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Conditions for synchronizability in arrays of coupled linear systems
S. Emre Tuna
TL;DR
The paper asks which conditions on system and output matrices guarantee synchronization across all connected interconnections in arrays of identical output-coupled linear systems. It analyzes sufficient and nonsufficient conditions and constructs synchronizing feedback laws where possible. Detectability suffices for neutrally stable systems, whereas critically unstable systems require full-state coupling in the stated general guarantee.
Problem
The paper asks what conditions on (C, A) guarantee one feedback law synchronizes arrays across a set of connected interconnections whose coupling matrix and size may be unknown.
Method
The paper proves sufficiency and nonsufficiency statements, constructs feedback laws for sufficient cases, and provides an algorithm for computing one such law.
Results
Detectability suffices when A is neutrally stable, while full column rank of C suffices when A is critically unstable; detectability is generally insufficient in the critically unstable case.
Takeaways & Limitations
For connected directed weighted interconnections, synchronizing feedback can be designed broadly for neutrally stable systems, but critically unstable systems need full-state coupling for the stated guarantee.
Takeaways & Limitations
The results are scoped to the paper’s specified interconnection sets and include an assumption that skew-symmetric neutrally stable systems use observable outputs and connected coupling.
Abstract
from arXiv · showhide
Synchronization control in arrays of identical output-coupled continuous-time linear systems is studied. Sufficiency of new conditions for the existence of a synchronizing feedback law are analyzed. It is shown that for neutrally stable systems that are detectable form their outputs, a linear feedback law exists under which any number of coupled systems synchronize provided that the (directed, weighted) graph describing the interconnection is fixed and connected. An algorithm generating one such feedback law is presented. It is also shown that for critically unstable systems detectability is not sufficient, whereas full-state coupling is, for the existence of a linear feedback law that is synchronizing for all connected coupling configurations.
1 Introduction
The paper studies when identical output-coupled linear systems can synchronize under unknown-size interconnections, focusing on fixed connected graphs without symmetry, balancedness, or coupling-strength assumptions. It establishes sufficient and nonsufficient conditions for synchronizing feedback laws, including detectability for neutrally stable systems and full-state coupling for critically unstable systems.
- Motivation: Synchronization is studied as a control problem for arrays whose interconnection and even size may be unknown to the feedback-law designer.The goal is to find conditions on (C, A) guaranteeing one feedback law synchronizes every configuration in a specified set.
- Problem setting: The paper focuses on all fixed interconnections with connected graphs, without assuming coupling strength, symmetry, or balancedness.This set of interconnections determines the scope of the synchronizability conditions investigated.
- Main results: For detectable (C, A) with neutrally stable A, a synchronizing linear feedback law exists for connected interconnections.The paper also provides an algorithm to explicitly compute such a law.
- Main results: If A is critically unstable and C has full column rank, a synchronizing linear feedback law exists.Full column rank allows the output coupling to recover full-state information through a suitable gain.
- Main results: If (C, A) is detectable while A is critically unstable, detectability alone does not generally guarantee a synchronizing linear feedback law.This nonsufficiency result contrasts with the neutrally stable case.
- Paper structure: The paper proves its conditions, constructs feedback laws for the sufficient cases, and gives an explicit synchronization trajectory.It also studies coupled harmonic oscillators and reports synchronization under connected graphs without symmetry, balancedness, or strong-coupling assumptions.
2 Notation and definitions
This section establishes notation, synchronization and graph concepts, and classes of system pairs and interconnections used throughout the paper.
- System properties: A pair (C, A) is detectable when an output-zero trajectory has state convergence to zero, and A is neutrally stable when imaginary-axis Jordan blocks are size one.Neutral stability also excludes eigenvalues with positive real parts.
- Graphs and interconnections: A directed graph is connected when one node is reachable from every other node, a condition weaker than strong connectivity and stronger than weak connectivity.Figure 1 contrasts a connected graph G1 with a disconnected graph G2.
- System classes: The paper distinguishes Hurwitz, neutrally stable, and non-positive-real-part system matrices, alongside full-column-rank and detectable output pairs.These classes satisfy AH ⊂ AN ⊂ AJ and OF ⊂ OP.
- Graphs and interconnections: The interconnection classes distinguish connected interconnections G>0 from all interconnections G≥0.For connected Γ, zero is a distinct eigenvalue and all other eigenvalues have strictly negative real parts.
3 Problem statement and main theorem
The paper asks when identical output-coupled linear systems can synchronize under one feedback law across all connected interconnections, and develops constructive sufficient and nonsufficient conditions.
- Problem statement: The array consists of identical linear systems with states x_i, inputs u_i, outputs y_i, and coupling signals z_i, governed by matrices A and C.The solution of each system is denoted x_i(t).
- Problem statement: Synchronizability requires a single linear feedback law L that makes all array solutions synchronize for every interconnection in a chosen set and every initial condition.The main question concerns the set G>0 of connected interconnections.
- Contribution: The authors use a constructive approach that computes L whenever possible and provides the explicit trajectory to which the systems synchronize.The results complete previously missing cases in a classification chart.
- Main theorem: Theorem 1 organizes the main conclusions as eight sufficient and nonsufficient synchronizability conditions.The paper follows the theorem with lemmas establishing each statement.
- Main theorem: Not all neutrally stable, full-column-rank pairs synchronize over all interconnections, and not all critically unstable detectable pairs synchronize over connected interconnections.These limitations are stated as Theorem 1 cases (e) and (f), respectively.
- Main theorem: Not all full-column-rank pairs are synchronizable with respect to connected interconnections.This is stated as Theorem 1 case (g).
4 Cases when synchronizing feedback exists
The paper constructs synchronizing feedback laws for several classes of linear systems and explicitly characterizes trajectories toward which the systems converge. For neutrally stable systems, detectability supports synchronization over all fixed connected interconnections, while stronger or more specialized conditions cover other cases.
- The authors compute a synchronizing feedback law and the explicit synchronized trajectory for each sufficiency case.The analysis therefore solves a feedback synthesis problem, not only an existence problem.
- For Hurwitz system matrices, the zero feedback law synchronizes every array over all interconnections by driving each state to zero.With L = 0, the systems decouple and each state converges to the origin.
- Neutrally stable systems: For skew-symmetric systems with observable output coupling and connected interconnections, solutions synchronize to a common trajectory.The proof uses a Lyapunov function, LaSalle’s invariance principle, and observability to show convergence to the synchronized set.
- Neutrally stable systems: For neutrally stable A and detectable (C, A), Algorithm 1 constructs L so arrays synchronize for every fixed connected interconnection.The algorithm separates imaginary-axis and stable modes before constructing the feedback law; the limiting trajectory depends on r satisfying r^TΓ = 0 and r^T1 = 1.
- Critically unstable systems: Full-state coupling suffices for critically unstable systems over all connected interconnections, using L = (C^TC)^−1C^T when C has full column rank.The paper explains that exponential coupling attraction dominates non-exponential divergence when A has no eigenvalues with positive real part.
- Arbitrary system matrices: For detectable pairs and interconnections with coupling strength at least δ, a Riccati-equation-based feedback law ensures synchronization.The gain is L = max{1, δ^−1}P C^T, where P solves the stated algebraic Riccati equation.
5 Cases when no synchronizing feedback exists
The paper gives counterexamples showing that several seemingly sufficient conditions fail for synchronization over broad interconnection sets. In particular, detectability alone does not suffice for critically unstable systems with partial-state coupling.
- There exists a neutrally stable full-state-observable pair that is not synchronizable over all nonnegative interconnections.
- Partial-state coupling is insufficient for critically unstable systems over all connected configurations, despite detectability sufficing in the neutrally stable case.The paper contrasts this nonexistence result with the full-state-coupling result that restores synchronizability.
- There exists a critically unstable observable pair that is not synchronizable over all connected interconnections.The proof constructs connected graphs whose eigenvalues enter a wedge incompatible with any stabilizing feedback gain.
6 Dual problem
The dual problem treats arrays with input coupling through controllability rather than output coupling through observability. A duality theorem transfers synchronizability conditions between the two formulations.
- The dual array consists of p identical linear systems with states, inputs, outputs, and system matrices A and B.
- A pair (B^T, A^T) synchronizable with respect to an interconnection set S if and only if a linear feedback gain K synchronizes every array in S.This follows from observability-controllability duality.
7 Conclusion
The conclusion reports that the paper completes the sufficiency picture for output-coupled linear arrays over connected interconnections and provides synchronizing feedback laws in the positive cases.
- The paper fills previously missing cases in the chart of conditions on (C, A) and interconnection sets governing synchronizability.Figure 2 summarizes sufficiency across different interconnection sets Γ.
- For every case marked as sufficient, the authors design a synchronizing feedback law.