Adaptive Leader-Following Consensus for Multiple Euler-Lagrange Systems with an Uncertain Leader System

I. INTRODUCTION

• Earlier leader-following approaches for uncertain Euler–Lagrange systems used distributed observers, but assumed the leader’s system matrix was known by at least some followers.The adaptive distributed observer relaxes this requirement; the earlier distributed observer requires every follower to know the matrix.Source: §I · p. 2
• Unknown leader dynamics arise in practical settings: for a sinusoidal signal, unknown frequency means the leader’s system matrix is unknown.Prior work also studied leaderless and leader-following consensus under several network and system conditions.Source: §I · pp. 2–3
• An adaptive distributed observer estimates and relays the leader’s state without its exact system matrix; with persistent excitation, it asymptotically learns that matrix’s parameters.Combined with an adaptive distributed control law, it solves leader-following consensus for multiple Euler–Lagrange systems, including multi-tone signals with unknown amplitudes, phases, and frequencies.Source: §I · p. 3

II. PROBLEM FORMULATION AND ASSUMPTIONS

• Each follower is modeled by Euler–Lagrange dynamics with positive-definite inertia, a skew-symmetric matrix property, and a linear parameterization of its uncertain dynamics.The model identifies generalized position, inertia, Coriolis and centripetal forces, and control torque.Source: §II · p. 3
• The leader generates the desired position through a linear exosystem whose matrix may depend on unknown parameters, and the communication graph must contain a leader-rooted spanning tree with an undirected follower subgraph.The leader matrix is assumed to have semisimple eigenvalues with zero real part.Source: §II · pp. 3–4
• The control objective is bounded follower signals and convergence of each follower’s position and velocity to the leader’s corresponding states.The control law must use only neighboring agents’ information, satisfying the communication constraints.Source: §II · pp. 3–4
• For multi-tone sinusoidal leader signals, assuming the matrix known is equivalent to assuming all signal frequencies are known exactly.The paper targets arbitrary unknown amplitudes, initial phases, and frequencies.Source: §II · p. 4

III. ADAPTIVE DISTRIBUTED OBSERVER WITH AN UNCERTAIN LEADER

• The earlier adaptive observer (7) uses only each follower’s and its neighbors’ information, but children of the leader still need its frequency.This motivates an observer that handles an unknown leader system matrix.Source: §III · p. 4
• For each i=1,...,N, η_i ∈ R^m and ω_i ∈ R^l, with S(ω_i)=(diag(ω_i) ⊗ a); η_0=v, and μ_1 and μ_2 are positive numbers.Source: §III · p. 4
• Observer (8) is independent of the leader’s unknown parameter vector and is designed to estimate and pass the leader’s state to each follower.The paper defines it as an adaptive distributed observer when each follower’s estimate converges to the leader state.Source: §III · p. 4
• Under Assumptions 1 and 3, if col(v2i−1(0), v2i(0)) ≠ 0 for all i = 1,…,l, then v(t) is persistently exciting.Source: §III · p. 6
• Compared with, observer (8) avoids a known compact parameter set and supports multiple-row C, using m/2 parameters and mN/2 equations instead of m and m×N.The paper notes that ’s common-matrix condition is unlikely to hold across Q when P generally depends on ρ.Source: §III · p. 7

IV. LEADER-FOLLOWING CONSENSUS OF MULTIPLE EL SYSTEMS

• Because the leader matrix S is uncertain, the control law estimates follower-specific vectors ωi rather than requiring exact S(ω), as in.The ωi estimates are updated by the control law’s fourth equations.Source: §IV · p. 7
• Under Assumptions 1 and 2, control law (33) solves the leader-following consensus problem for systems (2) and (3) over graph G.The proof establishes that the sliding variables and then the position and velocity tracking errors tend to zero.Source: §IV · pp. 7–8
• The adaptive control law uses a regressor YiΘi for desired-motion dynamics and updates estimates of the unknown EL parameters Θi.Positive-definite feedback and adaptation gains are specified in the control law.Source: §IV · p. 7
• The proof shows the sliding variables si tend to zero, after establishing boundedness of si and the EL parameter-estimation errors.It then uses the stable error dynamics and vanishing observer error to show position and velocity tracking errors tend to zero.Source: §IV · pp. 7–8

V. AN EXAMPLE

• The simulation uses six two-link robot arms and a leader whose two position components are sinusoids with unknown amplitudes, frequencies, and phases.The example states that the communication topology satisfies Assumption 1.Source: §V · p. 8
• Figures 2 and 3 plot the six followers’ position and velocity trajectories and tracking performance, respectively.The simulation initializes qi and Θhat_i at zero and randomly chooses ωi(0) in [0, 1].Source: §V · p. 8
• The observer guarantees ˜ωi → 0 for all six followers because the chosen leader signal is persistently exciting.The example uses leader initial state col(1, 0, 1, 0) and unknown frequencies ω01 = 4 and ω02 = 2.Source: §V · p. 8

VI. CONCLUSION

• The adaptive observer estimates and distributes the leader state without exact knowledge of its system matrix, and under persistent excitation it learns the unknown matrix parameters asymptotically.An adaptive distributed control law built on this observer solves the leader-following consensus problem.Source: §VI · p. 8
• The result permits leader-following consensus even when no follower knows the leader’s system matrix exactly.Source: §VI · p. 9
• The observer may also apply to cooperative output regulation; extending it to uncertain leaders under switching topologies remains future work.The paper points to for the cooperative output regulation problem.Source: §VI · p. 9

• A digraph G is defined by a finite node set V and an edge set E whose edges join distinct nodes.Source: §VII · p. 9
• A weighted directed graph models communication among N systems using directed edges and a nonnegative adjacency matrix C = [cij].An entry cij > 0 means (j, i) is an edge.Source: §VII · p. 10
• Node i’s in-degree is the sum of incoming adjacency weights, and the graph Laplacian is H = D − C.D is the diagonal matrix of node in-degrees.Source: §VII · p. 10
• A node is a neighbor when the corresponding directed edge exists, and reachability is defined by a path of consecutive edges.Source: §VII · p. 10