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We present an analysis framework for the study of synchronization of heterogeneous nonlinear systems interconnected over networks described by directed graphs. Heterogeneous systems may have totally different dynamical models, albeit of the same dimension, or may possess equal models with different lumped parameters. We show that their behavior, when network-interconnected, is fully characterized in terms of two properties whose study may be recasted in terms of the stability analysis of two corresponding interconnected dynamical systems that evolve in orthogonal spaces: on one hand, we have the so-called emergent dynamics and, on the other, the synchronization error dynamics. Based on this premise, we introduce the concept of dynamic consensus and we present results on robust stability which assess the conditions for practical asymptotic synchronization of networked systems and characterize their collective behavior. To illustrate our main theoretical findings we broach a brief case-study on mutual synchronization of heterogeneous chaotic oscillators.

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In previous papers we have introduced a sufficient condition for uniform attractivity of the origin for a class of nonlinear time-varying systems which is stated in terms of persistency of excitation (PE), a concept well known in the adaptive control and systems identification literature. The novelty of our condition, called uniform δ-PE, is that it is tailored for nonlinear functions of time and state and it allows us to prove uniform asymptotic stability. In this paper we present a new definition of uδ-PE which is conceptually similar to but technically different from its predecessors and give several useful characterizations. We make connections between this property and similar properties previously used in the literature. We also show when this condition is necessary and sufficient for uniform (global) asymptotic stability for a large class of nonlinear time-varying systems. Finally, we show the utility of our main results on some control applications regarding feedforward systems and systems with matching nonlinearities.