Leader Tracking of Euler-Lagrange Agents on Directed Switching Networks Using A Model-Independent Algorithm

TL;DR

A discontinuous distributed model-independent algorithm for a directed network of Euler–Lagrange agents to track the trajectory of a leader with nonconstant velocity achieves practical tracking with an adjustable tracking error and is stable for networks that switch with an explicitly computable dwell time.

Abstract

In this paper, we propose a discontinuous distributed model-independent algorithm for a directed network of Euler-Lagrange agents to track the trajectory of a leader with non-constant velocity. We initially study a fixed network and show that the leader tracking objective is achieved semi-globally exponentially fast if the graph contains a directed spanning tree. By model-independent, we mean that each agent executes its algorithm with no knowledge of the parameter values of any agent's dynamics. Certain bounds on the agent dynamics (including any disturbances) and network topology information are used to design the control gain. This fact, combined with the algorithm's model-independence, results in robustness to disturbances and modelling uncertainties. Next, a continuous approximation of the algorithm is proposed, which achieves practical tracking with an adjustable tracking error. Last, we show that the algorithm is stable for networks that switch with an explicitly computable dwell time. Numerical simulations are given to show the algorithm's effectiveness.

Figures (8)

• Figure 1: Diagram for Part 3 of the proof of Theorem \ref{['theorem:main_result']}. The red region is $\mathcal{S}$, in which $\dot{V}(t) < 0$ for all $t \geq 0$. The blue region is $\mathcal{T}$, in which $\dot{V}(t)$ is sign indefinite. A trajectory of \ref{['eq:network_system']} is shown with the black curve. We define $t = T_2$, if it exists, as the infimum of all $t$ values for which one of the inequalities $\Vert \widetilde{\boldsymbol{q}}(T_2) \Vert < \mathcal{X}$ or $\Vert \dot{\widetilde{\boldsymbol{q}}}(T_2) \Vert < \mathcal{Y}$ fails to hold, i.e. as the time at which the system \ref{['eq:network_system']} first leaves $\mathcal{S}$. By contradiction, it is shown in Part 3.2 that the trajectory of \ref{['eq:network_system']} satisfies $\Vert \widetilde{\boldsymbol{q}}(T_2) \Vert < \mathcal{X}, \Vert \dot{\widetilde{\boldsymbol{q}}}(T_2) \Vert < \mathcal{Y}$. I.e., $T_2$ does not exist and the trajectory remains in $\mathcal{T}\cup \mathcal{S}, \forall\,t$. The sign indefiniteness of $\dot{V}$ in $\mathcal{T}$ arises due to terms linear in $\Vert \widetilde{\boldsymbol{q}} \Vert$, $\Vert \dot{\widetilde{\boldsymbol{q}}} \Vert$ in \ref{['eq:Vdot_bound_T1']}. These terms disappear at $t = T_1$, when the finite-time observer converges. For all $t > T_1, \dot{V} < 0$ in $\mathcal{T}\cup \mathcal{S}$ as shown in Part 4. Exponential convergence to the origin follows.
• Figure 2: In the simulation, graph $\mathcal{G}_{A}(t)$ switches between the above three graphs periodically at a rate of $1\;\textrm{Hz}$.
• Figure 3: Graph $\mathcal{G}_{B}(t)$ switches between the above three graphs periodically at a rate of $1\;\textrm{Hz}$; if $\mathcal{G}_{A}(t) = \mathcal{G}_{A,i}$ then $\mathcal{G}_{B}(t) = \mathcal{G}_{B,i}$ for $i = 1, 2, 3$.
• Figure 4: Plot of generalised coordinates vs. time; the graph $\mathcal{G}_B(t)$ is disconnected for $t\in[10,20)$.
• Figure 5: Plot of generalised velocity vs. time; the graph $\mathcal{G}_B(t)$ is disconnected for $t\in[10,20)$.

Theorems & Definitions (20)

• Definition 1
• Lemma 1: The Schur Complement horn2012matrixbook
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1
• proof
• Lemma 4: zhang2015graph_lyapunov
• Remark 1: Comparison of this paper to recent leader tracking results