Source-linked AI summary
Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory
A. Isidori, L. Marconi, G. Casadei
TL;DR
The paper studies output synchronization for heterogeneous diffusively coupled nonlinear agents tracking a prescribed nonlinear exosystem. It synchronizes identical local reference generators, then applies decentralized nonlinear output regulation to make each heterogeneous agent track its synchronized reference. The reported theory establishes bounded closed-loop trajectories and asymptotic tracking under the stated assumptions, with simulations illustrating oscillator synchronization.
Problem
The paper addresses how heterogeneous nonlinear agents can reach consensus on a nontrivial output trajectory using decentralized exchange of relative output information.
Method
It synchronizes N diffusively coupled copies of a nonlinear exosystem as local references, then designs nonlinear internal-model-based regulators for the heterogeneous agents.
Results
The theory yields bounded closed-loop trajectories and asymptotic tracking errors converging to zero uniformly over compact sets of initial conditions.
Takeaways & Limitations
The two-stage design provides a decentralized route for heterogeneous nonlinear agents to track a common reference signal generated by a nonlinear exosystem.
Takeaways & Limitations
The analysis assumes a nontrivial compact invariant set with globally Lipschitz exosystem dynamics and agents with globally defined normal forms and nonzero high-frequency gains.
Abstract
from arXiv · showhide
In this paper, we consider the output synchronization problem for a network of heterogeneous diffusively-coupled nonlinear agents. Specifically, we show how the (non-identical) agents can be controlled in such a way that their outputs asymptotically track the output of a prescribed nonlinear exosystem. The problem is solved in two steps. In the first step, the problem of achieving consensus among (identical) nonlinear reference generators is addressed. In this respect, it is shown how the techniques recently developed to solve the consensus problem among linear agents can be extended to agents modeled by nonlinear d-dimensional differential equations, under the assumption that the communication graph is connected. In the second step, the theory of nonlinear output regulation is applied in a decentralized control mode, to force the output of each agent of the network to robustly track the (synchronized) output of a local reference model.
1 Introduction
The paper addresses output synchronization for heterogeneous nonlinear networks by extending consensus ideas to nonlinear reference generators and combining them with decentralized nonlinear output regulation.
- The paper presents further contributions to output synchronization beyond existing linear and nonlinear consensus results.It situates the work among prior studies of linear and nonlinear network consensus.
- The paper targets consensus of outputs in heterogeneous networked nonlinear systems.It frames this as tracking the output of a prescribed nonlinear exosystem.
- The approach uses N identical copies of the nonlinear exosystem as local reference generators.Their coupling gains are selected so the generators synchronize on a common trajectory.
- The synchronized references are then used by local regulators to control the heterogeneous agents.The paper applies nonlinear output-regulation theory in a decentralized control mode.
2 Problem statement
The problem is to design decentralized local output-feedback controllers so heterogeneous nonlinear agents reach uniform asymptotic output consensus through neighbor-relative information. The proposed structure separates synchronized reference generation from local tracking regulation.
- 2.1 Communication graphs.: A connected graph has exactly one zero Laplacian eigenvalue, while all remaining eigenvalues have positive real parts.The all-ones vector corresponds to the trivial zero eigenvalue.
- 2.1 Communication graphs.: Agents exchange only relative values of controlled outputs through a time-invariant communication graph.The graph consists of vertices, directed edges, and weighted information flows.
- 2.2 Problem formulation: Each nonlinear agent has a local output-feedback controller whose input and exchanged variables are designed locally.The exchanged signal is determined from local controller states and outputs, and communication uses relative measurements.
- 2.2 Problem formulation: The design goal is bounded closed-loop behavior and y_k(t) → y∗(t) for every agent, uniformly over admissible initial conditions.The common trajectory is required to be nontrivial.
- 2.2 Problem formulation: The consensus trajectory is modeled as the output of a nonlinear autonomous system with a nontrivial compact invariant set and globally Lipschitz dynamics.The globally Lipschitz condition can be enforced outside the compact invariant set when the original function is only locally Lipschitz.
- 2.3 Structure of local controllers and communication protocol: The controller architecture uses local reference generators followed by regulators driven by local tracking errors.The first stage synchronizes the generators; the second makes each agent track its own reference.
- 2.3 Structure of local controllers and communication protocol: The paper addresses nonlinear systems of dimension d > 1 that exchange relative values of one-dimensional outputs rather than full d-dimensional states.This communication setting is identified as not previously proposed in the cited discussion.
3 Achieving consensus in a homogeneous network of nonlinear systems
The paper establishes consensus for diffusively coupled identical nonlinear systems on a connected graph under an invariant-set and input-to-state stability assumption. A high-gain decentralized coupling, designed through the graph Laplacian, makes all states converge to a common trajectory.
- The analysis assumes a connected communication graph and an invariant compact set W for the uncoupled nonlinear dynamics.The forced system is required to be input-to-state stable relative to W.
- The coupling vector uses high-gain scaling with design parameter g and a vector K0 selected through a Riccati-equation-based construction.The Laplacian’s nontrivial eigenvalues have positive real parts, enabling a stabilizing choice of K0.
- For every initial condition, there exists a trajectory w*(t) such that each state w_k(t) converges to w*(t) as t tends to infinity.The result holds when g ≥ g* under the stated assumptions and controller choice.
- A coordinate transformation separates the synchronized component from disagreement variables, yielding a triangular network structure.The disagreement dynamics are analyzed after setting z_k = w_k − w_1 for k = 2, ..., N.
- For sufficiently large g, the disagreement subsystem is asymptotically stable uniformly in the synchronized coordinate, while the invariant-set dynamics are controlled using ISS.The proof combines a Hurwitz disagreement matrix, high-gain arguments, and a quadratic Lyapunov function independent of w_1.
4 Achieving consensus in a heterogenous network of nonlinear systems
The design proceeds in two stages: synchronize heterogeneous networked reference generators, then use decentralized nonlinear regulators to drive each agent’s tracking error to zero. Under internal-model and stability assumptions, bounded trajectories and asymptotic regulation are obtained.
- Local regulation: The second design stage constructs a local regulator for each agent after the networked reference-generator set has been made globally asymptotically stable.The agent is represented in normal form, with the exposition focusing on relative degree r = 1.
- Local regulation: Each agent’s tracking error is defined as e_k = y_k − Cw_k, and the regulator is designed to steer e_k to zero.The controller uses an internal-model state η_k and feedback v_k = κ_k(e_k).
- Assumptions: The local-regulator design assumes an asymptotically stable compact invariant set for the agent’s zero dynamics, with a domain of attraction covering the prescribed initial conditions.This is identified as the networked analogue of a weak minimum-phase assumption.
- Guarantee: If the regulator triplet has the asymptotic internal model property, a continuous feedback κ_k makes closed-loop trajectories bounded and ensures lim t→∞ e_k(t) = 0 uniformly over compact initial-condition sets.The result applies to the coupled reference-generator dynamics and the agent’s regulator states.
- Internal-model construction: The internal model can be designed with dimension m_k ≥ 2(d + n_k − 1) + 2 for almost all suitable controllable matrix pairs, while constructive designs require an additional assumption.A high-gain construction uses a bounded locally Lipschitz extension and a Hurwitz polynomial, with sufficiently large design parameter ℓ.
5 Simulation results
Simulations examine three networked nonlinear oscillators using Van der Pol and Duffing examples, with phase-plane and first-state trajectories reported for distinct initial conditions.
- Three agents are simulated for both Van der Pol and Duffing oscillator examples.The simulations use N = 3 agents and examine the theory presented for diffusively coupled nonlinear systems.
- Van der Pol oscillators: The Van der Pol model uses y⋆ = w1 as its output and the specified nonlinear oscillator dynamics.
- Van der Pol oscillators: The Van der Pol network uses an incidence-matrix topology with coupling parameters g = 10 and a = 1.The corresponding Laplacian eigenvalues satisfy the stated condition with µ = 1.4.
- Van der Pol oscillators: The Van der Pol phase-plane simulation starts the three oscillators at (1, 1), (3, 3), and (5, 5), while Figure 2 reports their first-state time behavior.
- Duffing oscillators: The Duffing simulation uses the same network topology and coupling parameters, with phase-plane and first-state behavior shown for the three oscillators.The Duffing oscillators use the same listed initial conditions: (1, 1), (3, 3), and (5, 5).
6 Conclusions
The paper develops output consensus for networked nonlinear systems by synchronizing nonlinear reference generators and then applying local nonlinear regulators to heterogeneous agents.
- The paper considers consensus among outputs of networked nonlinear systems driven by an autonomous nonlinear exosystem.
- Diffusively coupled nonlinear exosystems reach consensus over a trajectory of the prescribed exosystem.
- Nonlinear output regulation then designs local regulators that make heterogeneous-system outputs track a common reference signal.
- The Duffing example includes phase-plane and first-state time-behavior simulations for three networked oscillators.