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Synchronization in Networks of Identical Linear Systems
Luca Scardovi, Rodolphe Sepulchre
TL;DR
The paper asks how identical linear systems can synchronize over general directed and time-varying communication graphs. It constructs a dynamic output-feedback coupling and shows exponential synchronization under stabilizability, detectability, no exponentially unstable modes, and uniform connectivity; static diffusive coupling requires stronger graph conditions.
Problem
The problem is to synchronize identical systems while accounting simultaneously for their individual dynamics and general communication limitations.
Method
The paper constructs a distributed dynamic output-feedback controller using output and controller-state differences over a time-varying directed communication graph.
Results
The controller achieves exponential synchronization when A has no exponentially unstable mode, (A, B) is stabilizable, (A, C) is detectable, and the graph is uniformly connected.
Takeaways & Limitations
The result generalizes classical consensus algorithms and extends synchronization to nontrivial linear dynamics such as harmonic oscillators and chains of integrators.
Takeaways & Limitations
The main result excludes exponentially unstable modes; handling such modes requires graph connectivity strong enough to dominate their instability.
Abstract
from arXiv · showhide
The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient but to some extent, also necessary to ensure synchronization with the diffusive static output coupling often considered in the literature.
1 Introduction
The paper frames synchronization as a control problem involving both individual linear dynamics and limited, possibly time-varying communication. It constructs a dynamic output-feedback coupling under broad stability and connectivity assumptions, and contrasts it with more restrictive static diffusive coupling.
- Motivation: Consensus emphasizes communication constraints, whereas synchronization emphasizes individual dynamics that may oscillate or be chaotic without communication.Coordination problems can involve both aspects simultaneously.
- Problem setting: The paper studies N identical linear state-space agents and seeks synchronization to a common solution of the individual dynamics.The objective is posed for general interconnection structures rather than only complete graphs.
- Main result: A dynamic output-feedback controller ensures exponential synchronization when A has no exponentially unstable mode, (A, B) is stabilizable, (A, C) is detectable, and the graph is uniformly connected.The result covers time-varying and directed interconnection structures.
- Interpretation: The result generalizes classical consensus algorithms, recovered as the special case A = 0, and includes harmonic oscillators and chains of integrators.These examples show that the framework includes nontrivial agent dynamics.
- Comparison with static coupling: Static diffusive output coupling requires more stringent graph assumptions, and synchronization can fail when those assumptions are not satisfied.The paper provides sufficient conditions and examples illustrating this limitation.
2 Preliminaries
The preliminaries define the graph and consensus framework used to formulate synchronization for identical systems. They characterize uniformly connected time-varying digraphs and specify synchronization as convergence to a common open-loop trajectory under distributed coupling.
- Notation: The notation includes stacked vectors, identity matrices, the all-ones vector, and Kronecker products for representing networks compactly.The Kronecker product is associative and supports the paper’s stacked-system notation.
- Communication graphs: A time-varying weighted digraph encodes which systems exchange information, with bounded piecewise-continuous weights and nonzero links bounded between η and γ.The graph uses nodes, directed edges, an adjacency matrix, neighbors, paths, and an associated Laplacian.
- Connectivity: Uniform connectivity means that, over every interval [t, t + T], all nodes connect through directed paths to one common node.The definition permits time-varying and directed communication patterns.
- Consensus preliminaries: Classical continuous- and discrete-time consensus protocols achieve asymptotic convergence to a common value under uniformly connected, bounded, piecewise-continuous graph dynamics.The consensus equilibrium set is stated to be uniformly exponentially stable.
- Synchronization problem: The synchronization problem asks for a distributed control law that makes identical systems converge asymptotically to a solution of the open-loop system.The coupling uses output differences and controller-state differences, and dynamic laws contain an internal controller state.
3 Synchronization of linear systems with state feedback
The paper generalizes consensus to identical linear systems by using state-feedback couplings over uniformly connected graphs. Under stabilizability and nonpositive-real-part eigenvalue conditions, the resulting network exponentially synchronizes to an open-loop trajectory.
- State-feedback synchronization: For systems with imaginary-axis eigenvalues and nonsingular B, the proposed control law exponentially synchronizes all solutions to a trajectory of ˙x0 = Ax0.The result applies when the communication graph is uniformly connected and its Laplacian is bounded and piecewise continuous.
- State-feedback synchronization: Stable modes synchronize to zero without coupling, whereas exponentially unstable modes require graph connectivity strong enough to dominate their divergence.This bounds the result’s direct applicability to systems without exponentially unstable modes under the stated connectivity condition.
- State-feedback synchronization: Stabilizability replaces the stronger nonsingularity assumption on B by using a stabilizing feedback matrix K such that A+BK is Hurwitz.The resulting dynamic construction separates the closed-loop dynamics into decoupled subsystems, enabling synchronization under the theorem’s graph assumptions.
- State-feedback synchronization: Theorem 2 guarantees exponential synchronization when A has eigenvalues in the closed left-half plane, (A, B) is stabilizable, and the graph is uniformly connected.The synchronized trajectory is a solution of the open-loop system ˙x0 = Ax0.
4 Synchronization of linear systems with output feedback
The paper develops dynamic output-feedback controllers that exponentially synchronize identical linear systems under stabilizability, detectability, non-unstable dynamics, and uniformly connected communication graphs. Static diffusive output coupling is also analyzed, but it requires stronger graph and passivity conditions.
- Dynamic output feedback: The output-feedback controller uses an observer, and its estimation error dynamics are exponentially stable and decoupled from the consensus dynamics.The closed-loop system is analyzed as a cascade of the synchronization dynamics and stable estimation-error dynamics.
- Dynamic output feedback: Theorem 3 achieves exponential synchronization to a solution of ˙x0 = Ax when A has no exponentially unstable modes and the graph is uniformly connected.The system must be stabilizable and detectable, with gains making A + BK and A + HC Hurwitz.
- Dynamic output feedback: The general synchronization result requires a dynamic controller rather than simple static output feedback.Static diffusive coupling is treated separately under stronger assumptions.
- Static diffusive coupling: Static diffusive coupling is sufficient under passivity and observability together with connected balanced graphs or symmetric graphs satisfying uniform observability.The stated assumptions also require a bounded, piecewise-continuous Laplacian.
- Static diffusive coupling: Under the static-coupling assumptions, Lyapunov analysis yields exponential convergence to the synchronization subspace, and the vanishing coupling identifies the limit with an open-loop solution.The proof uses graph connectivity, observability, and passivity to establish decay before showing convergence to the uncoupled system’s limit set.
5 Extensions and Generalizations
The paper extends its synchronization results to discrete-time and periodic linear systems. Uniform graph connectivity remains central, while discrete-time stability uses the closed unit disk and periodic-system stability uses characteristic exponents in the closed left half-plane.
- 5.1 Discrete-Time Linear Systems: A discrete-time change of variables converts the system into a consensus problem and yields exponential convergence under the stated connectivity and spectral conditions.The convergence estimate uses constants γ > 0 and 0 < q < 1.
- 5.1 Discrete-Time Linear Systems: In discrete time, uniformly connected graphs with bounded piecewise-continuous Laplacians support exponential synchronization when A has eigenvalues in the closed unit disk.For suitable gains, A + BK and A + HC are Schur matrices.
- 5.1 Discrete-Time Linear Systems: The discrete-time dynamic controller synchronizes all solutions to a solution of x0(t + 1) = Ax0(t).The output-feedback formulation is obtained from the continuous-time results through the discrete-time counterpart of the synchronization lemma.
- 5.2 Periodic Linear systems: Uniformly connected graphs and characteristic exponents in the closed left half-plane yield exponential synchronization to a solution of ˙x0 = A(t)x0.The periodic extension follows the same proof pattern after the Floquet transformation.
- 5.2 Periodic Linear systems: For periodic systems, Floquet theory transforms the time-varying model into a constant system whose eigenvalues are the characteristic exponents.The transformation is continuous, nonsingular, and periodic.
6 Examples
The examples compare the proposed dynamic controller with static diffusive coupling on time-varying communication topologies for harmonic oscillators and double integrators. In both cases, dynamic control achieves exponential synchronization, whereas diffusive interconnection does not.
- Example 1: Synchronization of harmonic oscillators: The harmonic-oscillator example uses a time-varying topology and compares dynamic control with static diffusive coupling.The topology period is set to 7 sec for a group of 4 oscillators.
- Example 1: Synchronization of harmonic oscillators: Dynamic control ensures exponential synchronization of the harmonic oscillators to a solution of the harmonic oscillator.The example uses stabilizing gain K = (0 −1).
- Example 1: Synchronization of harmonic oscillators: Synchronization is not observed for the harmonic oscillators with the diffusive interconnection.The static comparison is made using coupling (25).
- Example 1: Synchronization of harmonic oscillators: The harmonic-oscillator example indicates that passivity of (A, B, −K) would support synchronization with diffusive coupling under stronger graph assumptions.The paper notes a discrete-time result for time-invariant connected graphs and diffusive coupling.
- Example 2: Consensus for double integrators: Dynamic control ensures exponential synchronization for the double integrators, whereas synchronization is not observed with the diffusive interconnection.The static comparison uses coupling (26).
7 Conclusion and future work
The paper addresses synchronization of identical linear systems under general interconnection topologies and provides a dynamic controller for exponential convergence under stated system and connectivity assumptions. It also characterizes stronger conditions for static diffusive coupling and identifies nonlinear synchronization as future work.
- Conclusion: The paper provides a dynamic controller that ensures exponential convergence to a synchronized solution when A has no exponentially unstable mode, (A, B) is stabilizable, (A, C) is detectable, and the graph is uniformly connected.The systems are identical and described by the state-space model (A, B, C).
- Conclusion: Static diffusive output coupling requires stronger conditions that are sufficient and, to some extent, necessary for synchronization.These conditions concern synchronization under the often-considered static diffusive coupling.
- Future work: Extending the proposed technique to synchronization of nonlinear systems remains future work.