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Abstract

This paper focuses on the cooperative control of multiple robots with unknown control directions. The commonly used Nussbaum-gain approach may result in large overshoot and oscillations. Therefore, we propose a new switching adaptive control method to solve the leaderless and leader-following consensus problem for multiple robots. The proposed switching strategy can make the controller quickly adapt to the unknown control directions, resulting in a better transient control performance. Moreover, the presented controller can steer multiple robots to a common value in finite time regardless of the heterogeneous unknown control directions and fully unknown nonlinearities in the robot dynamic models. This implies that the controller can achieve fast convergence rate and have strong robustness. In order to make the controller distributed, new local supervisory functions are constructed to design the logic-based switching rule. These supervisory functions are local, meaning that they are only dependent on local state information. No neigborhood/global information are needed. Simulations and experiments verify the obtained results.

Keywords

multiple robots logic-based switching rule distributed control unknown control direction

Introduction

Lately, cooperative control of multiple robots have attracted an increasing attention.^1–4 They have broad applications in various kinds of fields, such as library, manufacturing, mining, etc. For examples, in library, multiple robots can be programmed to locate and retrieve specific books from shelves, saving time for librarians and patrons. They can also help with shelf maintenance, ensuring that books are arranged correctly and in their designated locations. Cooperative control can be roughly classified into leaderless^5,6 and leader-following^7–9 consensus control. Leaderless consensus control means that multiple robots will reach a common value cooperatively, while leader-following control indicates that the followers will track a reference signal generated by the leader robot. Lots of excellent works and advanced control strategies, such as model predictive control,^10,11 adaptive control,¹² iterative learning control,¹³ have been presented for the leaderless/leader-following consensus of multiple robots.

Finite-time cooperative control means that multiple robots can reach consensus in finite time.^14–16 Compared with traditional asymptotic cooperative control, finite-time control indicates a fast convergence speed and strong robust to external uncertainties/nonlinearities.¹⁷ Moreover, finite-time convergence also means that the cooperative control can be decoupled from other tasks such as state estimation, fault detection etc. Lots of good works have been taken on finite-time cooperative control. The control methods can be divided into three classes: sliding mode control, adding a power integrator technique, and time-varying feedback.

For sliding mode control, the states of the multiple robots will first reach a sliding mode surface in finite time, and then the states will reach the equilibrium points.^18,19 For examples, Yao et al.¹⁹ proposed a fully distributed controller for disturbed multi-agent systems. Event-based communication was also considered. However, the sliding mode control may cause chattering problem in practical.

For adding a power integrator technique, the idea is to construct a special Lyapunov function and utilize the concept of homogeneous polynomial to construct the controller.^20,21 Li et al.²⁰ investigated the finite-time output consensus for mult-agent systems with mismatched disturbance. Compared with sliding mode control, the adding a power integrator technique does not introduce sign function and can avoid the chattering problem. Yet, it may have weaker disturbance rejection ability than the sliding mode control.

Time-varying feedback is recent new method to achieve finite time control. The controller often contains a well-constructed time-varying function.^22,23 Based on the properties of this time-varying function, the finite-time control purpose of multiple robots can be achieved. Song et al.²³ presented an extensive literature review for the finite time control. The advantages of the time-varying feedback is that it can prescribe the convergence time and transient performance. However, if it needs to regulate the tracking error toward exact zero, the time-varying gain may become infinite, which makes the control system unstable.

Unknown control directions is a common phenomenon in multiple robot systems.^24,25 It may be caused by the failure of the actuators or input nonlinearities. Since the control direction is unknown, the standard negative feedback control methods fail. A common method is the utilization of Nussbaum gain.^26,27 The Nussbaum gain contains sine/cosine functions that can adapt to the unknown control directions. For examples, Liang et al.²⁸ proposed a fractional-order Nussbaum gain method to deal with heterogeneous unknown control directions in complex mechatronic systems. However, a common drawback of the the Nussbaum gain is that it often causes larger control effort and oscillations. Meanwhile, the selections of the Nussbaum gain is also a difficult task in practical engineering.

Therefore, the logic-based switching control method has been proposed^29–31 to deal with unknown control direction. Based on a supervisory function, the controller can automatically switch the sign of the control gain to adapt to the unknown control directions. The logic-based switching rule has also been used to achieve finite time control for a class of nonlinear systems.³¹ It is also shown that by the switching adaptive control method, the control performance can be improved compared with Nussbaum gain method. However, the above logic-based switching control methods are only for a single system/agent. To the best of our knowledge, there is no works about the logic-based switching finite-time control for multiple robots with unknown control directions. The main difficulties lie on that it is not easy to construct the supervisory functions for multiple robots because the supervisory functions need to be distributed and cannot rely on global information.

According to the above literature review, this paper will take a study on the logic-based switching finite-time cooperative control of multiple robots with unknown control directions. The main contributions are:

First, new switching adaptive control methods are presented to solve the finite-time leaderless and leader-following consensus problem for multiple robots. Compared with the existing methods,^14,17,18 the proposed controller can steer multiple robots to a common value in finite time regardless of the heterogeneous unknown control directions and fully unknown nonlinearities in the robot dynamic models;

Second, new local supervisory functions are constructed for multiple robots. Based on these supervisory functions, logic-based switching rules can be constructed to handle the heterogeneous unknown control directions. Moreover, these supervisory functions are local, meaning that they are only dependent on local state information. No neigborhood/global information are needed. This feature will bring convenience for the implementation of the proposed controller;

Third, different from the traditional Nussbaum gain methods,^26,27 the proposed control method adopts the logic-based switching rule to adaptively tune the control directions. This switching strategy will make the controller quickly adapt to the unknown control directions, resulting in a better transient control performance.

The organization of the paper is as follows. Section II presents the finite-time cooperative control problem to be solved. Sections III and IV investigate the leaderless and leader-following finite-time consensus problem separately. Section V gives two application examples. These examples demonstrate the situation where multiple robots running in a library environment to help people arrange books. Section VI concludes the paper.

Problem formulation and preliminaries

Problem formulation

Consider a group of robots given by:

{\overset{\cdot}{p}}_{i} = v_{i}

(1)

M_{i} {\overset{\cdot}{v}}_{i} + H_{i} (p_{i}, v_{i}) v_{i} + Q_{i} (p_{i}) = τ_{i},

(2)

where $i = 1, 2, \dots, N$ , $p_{i} = [p_{i 1}, p_{i 2}, \dots, p_{in}]^{T}, v_{i} = [v_{i 1}, v_{i 2}, \dots, v_{in}]^{T} \in R^{n}$ denotes the generalized position vector and velocity vectors. $M_{i} \in R^{n \times n}, H_{i} (p_{i}, v_{i}) \in R^{n \times n}, Q_{i} (p_{i}) \in R^{n}$ are uncertain model parameters, $τ_{i} \in R^{n}$ is the generalized torque vector which is expressed as:

τ_{i} = g_{i} (t) u_{i},

where $g_{i} (t)$ is the unknown control direction satisfying $0 < {\underline{g}}_{i} \leq | g_{i} (t) | \leq {\bar{g}}_{i}$ with positive constants ${\underline{g}}_{i}, {\bar{g}}_{i}$ , and $u_{i}$ is the control effort to be designed. It is noted that the sign of $g_{i} (t)$ is unknown. The unknown control direction may be caused by the actuator failure or input nonlinearities.

The output of the leader is denoted by a time-varying function $p_{r} (t) = [p_{r 1} (t), p_{r 2} (t), \dots, p_{rn} (t)]^{T} \in R^{n}$ , and $p_{r} (t), {\overset{\cdot}{p}}_{r} (t), {\overset{\cdot\cdot}{p}}_{r} (t)$ are all bounded and known. We also use an undirected graph $G$ to represent the communication network of the $N$ robots. See References 5 and 7 for detail information about graph theory.

The aim of this paper is to solve the following control problem: Finite-time cooperative control problem.

1) (Leaderless finite-time consensus problem). Design a distributed controller $u_{i}$ for each agent such that: For $\forall i, j \in {1, 2, \dots, N}$ , $‖ p_{i} (t) - p_{j} (t) ‖ \to 0$ as $t \to T^{-}$ and $‖ p_{i} (t) - p_{j} (t) ‖ \equiv 0$ when $t \geq T$ where $T > 0$ is a finite constant;

2) (Leader-following practical finite-time consensus problem). Design a distributed controller $u_{i}$ for each agent such that: For $\forall i \in {1, 2, \dots, N}$ , $‖ p_{i} (t) - p_{r} (t) ‖ \leq ε$ as $t \to T^{-}$ and $‖ p_{i} (t) - p_{r} (t) ‖ < ε$ when $t \geq T$ where $T > 0$ is a finite constant, and $ε$ is a pre-specified small constant.

Preliminaries

The following are important lemmas that will be used in the controller design and stability analysis.

Lemma 1. ^31,32 (Young’s Inequality)

{| α |}^{a} {| β |}^{b} \leq \frac{a}{a + b} ϕ (α, β) {| α |}^{a + b} + \frac{b}{a + b} ϕ^{- a / b} (α, β) {| β |}^{a + b},

(3)

where $α, β \in R$ , $a, b > 0$ , $ϕ (α, β) > 0$ is any function.

Lemma 2. ^31,32

{| α - β |}^{p} \leq 2^{p - 1} | α^{p} - β^{p} |,

(4)

| α^{1 / p} - β^{1 / p} | \leq 2^{1 - 1 / p} {| α - β |}^{1 / p},

(5)

{(\sum_{i = 1}^{n} | ν_{i} |)}^{1 / p} \leq \sum_{i = 1}^{n} {| ν_{i} |}^{1 / p} \leq n^{1 - 1 / p} {(\sum_{i = 1}^{n} | ν_{i} |)}^{1 / p},

(6)

where $p \geq 1$ , $α, β, ν_{i} > 0$ are constants.

Lemma 3. ^31,32 Given a continuously differentiable nonlinear function $F (x) : R \to R$ where $x \in R$ and $F (0) \equiv 0$ , there exists a non-negative smooth function $ψ (x) : R \to R$ such that $| F (x) | \leq | x | ψ (x)$ .

Lemma 4.³¹(Comparison Principle) Let $α (t), β (t), b (t) : [0, + \infty) \to R$ be three continuous functions such that

\overset{\cdot}{α} (t) = - ρ α^{q} (t) + b (t),

(7)

\overset{\cdot}{β} (t) \leq - ρ β^{q} (t) + b (t),

(8)

for $\forall t \in [t_{0}, t_{1}) \subseteq [0, + \infty)$ where $α (t_{0}) = β (t_{0}) + ε$ . $ε, ρ > 0$ and $0 < q \leq 1$ are constants. Then, $α (t) \geq β (t)$ for $\forall t \in [t_{0}, t_{1})$ .

Leaderless finite-time consensus

This section will focus on the study of leaderless fixed-time consensus problem. Similar to¹⁵, we make the following assumption. $‖ H_{i} (p_{i}, v_{i}) v_{i} + Q_{i} (p_{i}) ‖ \leq F (‖ v_{i} ‖) (i = 1, 2, \dots, N)$ where $F (‖ v_{i} ‖)$ is an unknown continuous function with $F (0) = 0$ . $M_{i}$ can be written as $M_{i} = diag {m_{i 1}, m_{i 2}, \dots, m_{in}}$ with unknown constants $m_{ij} (i = 1, 2, \dots, N; j = 1, 2, \dots, n)$ . Next, we present our main results.

Main results

The controller is inspired by the backstepping technique. Consider the following change of coordinates

z_{ij} \overset{Δ}{=} \sum_{k = 1}^{N} a_{ik} (p_{ij} - p_{kj}),

(9)

ξ_{ij} \overset{Δ}{=} v_{ij}^{1 / q} - α_{ij}^{1 / q},

(10)

where $i = 1, 2, \dots, N$ , $j = 1, 2, \dots, n$ , $a_{ik}$ are the weights of the communication graph $G$ (see References 5 and 7), and $α_{ij}$ is the virtual control effort. Then, the controller is given by:

Controller structure

The virtual control effort $α_{ij} (i = 1, 2, \dots, N; j = 1, 2, \dots, n)$ is designed as:

α_{ij} = - K_{i 1} z_{ij}^{q},

(11)

where $q \in (1 / 2, 1)$ is ratio of odd integers, and $K_{i 1}$ is the control gain. The control effort $u_{ij} (i = 1, 2, . . . N; j = 1, 2, \dots, n)$ is designed as:

\begin{matrix} u_{ij} = - {\hat{γ}}_{i} [K_{i 2} ξ_{ij}^{q_{1}} + \frac{R_{i} ξ_{ij}^{q_{1}}}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{1 + 2 q}}], \end{matrix}

(12)

where $K_{i 2}, R_{i}, Z_{i}$ are positive design parameters, and $q_{1} = 2 q - 1$ .

Logic-based switching rule

As shown in Figure 1, ${\hat{γ}}_{i}$ is an adaptive parameter which is tuned by the following switching logic rule:

{\hat{γ}}_{i} (t) = {\begin{matrix} {\hat{γ}}_{i} (t_{is}), & if W_{i} (t) \leq χ_{i} (t) \\ - sgn ({\hat{γ}}_{i} (t_{s})) (| {\hat{γ}}_{i} (t_{s}) | + κ_{i}), & if W_{i} (t) > χ_{i} (t) \end{matrix}

(13)

where $κ_{i} > 0$ is a design parameter, $t_{is} (s = 1, 2, . . .)$ is the switching time instants for agent $i$ such that $t_{i, s + 1} = \inf {t > t_{is} | W_{i} (t) - χ_{i} (t) > 0}$ .

Figure 1.

Switching mechanism.

The Lyapunov function $W_{i}$ is given by:

W_{i} = \sum_{j = 1}^{n} \int_{α_{ij}}^{v_{ij}} \frac{ζ_{ij}^{2 - q} (τ)}{Z_{i}^{2} - ζ_{ij}^{2} (τ)} d τ

(14)

with $ζ_{ij} (τ) = τ^{1 / q} - α_{ij}^{1 / q}$ , and $Z_{i}$ is a positive parameter.

$χ_{i} (t)$ is determined by:

\begin{matrix} {\overset{\cdot}{χ}}_{i} = - ρ_{1} χ_{i}^{\frac{1 + q}{2}} - ρ_{2} \sum_{j = 1}^{n} ξ_{ij}^{1 + q} \\ + b_{1} \sum_{j = 1}^{n} z_{ij}^{1 + q} + b_{2} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} {| v_{kj} |}^{(1 + q) / q}, \end{matrix}

(15)

where $χ_{i} (t_{is}) = W_{i} (t_{is}) + ε$ , and $ε, ρ_{1}, ρ_{2}, b_{1}, b_{2}$ are positive design parameters.

Based on the above controller, the main result is as follows:

Theorem 5. Consider the multi-agent systems (1)-(2) under Assumption 0 and the initial conditions satisfying $| ξ_{ij} (0) | < Z_{i} (i = 1, 2, \dots, N; j = 0, 1, \dots, n)$ . Then, the controller law (11)-(13) can solve Problem 1-1).

The proof will be presented in the next subsection.

Proof of the main results

The proof is seperated into two parts.

Part I: Prove all signals in the multi-agent systems are bounded.

We will first show that for $\forall t \in [0, + \infty)$ , we have

| ξ_{ij} (t) | < Z_{i} (i = 1, 2, \dots, N, j = 1, 2, \dots, n) .

(16)

By contradiction, we can find a time instant $t_{0}$ , $i^{*} \in {1, 2, \dots, N}$ , and $j^{*} \in {1, 2, \dots, n}$ such that $| ξ_{i^{*} j^{*}} (t_{0}) | \geq Z_{i^{*}}$ . Notice that $ξ_{ij} (t)$ are all continuous and $| ξ_{ij} (0) | < Z_{i}$ . Thus, it can be concluded that on $[0, t_{0})$ , we have $| ξ_{ij} (t) | < Z_{i}$ for $i = 1, 2, \dots, N, j = 1, 2, \dots, n$ . Also when $t \to t_{0}^{-}$ , $ξ_{i^{*} j^{*}} \to Z_{i^{*}}$ . The following proof will be conducted on the interval $[0, t_{0})$ . It is divided into three steps.

Step 1). We will first prove $v_{ij} (i = 1, \dots, N; j = 1, 2, \dots, n)$ are all bounded on $[0, t_{0})$ .

Consider following Lyapunov function

V_{1} = x^{T} (I_{n} \otimes L) x = \sum_{j = 1}^{n} \sum_{i = 1}^{N} \sum_{k = 1}^{N} a_{ik} {(p_{ij} - p_{kj})}^{2} \geq 0 .

where

x_{j} \overset{Δ}{=} [p_{1 j}, p_{2 j}, \dots, p_{Nj}]^{T} (j = 1, 2, \dots, n),

(17)

x \overset{Δ}{=} [x_{1}^{T}, x_{2}^{T}, \dots, x_{n}^{T}]^{T},

(18)

and $L$ is the Laplacian matrix of the communication graph $G$ .

The derivative of $V_{1}$ is computed as:

\begin{matrix} {\overset{\cdot}{V}}_{1} = x^{T} (I_{n} \otimes L) \overset{\cdot}{x} = \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij} {\overset{\cdot}{p}}_{ij} = \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij} v_{ij} \\ = \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij} (v_{ij} - α_{ij}) + \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij} α_{ij} . \end{matrix}

(19)

Substituting (11) into (19) and using Lemmas 1-2, we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij} ξ_{ij}^{q} - K_{1} \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij}^{1 + q} \\ \leq - \frac{K_{1}}{2} \sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij}^{1 + q} + c_{1} \sum_{j = 1}^{n} \sum_{i = 1}^{N} ξ_{ij}^{1 + q}, \end{matrix}

(20)

where $c_{1}$ is a positive constant.

By the property of Laplacian matrix, we have $\sum_{j = 1}^{n} \sum_{i = 1}^{N} z_{ij}^{1 + q} \geq λ_{2} x^{T} L x$ where $λ_{2}$ is the second smallest eigenvalue of $L$ . Hence, it can be concluded that

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - c_{2} V_{1}^{\frac{1 + q}{2}} + c_{1} \sum_{j = 1}^{n} \sum_{i = 1}^{N} ξ_{ij}^{1 + q} \leq - c_{2} V_{1}^{\frac{1 + q}{2}} + {\bar{c}}_{1} \end{matrix}

where $c_{2}, {\bar{c}}_{1}$ are positive constants such that ${\bar{c}}_{1} \geq c_{1} \sum_{i = 1}^{N} {nZ}_{i}^{1 + q} \geq c_{1} \sum_{j = 1}^{n} \sum_{i = 1}^{N} ξ_{ij}^{1 + q}$ . This implies that $V_{1}$ is bounded. From (11), we can conclude that $α_{ij}$ are all bounded. Hence, $v_{ij}^{1 / q} = ξ_{ij} + α_{ij}^{1 / q}$ are bounded on $[0, t_{0})$ .

Step 2). We will present some properties of the Lyapunov function $W_{i}$ described by (14).

We know that $W_{i}$ has the following property.³¹

Proposition 6. Assume $| ξ_{ij} | < Z_{i}$ , then $W_{i}$ is positive definite such that $W_{i} \leq \frac{ξ_{ij}^{2}}{Z_{i}^{2} - ξ_{ij}^{2}}$ . Moreover, if $α_{i}$ is bounded, then when $| ξ_{ij} | \to Z_{i}$ , $W_{i} \to + \infty$ .

Then, differentiating $W_{i}$ with respect to time, we have

\begin{matrix} {\overset{\cdot}{W}}_{i} = \sum_{j = 1}^{n} \frac{\partial W_{i}}{\partial v_{ij}} {\overset{\cdot}{v}}_{ij} + \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} \frac{\partial W_{i}}{\partial p_{kj}} v_{kj} . \end{matrix}

(21)

For the term $\sum_{j = 1}^{n} \frac{\partial W_{i}}{\partial v_{ij}} {\overset{\cdot}{v}}_{ij}$ , by Assumption 0 and Lemma 3, we have

\begin{matrix} \sum_{j = 1}^{n} \frac{\partial W_{i}}{\partial v_{ij}} {\overset{\cdot}{v}}_{ij} = \sum_{j = 1}^{n} \frac{ξ_{ij}^{2 - q}}{Z_{i}^{2} - ξ_{ij}^{2}} {\overset{\cdot}{v}}_{ij} = {\bar{ξ}}_{i}^{T} {\overset{\cdot}{v}}_{i} \\ = {\bar{ξ}}_{i}^{T} [M_{i}^{- 1} (g_{i} (t) u_{i} - H_{i} (p_{i}, v_{i}) v_{i} - Q_{i} (p_{i}))] \\ \leq \sum_{j = 1}^{n} \frac{ξ_{ij}^{2 - q} m_{ij}^{- 1} g_{i} (t) u_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} + ‖ {\bar{ξ}}_{i} ‖ F (‖ v_{i} ‖) \\ \leq \sum_{j = 1}^{n} \frac{ξ_{ij}^{2 - q} {\bar{g}}_{i} (t) u_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} + ‖ {\bar{ξ}}_{i} ‖ \cdot ‖ v_{i} ‖ ψ_{i} (‖ v_{i} ‖), \end{matrix}

where ${\bar{ξ}}_{i} = {[\frac{ξ_{i 1}^{2 - q}}{Z_{i}^{2} - ξ_{i 1}^{2}}, \frac{ξ_{i 2}^{2 - q}}{Z_{i}^{2} - ξ_{i 2}^{2}}, \dots, \frac{ξ_{in}^{2 - q}}{Z_{i}^{2} - ξ_{in}^{2}}]}^{T}$ , and ${\bar{g}}_{i} (t) = m_{ij}^{- 1} g_{i} (t)$ .

By the result in Step 1), we know that $ψ_{i} (‖ v_{i} ‖)$ must be bounded. Therefore, we have

\begin{matrix} \sum_{j = 1}^{n} \frac{\partial W_{i}}{\partial v_{ij}} {\overset{\cdot}{v}}_{ij} \leq \sum_{j = 1}^{n} \frac{ξ_{ij}^{2 - q} m_{ij}^{- 1} {\bar{g}}_{i} (t) u_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} \\ + \sum_{j = 1}^{n} \frac{ξ_{ij}^{1 + q} Λ_{i 1}}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2 + 2 q}} + \sum_{j = 1}^{n} c_{3} z_{ij}^{1 + q}, \end{matrix}

(22)

where $Λ_{i 1}$ is an unknown positive constant, and $c_{3} > 0$ is a positive parameter.

On the other side, we have

\begin{matrix} \frac{\partial W_{i}}{\partial α_{ij}} = \frac{\partial α_{ij}^{1 / q}}{\partial α_{ij}} \int_{α_{ij}}^{v_{ij}} \frac{\partial (\frac{ζ_{ij}^{2 - q} (τ)}{Z_{i}^{2} - ζ_{ij}^{2} (τ)})}{\partial α_{ij}^{1 / q}} d τ \\ \leq | \frac{\partial α_{ij}^{1 / q}}{\partial α_{ij}} | | \int_{α_{ij}}^{v_{ij}} \frac{ζ_{ij}^{1 - q} (τ) [(2 - q) Z_{i}^{2} + q ζ_{ij}^{2} (τ)]}{{(Z_{i}^{2} - ζ_{ij}^{2} (τ))}^{2}} d τ | \\ \leq | \frac{\partial α_{ij}^{1 / q}}{\partial α_{ij}} | \frac{| ξ_{ij} |}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2}} C (ξ_{ij}), \end{matrix}

where $C (ξ_{ij})$ is an unknown smooth function.

It follows that

\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} \frac{\partial W_{i}}{\partial p_{kj}} v_{kj} \\ \leq \sum_{j = 1}^{n} | \frac{\partial α_{ij}^{1 / q}}{\partial α_{ij}} | \frac{| ξ_{ij} | C (ξ_{ij})}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2}} \sum_{k = 1}^{N} a_{ik} | \frac{\partial α_{ij}}{\partial p_{kj}} | | v_{kj} | \\ \leq \sum_{j = 1}^{n} \frac{| ξ_{ij} | C (ξ_{ij}) c_{4}}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2}} \sum_{k = 1}^{N} a_{ik} | v_{kj} | \\ \leq \sum_{j = 1}^{n} \frac{Λ_{i 2} ξ_{ij}^{1 + q}}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2 + 2 q}} + c_{5} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} {| v_{kj} |}^{(1 + q) / q}, \end{matrix}

(23)

where $Λ_{i 2}$ is an unknown positive constant, and $c_{4}, c_{5} > 0$ are positive parameters.

Substituting (12) into (22) and using (23), we obtain

\begin{matrix} {\overset{\cdot}{W}}_{i} \leq - \sum_{j = 1}^{n} \frac{K_{i 2}^{'} ξ_{ij}^{1 + q}}{Z_{i}^{2} - ξ_{ij}^{2}} + \sum_{j = 1}^{n} \frac{(K_{i 2}^{'} - {\bar{g}}_{i} (t) K_{i 2} \hat{γ_{i}}) ξ_{ij}^{1 + q}}{Z_{i}^{2} - ξ_{ij}^{2}} \\ + \sum_{j = 1}^{n} \frac{R_{i 2} ξ_{ij}^{1 + q} (Λ_{i} - {\bar{g}}_{i} (t) \hat{γ_{i}})}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2 + 2 q}} + \sum_{j = 1}^{n} c_{3} z_{ij}^{1 + q} \\ + c_{5} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} {| v_{kj} |}^{(1 + q) / q}, \end{matrix}

(24)

where $K_{i 2}^{'} > 0$ is a positive constant, and $Λ_{i} \overset{Δ}{=} Λ_{i}^{1} + Λ_{i}^{2}$ .

Step 3). We will show all the signals are bounded.

We first show ${\hat{γ}}_{i} (t)$ is bounded, $i . e$ ., the switching time is finite. In fact, if this is not true, since $| {\hat{γ}}_{i} (t) | \to + \infty$ when $s \to + \infty$ , we can find a time instant $t_{is}$ such that $K_{i 2}^{'} - \underline{g} K_{i 2} | \hat{γ_{i}} (t_{is}) | < 0$ and $Λ_{i} - \underline{g} \hat{γ_{i}} | (\hat{γ_{i}} (t_{is})) | < 0$ . Hence, we have

\begin{matrix} {\overset{\cdot}{W}}_{i} \leq - ρ_{1} W_{i}^{\frac{1 + q}{2}} - ρ_{2} \sum_{j = 1}^{n} ξ_{ij}^{1 + q} \\ + \sum_{j = 1}^{n} b_{1} z_{ij}^{1 + q} + b_{2} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} {| v_{kj} |}^{(1 + q) / q}, \end{matrix}

(25)

where $ρ_{1}, ρ_{2}$ are positive constants, and $b_{1} = c_{3}$ , $b_{2} = c_{5}$ .

Comparing (25) with (15), and noting that at time instant $t_{is}$ , we have $W_{i} (t_{is}) < χ_{i} (t_{is}) = W_{i} (t_{is}) + ε$ , this indicates that $W_{i} (t) \leq χ_{i} (t)$ always holds based on Lemma 4. Hence, the switching time is finite and ${\hat{γ}}_{i} (t)$ is bounded.

According to the above analysis, we known there is a time interval $[t_{s^{*} i}, t_{0})$ with finite integer $s^{*}$ such that $χ_{i} (t) \leq W_{i} (t)$ is satisfied. Note that $z_{ij}$ and $v_{ij}$ are bounded. Then on $[t_{s^{*} i}, t_{0})$ ,

\begin{matrix} {\overset{\cdot}{χ}}_{i} \leq - ρ_{1} χ_{i}^{\frac{1 + q}{2}} + \sum_{j = 1}^{n} b_{1} z_{ij}^{1 + q} + b_{2} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} {| v_{kj} |}^{(1 + q) / q} \\ \leq - ρ_{1} χ_{i}^{\frac{1 + q}{2}} + {\bar{b}}_{1}, \end{matrix}

where ${\bar{b}}_{1} \geq \sum_{j = 1}^{n} b_{1} z_{ij}^{1 + q} + b_{2} \sum_{j = 1}^{n} \sum_{k = 1}^{N} a_{ik} {| v_{kj} |}^{(1 + q) / q}$ . This shows $χ_{i} (t), W_{i} (t)$ are both bounded. According to Proposition 6, we can conclude that there is a $ι_{ij} > 0$ such that $| ξ_{ij} (t) | \leq Z_{i} - ι_{ij}$ for $\forall i$ . This contradicts the fact that when $t \to t_{0}^{-}$ , $ξ_{i^{*} j^{*}} \to Z_{i^{*}}$ . Hence, (16) holds.

Finally, by repeating the above procedures and using (16), we can show all the signals are bounded. It completes the first part of the proof.

Part II: The multiple robots system is finite-time stable.

Based on the above analysis, we know that ${\hat{γ}}_{i} (t)$ is bounded for $\forall i \in {1, 2, \dots, N}$ . Thus, there exists an uniform time interval $[t^{'}, + \infty)$ such that for all $i$ , we have $χ_{i} \leq W_{i}$ with $χ_{i}$ satisfy (15).

From (15), we have

\begin{matrix} {\overset{\cdot}{χ}}_{i} \leq - ρ_{1} χ_{i}^{\frac{1 + q}{2}} - ρ_{2} \sum_{j = 1}^{n} ξ_{ij}^{1 + q} + \sum_{j = 1}^{n} b_{1} z_{ij}^{1 + q} \\ + b_{3} \sum_{j = 1}^{n} \sum_{k = 1}^{N} (ξ_{kj}^{1 + q} + z_{kj}^{1 + q}), \end{matrix}

where $b_{3}$ is a positive constant.

Using (20), it follows that

\begin{matrix} {\overset{\cdot}{V}}_{1} + \sum_{i = 1}^{N} {\overset{\cdot}{χ}}_{i} \\ \leq - \sum_{i = 1}^{N} (\frac{K_{i 1}}{2} - b_{1} - N b_{3}) \sum_{j = 1}^{n} z_{ij}^{1 + q} \\ - ρ_{1} \sum_{i = 1}^{N} χ_{i}^{\frac{1 + q}{2}} - \sum_{i = 1}^{N} (ρ_{2} - c_{1} - N b_{3}) \sum_{j = 1}^{n} ξ_{ij}^{1 + q} . \end{matrix}

Therefore, when $K_{i 1}, ρ_{2}$ are sufficient large, we have

{\overset{\cdot}{V}}_{1} + \sum_{i = 1}^{N} {\overset{\cdot}{χ}}_{i} \leq - ρ_{3} {(V_{1} + \sum_{i = 1}^{N} χ_{i})}^{\frac{1 + q}{2}},

where $ρ_{3}$ is a positive constant. This means that $W_{i}, χ_{i}$ converges to zero in finite time. Because $0 \leq W_{i} \leq χ_{i} (t)$ on $[t^{'}, + \infty)$ , $W_{i} (t)$ will also become zero in finite time.

Leader-following finite-time consensus

This section will investigate the leader-folllowing finite-time consensus problem.

Main results

Consider the following change of coordinates

z_{ij} \overset{Δ}{=} \sum_{k = 1}^{N} a_{ik} (p_{ij} - p_{kj}) + a_{i 0} (p_{ij} - p_{rj}),

(26)

ξ_{ij} \overset{Δ}{=} v_{ij} - α_{ij},

(27)

where $i = 1, 2, \dots, N; j = 1, 2, \dots, n$ , $a_{ik}$ are the weights of the communication graph $G$ , $a_{i 0}$ denotes the weight between the leader and the followers, and $α_{ij}$ is the virtual control effort. Then, the controller is given by:

Controller structure

The virtual control effort $α_{ij}$ and $u_{ij} (i = 1, 2, \dots, N; j = 1, 2, \dots, n)$ are designed as:

α_{ij} = - K_{i 1} s_{ij},

(28)

\begin{matrix} u_{ij} = - {\hat{γ}}_{i} [K_{i 2} ξ_{ij} + \frac{R_{i} ξ_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}}], \end{matrix}

(29)

where $K_{i 1}, K_{i 2}$ are positive design parameters, and

s_{ij} = \ln (\frac{1 + δ (t) z_{ij}}{1 - δ (t) z_{ij}}) .

(30)

As shown in Figure, $δ (t)$ is a class of prescribed finite-time function given by the following definition:

Definition 7. $δ (t) \overset{Δ}{=} 1 / \bar{δ} (t)$ is smooth with respect to $t$ satisfying the following two conditions:

1) $0 < \underline{σ} \leq \bar{δ} (t) \leq \bar{σ}$ and $0 < \underline{ϱ} \leq \overset{\cdot}{\bar{δ}} (t) \leq \bar{ϱ}$ such that $\underline{σ}, \bar{σ}, \underline{ϱ}, \bar{ϱ}$ are positive constants;

2) $\bar{δ} (t) \to μ$ as $t \to T$ and $\bar{δ} (t) \equiv μ$ as $t \in [T, + \infty)$ where $T, μ$ are positive design parameters.

Logic-based switching rule

${\hat{γ}}_{i}$ is an adaptive parameter which is tuned by the following switching logic rule:

{\hat{γ}}_{i} (t) = {\begin{matrix} {\hat{γ}}_{i} (t_{is}), & if W_{i} (t) \leq χ_{i} (t) \\ - sgn ({\hat{γ}}_{i} (t_{s})) (| {\hat{γ}}_{i} (t_{s}) | + κ_{i}), & if W_{i} (t) > χ_{i} (t) \end{matrix}

(31)

where $κ_{i} > 0$ is a design parameter, $t_{is} (s = 1, 2, . . .)$ is the switching time instants for agent $i$ such that $t_{i, s + 1} = \inf {t > t_{is} | W_{i} (t) - χ_{i} (t) > 0}$ .

The Lyapunov function $W_{i}$ is given by:

W_{i} = \sum_{j = 1}^{n} \ln (\frac{Z_{i}^{2}}{Z_{i}^{2} - ξ_{ij}^{2}}),

(32)

where $Z_{i}$ is a positive parameter.

$χ_{i} (t)$ is determined by:

\begin{matrix} {\overset{\cdot}{χ}}_{i} = - ρ_{1} χ_{i} + b_{1}, \end{matrix}

(33)

where $χ_{i} (t_{is}) = W_{i} (t_{is}) + ε$ , and $ε, ρ_{1}, b_{1}$ are positive design parameters.

Based on the above controller, the main result is as follows:

Theorem 8. Consider the multi-agent systems (1)-(2) under Assumption and the initial conditions satisfying $| ξ_{ij} (0) | < Z_{i} (i = 1, 2, \dots, N; j = 0, 1, \dots, n)$ . Then, the controller law (28)-(29) can solve Problem 1-2).

The proof will be presented in the next subsection.

Proof of Theorem 5

We will show that for $\forall t \in [0, + \infty)$ , we have

| δ (t) z_{ij} (t) | < 1,

(34)

| ξ_{ij} (t) | < Z_{i},

(35)

for $i = 1, 2, \dots, N; j = 1, 2, \dots, n$ . By contradiction, we can find a time instant $t_{0}$ , $i^{*} \in {1, 2, \dots, N}$ , and $j^{*} \in {1, 2, \dots, n}$ such that $| δ (t_{0}) z_{i^{*} j^{*}} (t_{0}) | \geq 1$ or $| ξ_{i^{*} j^{*}} (t_{0}) | \geq Z_{i^{*}}$ . Notice that $δ (t)$ , $z_{ij} (t)$ , $ξ_{ij} (t)$ are all continuous and $| δ (0) z_{ij} (0) | < 1$ , $| ξ_{ij} (0) | < Z_{i}$ for $\forall i \in {1, 2, \dots, N}$ and $\forall j \in {1, 2, \dots, n}$ . Thus, it can be concluded that on $[0, t_{0})$ , we have $| δ (t) z_{ij} (t) | < 1$ and $| ξ_{ij} (t) | < Z_{i}$ for $\forall i = 1, 2, \dots, N$ and $\forall j \in {1, 2, \dots, n}$ . Also when $t \to t_{0}^{-}$ , $δ (t) z_{i^{*} j^{*}} (t) \to 1$ or $ξ_{i^{*} j^{*}} (t) \to Z_{i^{*}}$ . The following proof will be conducted on the interval $[0, t_{0})$ . It is divided into three steps.

Step 1). We will first prove there exists a positive constant $ι_{ij}$ such that $| δ (t) z_{ij} (t) | \leq 1 - ι_{ij} (i = 1, \dots, N; j = 1, 2, \dots, n)$ on $[0, t_{0})$ .

From (30), we have

{\overset{\cdot}{s}}_{ij} = h_{ij} (δ, z_{ij}) (δ {\overset{\cdot}{z}}_{ij} + \overset{\cdot}{δ} z_{ij}),

where $h_{i} (δ, z_{i 1}) = \frac{2}{(1 + δ z_{i 1}) (1 - δ z_{i 1})}$ . It follows that

{\overset{\cdot}{s}}_{i} = H_{i} (δ, z_{i}) (δ {\overset{\cdot}{z}}_{i} + \overset{\cdot}{δ} z_{i}),

\overset{\cdot}{s} = H (δ, z) (δ \overset{\cdot}{z} + \overset{\cdot}{δ} z),

(36)

where $H (δ, z) = diag (H_{1}, H_{2}, \dots, H_{n})$ with $H_{i} (δ, z_{i}) = diag (h_{i 1}, h_{i 2}, \dots, h_{in})$ , $z = [z_{1}^{T}, z_{2}^{T}, \dots, z_{n}^{T}]^{T}$ with $z_{i} = [z_{i 1}, z_{i 2}, \dots, z_{in}]^{T}$ , and $s = [s_{1}^{T}, s_{2}^{T}, \dots, s_{N}^{T}]^{T}$ with $s_{i} = [s_{i 1}, s_{i 2}, \dots, s_{in}]^{T}$ . In addition, we have the following relation:

z = [(L + B) \otimes I_{n}] (p - {\bar{p}}_{r}),

(37)

where $p = [p_{1}^{T}, p_{2}^{T}, \dots, p_{N}^{T}]^{T}$ , ${\bar{p}}_{r} = [p_{r}^{T}, p_{r}^{T}, \dots, p_{r}^{T}]^{T}$ , $L$ is the Laplacian matrix of the graph $G$ , and $B = diag (a_{10}, a_{20}, \dots, a_{N 0})$ .

Consider following Lyapunov function

V_{1} = \frac{1}{2} s^{T} (P \otimes I_{n}) s,

where $P \in R^{N \times N}$ is diagonal positive definite matrix such that $P (L + B) {\bar{K}}_{1} + {\bar{K}}_{1} (L + B) P < 0$ where ${\bar{K}}_{1} = diag (K_{11}, K_{21}, \dots, K_{N 1})$ . Note that according to Wang et al.,² $P$ exists.

Using (36)-(37), the derivative of $V_{1}$ is computed as:

\begin{matrix} {\overset{\cdot}{V}}_{1} = s^{T} (P \otimes I_{n}) H (δ, z) (δ \overset{\cdot}{z} + \overset{\cdot}{δ} z) \\ = s^{T} (P \otimes I_{n}) H (δ, z) (δ ((L + B) \otimes I_{n}) (v - {\overset{\cdot}{\bar{p}}}_{r}) + \overset{\cdot}{δ} z), \end{matrix}

where $v = [v_{1}^{T}, v_{2}^{T}, \dots, v_{N}^{T}]^{T}$ .

Using (27), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = - δ {(H (δ, z) s)}^{T} (P \otimes I_{n}) ((L + B) \otimes I_{n}) α \\ + δ {(H (δ, z) s)}^{T} (P \otimes I_{n}) ((L + B) \otimes I_{n}) (ξ - {\overset{\cdot}{\bar{p}}}_{r}) \\ + {(H (δ, z) s)}^{T} (P \otimes I_{n}) \overset{\cdot}{δ} z, \end{matrix}

(38)

where $α = [α_{1}^{T}, α_{2}^{T}, \dots, α_{N}^{T}]^{T}$ with $α_{i} = [α_{i 1}, α_{i 2}, \dots, α_{in}]^{T}$ .

Substituting (28) into (38), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = - δ {(H (δ, z) s)}^{T} [Q \otimes I_{n}] (H (δ, z) s) \\ + δ {(H (δ, z) s)}^{T} [P (L + B) \otimes I_{n}] (ξ - {\overset{\cdot}{\bar{p}}}_{r}) \\ + {(H (δ, z) s)}^{T} (P \otimes I_{n}) \overset{\cdot}{δ} z, \end{matrix}

where $Q = (P (L + B) {\bar{K}}_{1} + {\bar{K}}_{1} (L + B) P) / 2$ . Note that $ξ, {\overset{\cdot}{\bar{p}}}_{r}, z$ are bounded on $[0, t_{0})$ . Hence, by Young’s inequality,

{\overset{\cdot}{V}}_{1} \leq - c_{1} {‖ H (δ, z) s ‖}^{2} + c_{2}, \leq - c_{3} V_{1} + c_{2},

where $c_{1}, c_{2}, c_{3}$ are positive constants. This implies that $V_{1}$ is bounded. Hence, there exists a positive constant $ι_{ij}$ such that $| δ (t) z_{ij} (t) | \leq 1 - ι_{ij} (i = 1, \dots, N; j = 1, 2, \dots, n)$ on $[0, t_{0})$ . In addition, from (28), we can conclude that $α_{ij}$ are all bounded. Hence, $v_{ij} = ξ_{ij} + α_{ij}$ are bounded.

Step 2). We will present some properties of the Lyapunov function $W_{i}$ described by (32).

Consider the Lyapuonv function described by (32). We have:

{\overset{\cdot}{W}}_{i} = \sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{ξ}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} = \sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{v}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} - \sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{α}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} .

For $\sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{v}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}}$ , it can be written as:

\begin{matrix} \sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{v}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} = {\bar{ξ}}_{i}^{T} {\overset{\cdot}{v}}_{i} \\ = {\bar{ξ}}_{i}^{T} [M_{i}^{- 1} (g_{i} (t) u_{i} - H_{i} (p_{i}, v_{i}) v_{i} - Q_{i} (p_{i}))] \\ = \sum_{j = 1}^{n} \frac{ξ_{ij} {\bar{g}}_{i} (t) u_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} - {\bar{ξ}}_{i}^{T} M_{i}^{- 1} [H_{i} (p_{i}, v_{i}) v_{i} + Q_{i} (p_{i})] \end{matrix}

where ${\bar{ξ}}_{i} = [\frac{ξ_{i 1}}{Z_{i}^{2} - ξ_{i 1}^{2}}, \frac{ξ_{i 2}}{Z_{i}^{2} - ξ_{i 2}^{2}}, \dots, \frac{ξ_{i 2}}{Z_{i}^{2} - ξ_{in}^{2}}]^{T}$ , and ${\bar{g}}_{i} (t) = m_{ij}^{- 1} g_{i} (t)$ .

Noting that $p_{i}, v_{i}$ are bounded on $[0, t_{0})$ , we obtain

\begin{matrix} \sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{v}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} \\ \leq \sum_{j = 1}^{n} \frac{ξ_{ij} {\bar{g}}_{i} (t) u_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} + \sum_{j = 1}^{n} \frac{R_{i 2} ξ_{ij}^{2} Λ_{i 1}}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2}} + c_{4}, \end{matrix}

(39)

where $Λ_{i 1}$ is an unknown positive constant, and $c_{4}$ is a positive constant.

For $\sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{α}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}}$ , it can be written as:

\begin{matrix} \sum_{j = 1}^{n} \frac{ξ_{ij} {\overset{\cdot}{α}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} \\ = \sum_{j = 1}^{n} \frac{ξ_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} (\frac{\partial α_{ij}}{\partial z_{ij}} \frac{\partial z_{ij}}{\partial t} + \frac{\partial α_{ij}}{\partial δ} \frac{\partial δ}{\partial t}) \\ \leq \sum_{j = 1}^{n} \frac{R_{i 2} ξ_{ij}^{2} Λ_{i 2}}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2}} + c_{5}, \end{matrix}

(40)

where $c_{5}$ is a positive constant.

Substituting (29) into (39) and using (40), we get

\begin{matrix} {\overset{\cdot}{W}}_{i} \leq - \sum_{j = 1}^{n} \frac{K_{i 2}^{'} ξ_{ij}^{2}}{Z_{i}^{2} - ξ_{ij}^{2}} + \sum_{j = 1}^{n} \frac{(K_{i 2}^{'} - {\bar{g}}_{i} (t) K_{i 2} \hat{γ_{i}}) ξ_{ij}^{2}}{Z_{i}^{2} - ξ_{ij}^{2}} \\ + \sum_{j = 1}^{n} \frac{R_{i 2} ξ_{ij}^{2} (Λ_{i} - {\bar{g}}_{i} (t) \hat{γ_{i}})}{{(Z_{i}^{2} - ξ_{ij}^{2})}^{2}} + c_{4} + c_{5}, \end{matrix}

(41)

where $Λ_{i} = Λ_{i 1} + Λ_{i 2}$ is an unknown positive constant.

Step 3). We will show all the signals are bounded.

We first show ${\hat{γ}}_{i} (t)$ is bounded, $i . e$ ., the switching time is finite. In fact, if this is not true, since $| {\hat{γ}}_{i} (t) | \to + \infty$ when $s \to + \infty$ , there must exist a time instant $t_{is}$ such that $K_{i 2}^{'} - \underline{g} K_{i 2} | \hat{γ_{i}} (t_{is}) | < 0$ and $Λ_{i} - \underline{g} \hat{γ_{i}} | (\hat{γ_{i}} (t_{is})) | < 0$ . Hence, (41) becomes

\begin{matrix} {\overset{\cdot}{W}}_{i} \leq - ρ_{1} W_{i} + b_{1} \end{matrix}

(42)

where $a_{1}, b_{1}$ are positive constants with $b_{1} = c_{4} + c_{5}$ .

Comparing (33) with (42), and noting that at time instant $t_{is}$ , we have $W_{i} (t_{is}) < χ_{i} (t_{is}) = W_{i} (t_{is}) + ε$ , this indicates that $W_{i} (t) \leq χ_{i} (t)$ always holds based on Lemma 4. Hence, the switching time is finite and ${\hat{γ}}_{i} (t)$ is bounded.

According to the above analysis, we can conclude that there exists a time interval $[t_{s^{*} i}, t_{0})$ with finite integer $s^{*}$ such that $χ_{i} (t) \leq W_{i} (t)$ is satisfied. Then, on $[t_{s^{*} i}, t_{0})$ ,

\begin{matrix} {\overset{\cdot}{χ}}_{i} \leq - ρ_{1} χ_{i} + b_{1} . \end{matrix}

This shows $χ_{i} (t), W_{i} (t)$ are both bounded. Hence, we can conclude that there exist a positive constant $ι_{ij}$ such that $| ξ_{ij} (t) | \leq Z_{i} - ι_{ij}$ for $\forall i \in {1, 2, \dots, N}$ . This contradicts the fact that when $t \to t_{0}^{-}$ , $ξ_{i^{*} j^{*}} \to Z_{i^{*}}$ . Hence, (35) holds.

Finally, by repeating the above procedures and using (35), we can show all the signals in the multi-agent systems are bounded. Meanwhile, $z_{ij} (t)$ satisfy (34), indicating that $z_{ij} (t)$ is constrained in the tube $(- 1 / δ (t), 1 / δ (t))$ . Hence, the consensus error will converge to the neighborhood around the origin in prescribed finite time $T$ . It completes the proof.

Simulations

In this section, we will present two examples to demonstrate the effectiveness of the proposed method.

Example 1

Suppose four mobile robots running in a library. They need to retrieve books from four shelves to a common location. The communication graph of the four robots is depicted in Figure 2 without considering the leader 0. We use the proposed method in Section 3 to design the controller for these robots. The robot models are given by:

{\overset{\cdot}{p}}_{i} = v_{i},

(43)

M_{i} {\overset{\cdot}{v}}_{i} = τ_{i},

(44)

where $i = 1, 2, 3, 4$ , $p_{i}, v_{i} \in R^{2}$ , $M_{i} = diag (1, 1)$ , and $τ_{i} = g_{i} (t) u_{i} . g_{i} (t) = 1$ is an unknown control direction, $u_{i}$ is given by (11)-(12). The logic-based switching rule is given by (13). The parameters are set as: $K_{i 1} = 1, K_{i 2} = 4, R_{i} = 1, q = 0.95, Z_{i} = 4$ .

Figure 2.

Communication graph.

By the proposed method, the state trajectories of the four robots are shown in Figure 3. It can be seen that all the positions and velocities of the robots have reached a common value in finite time. Meanwhile, from Figure 4, we can see that four robots have moved from four different initial points to a common location. Figure 5 shows the variations of the adaptive gain ${\hat{γ}}_{i}$ and $W_{i}, χ_{i}$ , we can see that the adaptive gain ${\hat{γ}}_{i}$ switches to an appropriate value to guarantee the stability of of the multi-robot systems. Moreover, Figure 5 shows that $W_{i}$ is always smaller than $χ_{i}$ and finally converges to zero. The above results demonstrate the validity of the proposed controller.

Figure 3.

Control performance for four robots.

Figure 4.

State trajectories of four robots. These robots retrieve books from four shelves to a common book return location.

Figure 5.

Control efforts and variations of adaptive parameters.

To have a comparison, we consider the Nussbaum gain method. According to Liang et al.,⁸ the controller can be designed as:

α_{ij} = - K_{i 1} z_{ij},

(45)

{\bar{u}}_{ij} = K_{i 2} {\bar{ξ}}_{ij} + \frac{R_{i} {\bar{ξ}}_{ij}}{Z_{i}^{2} - {\bar{ξ}}_{ij}^{2}},

(46)

u_{ij} = N_{ij} (η_{ij}) {\bar{u}}_{ij},

(47)

where

\begin{matrix} {\bar{ξ}}_{ij} = v_{ij} - α_{ij}, \\ N_{ij} (η_{ij}) = η_{ij}^{2} \sin (η_{ij}), \\ {\overset{\cdot}{η}}_{ij} = \frac{R_{i} {\bar{ξ}}_{ij} {\bar{u}}_{ij}}{Z_{i}^{2} - {\bar{ξ}}_{ij}^{2}} . \end{matrix}

Meanwhile, we consider the following five cases:

Case I: $g_{1} = 1, g_{2} = 1, g_{3} = 1, g_{4} = 1$ ;

Case II: $g_{1} = 1, g_{2} = - 1, g_{3} = 1, g_{4} = 1$ ;

Case III: $g_{1} = 1, g_{2} = 1, g_{3} = - 1, g_{4} = - 1$ ;

Case IV: $g_{1} = - 1, g_{2} = - 1, g_{3} = - 1, g_{4} = 1$ ;

Case V: $g_{1} = - 1, g_{2} = - 1, g_{3} = - 1, g_{4} = - 1$ .

Figures 6 and 7 demonstrate the performance of the robots for the above five cases. We can see that the proposed logic-based controller is robust to the variations of $g_{i} (t)$ . In all five cases, the robots have a similar performance and can reach a steady state, while the Nussbaum gain method results in large oscillations. These results show the advantages of the proposed methods over existing Nussbaum gain methods.

Remark 9. We will give a detail analysis with Nussbaum-gain approach. The proposed method may obtain a smaller control overshoot than Nussbaum method. This is because the Nussbaum method utilizes a sine/cosine gain. The large overshoot phenomenon of Nussbaum method has been recognized in many references, the logic-based switching controller is more sensitive to the mismatch of the unknown control directions than Nussbaum method. Therefore, it may detect the right control directions more quickly than Nussbaum method, which will result in a smaller control overshoot.

Next, we take the following simple Nussbaum controller as an example to further explain this.

\begin{matrix} u = N (ξ) x, \overset{\cdot}{ξ} = V (x) \end{matrix}

(48)

where $N (ξ) = ξ^{2} \cos (ξ)$ or $N (ξ) = e^{ξ^{2}} \cos (ξ π / 2)$ . $V (x) = x^{2} \geq 0$ is a Lyapunov function. The Nussbaum gain $N (ξ)$ will switch its sign when $ξ$ increases to some threshold values, $i . e$ ., zeros of the cosine function in $N (ξ)$ .

Figure 6.

Control performance of four mobile robots for Cases I–V under the proposed method.

Figure 7.

Control performance of four mobile robots for Cases I-V under the Nussbaum gain method.

For the logic-based switching controller, the control direction is detected according to the supervisory function, which is selected as $S = V - η$ where $\overset{\cdot}{η} = - a η^{γ}$ and $η (0) = V (0) + ε$ with a small constant $ε$ . The control gain will be switched when $S > 0$ . At the initial phase, if the states of the system are driven away by a wrong control direction, $i . e . that is, V$ is increasing, then the supervisory function $S$ will quickly become larger than zero and the control gain will be switched. Thus, the right control direction will be found. If the control direction is right, then $V$ will decrease and we can also identify a right control direction.

However, for Nussbaum gain, since $ξ$ in (48) is an integration of the Lyapunov $V$ , it will always increase whether the states are driven away or converge to zero. This implies that a longer time may be needed to find the right control direction.

In addition, the polynomial or exponential increasing terms $ξ^{2}, e^{ξ^{2}}$ in $N (ξ)$ may lead to large control effort.

Example 2

In this example, we consider the control of four robot manipulators. These robot manipulators need to move the books from the shelves to the mobile robots, then the mobile robots will transport these books to the return point. The communication graph of the robots is also shown in Figure 2 with a virtual leader. The method presented in Section 4 is utilized to design the controller for these robots. The models are given by:

\begin{matrix} {\overset{\cdot}{p}}_{i} = v_{i} \\ M_{i} {\overset{\cdot}{v}}_{i} + H_{i} (p_{i}, v_{i}) v_{i} = τ_{i}, \end{matrix}

where $i = 1, 2, 3, 4$ , $p_{i}, v_{i} \in R^{2}$ , $M_{i} = diag (5, 5)$ ,

H_{i} (p_{i}, v_{i}) = [\begin{matrix} \cos p_{i 1} + 3 & 2 v_{i 2} \\ 2 v_{i 2} & \sin p_{i 2} + 3 \end{matrix}]

and $τ_{i} = g_{i} (t) u_{i} . g_{i} (t) = 1$ is an unknown control direction, $u_{i}$ is given by (28)-(29). The logic-based switching rule is given by (31). The leader signal is given by: $p_{r} (t) = [p_{r 1} (t) p_{r 2} (t)]^{T}$ with $p_{r 1} (t) = {\begin{matrix} \frac{π}{3} t, & 0 < t \leq 1, \\ \frac{π}{3}, & t \geq 1 . \end{matrix}$ and $p_{r 2} (t) = {\begin{matrix} \frac{π}{6} t, & 0 < t \leq 1, \\ \frac{π}{6}, & t \geq 1 . \end{matrix}$ The parameters are set as: $K_{i 1} = 4, K_{i 2} = 15, R_{i} = 1, Z_{i} = 5$ . $δ (t)$ is set as $δ (t) = 2 ρ (t) + 0.1$ where $ρ (t) = {\begin{matrix} {(\frac{T - t}{T})}^{3}, & 0 < t < T, \\ 0, & t \geq T, \end{matrix}$ with $T = 4$ .

Similar to Example 1, by the proposed method, the control performance of the four robots are shown in Figures 8 and 9. It can be seen that all the positions of the robot manipulators follow the leader signal $p_{r} (t)$ . Meanwhile, it can be seen from Figure 9 that the tracking error $z_{ij}$ is constrained in $(- 1 / δ (t), 1 / δ (t))$ and converge to zero in prescribed finite time. Figure 10 shows the variations of the adaptive gain ${\hat{γ}}_{i}$ and $W_{i}, χ_{i}$ . We can also see that $W_{i}$ is always smaller than $χ_{i}$ and finally converges to a small value around the neighborhood of the origin.

Figure 8.

Control performance for four robot manipulators in Case I.

Figure 9.

Tracking error and prescribed finite-time function.

Figure 10.

Control efforts and variations of adaptive parameters.

For comparison, we also consider the Nussbaum gain method. The controller is given by (45)-(47), and the same five cases are considered. Figure 11 illustrates the performance of the robots. Still in all five cases, the robots can reach a steady state by the proposed method, while robots using the Nussbaum gain method are unstable as shown in Figure 12. These verify the advantages of the proposed controller.

Figure 11.

Control performance of four robot manipulators for Cases I-V under the proposed method.

Figure 12.

Control performance of four robot manipulators for Cases I-V under the Nussbaum method.

In order to further verify the effectiveness of the proposed method, we consider a class of adaptive fuzzy control method.³³ It is expressed as:

α_{ij} = - K_{i 1} s_{ij},

(49)

\begin{matrix} u_{ij} = - {\hat{γ}}_{i} [K_{i 2} ξ_{ij} + \frac{R_{i} ξ_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}} + \frac{H_{i} ξ_{ij} ‖ φ_{ij} (s_{ij}, ξ_{ij}) ‖ {\hat{ϑ}}_{ij}}{Z_{i}^{2} - ξ_{ij}^{2}}], \end{matrix}

(50)

where

{\overset{\cdot}{\hat{ϑ}}}_{ij} = \frac{H_{i} ξ_{ij} {‖ φ_{ij} (s_{ij}, ξ_{ij}) ‖}^{2} ξ_{ij}^{2}}{2} - η_{i} {\hat{ϑ}}_{ij},

$η_{i}$ is a positive constant, $φ_{ij} (s_{ij}, ξ_{ij})$ is a fuzzy membership function given by:

φ_{ij} (s_{ij}, ξ_{ij}) = [φ_{ij}^{1} (s_{ij}, ξ_{ij}), φ_{ij}^{2} (s_{ij}, ξ_{ij}), \dots, φ_{ij}^{5} (s_{ij}, ξ_{ij})]^{T}

with

φ_{ij}^{l} (s_{ij}, ξ_{ij}) = \frac{μ_{F_{m}^{l}} (s_{ij}) μ_{F_{m}^{l}} (ξ_{ij})}{\sum_{l = 1}^{5} μ_{F_{m}^{l}} (s_{ij}) μ_{F_{m}^{l}} (ξ_{ij})},

μ_{F_{m}^{l}} (•) = \exp [- 0.5 {(• - 3 + l)}^{2}], l = 1, \dots, 5 .

${\hat{γ}}_{i}, K_{i 1}, K_{i 2}, R_{i}$ are the same as (28)-(29). $H_{i} = 1$ , $η_{i} = 1$ .

The control performance is shown in Figure 13. We can see that the performance of the above adaptive fuzzy controller is a little better than the proposed controller due to the approximation ability of the fuzzy logic. Meanwhile, we can see that for both controllers, the tracking error is constrained in the prescribed tube. However, the adaptive fuzzy controller has a much more complex structure than the proposed controller.

Figure 13.

Tracking performance comparison with adaptive fuzzy controller.

Discussion

Finite-time cooperative control means that multiple robots can reach consensus in finite time. That is finite time control guarantees that system states converge to equilibrium within a pre-defined finite time, whereas asymptotic control only ensures convergence as time approaches infinity. This indicates a fast convergence speed, reduced cumulative tracking error, and strong robustness to external uncertainties/nonlinearities. Moreover, finite-time convergence also means that the cooperative control can be decoupled from other tasks such as state estimation, fault detection, etc.

It is noted that theoretically the states or tracking error will converge to zero in finite time if all the design parameters are positive. This is due to the adaptive feature of our proposed method. The theoretical results imply that the parameters selection can be very flexible and straightforward. This can be also seen from the design procedures and the proofs of the main results where Young’s inequalities are extensively used. In fact, from the simulations, we can see that many of the design parameters are the same. It is also noted that appropriate design parameters can result in smaller settling time and better transient performance. Next, we present some guidelines for the parameters selections.

Basically, larger control gain $K_{i 1}, K_{i 2}, R$ and smaller $q$ may result in a faster convergence speed, but bigger overshoot and control effort. By increasing the value $ρ_{1}$ in the supervisory functions (12), we can obtain a smaller settling time. The initial value ${\hat{γ}}_{i} (0)$ should be set small to prevent the control effort from becoming very large in the beginning.

Moreover, for the leader-following finite-time consensus, one can prescribe a convergence time or transient performance for the closed loop systems. In fact, from Theorem 8, we know the output satisfies $| z_{ij} | < 1 / δ (t)$ where $δ (t)$ is a class of prescribed finite-time function. This will bring more convenience to the parameters selection.

We will use some examples to show the above facts. Consider the following five cases:

Case A: $K_{i 1} = 2, K_{i 2} = 10, R_{i} = 1, Z_{i} = 5; T = 4$ ;

Case B: $K_{i 1} = 4, K_{i 2} = 15, R_{i} = 1, Z_{i} = 5; T = 4$ ;

Case C: $K_{i 1} = 6, K_{i 2} = 20, R_{i} = 1, Z_{i} = 10; T = 4$ ;

Case D: $K_{i 1} = 4, K_{i 2} = 15, R_{i} = 1, Z_{i} = 5; T = 2$ ;

Case E: $K_{i 1} = 4, K_{i 2} = 15, R_{i} = 1, Z_{i} = 5; T = 6$ ;

For Case A, Case B and Case C, the control gains are increasing. The performance are shown in Figures 14 and 15. We can see from Figure 14 that the convergence rate becomes faster when the control gains are larger. Meanwhile, the control effort becomes bigger for larger control gains as demonstrated in Figure 15.

Figure 14.

Control performance comparison in Cases A, B and C.

Figure 15.

Control efforts comparison in Cases A, B and C.

For Case D, Case B and Case E, the prescribed finite time T becomes larger. As shown in Figure 16, we can see that for a smaller T, it means a faster convergence rate and a smaller transient time. We can see from Figure 17, when we increase the value of T, the control effort becomes smaller.

Figure 16.

Control performance comparison in Cases D, B and E.

Figure 17.

Control efforts comparison in Cases D, B and E.

Experiments

In this section, we will conduct an experiment to further verify the proposed method. The experimental platform is shown in Figure 18. The mobile robot has four Mecanum wheels driven by servo motors. Hence, it can move in four directions. A motion control board consisting of STM32F103 and Raspberry Pi 3B is used to receive/send data to the servo motors. The motion control board can transform PWM commands from the Raspberry Pi to motor voltage for speed regulation. An IMU is integrated into the control board to detect the pose of the robot. Therefore, closed-loop control for the mobile robot can be realized. Ubuntu 18.04, ROS (Robot Operating System) and MATLAB/Simulink are utilized in the host computer to control the mobile robot. They can simulate the environment of the mobile robot. Cameras and laser radar are integrated in the mobile robots to detect the surrounding environment. Motors, control board, cameras and laser radar are connected by serial ports.

Figure 18.

Experimental platform.

Three robots are considered in the experiment. The communication graph is shown in Figure 19. The system model is given by (44) where $g_{i} (t) = 1$ is assumed to be an unknown control direction. $u_{i}$ is given by (28)-(29) with some modification to guarantee the triangle formation of the three robots. That is

\begin{matrix} z_{11} = a_{10} (p_{11} - p_{r 1}), \\ z_{12} = a_{10} (p_{12} - p_{r 2}), \\ z_{21} = a_{21} (p_{21} - p_{11} - 0.5), \\ z_{22} = a_{21} (p_{22} - p_{12} - 0.8), \\ z_{31} = a_{31} (p_{31} - p_{11} + 0.5), \\ z_{32} = a_{31} (p_{32} - p_{12} - 0.8), \end{matrix}

The logic-based switching rule is given by (31). The leader signal is given by: $p_{r} (t) = [p_{r 1} (t) p_{r 2} (t)]^{T}$ with $p_{r 1} (t) = {\begin{matrix} 2 + t, & 0 < t \leq 3, \\ 5, & t \geq 3 . \end{matrix}$ and $p_{r 2} (t) = {\begin{matrix} 5.5 - t, & 0 < t \leq 3, \\ 2.5, & t \geq 3 . \end{matrix}$ The parameters are set as: $K_{i 1} = 2, K_{i 2} = 10, R_{i} = 1, Z_{i} = 5$ . $δ (t)$ is set as $δ (t) = 30 ρ (t) + 0.2$ where $ρ (t) = {\begin{matrix} {(\frac{T - t}{T})}^{3}, & 0 < t < T, \\ 0, & t \geq T, \end{matrix}$ with $T = 8$ .

Figure 19.

Communication graph.

The experimental results are shown in Figure 20. We can see that the three robots quickly reach a steady state despite the unknown control direction. Meanwhile, we can see that $z_{i 1}, z_{i 2}$ are evolving in the constrained tube in Figure 21. Figures 22 and 23 demonstrate the state trajectories of three robots. We can see that they form a triangle in finite time. These show the effectiveness of the proposed method.

Figure 20.

Tracking performance in experiment.

Figure 21.

Tracking error in experiment.

Figure 22.

State trajectories in experiment.

Figure 23.

Three robots in experiment.

Conclusions

In this paper, we propose two kinds of switching adaptive control for leaderless and leader-following consensus of multiple robots. The proposed leaderless consensus method can steer all the agents to a comment value in finite time, while the leader-following consensus method can make all the agents track a desired reference value in a prescribed finite time. New supervisory functions and switching rules are constructed to adaptively tune the control gains. It is shown that the proposed method can deal with heterogeneous unknown control directions and complex nonlinearities. A key feature of the constructed supervisory function is that it is local and only dependent on local state information. No neigborhood/global information are needed. Simulations and comparison with traditional Nussbaum gain methods have demonstrated the advantages of the designed controller.

The proposed method has several limitations: First, the implementation of the logic-based switching rule may need some extra computation resource. Second, when the number of the agent is increasing, the complexity of the proposed method may increase. Last, the proposed method has not been used in real engineering environment. These limitations will be considered in our future work.

Footnotes

ORCID iD

Shiqi Zheng

Consent to participate

This article does not contain any studies with human or animal participants.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Natural Science Foundation of China (52375520)

Declaration of conflicting interests

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

References

Yang

, et al. Anti-attack event-triggered control for nonlinear multi-agent systems with input quantization. IEEE Trans Neural Netw Learn Syst 2023; 34(12): 10105–10115.

Wang

Cao

, et al. A hierarchical design framework for distributed control of multi-agent systems. Automatica 2024; 160: 111402.

Zeng

Liu

Sampled-data robust practical tracking of Euler-Lagrange systems with an uncertain exosystem. Int J Robust Nonlinear Control 2024; 34(3): 1629–1647.

Huang

Chen

. Air-ground cooperative multi-target searching under an unknown urban environment. Trans Inst Meas Control. Epub ahead of print. DOI: 10.1177/01423312241239386

Long

Wang

Huang

, et al. Adaptive leaderless consensus for uncertain high-order nonlinear multiagent systems with event-triggered communication. IEEE Trans Syst Man Cybern Syst 2022; 52(11): 7101–7111.

Zhang

Wang

, et al. Leader-following and leaderless consensus of linear multiagent systems under directed graphs by double dynamic event-triggered mechanism. IEEE Trans Syst Man Cybern Syst 2022; 52(10): 6426–6438.

Yao

, et al. Event-triggered guaranteed cost leader-following consensus control of second-order nonlinear multiagent systems. IEEE Trans Syst Man Cybern Syst 2022; 52(4): 2615–2624.

Liang

Wang

Liu

, et al. Necessary and sufficient conditions for leader-follower consensus of discrete-time multiagent systems with smart leader. IEEE Trans Syst Man Cybern Syst 2022; 52(5): 2779–2788.

Yang

Bipartite containment control of multi-agent systems with multiple leader groups. Trans Inst Meas Control 2022; 44(16): 3132–3140.

10.

Zhang

Chai

Zhang

, et al. Distributed model-free sliding-mode predictive control of discrete-time second-order nonlinear multiagent systems with delays. IEEE Trans Cybern 2022; 52(11): 12403–12413.

11.

Yuan

, et al. Robust vision-based tube model predictive control of multiple mobile robots for leader-follower formation. IEEE Trans Ind Electron 2020; 67(4): 3096–3106.

12.

Wen

Liang

Zhou

, et al. Distributed adaptive consensus protocol for heterogeneous nonlinear uncertain multi-agent system with disturbance. Int J Robust Nonlinear Control 2024; 34(3): 1475–1492.

13.

Chen

Xie

, et al. Human-in-the-loop fuzzy iterative learning control of consensus for unknown mixed-order nonlinear multi-agent systems. IEEE Trans Fuzzy Syst 2024; 32(1): 255–265.

14.

Liu

Zhan

Han

, et al. Distributed adaptive finite-time bipartite containment control of linear multi-agent systems. IEEE Trans Circuits Syst II Express Briefs 2022; 69(11): 4354–4358.

15.

Huang

Wen

Wang

, et al. Adaptive finite-time consensus control of a group of uncertain nonlinear mechanical systems. Automatica 2015; 51: 292–301.

16.

Zhang

Adaptive finite-time optimal formation control for second-order nonlinear multiagent systems. IEEE Trans Syst Man Cybern Syst 2023; 53(10): 6132–6144.

17.

Mehdifar

Hashemzadeh

Baradarannia

, et al. Finite-time rigidity-based formation maneuvering of multiagent systems using distributed finite-time velocity estimators. IEEE Trans Cybern 2019; 49(12): 4473–4484.

18.

Chen

Han

Zhu

, et al. Distributed fixed-time tracking and containment control for second-order multi-agent systems: a nonsingular sliding-mode control approach. IEEE Trans Netw Sci Eng 2023; 10(2): 687–697.

19.

Yao

Shi

Event-based average consensus of disturbed mass via fully distributed sliding mode control. IEEE Trans Autom Control 2024; 69(3): 2015–2022.

20.

Wang

Finite-time output consensus of higher-order multiagent systems with mismatched disturbances and unknown state elements. IEEE Trans Syst Man Cybern Syst 2019; 49(12): 2571–2581.

21.

Wang

Finite-time distributed approximate optimization algorithms of higher order multiagent systems via penalty-function-based method. IEEE Trans Syst Man Cybern Syst 2022; 52(10): 6174–6182.

22.

Ren

Zhou

, et al. Prescribed-time consensus tracking of multiagent systems with nonlinear dynamics satisfying time-varying lipschitz growth rates. IEEE Trans Cybern 2023; 53(4): 2097–2109.

23.

Song

Lewis

FL.

Prescribed-time control and its latest developments. IEEE Trans Syst Man Cybern Syst 2023; 53(7): 4102–4116.

24.

Guo

X-G

W-D

Wang

J-L

, et al. Distributed neuroadaptive fault-tolerant sliding-mode control for 2-d plane vehicular platoon systems with spacing constraints and unknown direction faults. Automatica 2021; 129: 109675.

25.

Huang

Long

, et al. Improved prescribed performance control for nonlinear systems with unknown control direction and input dead-zone. Int J Robust Nonlinear Control 2024; 34(7): 4489–4508

26.

Liu

Yang

Zhao

, et al. Novel nussbaum design for nonlinear systems with unknown switching control directions. IEEE Trans Circuits Syst II Express Briefs 2023; 70(9): 3529–3533.

27.

Chen

Liu

Zhang

, et al. Saturated nussbaum function based approach for robotic systems with unknown actuator dynamics. IEEE Trans Cybern 2016; 46(10): 2311–2322.

28.

Liang

Zheng

Ahn

, et al. Adaptive fuzzy control for fractional-order interconnected systems with unknown control directions. IEEE Trans Fuzzy Syst 2022; 30(1): 75–87.

29.

De Schutter

Shi

, et al. Logic-based distributed switching control for agents in power-chained form with multiple unknown control directions. Automatica 2022; 137: 110143.

30.

Chen

Global finite-time adaptive stabilization for nonlinear systems with multiple unknown control directions. Automatica 2016; 69: 298–307.

31.

Zheng

Wang

Chen

, et al. Logic-based switching finite-time stabilization with applications in mechatronic systems. Int J Robust Nonlinear Control 2023; 33(5): 3064–3085.

32.

Chen

Zheng

Ahn

, et al. Constrained fast finite-time exact tracking for disturbed nonlinear systems. Int J Robust Nonlinear Control 2022; 32(7): 4376–4400.

33.

Zheng

Adaptive control for switched nonlinear systems with coupled input nonlinearities and state constraints. Inf Sci 2018; 462: 331–356.

Logic-based switching finite-time cooperative control for multiple robots with applications in library

Abstract

Keywords

Introduction

Problem formulation and preliminaries

Problem formulation

Preliminaries

Leaderless finite-time consensus

Main results

Controller structure

Logic-based switching rule

Leader-following finite-time consensus

Main results

Controller structure

Logic-based switching rule

Simulations

Example 1

Example 2

Discussion

Experiments

Conclusions

Footnotes

ORCID iD

Consent to participate

Funding

Declaration of conflicting interests

References