Event-Triggered Algorithms for Leader-Follower Consensus of Networked Euler-Lagrange Agents

TL;DR

Three different distributed event-triggered control algorithms to achieve leader–follower consensus for a network of Euler–Lagrange agents are proposed and a trigger function is proposed to govern the event update times.

Abstract

This paper proposes three different distributed event-triggered control algorithms to achieve leader-follower consensus for a network of Euler-Lagrange agents. We firstly propose two model-independent algorithms for a subclass of Euler-Lagrange agents without the vector of gravitational potential forces. By model-independent, we mean that each agent can execute its algorithm with no knowledge of the agent self-dynamics. A variable-gain algorithm is employed when the sensing graph is undirected; algorithm parameters are selected in a fully distributed manner with much greater flexibility compared to all previous work concerning event-triggered consensus problems. When the sensing graph is directed, a constant-gain algorithm is employed. The control gains must be centrally designed to exceed several lower bounding inequalities which require limited knowledge of bounds on the matrices describing the agent dynamics, bounds on network topology information and bounds on the initial conditions. When the Euler-Lagrange agents have dynamics which include the vector of gravitational potential forces, an adaptive algorithm is proposed which requires more information about the agent dynamics but can estimate uncertain agent parameters. For each algorithm, a trigger function is proposed to govern the event update times. At each event, the controller is updated, which ensures that the control input is piecewise constant and saves energy resources. We analyse each controllers and trigger function and exclude Zeno behaviour. Extensive simulations show 1) the advantages of our proposed trigger function as compared to those in existing literature, and 2) the effectiveness of our proposed controllers.

Figures (5)

• Figure 6: Two-link manipulator, generalized coordinates $\boldsymbol{q}=[q^1,q^2]^\top$
• Figure 7: Simulation results for controller \ref{['eq:control_input_1']} under trigger function \ref{['eq:trigger_function_1']}. From top to bottom: the plots the generalized coordinates; the plots of generalised velocities of all the follower manipulators; the plot of variable gain $\mu_i(t)$; the plot of trigger events
• Figure 8: Simulation results for controller \ref{['eq:agent_control_law_direct']} under trigger function \ref{['eq:agent_trigger_direct']}. From top to bottom: the plots the generalized coordinates; the plots of generalised velocities of all the follower manipulators; the plot of trigger events
• Figure 9: Simulation results for controller \ref{['eq:r9']} under trigger function \ref{['eq:trigger_function_3']}. From top to bottom: the plots the generalized coordinates; the plots of generalised velocities of all the follower manipulators; the plot of trigger events
• Figure 10: Diagram for proof of Theorem\ref{['theorem:directed_main_MI']}. The red region is $\mathcal{S}(t)$, in which $\dot{V}(t) < 0$ for all $t \geq 0$. The blue region is $\mathcal{T}(t)$, in which $\dot{V}(t)$ is sign indefinite. A trajectory of \ref{['eq:euler_Lagrange_nonautonomous_network']} is shown with the black curve. At $t = T_1$, it is shown in Part 2 that the trajectory of \ref{['eq:euler_Lagrange_nonautonomous_network']} is such that $\Vert \boldsymbol{u}(T_1) \Vert < \mathcal{X}, \Vert \boldsymbol{v}(T_1) \Vert < \mathcal{Y}$ and thus the trajectory does not leave $\mathcal{S}(t)$. The sign indefiniteness of $\dot{V}(t)$ in $\mathcal{T}(t)$ arises due to the terms linear in $\Vert \boldsymbol{u} \Vert$ and $\Vert \boldsymbol{v} \Vert$ in \ref{['eq:Vdot_derivative_boundb']}, i.e. the terms containing $\bar{\omega}(t)$ (coefficients $A_5(t)$ and $A_6(\mu,t)$ in \ref{['eq:p_uv_def']}). Because $\bar{\omega}(t)$ goes to zero at an exponential rate, so do the coefficients $A_5(t)$ and $A_6(\mu,t)$. Examining the inequalities detailed in Corollary\ref{['cor:bounded_pd_function_2']} as applied to $p(\Vert \boldsymbol{u} \Vert, \Vert \boldsymbol{v} \Vert)$ in \ref{['eq:p_uv_def']}, it is straightforward to conclude that for a fixed $\mu_6^*$, the exponential decay of $A_5, A_6$ implies that the region $\mathcal{T}(t)$ shrinks towards the origin at an exponential rate. In other words, $\vartheta(t)$ and $\varphi(t)$ monotonically increase until $\vartheta(t) = \mathcal{X}$ and $\varphi(t) = \mathcal{Y}$, at which point $\mathcal{T}(t) = [0,0]$. This corresponds to the dotted red and blue lines, which show, respectively, the time-varying boundaries of $\mathcal{S}(t)$ and $\mathcal{T}(t)$. The solid red and blue lines show respectively, the boundaries of $\mathcal{S}(t)$ and $\mathcal{T}(t)$, which are time-invariant. Exponential convergence to the leader-follower objective is discussed in Part 3 making using of $T_2$.

Theorems & Definitions (26)

• Theorem 1: Mean Value Theorem for Vector-Valued Functions rudin1964principles
• Theorem 2: The Schur Complement horn2012matrixbook
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1
• proof
• Lemma 4