State Observation and Sensor Selection for Nonlinear Networks

Abstract

A large variety of dynamical systems, such as chemical and biomolecular systems, can be seen as networks of nonlinear entities. Prediction, control, and identification of such nonlinear networks require knowledge of the state of the system. However, network states are usually unknown, and only a fraction of the state variables are directly measurable. The observability problem concerns reconstructing the network state from this limited information. Here, we propose a general optimization-based approach for observing the states of nonlinear networks and for optimally selecting the observed variables. Our results reveal several fundamental limitations in network observability, such as the trade-off between the fraction of observed variables and the observation length on one side, and the estimation error on the other side. We also show that, owing to the crucial role played by the dynamics, purely graphtheoretic observability approaches cannot provide conclusions about one's practical ability to estimate the states. We demonstrate the effectiveness of our methods by finding the key components in biological and combustion reaction networks from which we determine the full system state. Our results can lead to the design of novel sensing principles that can greatly advance prediction and control of the dynamics of such networks.

Recovering the Structural Observability of Composite Networks via Cartesian Product

“…The numerical value of nonzero entries of J depends on the operating (linearization) point, however the zero-nonzero pattern of J is irrespective of the operating point [41]. Therefore, the structural observability results are valid for linearization of nonlinear systems [40].…”

Sensor Placement Strategies for Some Classes of Nonlinear Dynamic Systems via Lyapunov Theory

“…First, Qi et al [12] utilize the concept of empirical Gramians to obtain a measure towards observability in NDS, which is then used to determine the location of phasor measurement unit in power networks. Next, Haber et al [13] present a method for reconstructing the initial states x(t 0 ) of NDS while optimally selecting sensors for a given observation window. It has been argued in [13] that the proposed approach is computationally more tractable than the one in [12].…”

Particle filtering of dynamical networks: Highlighting observability issues

“…For instance, it is arguable that works grounded on crisp definitions of observability (controllability), such as Lin's structural definition [33,9] and following works [35,36], are not really apt to classify if a certain set of sensor nodes is really the best one or even if its feasible. Indeed, the pioneering methods of Liu et al [35,36] have been shown to underestimate the required set of sensor [23,32] and driver [20,27,47] nodes. A natural approach to circumvent this problem is to extend metrics that gradually quantify system observability (controllability), under a specific set of sensor (driver) nodes, to a network context [39,21,45].…”

“…Cowan et al [14] show that significantly different results could be derived by considering the presence of self-edges that dictate the independent dynamic behaviour of any given node. This includes a new level of depth to the observability problem of network systems, specially when nonlinear dynamics are considered [14,39,21,45,44,27,23,5,32].…”

“…Figure 1 illustrates the corresponding graph of a given network topology when nodal dynamics are considered. Despite recent investigations using small motifs [38,50,44], the interplay between observability, nodal dynamics, especially with nonlinearities, and network topology is still not clear [23,5].…”