Fixed-time consensus tracking control of multiple nonholonomic dynamics under disturbance

Abstract

In this paper, the fixed-time consensus tracking problem is investigated for a multi-agent system with nonholonomic dynamics under the disturbances with unknown upper bounds. For an undirected communication graph, fixed-time distributed state observers and input observers are respectively proposed to estimate state and input of leader system. To deal with the unknown upper-bound disturbance, a second-order sliding mode super-twisting structure adaptive disturbance observer is developed to estimate disturbance in fixed time. Then, by resorting to the estimated information and adding a power integrator method, a novel nonlinear controller is designed for tracking the leader state in fixed-time under disturbance. Moreover, the fixed-time consensus analysis is showed, and the upper bound of settling time is calculated without dependence on initial states even in the presence of disturbance. Finally, numerical examples are included to demonstrate the effectiveness of the proposed procedure.

Keywords

Consensus tracking observer design nonholonomic systems fixed-time consensus

Introduction

In the last few years, there has been a great deal of research interest in the problem of distributed control of multi-agent systems due to its wide range of real-world applications, such as sensor networks, automotive network, and quadrotor formation (Dong et al., 2017; Liu et al., 2020; Pham et al., 2020). As one of the fundamental research topics, consensus tracking generally requires each follower to track the leader’s information. Some consensus algorithms have been proposed to obtain asymptotic consensus results for multi-agent systems, which means the system states achieve an agreement in an infinite time (Li et al., 2019; Li and Huang, 2023; Sun et al., 2023).

In reality, the convergence time should be considered in the algorithm design. To shorten the settling time, finite-time stabilization has received much attention due to faster convergence rates. In particular, by utilizing the input–output information of each agent as well as its neighboring agent, a finite-time distributed data learning consensus algorithm has been constructed for a class of unknown linear multi-agent systems (Bu et al., 2019). A finite-time control protocol has been first proposed based on an observer with converging at a finite-time rate with homogeneous systems theory (Du et al., 2014). Note that the calculation of settling time for finite-time consensus algorithm relies on the initial states of the system. However, it is not easy to obtain the initial states of all agents in advance for calculating convergence time. As such, it becomes a nature to investigate the so-called fixed-time control problem, where the upper bound on the convergence time can be calculated explicitly. For instance, in Ni et al. (2020), an event-triggered fixed-time controller based on the adaptive dynamic surface has been proposed for the high-order multi-agent systems under directed interaction topology. Then, a nonlinear distributed controller with the distributed observer has been proposed for fixed-time consensus for multi-agent systems that consist of the first- and second-order individuals (Sun et al., 2021). In Pan et al. (2024), an adaptive neural network predefined-time tracking control method has been proposed to predefined-time stability of all signals in a class of high-order nonlinear multi-agent systems. However, the above fixed-time results are mainly addressed for first-order or a special class of higher-order nonlinear systems and cannot adequately describe complex individuals in multi-agent systems, such as wheeled robots, which motivates our efforts in this paper.

It is worth mentioning that various vehicle systems can be considered to be in pure rolling motion at each wheel when traveling normally on the road. Thus, the velocity at the point of contact of the wheel with the road is zero, which gives rise to a nonholonomic constraint. A system containing nonholonomic constraints or nonintegrated constraints is called a nonholonomic system. Conversely, it is called a holonomic system (Teel et al., 1995). A nonholonomic constraint is a kind of constraint of that system with respect to spatial position and velocity of motion that cannot be converted from a constraint on velocity of motion to a constraint on spatial position by integration. Therefore, nonholonomic dynamics can better describe some mechanical systems compared to linear second-order systems, such as wheeled robots (Hassan and Hammuda, 2020). For single nonholonomic system control issues, a fuzzy logic system and hybrid adaptive-robust controller have been developed for both holonomic and nonholonomic mechanical systems under uncertainties and disturbances (Chang and Chen, 2000). In Gao et al. (2021), based on the universal barrier Lyapunov function, an output feedback controller has been developed by employing the bi-limit control techniques for the nonholonomic chained-form systems with time-varying output constraints. For multiple nonholonomic system consensus, in Ning and Han (2018), by the distributed observer, a nonlinear protocol for nonholonomic chained-form systems without nonlinear uncertainties has been designed to track the estimated leader state in fixed time. In Shi et al. (2019), a robust fixed-time state feedback consensus tracking controller has been designed for chained-form systems with nonholonomic dynamics under unknown parameters and uncertainties. However, Hassan and Hammuda (2020) do not fully take into account the effect of uncertainty on the system. The proposed control methods for uncertainty in Hassan and Hammuda (2020) and Shi et al. (2019) require the assumption that the upper bound on uncertainty is known. To relax these constraints, it is an interesting idea to estimate a disturbance with an unknown upper bound in fixed time.

In this paper, we consider the problem of fixed-time consensus for nonholonomic multi-agent systems subject to disturbance with unknown upper bound. Specifically, the main contributions can be highlighted as follows:

the fixed-time distributed observers are designed to estimate the state and input of leader in fixed time, and the upper bound of settling time is only given dependent on the observers parameters;

by developing the adaptive disturbance observer, assumption on the upper bound of disturbance can be relaxed to unknown constants;

the sufficient conditions are established such that the adaptive disturbance observer and distributed observer are convergent as well as the nonholonomic multi-agent system achieve leader–following consensus in fixed time.

Notation

In this paper, $I_{n}$ is the $n$ dimension identity matrix. ∥·∥ stands for the Euclidean norm. $1_{n}$ is a vector that all elements are $1$ . Note that ${⌊ x ⌉}^{σ} = [{| x_{1} |}^{σ} sign (x_{1}), \dots, {| x_{n} |}^{σ} sign (x_{n})]^{T}$ for $x \in R^{n}$ , where $σ \in (0, 1)$ is a parameter. In particular, ${⌊ \bar{x} ⌉}^{0} = [sign (x_{1}), \dots, sign (x_{n})]^{T}$ is defined. $p$ -norm is defined as $∥ x_{i} ∥_{p} = {({| x_{i 1} |}^{p} + {| x_{i 2} |}^{p} + \dots + {| x_{in} |}^{p})}^{\frac{1}{p}}$ . Euclidean norm is defined as $∥ x_{i} ∥_{2} = {({| x_{i 1} |}^{2} + {| x_{i 2} |}^{2} + \dots + {| x_{in} |}^{2})}^{\frac{1}{2}}$ .

Preliminaries and problem statement

Graph topology

In this section, a directed graph $G = (V, E)$ is utilized in which $V = v_{0}, \dots, v_{N}$ is the set of indexes corresponding to each agent. We regard $i = 0$ as the leader and $i = 1, 2, 3, \dots, N$ as the followers. $E \subseteq V \times V$ is the set of edges. The connectivity matrix is denoted as $A = [a_{il}] \in R^{N \times N}$ , where $a_{il} > 0$ if $(v_{i}, v_{l}) \in E$ and $a_{il} = 0$ otherwise. Note that the diagonal elements $a_{ii} = 0$ . The set of neighborhood for agent $i$ is denoted by $N_{i} = {l \in V : (v_{i}, v_{l}) \in E}$ . The Laplacian matrix $L = [L_{il}] \in R^{N \times N}$ is given as $L_{il} = Σ_{l \in N_{i}} a_{il}$ for $i = l$ and $L_{il} = - a_{il}$ for $i \neq l$ . For an undirected graph, $a_{il} = a_{li}$ . $b_{i} > 0$ holds if and only if there exists an edge from the leader to the $i$ th follower. Denote $B = diag {b_{1}, b_{2} \dots, b_{N}}$ .

Assumption 1. (You et al., 2018) The graph $G$ is undirected. Each follower can connect directly or indirectly to the leader via topology.

Lemma 1. (Yang et al., 2024) If Assumption 1 holds, matrix $\bar{L} = L + B$ is positive definite.

Problem formulation

In this paper, consider a group of mobile robots with a leader subscripted as $0$ and $N$ followers. For the nonholonomic constraint of the nonslipping condition ${\overset{\cdot}{x}}_{i} \sin (Θ_{i} (t)) - {\overset{\cdot}{y}}_{i} \cos (Θ_{i} (t)) = 0$ , the model of the $i$ th wheeled mobile robot is described by Dong and Farrell (2008)

{\overset{\cdot}{x}}_{i} = (v_{i} (t) + f_{i} (t)) \cos (Θ_{i} (t))

(1)

{\overset{\cdot}{y}}_{i} = (v_{i} (t) + f_{i} (t)) \sin (Θ_{i} (t))

(2)

{\overset{\cdot}{Θ}}_{i} (t) = ω_{i} (t)

(3)

where $(x_{i}, y_{i})$ is the Cartesian position of the $i$ th robot center; $Θ_{i} (t)$ represents the orientation; $v_{i} (t)$ and $ω_{i} (t)$ are the linear and the angular velocity, respectively. It is well known that many mechanical systems with nonholonomic constraints can be locally, or globally, converted to the chained form under coordinate change and state feedback. By defining $x_{i 1} (t) = - Θ_{i} (t)$ , $x_{i 2} (t) = - x_{i} (t) \sin (Θ_{i} (t)) + y_{i} (t) \cos (Θ_{i} (t))$ , $x_{i 3} (t) = x_{i} (t) \cos (Θ_{i} (t)) + y_{i} (t) \sin (Θ_{i} (t))$ , $u_{i 1} (t) = - ω_{i} (t)$ as well as $u_{i 2} (t) = v_{i} (t) + x_{i 2} (t) ω_{i} (t)$ hold for $i = 0, 1, 2, 3$ and $f_{0} (t) = 0$ . Then, systems (1)–(3) can be transformed as

{\overset{\cdot}{x}}_{i 1} (t) = u_{i 1} (t) + d_{i 1} (t)

(4)

{\overset{\cdot}{x}}_{i 2} (t) = u_{i 1} (t) x_{i 3} (t)

(5)

{\overset{\cdot}{x}}_{i 3} (t) = u_{i 2} (t) + d_{i 2} (t)

(6)

where $x_{i} (t) = [x_{i 1} (t), x_{i 2} (t), x_{i 3} (t)]^{T}$ and $u_{i} (t) = [u_{i 1} (t), u_{i 2}$ $(t)]^{T}$ are the state and control input, respectively. $d_{i 1} (t)$ and $d_{i 2} (t)$ denote the unknown external disturbances.

The dynamics of the leader is modeled as

{\overset{\cdot}{x}}_{01} (t) = u_{01} (t)

(7)

{\overset{\cdot}{x}}_{02} (t) = u_{01} (t) x_{03} (t)

(8)

{\overset{\cdot}{x}}_{03} (t) = u_{02} (t)

(9)

where $x_{0} (t) = [x_{01} (t), x_{02} (t), x_{03} (t)]^{T}$ and $u_{0} (t) = [u_{01} (t), u_{02}$ $(t)]^{T}$ are respectively the state and input of leader.

In this paper, only some of the followers are all able to receive the leader’s state and input information due to the communication topology. It is aimed to design an appropriate control algorithm $u_{i} (t)$ for multi-agent systems (7) and (6) composed of leader $0$ and $N$ follower mobile robots such that ${\tilde{x}}_{i} (t) = x_{i} (t) - x_{0} (t)$ can be convergent to zero in a fixed period of time, where ${\tilde{x}}_{i} (t)$ is the leader–following consensus error. In addition, the system settling time could be arbitrarily adjusted by choosing different parameters of observer and controller. Then, the definition on fixed-time leader–following consensus tracking will be featured.

Definition 1. It is claimed to achieve the leader–follower consensus of the multi-agent systems (6) and (7), for any initial condition, there exists a fixed period of time $T$ such that

\lim_{t \to T} {\tilde{x}}_{i} (t) = 0

holds for $i = 1, \dots, N$ .

To achieve this control objective, the following assumptions are made.

Assumption 2. (Huang et al., 2024) For the unknown disturbances $d_{i 1} (t)$ and $d_{i 2} (t)$ in the follower $i$ systems (4)–(6), there are two positive unknown constants ${\bar{d}}_{i 1}$ and ${\bar{d}}_{i 2}$ satisfying that

| {\overset{\cdot}{d}}_{i 1} (t) | \leq {\bar{d}}_{i 1}

(10)

and

| {\overset{\cdot}{d}}_{i 2} (t) | \leq {\bar{d}}_{i 2}

(11)

hold.

Assumption 3. (Pan et al., 2024) For the inputs of the leader $u_{01} (t)$ and $u_{02} (t)$ , there are four positive constants $ϱ_{1}$ , $ϱ_{2}$ , $ū_{01}$ , and $ū_{02}$ satisfying that

| {\dot{u}}_{01} (t) | \leq ū_{01}, ϱ_{1} \leq | u_{01} (t) | \leq ϱ_{2}

(12)

and

| {\dot{u}}_{02} (t) | \leq ū_{02}

(13)

hold.

Assumption 4. (Sarrafan and Zarei, 2021) The state of leader $x_{0 k}$ is assumed as bounded.

The following lemmas are provided for subsequent analysis.

Lemma 2. (Li et al., 2017) Consider the following differential equation

\overset{\cdot}{y} = - α y^{p} - β y^{q}, y (0) = y_{0}

(14)

where $y \in R^{+} \cup^{{} 0}$ ; $α > 0$ , $β > 0$ , $p > 1$ and $0 < q < 1$ are four parameters. Then, the differential system (14) can be fixed-time stable. In addition, the upper bound of convergence time satisfies

T \leq \frac{1}{α (p - 1)} + \frac{1}{β (1 - q)}

(15)

Lemma 3. (Li et al., 2009) For $x \in R$ and $y \in R$ , if $0 < p < 1$ is a fraction number, the inequality holds:

| x^{p} - y^{p} | \leq 2^{1 - p} {| x - y |}^{p}

Lemma 4. (Li et al., 2021) For any vector $x \in R^{n}$ and parameters $0 < p < q$ , then the inequality $∥ x ∥_{q} \leq ∥ x ∥_{p} \leq n^{\frac{1}{p} - \frac{1}{q}} ∥ x ∥_{q}$ holds.

Main results

In this section, we will construct distributed fixed-time observers to estimate the leader’s input and state for each follower. Moreover, in order to deal with the disturbance in each agent, an adaptive fixed-time disturbance observer is proposed by constructing the adaptive parameter. Then, based on the estimation of observers, a compound nonlinear controller is designed to achieve the leader–following consensus.

Distributed fixed-time observer

In this section, in order to estimate the leader’s state, a distributed fixed-time state observer for the $i$ th follower agent is proposed as

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i 1} (t) = {\hat{u}}_{i 1} (t) - α_{1} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01}) |}^{\frac{3}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01})) \\ - α_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01}) |}^{\frac{7}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01})) \end{matrix}

(16)

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i 2} (t) = {\hat{u}}_{i 1} (t) {\hat{x}}_{i 3} - α_{1} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 2} - {\hat{x}}_{l 2}) + b_{i} ({\hat{x}}_{i 2} - x_{02}) |}^{\frac{3}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 2} - {\hat{x}}_{l 2}) + b_{i} ({\hat{x}}_{i 2} - x_{02})) \\ - α_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 2} - {\hat{x}}_{l 2}) + b_{i} ({\hat{x}}_{i 2} - x_{02}) |}^{\frac{7}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 2} - {\hat{x}}_{l 2}) + b_{i} ({\hat{x}}_{i 2} - x_{02})) \end{matrix}

(17)

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i 3} (t) = {\hat{u}}_{i 2} (t) - α_{1} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 3} - {\hat{x}}_{l 3}) + b_{i} ({\hat{x}}_{i 3} - x_{03}) |}^{\frac{3}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 3} - {\hat{x}}_{l 3}) + b_{i} ({\hat{x}}_{i 3} - x_{03})) \\ - α_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 3} - {\hat{x}}_{l 3}) + b_{i} ({\hat{x}}_{i 3} - x_{03}) |}^{\frac{7}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 3} - {\hat{x}}_{l 3}) + b_{i} ({\hat{x}}_{i 3} - x_{03})) \end{matrix}

(18)

where $α_{1}$ and $α_{2}$ are the positive parameters; ${\hat{x}}_{i 1}$ , ${\hat{x}}_{i 2}$ , and ${\hat{x}}_{i 3}$ are the estimations of $x_{01}$ , $x_{02}$ , and $x_{03}$ for agent $i$ , respectively. For $k = 1, 2$ , a distributed fixed-time input observer is developed as

\begin{matrix} {\overset{\cdot}{\hat{u}}}_{ik} = - ρ_{1} sign (\sum_{l = 1}^{N} a_{il} ({\hat{u}}_{ik} - {\hat{u}}_{lk}) + b_{i} ({\hat{u}}_{ik} - u_{0 k})) \\ - ρ_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{u}}_{ik} - {\hat{u}}_{lk}) + b_{i} ({\hat{u}}_{ik} - u_{0 k}) |}^{\frac{3}{2}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{u}}_{ik} - {\hat{u}}_{lk}) + b_{i} ({\hat{u}}_{ik} - u_{0 k})) \end{matrix}

(19)

where $ρ_{1}$ and $ρ_{2}$ are positive parameters; ${\hat{u}}_{ik}$ with $k = 1, 2$ is the estimation of $u_{0 k}$ .

Theorem 1. Consider the multiple mobile robot systems (4)–(6) with the distributed fixed-time observers (16)–(19). Under Assumptions 1, 2, and 4, if there are the positive parameters $α_{1}$ , $α_{2}$ , $ρ_{1}$ , and $ρ_{2}$ satisfying that

ρ_{1} - ū_{01} > 0, ρ_{1} - ū_{02} > 0, α_{1} > 0, α_{2} > 0, ρ_{2} > 0

(20)

then observers (16)–(19) can estimate the leader’s state and inputs in fixed time $2 T_{\max 2}$ , where $T_{\max 2} \leq \frac{5}{α_{1} {(2 λ_{\min} (L))}^{\frac{4}{5}}} + \frac{5 N^{\frac{1}{10}}}{α_{2} {(2 λ_{\min} (L))}^{\frac{6}{5}}} + T_{\max 1}$ holds with $T_{m a x 1} \leq \frac{1}{(ρ_{1} - ū_{01})} +$ $\frac{4}{N^{- \frac{3}{2}} ρ_{2} {(2 λ_{\min} (\bar{L}))}^{\frac{5}{4}}}$ .

Proof. First, it will be shown that each follower can have an accurate estimation of the input of the leader in fixed time. Taking the theoretical analysis of ${\tilde{u}}_{i 1} = {\hat{u}}_{i 1} - u_{01}$ as an example, the effectiveness analysis of the observer is given as follows.

The time derivative of ${\tilde{u}}_{i 1}$ is given as

\begin{matrix} {\overset{\cdot}{\tilde{u}}}_{i 1} = - {\overset{\cdot}{u}}_{01} - ρ_{1} sign (\sum_{l = 1}^{N} a_{il} ({\hat{u}}_{i 1} - {\hat{u}}_{l 1}) + b_{i} ({\hat{u}}_{i 1} - u_{01})) \\ - ρ_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{u}}_{i 1} - {\hat{u}}_{l 1}) + b_{i} ({\hat{u}}_{i 1} - u_{01}) |}^{\frac{3}{2}} \\ \times sign (\sum_{i = 1}^{N} a_{il} ({\hat{u}}_{i 1} - {\hat{u}}_{l 1}) + b_{i} ({\hat{u}}_{i 1} - u_{01})) = - {\overset{\cdot}{u}}_{01} \\ - ρ_{1} sign (\sum_{l = 1}^{N} a_{il} ({\hat{u}}_{i 1} - u_{01} + u_{01} - {\hat{u}}_{l 1}) + b_{i} ({\hat{u}}_{i 1} - u_{01})) \\ - ρ_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{u}}_{i 1} - u_{01} + u_{01} - {\hat{u}}_{l 1}) + b_{i} ({\hat{u}}_{i 1} - u_{01}) |}^{\frac{3}{2}} \\ \times sign (\sum_{i = 1}^{N} a_{il} ({\hat{u}}_{i 1} - u_{01} + u_{01} - {\hat{u}}_{l 1}) + b_{i} ({\hat{u}}_{i 1} - u_{01})) \\ = - {\overset{\cdot}{u}}_{01} - ρ_{1} sign (\sum_{l = 1}^{N} a_{il} ({\tilde{u}}_{i 1} - {\tilde{u}}_{l 1}) + b_{i} {\tilde{u}}_{i 1}) \\ - ρ_{2} {| \sum_{l = 1}^{N} a_{il} ({\tilde{u}}_{i 1} - {\tilde{u}}_{l 1}) + b_{i} {\tilde{u}}_{i 1} |}^{\frac{3}{2}} \\ sign (\sum_{i = 1}^{N} a_{il} ({\tilde{u}}_{i 1} - {\tilde{u}}_{l 1}) + b_{i} {\tilde{u}}_{i 1}) \end{matrix}

(21)

where $ρ_{1}$ and $ρ_{2}$ are the two positive parameters in observer error system (21). Letting ${\tilde{U}}_{1} = [{\tilde{u}}_{11}, \dots, {\tilde{u}}_{N 1}]^{T}$ , the dynamics of ${\tilde{U}}_{1}$ is given as

{\overset{\cdot}{\tilde{U}}}_{1} = - ρ_{1} ⌊ \bar{L} {{\tilde{U}}_{1} ⌉}^{0} - ρ_{2} ⌊ \bar{L} {{\tilde{U}}_{1} ⌉}^{\frac{3}{2}} - {\overset{\cdot}{u}}_{01} 1_{N}

(22)

where $\bar{L} = L + B$ is a positive-definite matrix by Lemma 1. Then, select the Lyapunov function as follows

V_{1} = \frac{1}{2} {\tilde{U}}_{1}^{T} \bar{L} {\tilde{U}}_{1}

(23)

Taking the derivative of equation (23), from inequality (12) in Assumption 3, it is obtained as

\begin{array}{l} {\dot{V}}_{1} = {\tilde{U}}_{1}^{T} \bar{ℒ} (- ρ_{1} ⌊ \bar{ℒ} {{\tilde{U}}_{1} ⌉}^{0} - ρ_{2} {⌊ \bar{ℒ} {\tilde{U}}_{1} ⌉}^{\frac{3}{2}} - {\dot{u}}_{01} 1_{N}) \\ \leq - (ρ_{1} - ū_{01}) ∥ \bar{ℒ} {\tilde{U}}_{1} ∥ - ρ_{2} ∥ \bar{ℒ} {\tilde{U}}_{1} ∥_{\frac{5}{2}}^{\frac{5}{2}} \end{array}

(24)

where $ū_{01}$ is a positive constant and given in Assumption 3. $ū_{01}$ is the upper bound of ${\overset{\cdot}{u}}_{01}$ . Letting $p = 2$ and $q = \frac{5}{2}$ , based on Lemma 4, one has that

∥ \bar{L} {\tilde{U}}_{1} ∥_{\frac{5}{2}}^{\frac{5}{2}} \geq N^{- \frac{1}{4}} ∥ \bar{L} {\tilde{U}}_{1} ∥_{2}^{\frac{5}{2}}

(25)

By inequality equation (25), the derivative of $V_{1}$ can be written as

{\dot{V}}_{1} \leq - (ρ_{1} - ū_{01}) ∥ \bar{ℒ} {\tilde{U}}_{1} ∥ - N^{- \frac{1}{4}} ρ_{2} ∥ \bar{ℒ} {\tilde{U}}_{1} ∥_{2}^{\frac{5}{2}}

(26)

Due to $∥ \bar{L} {\tilde{U}}_{1} ∥_{2}^{2} \geq 2 λ_{\min} (\bar{L}) V_{1}$ , inequality (26) is given as

{\dot{V}}_{1} \leq - 2 (ρ_{1} - ū_{01}) V_{1}^{\frac{1}{2}} - N^{- \frac{1}{4}} ρ_{2} {(2 λ_{m i n} (\bar{ℒ}))}^{\frac{5}{4}} V_{1}^{\frac{5}{4}}

(27)

By using Lemma 2, function $V_{1}$ can be convergent to $V_{1} = 0$ in fixed time. The required time is calculated as

T_{m a x 1} \leq \frac{1}{(ρ_{1} - ū_{01})} + \frac{4}{N^{- \frac{3}{2}} ρ_{2} {(2 λ_{m i n} (\bar{ℒ}))}^{\frac{5}{4}}}

(28)

Following the above analysis procedure, it is concluded that ${\tilde{u}}_{i 2}$ is convergent to zero in fixed time $T_{\max 1}$ .

Then, we will prove that each follower can estimate the leader’s state in fixed time. Defining ${\tilde{x}}_{ik} = {\hat{x}}_{ik} - x_{0 k}$ with $k = 1, 2, 3$ , considering the case of $k = 1$ and $t \geq T_{\max 1}$ , the dynamics of ${\tilde{x}}_{i 1}$ is written as

\begin{matrix} {\overset{\cdot}{\tilde{x}}}_{i 1} (t) = - α_{1} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - x_{01} + x_{01} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01}) |}^{\frac{3}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - x_{01} + x_{01} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01})) \\ - α_{2} {| \sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01}) |}^{\frac{7}{5}} \\ \times sign (\sum_{l = 1}^{N} a_{il} ({\hat{x}}_{i 1} - x_{01} + x_{01} - {\hat{x}}_{l 1}) + b_{i} ({\hat{x}}_{i 1} - x_{01})) \\ = - α_{1} {| \sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 1} - {\tilde{x}}_{l 1}) + b_{i} {\tilde{x}}_{i 1} |}^{\frac{3}{5}} \\ sign (\sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 1} - {\tilde{x}}_{l 1}) + b_{i} {\tilde{x}}_{i 1}) \\ - α_{2} {| \sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 1} - {\tilde{x}}_{l 1}) + b_{i} {\tilde{x}}_{i 1} |}^{\frac{7}{5}} \\ sign (\sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 1} - {\tilde{x}}_{l 1}) + b_{i} {\tilde{x}}_{i 1}) \end{matrix}

(29)

where $α_{1}$ and $α_{2}$ are the two positive parameters in the state observer error system (29).

Letting ${\tilde{x}}_{1} = [{\tilde{x}}_{11}, {\tilde{x}}_{21}, \dots, {\tilde{x}}_{N 1}]^{T}$ , the dynamics of ${\tilde{x}}_{1}$ is described as

{\overset{\cdot}{\tilde{x}}}_{1} (t) = - α_{1} ⌊ \bar{L} {{\tilde{x}}_{1} ⌉}^{\frac{3}{5}} - α_{2} ⌊ \bar{L} {{\tilde{x}}_{1} ⌉}^{\frac{7}{5}}

(30)

Choose the following Lyapunov function

V_{2} = \frac{1}{2} {\tilde{x}}_{1}^{T} \bar{L} {\tilde{x}}_{1}

(31)

By resorting to system (30), the time derivatives of equation (31) is given as

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\tilde{x}}_{1}^{T} \bar{L} (- α_{1} ⌊ \bar{L} {{\tilde{x}}_{1} ⌉}^{\frac{3}{5}} - α_{2} ⌊ \bar{L} {{\tilde{x}}_{1} ⌉}^{\frac{7}{5}}) \\ \leq - α_{1} ∥ \bar{L} {\tilde{x}}_{1} ∥_{\frac{8}{5}}^{\frac{8}{5}} - α_{2} ∥ \bar{L} {\tilde{x}}_{1} ∥_{\frac{12}{5}}^{\frac{12}{5}} \end{matrix}

(32)

Based on the Lemma 4, defining $p_{1} = \frac{8}{5}$ , $p_{2} = 2$ and $q = \frac{12}{5}$ , one has that $∥ \bar{L} {\tilde{x}}_{1} ∥_{\frac{8}{5}} \geq ∥ \bar{L} {\tilde{x}}_{1} ∥_{2}$ and $∥ \bar{L} {\tilde{x}}_{1} ∥_{\frac{12}{5}} \geq N^{- \frac{1}{12}} ∥ \bar{L} {\tilde{x}}_{1} ∥_{2}$ . Therefore, inequality (32) is rewritten as

{\overset{\cdot}{V}}_{2} \leq - α_{1} ∥ \bar{L} {\tilde{x}}_{1} ∥_{\frac{8}{5}}^{2} - α_{2} N^{- \frac{1}{10}} ∥ \bar{L} {\tilde{x}}_{1} ∥_{\frac{12}{5}}^{2}

(33)

According to $∥ \bar{L} {\tilde{x}}_{1} ∥_{2}^{2} \geq 2 λ_{\min} (\bar{L}) V_{2}$ , inequality (33) can be obtained as

{\overset{\cdot}{V}}_{2} \leq - α_{1} {(2 λ_{\min} (\bar{L}))}^{\frac{4}{5}} V_{2}^{\frac{4}{5}} - α_{2} N^{- \frac{1}{10}} {(2 λ_{\min} (\bar{L}))}^{\frac{6}{5}} V_{2}^{\frac{6}{5}}

(34)

By using Lemma 2, under Assumption 4, function $V_{2}$ can be convergent to $V_{2} = 0$ in fixed-time. The required time $T_{\max 2}$ is calculated as

T_{\max 2} \leq \frac{5}{α_{1} {(2 λ_{\min} (\bar{L}))}^{\frac{4}{5}}} + \frac{5 N^{\frac{1}{10}}}{α_{2} {(2 λ_{\min} (\bar{L}))}^{\frac{6}{5}}} + T_{\max 1}

(35)

Similar to the previous step, it can be given that $∥ {\tilde{x}}_{3} ∥ = 0$ with ${\tilde{x}}_{3} = [{\tilde{x}}_{13}, {\tilde{x}}_{23}, \dots, {\tilde{x}}_{N 3}]^{T}$ holds in fixed-time $T_{\max 2}$ .

Then, the dynamics of ${\tilde{x}}_{i 2}$ is modeled as

\begin{matrix} {\overset{\cdot}{\tilde{x}}}_{i 2} (t) = - α_{1} {| \sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 2} - {\tilde{x}}_{l 2}) + b_{i} {\tilde{x}}_{i 2} |}^{\frac{3}{5}} \\ sign (\sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 2} - {\tilde{x}}_{l 2}) + b_{i} {\tilde{x}}_{i 2}) \\ - α_{2} {| \sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 2} - {\tilde{x}}_{l 2}) + b_{i} {\tilde{x}}_{i 2} |}^{\frac{7}{5}} \\ sign (\sum_{l = 1}^{N} a_{il} ({\tilde{x}}_{i 2} - {\tilde{x}}_{l 2}) + b_{i} {\tilde{x}}_{i 2}) + {\hat{x}}_{i 3} {\hat{u}}_{i 1} - x_{03} u_{01} \end{matrix}

(36)

when $t \geq T_{\max 2}$ , ${\hat{x}}_{i 3} {\hat{u}}_{i 1} - x_{03} u_{01} = 0$ holds. According to the above previous step, we can prove that ${\tilde{x}}_{i 2}$ converges to zero in a fixed-time $2 T_{\max 2}$ .

Therefore, it is proved that the leader state and input can be estimated by all followers within $2 T_{\max 2}$ . □

Remark 1. In this paper, only some of the followers are all able to receive the leader’s input information due to the communication topology. For the subsequent control protocol design, it is necessary to design observer (19) to estimate the leader’s input information. In the distributed observer (19), if agent $i$ is unable to obtain input information from the leader, $b_{i} = 0$ holds. Therefore, the leader’s input information is unknown for each agent.

Adaptive fixed-time disturbance observer

In order to estimate the disturbances $d_{i 1}$ and $d_{i 2}$ of the mobile robot follower $i$ , an adaptive fixed-time disturbance observer is developed as

{\overset{\cdot}{z}}_{i 1} = z_{i 2} + u_{i 1} (t) - β_{i 1} (t) (⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{1}{2}} + θ_{i 1} ⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{3}{2}}) + ϕ_{i 1} ({\tilde{z}}_{i 1})

(37)

{\overset{\cdot}{z}}_{i 2} = - β_{i 2} (t) (\frac{1}{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{0} + 2 θ_{i 1} {\tilde{z}}_{i 1} + + \frac{3}{2} θ_{i 1}^{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{2})

(38)

{\overset{\cdot}{z}}_{i 3} = z_{i 4} + u_{i 2} (t) - β_{i 3} (t) (⌊ {{\tilde{z}}_{i 3} ⌉}^{\frac{1}{2}} + θ_{i 2} ⌊ {{\tilde{z}}_{i 3} ⌉}^{\frac{3}{2}}) + ϕ_{i 2} ({\tilde{z}}_{i 3})

(39)

{\overset{\cdot}{z}}_{i 4} = - β_{i 4} (t) (\frac{1}{2} {{⌊ \tilde{z}}_{i 3} ⌉}^{0} + 2 θ_{i 2} {\tilde{z}}_{i 3} + + \frac{3}{2} θ_{i 2}^{2} ⌊ {{\tilde{z}}_{i 3} ⌉}^{2})

(40)

where $z_{i 1}$ , $z_{i 2}$ , $z_{i 3}$ , and $z_{i 4}$ are the estimations of $x_{i 1}$ , $d_{i 1}$ , $x_{i 3}$ , and $d_{i 2}$ , respectively; ${\tilde{z}}_{i 1} = z_{i 1} - x_{i 1}$ and ${\tilde{z}}_{i 3} = z_{i 3} - x_{i 3}$ are the estimated errors; $θ_{i 1}$ and $θ_{i 2}$ are the two positive parameters; the adaptive parameters $β_{i 1} (t)$ , $β_{i 2} (t)$ , $β_{i 3} (t)$ , and $β_{i 4} (t)$ are respectively given as

\begin{matrix} β_{i 1} (t) = β_{i 01} \sqrt{M_{i 1} (t)} β_{i 2} (t) = β_{i 02} M_{i 1} (t), \\ β_{i 3} (t) = β_{i 03} \sqrt{M_{i 2} (t)}, β_{i 4} (t) = β_{i 04} M_{i 2} (t) \end{matrix}

(41)

where $β_{i 01}$ , $β_{i 02}$ , $β_{i 03}$ , and $β_{i 04}$ are the four positive parameters; the adaptive functions $ϕ_{i 1} ({\tilde{z}}_{i 1})$ and $ϕ_{i 2} ({\tilde{z}}_{i 3})$ are defined as

ϕ_{i 1} ({\tilde{z}}_{i 1}) = \frac{{\overset{\cdot}{M}}_{i 1} (t) (⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{1}{2}} + θ_{i 1} ⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{3}{2}})}{2 M_{i 1} (t) (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + θ_{i 1} \frac{3}{2} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}})}

(42)

ϕ_{i 2} ({\tilde{z}}_{i 3}) = \frac{{\overset{\cdot}{M}}_{i 2} (t) (⌊ {{\tilde{z}}_{i 3} ⌉}^{\frac{1}{2}} + θ_{i 2} ⌊ {{\tilde{z}}_{i 3} ⌉}^{\frac{3}{2}})}{2 M_{i 2} (t) (\frac{1}{2 {| {\tilde{z}}_{i 3} |}^{\frac{1}{2}}} + θ_{i 2} \frac{3}{2} {| {\tilde{z}}_{i 3} |}^{\frac{1}{2}})}

(43)

Taking, for example, the estimation of $d_{i 1}$ , by defining error ${\tilde{z}}_{i 2} = z_{i 2} - d_{i 1}$ , the observer error dynamics is described as

{\overset{\cdot}{\tilde{z}}}_{i 1} = {\tilde{z}}_{i 2} - β_{i 1} (t) (⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{1}{2}} + θ_{i 1} ⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{3}{2}}) + ϕ_{i 1} ({\tilde{z}}_{i 1})

(44)

{\dot{\tilde{z}}}_{i 2} = - β_{i 2} (t) (\frac{1}{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{0} + 2 θ_{i 1} {\tilde{z}}_{i 1} + + \frac{3}{2} θ_{i 1}^{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{2}) - {\overset{\cdot}{d}}_{i 1}

(45)

Theorem 2. Under Assumption 3, consider the adaptive disturbance observer error systems (44) and (45). Given the adaptive parameter $M_{i 1} (t)$ , both ${\tilde{z}}_{i 1} = 0$ and ${\tilde{z}}_{i 2} = 0$ hold in fixed time if there exist parameters $β_{i 01}$ , $β_{i 02}$ , $m_{i 0}$ , $θ_{i 1}$ , and $ϵ_{i}$ , a positive definite symmetric matrix $P_{i}$ such that

P_{i} A_{i} + A_{i}^{T} P_{i} + ϵ_{i} P_{i} + P_{i} B B^{T} P_{i} + 4 C^{T} C < 0

(46)

where

A_{i} = [\begin{matrix} - β_{i 01} & 1 \\ - β_{i 02} & 0 \end{matrix}], B = [\begin{matrix} 0 \\ 1 \end{matrix}], C = {[\begin{matrix} 1 \\ 0 \end{matrix}]}^{T}

Moreover, the settling time is given as $T_{io} \leq \frac{4}{ϵ_{i} \sqrt{m_{i 0} λ_{\min} (P_{i})}} + \frac{4}{3 θ_{i 1} ϵ_{i} \sqrt{m_{i 0}} λ_{\min} (P_{i})}$ .

Proof. When ${\tilde{z}}_{i 2} \equiv 0$ holds, the switching signal $- β_{i 2} (t)$ $(\frac{1}{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{0} + 2 θ_{i 1} {\tilde{z}}_{i 1} + \frac{3}{2} θ_{i 1}^{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{2})$ is called as equivalent control $u_{eqi 1} (t)$ with the positive parameter $θ_{i 1}$ . A close approximation of the $u_{eqi 1} (t)$ can be obtained in real-time by filters. A nonlinear low-pass filter is developed as

\begin{matrix} τ_{i 1} {\overset{\cdot}{u}}_{eqi 1} (t) = ⌊ \frac{1}{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{0} + 2 θ_{i 1} {\tilde{z}}_{i 1} + + \frac{3}{2} θ_{i 1}^{2} {{⌊ \tilde{z}}_{i 1} ⌉}^{2} - u_{eqi 1} {(t) ⌉}^{\frac{5}{7}} \\ + ⌊ \frac{1}{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{0} + 2 θ_{i 1} {\tilde{z}}_{i 1} + + \frac{3}{2} θ_{i 1}^{2} ⌊ {{\tilde{z}}_{i 1} ⌉}^{2} - u_{eqi 1} {(t) ⌉}^{\frac{7}{5}} \end{matrix}

(47)

where $τ_{i 1}$ is a positive parameter. Define a new variable $δ_{i 1}$ as

δ_{i 1} = M_{i 1} (t) - \frac{1}{r_{i 1} β_{i 02}} | u_{eqi 1} (t) | - ϵ_{i}

(48)

where $M_{i 1} (t)$ is the adaptive parameter; $β_{i 02}$ is the parameter associated with the disturbance observer (45); $ϵ_{i}$ and $r_{i 1}$ are the two positive parameters with $0 < r_{i 1} < β_{i 02}^{- 1} < 1$ . Then, the adaptive term $M_{i 1} (t)$ is given as

M_{i 1} (t) = m_{i 0} + m_{i 1} (t)

(49)

{\overset{\cdot}{m}}_{i 1} (t) = - ρ_{i 1} (t) (⌊ {δ_{i 1} ⌉}^{\frac{5}{7}} + ⌊ {δ_{i 1} ⌉}^{\frac{5}{3}})

(50)

where $m_{i 0}$ is a positive parameter, and $m_{i 1} (t)$ is a adaptive component. The adaptive parameter $ρ_{i 1} (t)$ is defined as

ρ_{i 1} (t) = c_{i 0} + c_{i 1} (t)

(51)

where $c_{i 0}$ is a fixed positive parameter and $c_{i 1} (t)$ is a time-varying term. The derivative of $c_{i 1} (t)$ is obtained as

{\overset{\cdot}{c}}_{i 1} (t) = {\begin{matrix} ξ_{0 i} | δ_{i 1} (t) |, | δ_{i} (t) | > \frac{\in_{i}}{2} \\ 0, & otherwise \end{matrix}

(52)

where $ξ_{0 i}$ is a positive parameter. Defining the auxiliary variable $ψ_{i 1} (t) = \frac{{\bar{d}}_{i 1}}{r_{i 1} β_{i 02}} - c_{i 1} (t)$ , the Lyapunov function is selected as

V_{i 2} (t) = \frac{1}{2} δ_{i 1} {(t)}^{2} + \frac{1}{2 ξ_{0 i}} ψ_{i 1}^{2} (t)

(53)

For the bound of $δ_{i 1} (t)$ , the following derivation process of $V_{i 2}$ is divided into two cases.

Case 1: $| δ_{i 1} (t) | > \frac{ϵ_{i}}{2}$ and $ψ_{i 1} (t) \in R$ . From conditions (48)–(52), the derivative of $V_{i 2} (t)$ is given as

\begin{matrix} {\dot{V}}_{i 2} (t) = δ_{i 1} (t) ({\dot{m}}_{i 1} (t) - \frac{1}{r_{i 1} λ_{2 i}} \frac{d}{d t} | ū_{e q} (t) |) - \frac{1}{ξ_{0 i}} ψ_{i 1} (t) {\dot{c}}_{i 1} (t) \\ \leq δ_{i 1} (t) {\dot{m}}_{i 1} (t) + | δ_{i 1} (t) | \frac{{\bar{d}}_{i 1}}{r_{i 1} λ_{2 i}} - \frac{1}{ξ_{0 i}} ψ_{i 1} (t) {\dot{c}}_{i 1} (t) \\ = - ρ_{i 1} (t) ({| δ_{i 1} |}^{\frac{12}{7}} + {| δ_{i 1} |}^{\frac{8}{3}}) + | δ_{i 1} (t) | \frac{{\bar{d}}_{i 1}}{r_{i 1} λ_{2 i}} \\ - (\frac{{\bar{d}}_{i 1}}{r_{i 1} β_{i 02}} - c_{i 1} (t)) | δ_{i 1} | = - c_{i 0} ({| δ_{i 1} |}^{\frac{12}{7}} + {| δ_{i 1} |}^{\frac{8}{3}}) \end{matrix}

(54)

Therefore, $δ_{i 1} (t)$ is convergent to zero in fixed time.

Case 2: $| δ_{i 1} (t) | \leq \frac{ϵ_{i}}{2}$ . When $| δ_{i 1} (t) | < \frac{ϵ_{i}}{2}$ holds in fixed time, it is obtained as

M_{i 1} (t) - \frac{1}{r_{i 1} β_{i 02}} | u_{eqi 1} (t) | - ϵ_{i} > - \frac{ϵ_{i}}{2}

(55)

Since $r_{i 1} β_{i 02} < 1$ holds, it follows that

M_{i 1} (t) > | u_{eqi 1} (t) | + \frac{ϵ_{i}}{2} > | {\overset{\cdot}{d}}_{i 1} (t) |

(56)

From equation (48), it is also obtained that

\begin{matrix} M_{i 1} (t) < | δ_{i 1} | + \frac{1}{r_{i 1} β_{i 02}} | u_{eqi 1} (t) | + ϵ_{i} \\ < | δ_{i 1} | + \frac{1}{r_{i 1} β_{i 02}} {\bar{d}}_{i 1} + ϵ_{i} \end{matrix}

(57)

Therefore, one has that

| {\overset{\cdot}{d}}_{i 1} | < M_{i 1} (t) < | δ_{i 1} | + \frac{1}{r_{i 1} β_{i 02}} {\bar{d}}_{i 1} + ϵ_{i}

(58)

Therefore, it is concluded that $M_{i 1} (t)$ is always greater than the derivative of disturbance $d_{i 1} (t)$ and less than a constant associated with the upper bound ${\bar{d}}_{i 1}$ . Then, defining $e_{i 1} = \sqrt{M_{i 1} (t)} (⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{1}{2}} + θ_{i 1} ⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{3}{2}})$ , based on the definition of $ϕ_{i 1} ({\tilde{z}}_{i 1})$ , the dynamics of $e_{i 1}$ is written as

\begin{matrix} ė_{i 1} (t) = \sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + θ_{i 1} \frac{3 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) \\ (- β_{i 01} ({⌊ {\tilde{z}}_{i 1} ⌉}^{\frac{1}{2}} + θ_{i 1} {⌊ {\tilde{z}}_{i 1} ⌉}^{\frac{3}{2}}) + {\tilde{z}}_{i 2}) \\ = \sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + θ_{i 1} \frac{3 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) (- β_{i 01} e_{i 1} + {\tilde{z}}_{i 2}) \end{matrix}

(59)

where $β_{i 01}$ is the positive parameter of the disturbance observer error system (44).

According to equation (45), one has that

{\overset{\cdot}{\tilde{z}}}_{i 2} (t) = \sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + θ_{i 1} \frac{3 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) (- β_{i 02} e_{i 1} + ψ_{i} (t))

(60)

where $ψ_{i} (t) = \frac{{\overset{\cdot}{d}}_{i 1} (t)}{\sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + θ_{i 1} \frac{3 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2})}$ . From Assumption 3, one has that $| {\overset{\cdot}{d}}_{i 1} (t) | \leq {\bar{d}}_{i 1}$ , the following inequality is obtained as

| ψ_{i} (t) | \leq \frac{{\bar{d}}_{i 1} e_{i 1}}{M_{i 1} (t) (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + \frac{3 θ_{i 1} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) (⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{1}{2}} + θ_{i 1} ⌊ {{\tilde{z}}_{i 1} ⌉}^{\frac{3}{2}})}

(61)

Defining ${\tilde{e}}_{i} = [e_{i 1} (t), {\tilde{z}}_{i 2}]^{T}$ , the dynamics of ${\tilde{e}}_{i}$ is given as

{\overset{\cdot}{\tilde{e}}}_{i} = \sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + \frac{3 θ_{i 1} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) (A_{i} {\tilde{e}}_{i} + B ψ_{i} (t))

(62)

where

A_{i} = [\begin{matrix} - β_{i 01} & 1 \\ - β_{i 02} & 0 \end{matrix}], B = [\begin{matrix} 0 \\ 1 \end{matrix}]

(63)

Select the following candidate Lyapunov function

V_{i 1} = {\tilde{e}}_{i}^{T} P_{i} {\tilde{e}}_{i}

(64)

where $P_{i}$ is a positive definite matrix. By the error systems (59)–(62), the derivative of $V_{i 1}$ is obtained as

\begin{matrix} {\overset{\cdot}{V}}_{i 1} = \sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + \frac{3 θ_{i 1} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) \\ ({\tilde{e}}_{i}^{T} (A_{i}^{T} P_{i} + P_{i} A_{i}) {\tilde{e}}_{i} + 2 {\tilde{e}}_{i}^{T} P_{i} B ψ_{i} (t)) \\ \leq \sqrt{M_{i 1} (t)} (\frac{1 + 3 θ_{i 1} | {\tilde{z}}_{i 1} |}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}) \\ ({\tilde{e}}_{i}^{T} (A_{i}^{T} P_{i} + P_{i} A_{i} + P_{i} B B^{T} P_{i} + 4 C^{T} C) {\tilde{e}}_{i}) \\ \leq - ϵ_{i} \sqrt{M_{i 1} (t)} (\frac{1}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} + \frac{3 θ_{i 1} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}}{2}) V_{i 1} \\ \leq - \frac{ϵ_{i} \sqrt{m_{i 0}}}{2 {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}}} V_{i 1} - \frac{3 θ_{i 1} ϵ_{i} \sqrt{m_{i 0}}}{2} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}} V_{i 1} \\ \leq - \frac{ϵ_{i} \sqrt{m_{i 0} λ_{\min} (P_{i})}}{2} V_{i 1}^{\frac{1}{2}} - \frac{3 θ_{i 1} ϵ_{i} \sqrt{m_{i 0}}}{2} {| {\tilde{z}}_{i 1} |}^{\frac{1}{2}} V_{i 1} \end{matrix}

(65)

Based on Lemma 2, ${\tilde{z}}_{i 1} (t)$ can be convergent to zero in the fixed time $T_{io} \leq \frac{4}{ϵ_{i} \sqrt{m_{i 0} λ_{\min} (P_{i})}} + \frac{4}{3 θ_{i 1} ϵ_{i} \sqrt{m_{i 0}} λ_{\min} (P_{i})}$ . Considering the systems (37) and (38), if there exists ${\tilde{z}}_{i 1} (t) = 0$ holds in fixed time, ${\tilde{z}}_{i 1}^{.} (t)$ can also converge to zero in the fixed time. Therefore, ${\tilde{z}}_{i 2} (t)$ are convergent to zero in fixed time $T_{io}$ .

The proof is completed. □

Remark 2. The adaptive disturbance observer can realize the estimation of the disturbance in a fixed time with an unknown upper bound of the disturbance. Moreover, the adaptive parameters can be changed with the disturbance, which reduces the chatter of the sliding mode observer.

Remark 3. In this paper, we utilize the idea of equivalent control to construct a nonlinear low-pass filter to obtain the error function between the adaptive parameters and the equivalent control state. The adaptive algorithm is proposed to ensure that the error function converges to the origin in a fixed time, so as to guarantee that the adaptive parameters can change with the disturbance. Moreover, the parameter selection of the traditional fixed-time disturbance observer usually requires the known upper bound of the disturbance, which will lead to difficulty in selecting the parameters of the traditional disturbance observer when the upper bound of the disturbance is unknown. In addition, when the difference between the upper bound of the perturbation and the mean value of the disturbance is too large, it will also lead to more conservative parameter selection resulting in increased chatter of the traditional disturbance observer.

Remark 4. In this paper, ${\hat{x}}_{i 1}$ is the follower’s estimation of the leader’s state, and since there is no disturbance in the leader system considered in this paper, it is not possible to use ${\hat{x}}_{i 1}$ to realize the estimation of the disturbance $d_{i 1}$ . In addition, the disturbances $d_{i 1}$ and $d_{i 2}$ mainly affect the position and velocity of the follower; therefore, it is necessary to use the adaptive fixed-time disturbance observer to estimate the disturbance $d_{i 1}$ .

Fixed-time controller

In the previous section, according to Theorem 1, it is obvious that the estimated state errors ${\tilde{x}}_{i 1}$ and ${\tilde{x}}_{i 2}$ will convergent to zero for $t \geq 2 T_{\max 2}$ . Moreover, the estimation of disturbances ${\hat{d}}_{i 1} (t)$ and ${\hat{d}}_{i 2} (t)$ has been obtained in the disturbance observers (37) and -(38). In this section, the fixed-time compound control protocol will be constructed.

By defining $η_{i 1} (t) = x_{i 1} (t) - {\hat{x}}_{i 1} (t)$ , $η_{i 2} (t) = x_{i 2} (t) - {\hat{x}}_{i 2} (t)$ , and $η_{i 3} (t) = x_{i 3} (t) - {\hat{x}}_{i 3} (t)$ as the track error, after $t \geq 2 T_{\max 2}$ , the time derivative of $η_{ik} (t)$ with $k = 1, 2, 3$ is given as

{\overset{\cdot}{η}}_{i 1} (t) = u_{i 1} (t) - {\hat{u}}_{i 1} (t) + d_{i 1} (t)

(66)

{\overset{\cdot}{η}}_{i 2} (t) = u_{i 1} (t) x_{i 3} (t) - {\hat{u}}_{i 1} (t) {\hat{x}}_{i 3} (t)

(67)

{\overset{\cdot}{η}}_{i 3} (t) = u_{i 2} (t) - {\hat{u}}_{i 2} (t) + d_{i 2} (t)

(68)

The control objective is to design the control signals $u_{i 1} (t)$ and $u_{i 2} (t)$ such that the track error $η_{ik} (t)$ with $k = 1, 2, 3$ is convergent to zero in fixed time. The controllers $u_{i 1} (t)$ and $u_{i 2} (t)$ will be designed separately.

First, it is our aims to ensure that error $η_{i 1} (t)$ converges to the origin within a fixed time in the presence of disturbance $d_{i 1} (t)$ by designing controller $u_{i 1} (t)$ . Based on the fixed time stability theory, the controller $u_{i 1} (t)$ for system (66) is designed as

\begin{matrix} u_{i 1} (t) = {\hat{u}}_{i 1} (t) - z_{i 2} (t) - k_{i 1} {| η_{i 1} (t) |}^{p} sign (η_{i 1} (t)) \\ - k_{i 2} {| η_{i 1} (t) |}^{q} sign (η_{i 1} (t)) \end{matrix}

(69)

where ${\hat{u}}_{i 1} (t)$ and ${\hat{d}}_{i 1} (t)$ are the estimates of $u_{01} (t)$ and $d_{i 1} (t)$ , as well as the real-time compensation for $u_{01} (t)$ and $d_{i 1} (t)$ is implemented in the controller, respectively; $k_{i 1}$ , $k_{i 2}$ , $q > 1$ , and $p < 1$ are the four positive parameters. Term $k_{i 2} {| η_{i 1} (t) |}^{q} sign (η_{i 1} (t))$ is used to ensure that the system error $η_{i 1} (t)$ can converge to $1$ within a fixed time when the error is greater than $1$ . Then, when the absolute value of the error $η_{i 1} (t)$ is less than $1$ . Choose a Lyapunov function $W_{i 1} = \frac{1}{2} η_{i 1}^{2} (t)$ . Based on system (66) and the fixed-time controller (69), after $t \geq T_{io} + T_{\max 2}$ , taking the time derivative of $W_{i 1}$ is obtained as

\begin{matrix} {\overset{\cdot}{W}}_{i 1} = - k_{i 1} {| η_{i 1} (t) |}^{p + 1} - k_{i 2} {| η_{i 1} (t) |}^{q + 1} \\ \leq - 2^{\frac{p + 1}{2}} k_{i 1} W_{i 1}^{\frac{p + 1}{2}} - 2^{\frac{q + 1}{2}} k_{i 2} W_{i 1}^{\frac{q + 1}{2}} \end{matrix}

(70)

According to Lemma 2, $∥ η_{i 1} (t) ∥$ can converge to the origin within a fixed time. One has that the tracking error $η_{i 1} (t)$ is convergent to zero in fixed time $T_{i 1} \leq \frac{2}{2^{\frac{p + 1}{2}} k_{i 1} (p - 1)} + \frac{2}{2^{\frac{q + 1}{2}} k_{i 2} (1 - q)}$ . After the fixed time $T_{i 1}$ , the equations (67) and (68) are rewritten as

{\overset{\cdot}{η}}_{i 2} (t) = {\hat{u}}_{i 1} (t) η_{i 3} (t) - z_{i 2} (t) x_{i 3} (t)

(71)

{\overset{\cdot}{η}}_{i 3} (t) = u_{i 2} (t) - {\hat{u}}_{i 2} (t) + d_{i 2} (t)

(72)

Since systems (71) and (72) are a special class of second-order systems, it is not possible to design the controller $u_{i 2} (t)$ directly by Lemma 2 to ensure that the error converges to the origin in fixed time, so we design the controller by backstepping as well as by adding the power integral method. The nonlinear compound controller $u_{i 2} (t)$ is designed by the following step:

Step 1: Choose the Lyapunov function as $U_{i 1} = \frac{1}{2} η_{i 2}^{2}$ , whose time derivative along equation (71) yields

\begin{matrix} {\overset{\cdot}{U}}_{i 1} = η_{i 2} {\overset{\cdot}{η}}_{i 2} = {\hat{u}}_{i 1} (t) η_{i 2} η_{i 3} - z_{i 2} η_{i 2} x_{i 3} \\ = {\hat{u}}_{i 1} (t) η_{i 2} η_{i 3}^{*} + {\hat{u}}_{i 1} (t) η_{i 2} (η_{i 3} - η_{i 3}^{*}) - z_{i 2} η_{i 2} x_{i 3} \end{matrix}

(73)

where $η_{i 3}^{*}$ is a virtual controller and designed as

η_{i 3}^{*} = - sign ({\hat{u}}_{i 1} (t)) (η_{i 2}^{q_{2}} + ν_{i} η_{i 2}^{2 - q_{2}}) + ϒ_{i}

(74)

where $q_{2} = σ$ is a positive parameter with $σ < 1$ ; $ν_{i}$ is a positive parameter; and $ϒ_{i} = \frac{z_{i 2} η_{i 2} x_{i 3}}{{\hat{u}}_{i 1} (t)}$ . Then, by the virtual controller (74), equation (73) is further obtained as

{\overset{\cdot}{U}}_{i 1} = - | {\hat{u}}_{i 1} (t) | ({| η_{i 2} |}^{q_{2} + 1} - ν_{i} {| η_{i 2} |}^{3 - q_{2}}) + {\hat{u}}_{i 1} (t) η_{i 2} (η_{i 3} - η_{i 3}^{*})

(75)

Step 2: Select the following Lyapunov function $U_{i 2} = U_{i 1} + \int_{η_{i 3}^{*}}^{η_{i 3}} {(s^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{2 - q_{2}} d s$ , whose time derivative along equation (72) yields

\begin{matrix} {\overset{\cdot}{U}}_{i 2} = - | {\hat{u}}_{i 1} (t) | ({| η_{i 2} |}^{q_{2} + 1} - ν_{i} {| η_{i 2} |}^{3 - q_{2}}) + {\hat{u}}_{i 1} (t) η_{i 2} (η_{i 3} - η_{i 3}^{*}) \\ + {(η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{2 - q_{2}} (u_{i 2} (t) - {\hat{u}}_{i 2} (t) + d_{i 2} (t)) \\ - (2 - q_{2}) \frac{d {η_{i 3}^{*}}^{1 / q_{2}}}{dt} \int_{η_{i 3}^{*}}^{η_{i 3}} {(s^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{1 - q_{2}} \end{matrix}

(76)

For term ${\hat{u}}_{i 1} (t) η_{i 2} (η_{i 3} - η_{i 3}^{*})$ , by Lemma 3 and Young’s inequality, it is obtained as

\begin{matrix} {\hat{u}}_{i 1} (t) η_{i 2} (η_{i 3} - η_{i 3}^{*}) \leq | {\hat{u}}_{i 1} (t) | | η_{i 2} | | ({(η_{i 3}^{1 / q_{2}})}^{q_{2}} - {({η_{i 3}^{*}}^{1 / q_{2}})}^{q_{2}}) | \\ \leq 2^{1 - q_{2}} | {\hat{u}}_{i 1} (t) | | η_{i 2} | {| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{q_{2}} \\ \leq \frac{q_{2}}{q_{2} + 1} 2^{1 - q_{2}} | {\hat{u}}_{i 1} (t) | {| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{q_{2} + 1} \\ + \frac{1}{q_{2} + 1} 2^{1 - q_{2}} | {\hat{u}}_{i 1} (t) | {| η_{i 2} |}^{q_{2} + 1} \end{matrix}

(77)

For term $\frac{d {η_{i 3}^{*}}^{1 / q_{2}}}{dt}$ , it is obtained as

\begin{matrix} | \frac{d {η_{i 3}^{*}}^{1 / q_{2}}}{dt} | \leq | {\hat{u}}_{i 1} (t) η_{i 3} (t) | (1 + ν_{i} (2 - q_{2}) / q_{2} η_{i 2}^{(2 - 2 q_{2}) / q_{2}}) \\ + {| η_{i 3}^{*} |}^{(1 - q_{2}) / q_{2}} | {\overset{\cdot}{ϒ}}_{i} | \leq | {\hat{u}}_{i 1} (t) | ({| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{q_{2}} + | η_{i 3}^{*} |) \\ (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) + {| η_{i 3}^{*} |}^{(1 - q_{2}) / q_{2}} | {\overset{\cdot}{ϒ}}_{i} | \end{matrix}

(78)

Based on equation (78), one has that

\begin{matrix} | (2 - q_{2}) \frac{d {η_{i 3}^{*}}^{1 / q_{2}}}{dt} \int_{η_{i 3}^{*}}^{η_{i 3}} {(s^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{1 - q_{2}} | \\ \leq (2 - q_{2}) | {\hat{u}}_{i 1} (t) | ({| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{q_{2}} + | η_{i 3}^{*} |) \\ \times (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) | η_{i 3} - η_{i 3}^{*} | \\ | (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |^{1 - q_{2}} + (2 - q_{2}) {| η_{i 3}^{*} |}^{(1 - q_{2}) / q_{2}} | {\overset{\cdot}{ϒ}}_{i} | \\ | η_{i 3} - η_{i 3}^{*} | {| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{1 - q_{2}} \\ \leq 2^{1 - q_{2}} (2 - q_{2}) | {\hat{u}}_{i 1} (t) | ({| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{q_{2}} + | η_{i 3}^{*} |) \\ \times (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) | (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) | \\ + 2^{1 - q_{2}} (2 - q_{2}) {| η_{i 3}^{*} |}^{(1 - q_{2}) / q_{2}} | {\overset{\cdot}{ϒ}}_{i} | | (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) | \\ \leq 2^{1 - q_{2}} (2 - q_{2}) (1 + \frac{1}{2}) | {\hat{u}}_{i 1} (t) | {| η_{i 3}^{*} |}^{1 - q_{2}} \\ {| (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) |}^{1 + q_{2}} (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) \\ + 2^{- q_{2}} (2 - q_{2}) | {\hat{u}}_{i 1} (t) | (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) \\ \times | (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) | {| η_{i 3}^{*} |}^{2} \\ + 2^{1 - q_{2}} (2 - q_{2}) {| η_{i 3}^{*} |}^{(1 - q_{2}) / q_{2}} | {\overset{\cdot}{ϒ}}_{i} | | (η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}}) | \end{matrix}

(79)

Then, the nonlinear compound controller $u_{i 2} (t)$ for the $i$ th follower-wheeled robot is constructed as

\begin{matrix} u_{i 2} (t) = - {(η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{q_{3}} (Γ_{i 2} + k_{i 3} + k_{i 4} \\ {| η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}} |}^{2 - 2 q_{2}}) + Γ_{i 1} + {\hat{u}}_{i 2} (t) - z_{i 4} (t) \end{matrix}

(80)

where $q_{3} = 2 σ - 1$ is a positive parameter; and $k_{i 3}$ and $k_{i 4}$ are the two adjustable parameters. $Γ_{i 1}$ and $Γ_{i 2}$ are respectively defined as

Γ_{i 1} = {\begin{array}{l} \frac{2^{- q_{2}} (2 - q_{2}) | û_{i 1} (t) | (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) {| η_{i 3}^{*} |}^{2}}{{(η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{1 - q_{2}}} \\ + \frac{1}{{(η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{1 - q_{2}}} 2^{- q_{2}} (2 - q_{2}) {| η_{i 3}^{*} |}^{(1 - q_{2}) / q_{2}} | {\dot{ϒ}}_{i} |, \\ | η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}} | > 0 \\ 0, | η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}} | = 0 \end{array}

(81)

and

\begin{matrix} Γ_{i 2} = 2^{1 - q_{2}} \frac{3}{2} (2 - q_{2}) | {\hat{u}}_{i 1} (t) | {| η_{i 3}^{*} |}^{1 - q_{2}} \\ (1 + ν_{i} (2 - q_{2}) / q_{2} {| η_{i 2} |}^{(2 - 2 q_{2}) / q_{2}}) \end{matrix}

(82)

Based on the nonlinear compound controller (80), ${\overset{\cdot}{U}}_{i 2}$ can be rewritten as

\begin{matrix} {\overset{\cdot}{U}}_{i 2} \leq - (1 - \frac{1}{q_{2} + 1} 2^{1 - q_{2}}) | {\hat{u}}_{i 1} (t) | ({| η_{i 2} |}^{q_{2} + 1} + ν_{i} {| η_{i 2} |}^{3 - q_{2}}) \\ - (k_{i 3} - \frac{q_{2}}{q_{2} + 1} 2^{1 - q_{2}}) {| η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}} |}^{q_{2} + 1} \\ - k_{i 4} {| η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}} |}^{3 - q_{2}} \end{matrix}

(83)

Theorem 3. Under Assumptions 1–3, considering the consensus tracking error systems (71) and (72) with controller (80). If there exist positive parameters $k_{i 3}$ , $k_{i 4}$ , $q_{2}$ , $q_{3}$ , and $μ_{i}$ such that

q_{3} = 2 q_{2} - 1, k_{i 3} - \frac{q_{2}}{q_{2} + 1} 2^{1 - q_{2}} > 0, k_{i 4} > 0, μ_{i} > 0

(84)

where $0 < q_{2} < 1$ is a fraction number whose numerator and denominator are odd numbers, and then it is obtained that $η_{i 2} (t)$ and $η_{i 3} (t)$ can converge to zero in fixed time.

Proof. According to Lemma 3, one has that

\begin{matrix} \int_{η_{i 3}^{*}}^{η_{i 3}} {(s^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{2 - q_{2}} \leq 2^{1 - q_{2}} {(η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{2} \\ \leq 2 {(η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}})}^{2} \end{matrix}

(85)

Therefore, considering Lyapunov function $U_{i 2}$ , it is easy to obtain that

U_{i 2} \leq 2 (η_{i 2}^{2} + {| η_{i 3}^{1 / q_{2}} - {η_{i 3}^{*}}^{1 / q_{2}} |}^{2})

(86)

Combining equations (83) and (86), by choosing $q_{2}$ , $k_{i 3}$ , $k_{i 4}$ , and $μ_{i}$ , ${\overset{\cdot}{U}}_{i 2}$ is further obtained as

{\overset{\cdot}{U}}_{i 2} \leq - 2^{- \frac{1 + q_{2}}{2}} τ_{i 1} | U_{i 2}^{\frac{1 + q_{2}}{2}} - 2^{- \frac{3 - q_{2}}{2}} τ_{i 2} | U_{i 2}^{\frac{3 - q_{2}}{2}} \leq 0

(87)

where $τ_{i 1}$ and $τ_{i 2}$ are the two positive parameters and defined as

\begin{matrix} τ_{i 1} = \min {(k_{i 3} - \frac{q_{2}}{q_{2} + 1} 2^{1 - q_{2}}), (1 - \frac{1}{q_{2} + 1} 2^{1 - q_{2}}) | {\hat{u}}_{i 1} (t) |} \\ τ_{i 2} = \min {k_{i 4}, (1 - \frac{1}{q_{2} + 1} 2^{1 - q_{2}}) ν_{i} | {\hat{u}}_{i 1} (t) |} \end{matrix}

By resorting to Lemma 2, then the equilibrium of $U_{i 2}$ is fixed-time stable. Thus, the leader–following consensus errors $η_{i 2}$ and $η_{i 3}$ are convergent to zero in the fixed time. □

Remark 5. In this paper, the fixed-time distributed control scheme is proposed such that each follower-wheeled robot can track the leader with the convergence time that does not rely on the initial state. The disturbance with unknown upper bound is estimated in fixed time by the adaptive disturbance observers (37) and (38). Moreover, the states and inputs of the leader are estimated by the distributed observers (16)–(19). There are three distinctive features that complicate the design process, that is, nonholonomic system, disturbance with unknown upper bound, and fixed-time convergence requirement. The sufficient conditions in Theorems 1–3 are presented reflecting these three pieces of information.

Numerical examples

The simulation results are given to show the effectiveness of the theoretical results provided in this paper. We consider the case where the wheeled mobile robot system consists of four wheeled mobile robots (one leader as well as three followers). The disturbance is given as $f_{i} (t) = 0.9 \sin (t)$ . The Laplacian matrix of topology is described by

L = [\begin{matrix} 0 & - 1 & 0 \\ - 1 & 1 & - 1 \\ 0 & - 1 & 1 \end{matrix}], B = diag {1, 0, 0}

(88)

In Theorem 1, the sufficient conditions are given as

ρ_{1} - ū_{01} > 0, ρ_{1} - ū_{02} > 0, α_{1} > 0, α_{2} > 0, ρ_{2} > 0

(89)

where $α_{1}$ , $α_{2}$ , $ρ_{1}$ , and $ρ_{2}$ are the positive parameters. In simulation, the model of the leader wheeled mobile robot is given as

{\overset{\cdot}{x}}_{01} (t) = u_{01} (t)

(90)

{\overset{\cdot}{x}}_{02} (t) = u_{01} (t) x_{03} (t)

(91)

{\overset{\cdot}{x}}_{03} (t) = u_{02} (t)

(92)

with $u_{01} (t) = \sin (0.5 t) + 2.5$ and $u_{02} (t) = \sin (0.5 t) + 2.5$ . Based on inequality (89), the positive parameters $α_{1}$ , $α_{2}$ , $ρ_{1}$ , and $ρ_{2}$ are chosen as $α_{1} = α_{2} = 2 > 0$ and $ρ_{1} = ρ_{2} = 1.5$ , respectively.

In Theorem 2, the sufficient conditions are shown as

P_{i} A_{i} + A_{i}^{T} P_{i} + ϵ_{i} P_{i} + P_{i} B B^{T} P_{i} + 4 C^{T} C < 0

(93)

Moreover, $m_{i 0}$ , $ϵ_{i} = 0.1$ , and $c_{i 0}$ are the three small positive parameters. For the margin of observer, $\frac{1}{r_{i 1} β_{i 02}} > 1$ should be ensured. $τ_{i 1}$ is a suitable small parameter. $ξ_{0 i}$ is related to the rate of change of the adaptive parameter $c_{i 1} (t)$ . The larger is parameter $ξ_{0 i}$ , the faster is the rate of change $c_{i 1} (t)$ . In simulation, the relevant parameter is selected as

\begin{matrix} β_{i 01} = 2, β_{i 02} = 2, τ_{i 1} = 0.05, r_{i 1} β_{i 02} = 0.95 \\ ϵ_{i} = 0.1, m_{i 0} = c_{i 0} = 0.1, ξ_{0 i} = 8, θ_{i 1} = 1 \end{matrix}

(94)

Substituting equation (94) into equation (93), it can be established that the inequality (93) holds. Then, it is verified that the chosen parameters satisfy the inequality (93).

In Theorem 3, the sufficient conditions are given as

q_{3} = 2 q_{2} - 1, k_{i 3} - \frac{q_{2}}{q_{2} + 1} 2^{1 - q_{2}} > 0, k_{i 4} > 0

(95)

where $0 < q_{2} < 1$ is a fraction number whose numerator and denominator are odd numbers. First, $q_{2} = \frac{5}{7}$ is fixed; therefore, $q_{3} = 2 q_{2} - 1 = \frac{3}{7}$ is constituted. Then, based on $k_{i 3} - \frac{q_{2}}{q_{2} + 1} 2^{1 - q_{2}} = k_{i 3} - \frac{5}{12} 2^{\frac{2}{7}} > 0$ , one has that $k_{i 3} = 2$ and $k_{i 4}$ $= 2 > 0$ .

The simulation results for the adaptive disturbance observers (37) and (38) taking agent 1 as an example are plotted in Figures 1 and 2. The adaptive observer errors are shown in Figure 1. It is seen that errors converge to zero within a short time. Moreover, in Figure 2, we can obtain that the adaptive parameters $β_{11} (t)$ and $β_{12} (t)$ can be adjusted based on the absolute value of the disturbance $f_{i} (t) = 0.9 \sin (t)$ . In order to show the effectiveness of the adaptive fixed-time disturbance observer proposed in this paper, the traditional fixed-time super-twisting observer (Liu et al., 2021) is used to compare with the adaptive observer in this paper. The convergence curve on the estimation errors of the traditional observer is shown in Figure 1, and it can be seen that the chatter of the error curve of the observer proposed in this paper is smaller compared to the traditional observer (Liu et al., 2021) from Figures 1 and 3.

Figure 1.

The adaptive observer error.

Figure 2.

The adaptive parameter.

Figure 3.

The traditional observer error (Liu et al., 2021).

Since the agent 1 can receive the leader information directly, taking agents 2 and 3 as an example, the estimation error on the distributed observers (16) and (18) is depicted in Figures 4 and 5, which shows that the observer error could converge to the neighborhood of zero in finite time. In addition, the comparison results with the finite-time state observer (Du et al., 2015) are given. Figures 6 and 7 present the estimation errors ${\tilde{x}}_{2}$ and ${\tilde{x}}_{3}$ , respectively. It can be seen that with the same parameters, the estimation errors of fixed-time state observer proposed in this paper require shorter convergence time and have less chatter after convergence. The consensus errors of the follower-wheeled robot with different initial conditions are shown in Figures 8 –16, respectively. It is obvious from Figures 8 –10 and 14 –16 that the consensus errors are capable of converging to a neighborhood of the origin within the settling times of about $T = 3$ seconds even with different initial conditions. In addition, the comparison results with the finite-time backstepping control strategy under the identical initial cases are given with the same parameters for fairness. Figures 11 –13 represent the leader–following consensus error plots for each of the three followers. It can be obvious from Figures 11 –13 that the convergence times of the leader–following consensus errors of the three followers are different due to the different initial values. The convergence time for the consensus error of follower 3 is the longest, reaching 4.5 seconds. From Figures 8 –13, it can be concluded that the control strategy proposed in this paper can ensure that the convergence time required for the leader–following consistency error is shorter, which proves the superiority of the control strategy proposed in this paper.

Figure 4.

The distributed observer error ${\tilde{x}}_{2}$ .

Figure 5.

The distributed observer error ${\tilde{x}}_{3}$ .

Figure 6.

The distributed observer error ${\tilde{x}}_{2}$ (Du et al., 2015).

Figure 7.

The distributed observer error ${\tilde{x}}_{3}$ (Du et al., 2015).

Figure 8.

The consensus error of agent 1 with initial condition $(0.1; 0.1; 0.1)$ .

Figure 9.

The consensus error of agent 2 with initial condition $(1; 1; 0.2)$ .

Figure 10.

The consensus error of agent 3 with initial condition $(- 1; 3; - 0.1)$ .

Figure 11.

The consensus error of agent 1 with initial condition $(0.1; 0.1; 0.1)$ (Du et al., 2015).

Figure 12.

The consensus error of agent 2 with initial condition $(1; 1; 0.2)$ (Du et al., 2015).

Figure 13.

The consensus error of agent 3 with initial condition $(- 1; 3; - 0.1)$ (Du et al., 2015).

Figure 14.

The consensus error of agent 1 with initial condition $(1; 0.5; 3)$ .

Figure 15.

The consensus error of agent 2 with initial condition $(2; 1; 0.3)$ .

Figure 16.

The consensus error of agent 3 with initial condition $(- 4; 0; - 0.2)$ .

Conclusion

In this paper, the fixed-time consensus tracking problem has been investigated for a multi-agent nonholonomic system with disturbance. A fixed-time distributed observer has been developed for each follower to estimate the leader state and input. Moreover, a second-order adaptive disturbance observer has been proposed to estimate the unknown upper-bound disturbance. By the proposed observers, a nonlinear compound controller has been used to achieve tracking the leader state in fixed time. Finally, numerical examples have been provided to illustrate effectiveness of the proposed methodology.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Juntao Jia

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article (and/or) its supplementary materials.

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