_{i, n} + h_{i, n} ϖ_{i, n - 1}] \\ + \frac{1}{2 η_{i, n}^{2}} z_{i, n}^{2} θ_{i, n} ∥ φ_{i, n} ({\bar{x}}_{i, n}) ∥^{2} + z_{i, n}^{2} + \frac{η_{i, n}^{2}}{2} \\ + \frac{ϵ_{i, n}^{2}}{2} + γ_{i}^{2} ξ_{i}^{r} - {\tilde{θ}}_{i, n} {\overset{\cdot}{\hat{θ}}}_{i, n} \end{matrix}

(51)

in which it is not hard to verify that

(1 - δ_{i}) z_{i, n} u_{i} = - \frac{z_{i, n}^{2} {\bar{u}}_{i}^{2}}{\sqrt{z_{i, n}^{2} {\bar{u}}_{i}^{2} + χ_{i}^{2}}} < χ_{i} - z_{i, n} {\bar{u}}_{i}

(1 - δ_{i}) z_{i, n} u_{i, min} sgn (z_{i, n}) \leq \frac{1}{2} z_{i, n}^{2} + \frac{{(1 - δ_{i})}^{2}}{2} u_{i, min}^{2}

Considering the above inequalities and substituting control laws (47)–(50) into equation (51), one can get

\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq \sum_{k = 1}^{n} [- λ_{i, k} z_{i, k}^{2} - β_{i, k} ζ_{i, k} (z_{i, k}) z_{i, k} + p_{i, k} {\tilde{θ}}_{i, k} {\hat{θ}}_{i, k} + q_{i, k} {\tilde{θ}}_{i, k} {\hat{θ}}_{i, k}^{r} + \frac{1}{2} (η_{i, k}^{2} + ϵ_{i, k}^{2})] + n γ_{i}^{2} ξ_{i}^{r} + \frac{{(1 - δ_{i})}^{2}}{2} u_{i, min}^{2} + χ_{i} \end{matrix}

(52)

Based on Lemma 4, there are

p_{i, k} {\tilde{θ}}_{i, k} {\hat{θ}}_{i, k} \leq - \frac{p_{i, k}}{2} {\tilde{θ}}_{i, k}^{2} + \frac{p_{i, k}}{2} θ_{i, k}^{2}

(53)

q_{i, k} {\tilde{θ}}_{i, k} {\hat{θ}}_{i, k}^{r} \leq - q_{i, k} σ_{1} {\tilde{θ}}_{i, k}^{1 + r} + q_{i, k} σ_{2} θ_{i, k}^{1 + r}

(54)

where the specific expressions of $σ_{1}$ and $σ_{2}$ can refer to Lemma 4. Combining equations (53) and (54), equation (52) can be further derived as

{\overset{\cdot}{V}}_{i, n} \leq \sum_{k = 1}^{n} (- λ_{i, k} z_{i, k}^{2} - \frac{p_{i, k}}{2} {\tilde{θ}}_{i, k}^{2} - q_{i, k} σ_{1} {\tilde{θ}}_{i, k}^{1 + r} - β_{i, k} ζ_{i, k} (z_{i, k}) z_{i, k}) + c_{i}

(55)

where $c_{i} = \sum_{k = 1}^{n} (p_{i, k} θ_{i, k}^{2} / 2 + q_{i, k} σ_{2} θ_{i, k}^{1 + r} + η_{i, k}^{2} + ϵ_{i, k}^{2}) / 2 + n γ_{i}^{2} ξ_{i}^{r} + (1 - δ_{i})^{2} u_{i, \min}^{2} / 2 + χ_{i} \in R_{+}$ .

Considering the definition of $ζ_{i, 1} (z_{i, 1}), \dots, ζ_{i, n} (z_{i, n})$ in equations (26), (37), and (50), the following analysis is necessary:

Case 1: When $| z_{i, k} | < τ_{i, k}$ $(k = 1, \dots, n)$ , substituting $ζ_{i, k} (z_{i, k}) = ρ_{i, k} z_{i, k} + ϱ_{i, k} z_{i, k}^{3}$ into equation (55) yields

\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq - \sum_{k = 1}^{n} [(λ_{i, k} + β_{i, k} ρ_{i, k}) z_{i, k}^{2} + \frac{p_{i, k}}{2} {\tilde{θ}}_{i, k}^{2}] + {\bar{c}}_{i} \end{matrix}

(56)

where ${\bar{c}}_{i} = c_{i} + 0.5 (1 - r) \sum_{k = 1}^{n} β_{i, k} τ_{i, k}^{r + 1} \in R_{+}$ . Recalling equation (41), we can get

\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq - g_{i} V_{i, n} + {\bar{c}}_{i} \end{matrix}

(57)

where $g_{i} = \min {2 (λ_{i, k} + β_{i, k} ρ_{i, k}), p_{i, k}, k = 1, \dots, n}$ , which means that all signals in are UUB.

Case 2: For $| z_{i, k} | \geq τ_{i, k}$ $(k = 1, \dots, n)$ , substituting $ζ_{i, k} (z_{i, k}) = z_{i, k}^{r}$ into equation (55), one has

{\overset{\cdot}{V}}_{i, n} \leq \sum_{k = 1}^{n} (- λ_{i, k} z_{i, k}^{2} - β_{i, k} z_{i, k}^{1 + r} - \frac{p_{i, k}}{2} {\tilde{θ}}_{i, k}^{2} - q_{i, k} σ_{1} {\tilde{θ}}_{i, k}^{1 + r}) + c_{i}

(58)

Recalling equation (41) and Lemma 5, there is

\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq & - {\bar{g}}_{i} V_{i, n} - ℏ_{i} V_{i, n}^{\frac{1 + r}{2}} + c_{i} . \end{matrix}

(59)

where ${\bar{g}}_{i} = \min {2 λ_{i, k}, p_{i, k}, k = 1, \dots, n} \in R_{+}$ and $ℏ_{i} = \min {2^{(1 + r) / 2} β_{i, k}, 2^{(1 + r) / 2} q_{i, k} σ_{1}} \in R_{+}$ . Inspired by Corollary 1 proposed by Yu et al. (2018), equation (59) can be rewritten as the following form with an analytical parameter $s_{i} \in (0, 1)$

{\overset{\cdot}{V}}_{i, n} \leq - s_{i} {\bar{g}}_{i} V_{i, n} - (1 - s_{i}) {\bar{g}}_{i} V_{i, n} - s_{i} ℏ_{i} V_{i, n}^{\frac{1 + r}{2}} - (1 - s_{i}) ℏ_{i} V_{i, n}^{\frac{1 + r}{2}} + c_{i}

(60)

Further, there are

{\begin{matrix} {\overset{\cdot}{V}}_{i, n} \leq - s_{i} {\bar{g}}_{i} V_{i, n} - ℏ_{i} V_{i, n}^{\frac{1 + r}{2}}, if V_{i, n} \geq \frac{c_{i}}{(1 - s_{i}) {\bar{g}}_{i}} \\ {\overset{\cdot}{V}}_{i, n} \leq - {\bar{g}}_{i} V_{i, n} - s_{i} ℏ_{i} V_{i, n}^{\frac{1 + r}{2}}, if V_{i, n} \geq {(\frac{c_{i}}{(1 - s_{i}) ℏ_{i}})}^{\frac{2}{1 + r}} \end{matrix}

(61)

Based on Lemma 1, $V_{i, n}$ can converge to the set $Ω_{i} = \min {c_{i} / [(1 - s_{i}) {\bar{g}}_{i}], [c_{i} / ((1 - s_{i}) ℏ_{i} {)]}^{2 / (1 + r)}}$ in $T_{i} = \max {1 / [s_{i} {\bar{g}}_{i} (1 - r)] \ln [(s_{i} {\bar{g}}_{i} V_{i, n}^{1 - r} (0) + ℏ_{i}) / ℏ_{i}], 1 / [{\bar{g}}_{i} (1 - r)] \ln [({\bar{g}}_{i} V_{i, n}^{1 - r} (0) + s_{i} ℏ_{i}) / (s_{i} ℏ_{i})]}$ . Furthermore, it can be deduced that when $t \geq T_{i}$

| z_{i, k} (t) | \leq min {\sqrt{\frac{2 c_{i}}{(1 - s_{i}) {\bar{g}}_{i}}}, \sqrt{2} {[\frac{c_{i}}{(1 - s_{i}) ℏ}]}^{1 / (1 + r)}}

| {\tilde{θ}}_{i, k} (t) | \leq min {\sqrt{\frac{2 c_{i}}{(1 - s_{i}) {\bar{g}}_{i}}}, \sqrt{2} {[\frac{c_{i}}{(1 - s_{i}) ℏ}]}^{1 / (1 + r)}}

where $k = 1, \dots, n$ and $i = 1, \dots, N$ . Note that $s_{i}$ is only used for analysis, and any $s_{i}$ selected corresponds to only one $Ω_{i}$ and one $T_{i}$ .

Considering the above two cases comprehensively, It can be concluded that the compensated error $z_{i, 1} (t)$ can converge to the compact set $Ω_{i, 1} : = \max {τ_{i, 1}, Ω_{z, i}}$ within a finite time, where $Ω_{z, i} : = \min {\sqrt{2 c_{i} / [(1 - s_{i}) {\bar{g}}_{i}]}, \sqrt{2} {[c_{i} / ((1 - s_{i}) ℏ)]}^{1 / (1 + r)}}$ . According to the above analysis, the error signals $z_{i, k}$ and ${\tilde{θ}}_{i, k}$ ( $k = 1, \dots, n$ and $i = 1, \dots, N$ ) are both bounded and given finite-time convergence properties. Therefore, the local consensus error $v_{i, k}$ $(v_{i, k} = z_{i, k} + ϖ_{i, k})$ is bounded and convergent in finite time since the compensation signal $ϖ_{i, k}$ is bounded as analyzed in Appendix A. Moreover, control signals $α_{i, 1}, \dots, α_{i, n_{i} - 1}$ and $u_{i}$ are bounded. At this point, the proof of Theorem 1 is complete.

Remark 8. The proposed NAFTC can generate a faster convergence rate than UUB control, since the terms $β_{i, k} ζ_{i, k} (z_{i, k})$ and $q_{i, m} {\hat{θ}}_{i, m}^{r}$ plays a major role when $z_{i, k}$ and ${\hat{θ}}_{i, m}$ approach zero, respectively, $k = 1, \dots, n$ and $i = 1, \dots, N$ . Specifically, if we choose $β_{i, k} = q_{i, k} = 0$ for control laws (24), (35), and (48), then ${\overset{\cdot}{V}}_{i, n} \leq - {\bar{g}}_{i} V_{i, n} + c_{i}^{*}$ $(c_{i}^{*} \in R_{+})$ can be obtained, which means that the UUB result is achieved. Obviously, the result equation (59) implies a faster convergence of adaptive signals and tracking errors when $| z_{i, k} | \geq τ_{i, k}$ . For the case of $| z_{i, k} | < τ_{i, k}$ , there is also $g_{i} \geq {\bar{g}}_{i}$ . That is, our controller can achieve relatively faster convergence throughout the whole control process. Incidentally, $τ_{i, k}$ is recommended to be chosen within the interval $(0, 1)$ . The smaller $τ_{i, k}$ , the better the control effect near the equilibrium point, and the greater the actuator burden.

Remark 9. Instead of performing direct differential calculation as in Chen et al. (2016), Huang et al. (2015), and Sui et al. (2020), we use a command filter to obtain an approximation of the first-order derivative of virtual control signals, so as to reduce the calculation effort and even avoid the “differential expansion” problem. Further, compared to the first-order filters used in Shahvali and Askari (2016), Cui et al. (2020), and Wu et al. (2021), an adaptive filtering compensation is introduced herein to reduce the influence of filtering errors. Note that the above two advantages are especially important for high-order systems. Unlike Li et al. (2015), Yin et al. (2020), and You et al. (2020), in which each agent requires all states from its neighbors, in our control design, each agent uses only first- and second-order state information from its neighbors. It is worth mentioning that the reduction of information transmission not only reduces the communication burden but also improves the information security of the MAS.

Simulation

In this part, a numerical case is presented to demonstrate the efficacy of the proposed controller.

Consider a MAS composed of a leader labeled by $0$ and five followers labeled by $1, \dots, 5$ . The dynamics of the leader with unknown input are described as

\begin{matrix} {\overset{\cdot}{x}}_{0, 1} & = f_{0, 1} (x_{0, 1}) + x_{0, 2} \\ {\overset{\cdot}{x}}_{0, 2} & = f_{0, 2} ({\bar{x}}_{0, 2}) + u_{0} \end{matrix}

(62)

The dynamics of the $i th$ $(i = 1, \dots, 5)$ follower with a hysteresis input quantizer equation (2) are described as follows

\begin{matrix} {\overset{\cdot}{x}}_{i, 1} & = f_{i, 1} (x_{i, 1}) + x_{i, 2} \\ {\overset{\cdot}{x}}_{i, 2} & = f_{i, 2} ({\bar{x}}_{i, 2}) + q_{i} (u_{i}) \end{matrix}

(63)

where the system order $n = 3$ . For equations (62) and (63), $f_{i, 1} (x_{i, 1}) = 1 - \sin (x_{i, 1})$ , $f_{i, 2} ({\bar{x}}_{i, 2}) = c os (x_{i, 1} x_{i, 2}) + x_{i, 2} / (2 + e^{- 0.5 x_{i, 2}})$ , $i = 0, 1, \dots, 5$ . Moreover, the interactive topology of the considered MAS is depicted in Figure 3.

Figure 3.

Communication topology of the MAS.

The dynamic input of the leader is set as $u_{0} = - f_{0, 2} - {\overset{\cdot}{f}}_{0, 1} - 0.5 [x_{0, 2} + f_{0, 1} + 1.5 \sin (t)] - 1.5 \sin (t) - 8 [x_{0, 2} + f_{0, 1} + 0.5 (x_{0, 1} - 1.5 \sin (t)) - 1.5 \cos (t)]$ , which is completely unavailable to all followers. The initial states of five followers are set to: $x_{0, 1} (0) = 0$ , $x_{1, 1} (0) = 0.8$ , $x_{2, 1} (0) = - 0.4$ , $x_{3, 1} (0) = 0.4$ , $x_{4, 1} (0) = - 0.6$ , $x_{5, 1} (0) = 0.7$ , $x_{i, 2} (0) = 0$ $(i = 0, 1, \dots, 5)$ , and $r = 3 / 5$ . According to equation (63), we know that the control signal $u_{i}$ $(i = 1, \dots, 5)$ with $α_{i, 1}$ for each follower is required. The control parameters are chosen as $λ_{i, 1} = 0.2$ , $λ_{i, 2} = 5$ , $γ_{i} = 0.2$ , $β_{i, 1} = 0.15$ , $β_{i, 2} = 1.2$ , $τ_{i, 1} = 0.1$ , $τ_{i, 2} = 0.05$ , $χ_{i} = 0.05$ , $η_{1, 1} = 0.8$ , $η_{2, 1} = 0.4$ , $η_{3, 1} = 0.6$ , $η_{4, 1} = η_{5, 1} = 0.9$ , $η_{1, 2} = η_{3, 2} = 0.5$ , $η_{2, 2} = 0.6$ , $η_{4, 2} = η_{5, 2} = 0.8$ , $p_{i, 1} = 0.6$ , $p_{i, 2} = 0.01$ , $q_{i, 1} = 0.5$ , and $q_{i, 2} = 0.01$ , $i = 1, \dots, 5$ . The first RBF-NN of each follower contains 25 nodes (i.e. $ℓ_{i, 1} = 25$ ) with centers evenly spaced in $[- 2, 2] \times [- 2, 2]$ for followers 1, 2, and 5, and $[- 2, 2] \times [- 2, 2] \times [- 2, 2]$ for followers 3 and 4, respectively. The second RBF-NN of each follower has 25 nodes (i.e. $ℓ_{i, 2} = 25$ ) with centers evenly spaced in $[- 2, 2] \times [- 3, 3]$ . Moreover, ${\hat{θ}}_{i, m} (0) = 0$ , $m = 1, \dots, n$ and $i = 1, \dots, N$ . The parameters of command filter and input quantizers are selected as $μ_{1} = 80$ , $μ_{2} = 0.5$ , $κ_{i} = 0.3$ , $u_{i, m in} = 0.5$ , $i = 1, \dots, 5$ and $ι = 1, \dots, 50$ .

In the context of this paper, regardless of input quantization, a UUB controller inspired by Wu et al. (2022) is designed as follows

{\begin{matrix} α_{i, 1} = \frac{1}{d_{i} + b_{i}} (- λ_{i, 1} v_{i, 1} + \sum_{j \in N_{i}} a_{ij} x_{j, 2} + b_{i} x_{0, 2} - {\hat{θ}}_{i, 1} \tanh (\frac{v_{i, 1}}{σ_{i, 1}})) \\ u_{i} = - λ_{i, 1} v_{i, 1} - z_{i, 1} - {\hat{θ}}_{i, 2} \tanh (\frac{v_{i, 2}}{σ_{i, 2}}) \\ {\overset{\cdot}{\hat{θ}}}_{i, m} = z_{i, m} \tanh (\frac{v_{i, m}}{σ_{i, m}}) - ρ_{i, m} {\hat{θ}}_{i, m}, m = 1, 2 \end{matrix}

(64)

where $λ_{i, 1} = 0.3$ , $λ_{i, 2} = 10$ , $σ_{i, 1} = σ_{i, 2} = 0.1$ , $ρ_{i, 1} = ρ_{i, 1} = 0.5$ , ${\hat{θ}}_{i, 1} (0) = 0$ , and ${\hat{θ}}_{i, 2} (0) = 0$ . In addition, the settings of NN and command filter in the UUB controller are the same as those in the proposed control scheme. The difference is that ${\hat{θ}}_{i, m}$ is the estimate of constant $| | W_{i, m} | |^{2} S_{i, m}$ , where $φ_{i, m}^{T} ({\bar{x}}_{i, m}) φ_{i, m} ({\bar{x}}_{i, m}) \leq S_{i, m}$ .

The simulation result is shown in Figures 4 –7. As can be seen from Figure 4(a), the states of all followers quickly track that of the leader driven by the proposed control scheme, that is, the consensus tracking of the considered MASs is achieved. Compared with the UUB control results shown in Figure 4(b), the proposed control scheme shows better tracking performance. Moreover, the comparison between Figure 5(a) and (b) shows that the designed controller has better rapidity and steady-state control accuracy than the UUB controller. The response of neuro-parameters is shown in Figure 6. Finally, Figure 7 gives the control signals and corresponding quantization signals provided to the five followers. As shown in Figure 7(a)–(e), it can be seen that the proposed quantized control signal is insensitive to tracking error compared to the UUB control signal. In addition, it is worth emphasizing that the quantized control can not only reduce bandwidth requirements but also reduce the execution frequency of the actuator.

Figure 4.

The state responses of the MASs driven by the proposed controller and a UUB controller (a) the states of the MAS with the proposed FTC scheme and (b) the states of the MAS with the UUB controller.

Figure 5.

The local errors of the MASs driven by the proposed FTC scheme and a UUB control scheme (a) the local errors of the MAS with the FTC scheme and (b) the local errors of the MAS with the UUB control scheme.

Figure 6.

The response of adaptive parameters of the proposed FTC scheme and a UUB control scheme (a) adaptive parameters in the proposed FTC scheme and (b) adaptive parameters in the UUB control scheme.

Figure 7.

The control signal of the five agents generated by UUB scheme and the proposed FTC scheme (a) the control signal and its quantization for agent 1, (b) the control signal and its quantization for agent 2, (c) the control signal and its quantization for agent 3, (d) the control signal and its quantization for agent 4, and (e) the control signal and its quantization for agent 5.

Conclusion

In this paper, the nonsingular finite-time consensus control problem of uncertain strict-feedback nonlinear MASs with input quantization is studied. By combining the filtered backstepping method, neural networks, and the FTC technology, a $C^{1}$ finite-time distributed control scheme is first proposed. The singularity problem prevalent in FTC is fundamentally avoided in the proposed scheme, so that the control signals can be filtered and quantized. It is worth mentioning that some strong assumptions and favorable default settings commonly used in many existing results have been removed from our work. However, this work only focuses on input quantization without considering state quantization, and its results are inevitably somewhat conservative. Our future work will focus on the design of quantized state-feedback control protocols for nonlinear MASs with intermittent communication and even consider the encoding and decoding of communication information.

Footnotes

Appendix A

Appendix B

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 Natural Science Foundation of Tianjin City (Grant No. 20JCYBJC01060), the National Natural Science Foundation of China (Grant Nos. 62103203 and 61973175), and the General Terminal IC Interdisciplinary Science Center of Nankai University.

ORCID iDs

Yunbiao Jiang

Fuyong Wang

Zhongxin Liu

Zengqiang Chen

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

Bhat

Bernstein

(2000) Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 38(3): 751–766.

Bikas

Rovithakis

(2019) Combining prescribed tracking performance and controller simplicity for a class of uncertain MIMO nonlinear systems with input quantization. IEEE Transactions on Automatic Control 64(3): 1228–1235.

Cao

Ren

, et al. (2013) An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial Informatics 9(1): 427–438.

Chen

CLP

Wen

Liu

, et al. (2016) Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Transactions on Cybernetics 46(7): 1591–1601.

Cui

Xia

Liu

, et al. (2020) Finite-time tracking control for a class of uncertain strict-feedback nonlinear systems with state constraints: A smooth control approach. IEEE Transactions on Neural Networks and Learning Systems 31(11): 4920–4932.

Davila

Pisano

(2020) On the fixed-time consensus problem for nonlinear uncertain multiagent systems under switching topology. International Journal of Robust and Nonlinear Control 31(9): 3841–3858.

De Persis

(2009) Robust stabilization of nonlinear systems by quantized and ternary control. Systems & Control Letters 58(8): 602–608.

Dehkordi

Baghaee

Sadati

, et al. (2019) Distributed noise-resilient secondary voltage and frequency control for islanded microgrids. IEEE Transactions on Smart Grid 10(4): 3780–3790.

Farrell

Polycarpou

Sharma

, et al. (2009) Command filtered backstepping. IEEE Transactions on Automatic Control 54(6): 1391–1395.

10.

Ferrari-Trecate

Galbusera

Marciandi

MPE

, et al. (2009) Model predictive control schemes for consensus in multi-agent systems with single- and double-integrator dynamics. IEEE Transactions on Automatic Control 54(11): 2560–2572.

11.

Wang

(2002) Robust adaptive neural control for a class of perturbed strict feedback nonlinear systems. IEEE Transactions on Neural Networks 13(6): 1409–1419.

12.

Hardy

Littlewood

Polya

(1952) Inequalities. Cambridge, UK: Cambridge University Press.

13.

Hayakawa

Ishii

Tsumura

(2009) Adaptive quantized control for nonlinear uncertain systems. Systems & Control Letters 58(9): 625–632.

14.

Hong

Wen

, et al. (2017) Distributed robust fixed-time consensus for nonlinear and disturbed multiagent systems. IEEE Transactions on Systems, Man and Cybernetics: Systems 47(7): 1464–1473.

15.

Huang

Dou

Fang

, et al. (2015) Distributed backstepping-based adaptive fuzzy control of multiple high-order nonlinear dynamics. Nonlinear Dynamics 81: 63–75.

16.

Jiang

Liu

Chen

(2023) Fuzzy adaptive finite-time tracking control for a class of nonlinear systems: An event-triggered quantized control scheme. IEEE Transactions on Fuzzy Systems 31(11): 4137–4144.

17.

Jiang

Wang

Liu

, et al. (2022) Composite learning adaptive tracking control for full-state constrained multiagent systems without using the feasibility condition. IEEE Transactions on Neural Networks and Learning Systems 35: 2460–2472.

18.

Kechagias

Rovithakis

(2020) Prescribed performance consensus of heterogeneous quantized high-order uncertain multi-agent systems. IFAC-PapersOnLine 53(2): 2938–2943.

19.

Wen

Duan

, et al. (2015) Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs. IEEE Transactions on Automatic Control 60(4): 1152–1157.

20.

Liang

Zhang

Huang

, et al. (2020) Prescribed performance cooperative control for multiagent systems with input quantization. IEEE Transactions on Cybernetics 50(5): 1810–1819.

21.

Liang

Wang

(2023) Adaptive platoon finite-time control for marine surface vehicles with connectivity preservation and collision avoidance. Transactions of the Institute of Measurement and Control. Epub ahead of print 10 February. DOI: 10.1177/01423312231152639.

22.

Liu

Jiang

Hill

(2012) Quantized stabilization of strict-feedback nonlinear systems based on ISS cyclic-small-gain theorem. Mathematics of Control Signals and Systems 24(1–2): 75–110.

23.

Mirzaei

Aslmostafa

Asadollahi

, et al. (2022) Finite-time synchronization control for a class of perturbed nonlinear systems with fixed convergence time and hysteresis quantizer: Applied to Genesio–Tesi chaotic system. Nonlinear Dynamics 107: 2327–2343.

24.

Mozaffari

Safarinejadian

Shasadeghi

(2021) A novel mobile agent-based distributed evidential expectation maximization algorithm for uncertain sensor networks. Transactions of the Institute of Measurement and Control 43(7): 1609–1619.

25.

Olfati-Saber

Murray

(2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control 49(9): 1520–1533.

26.

Sakthivel

Kanakalakshmi

Kaviarasan

, et al. (2019) Finite-time consensus of input delayed multi-agent systems via non-fragile controller subject to switching topology. Neurocomputing 325: 225–233.

27.

Sanner

Slotine

JJE

(1992) Gaussian networks for direct adaptive control. IEEE Transactions on Neural Networks 3(6): 837–863.

28.

Sedghi

Arefi

Abooee

, et al. (2022) Distributed adaptive-neural finite-time consensus control for stochastic nonlinear multiagent systems subject to saturated inputs. IEEE Transactions on Neural Networks and Learning Systems 34: 7704–7718.

29.

Shafiqul

Jorge

Gurdial

, et al. (2021) Distributed adaptive protocol for asymptotic consensus for a networked Euler-Lagrange systems with uncertainty. In: Proceedings of the 47th annual conference of the IEEE industrial electronics society (IECON), Toronto, ON, Canada, 13–16 October, pp. 1–8. New York: IEEE.

30.

Shahvali

Askari

(2016) Cooperative adaptive neural partial tracking errors constrained control for nonlinear multi-agent systems. International Journal of Adaptive Control and Signal Processing 30(7): 1019–1042.

31.

Song

Wang

(2021) Globally exponentially stable tracking control of self-restructuring nonlinear systems. IEEE Transactions on Cybernetics 51(9): 4755–4765.

32.

Sui

Chen

CLP

Tong

, et al. (2020) Finite-time adaptive quantized control of stochastic nonlinear systems with input quantization: A broad learning system based identification method. IEEE Transactions on Industrial Electronics 67(10): 8555–8565.

33.

Wang

Chen

Lin

, et al. (2018) Adaptive neural network finite-time output feedback control of quantized nonlinear systems. IEEE Transactions on Cybernetics 48(6): 1839–1848.

34.

Wang

Song

Krstic

, et al. (2016) Fault-tolerant finite time consensus for multiple uncertain nonlinear mechanical systems under single-way directed communication interactions and actuation failures. Automatica 63: 374–383.

35.

Wen

, et al. (2013) Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs. Systems & Control Letters 62(12): 1151–1158.

36.

Yang

, et al. (2022) Command-filter-based adaptive output feedback quantized tracking control for switched stochastic nonlinear time-delay systems with actuator saturation. International Journal of Robust and Nonlinear Control 32: 6866–6887.

37.

Pan

Chen

, et al. (2021) Quantized adaptive finite-time bipartite NN tracking control for stochastic multiagent systems. IEEE Transactions on Cybernetics 51(6): 2870–2881.

38.

Xiang

Jia

, et al. (2022) Energy-efficient distributed formation control of sampled-data multiagent systems with packet losses. IEEE Transactions on Cybernetics. Epub ahead of print 25 October. DOI: 10.1109/TCYB.2022.3213597.

39.

Yin

Wang

Liu

, et al. (2020) Finite/fixed-time consensus of nonlinear multi-agent systems against actuator faults and disturbances. Transactions of the Institute of Measurement and Control 42(16): 3254–3266.

40.

You

Hua

, et al. (2020) Fixed-time leader-following consensus for high-order time-varying nonlinear multiagent systems. IEEE Transactions on Automatic Control 65(12): 5510–5516.

41.

Shi

Zhao

(2018) Finite-time command filtered backstepping control for a class of nonlinear systems. Automatica 92: 173–180.

42.

Long

(2015) Finite-time consensus for second-order multi-agent systems with disturbances by integral sliding mode. Automatica 54: 158–165.

43.

Shirinzadeh

, et al. (2005) Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41(11): 1957–1964.

44.

Chen

Ren

, et al. (2011) Distributed higher order consensus protocols in multiagent dynamical systems. IEEE Transactions on Circuits and Systems I: Regular Papers 58(8): 1924–1932.

45.

Zamanian

Abdollahi

Nikravesh

SKY

(2022) Adaptive consensus for heterogeneous unknown nonlinear multi-agent systems with asymmetric input dead-zone: A finite-time approach. Transactions of the Institute of Measurement and Control 44(1): 105–120.

46.

Zhang

Sun

, et al. (2019) Cooperative adaptive finite-time control for stochastic multi-agent systems with input quantisation. IET Control Theory & Applications 13(6): 746–754.

47.

Zhao

Liu

Wen

, et al. (2018) Edge-based finite-time protocol analysis with final consensus value and settling time estimations. IEEE Transactions on Cybernetics 50(4): 1450–1459.

48.

Zhou

Wen

Yang

(2014) Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal. IEEE Transactions on Automatic Control 59(2): 460–464.

49.

Zhu

Xia

(2011) Attitude stabilization of rigid spacecraft with finite-time convergence. International Journal of Robust and Nonlinear Control 21(6): 686–702.

50.

Zou

Qian

Xiang

(2020) Fixed-time consensus for a class of heterogeneous nonlinear multiagent systems. IEEE Transactions on Circuits and Systems II: Express Briefs 67(7): 1279–1283.