Scalable Practical Synchronization of Heterogeneous Linear Systems

We study practical state synchronization of heterogeneous linear multi-agent systems over directed graphs using static diffusive state feedback. Persistent mismatches between the agent dynamics generally preclude exact synchronization and are treated as bounded disturbances within an Input-to-State Stability (ISS) framework. We derive a scalable synthesis procedure based on fixed-dimension linear matrix inequalities (LMIs), whose dimension is independent of the number of agents. Local robust dissipation conditions are combined through a small-gain argument to obtain global ISS guarantees and explicit bounds on the synchronization error. The framework is formulated in a unified manner for continuous- and discrete-time systems, while highlighting a fundamental difference in disturbance attenuation between the two domains. In particular, exact ZOH discretization of a continuous-time controller need not preserve closed-loop stability, motivating a native discrete-time synthesis. Numerical experiments on heterogeneous harmonic oscillators illustrate the proposed approach and the continuous/discrete-time distinction.