Exponential Consensus of Linear Systems Over Switching Network: A Subspace Method to Establish Necessity and...

Area of Science:

• Control Theory
• Networked Systems
• Systems Engineering

Background:

• The consensus problem in linear systems is crucial for distributed control and coordination.
• Existing methods often require strong network connectivity assumptions.
• Marginally stable systems with non-full-row rank input matrices present unique challenges.

Purpose of the Study:

• To revisit the consensus problem of linear systems from a novel geometric perspective.
• To establish necessary and sufficient conditions for exponential consensus under mild connectivity assumptions.
• To characterize the convergence rate lower bound for such systems.

Main Methods:

• Geometric analysis of system subspaces determined by the interaction network.
• Utilizing an observability condition to establish initial convergence.
• Leveraging joint connectivity properties to extend convergence uniformly.
• Deriving conditions based on joint connectivity and system/input matrices.

Main Results:

• Convergence is established by examining network-determined subspaces and an observability condition.
• Uniform convergence is extended to the orthogonal complement of the consensus manifold.
• Necessary and sufficient conditions for exponential consensus are derived.
• The lower bound of the convergence rate is characterized.

Conclusions:

• Exponential consensus can be achieved globally and uniformly if a jointly (δ,T)-connected condition and specific observability conditions are met.
• The geometric perspective offers new insights into consensus control for challenging systems.
• The findings provide a theoretical foundation for designing robust consensus protocols.