Differentiation is an important task in control, observation and fault detection. Levants differentiator is unique, since it is able to estimate exactly and robustly the derivatives of a signal with a bounded high-order derivative. However, the conve
rgence time, although finite, grows unboundedly with the norm of the initial differentiation error, making it uncertain when the estimated derivative is exact. In this paper we propose an extension of Levants differentiator so that the worst case convergence time can be arbitrarily assigned independently of the initial condition, i.e. the estimation converges in emph{Fixed-Time}. We propose also a family of continuous differentiators and provide a unified Lyapunov framework for analysis and design.
Network reconstruction is the first step towards understanding, diagnosing and controlling the dynamics of complex networked systems. It allows us to infer properties of the interaction matrix, which characterizes how nodes in a system directly inter
act with each other. Despite a decade of extensive studies, network reconstruction remains an outstanding challenge. The fundamental limitations governing which properties of the interaction matrix (e.g., adjacency pattern, sign pattern and degree sequence) can be inferred from given temporal data of individual nodes remain unknown. Here we rigorously derive necessary conditions to reconstruct any property of the interaction matrix. These conditions characterize how uncertain can we be about the coupling functions that characterize the interactions between nodes, and how informative does the measured temporal data need to be; rendering two classes of fundamental limitations of network reconstruction. Counterintuitively, we find that reconstructing any property of the interaction matrix is generically as difficult as reconstructing the interaction matrix itself, requiring equally informative temporal data. Revealing these fundamental limitations shed light on the design of better network reconstruction algorithms, which offer practical improvements over existing methods.
