Structural Invertibility and Optimal Sensor Node Placement for Error and Input Reconstruction in Dynamic Systems

Abstract

Despite recent progress in our understanding of complex dynamic networks, it remains challenging to devise sufficiently accurate models to observe, control or predict the state of real systems in biology, economics or other fields. A largely overlooked fact is that these systems are typically open and receive unknown inputs from their environment. A further fundamental obstacle are structural model errors caused by insufficient or inaccurate knowledge about the quantitative interactions in the real system.Here, we show that unknown inputs to open systems and model errors can be treated under the common framework of invertibility, which is a requirement for reconstructing these disturbances from output measurements. By exploiting the fact that invertibility can be decided from the influence graph of the system, we analyse the relationship between structural network properties and invertibility under different realistic scenarios. We show that sparsely connected scale free networks are the most difficult to invert. We introduce a new sensor node placement algorithm to select a minimum set of measurement positions in the network required for invertibility. This algorithm facilitates optimal experimental design for the reconstruction of inputs or model errors from output measurements. Our results have both fundamental and practical implications for nonlinear systems analysis, modelling and design.

Development of a hybrid model for a partially known intracellular signaling pathway through correction term estimation and neural network modeling

“…Specifically, we choose x c from x so that the resulted system is close to be invertible [40]. If a system is invertible, for a given value of x 0 , unique values of y will correspond to unique values of inputs, so one could reconstruct the values of inputs from available output measurements [41,42]. If w c in the hybrid model (Eqs 5b and 5c) is viewed as an input to the system and the hybrid model is invertible, the values of w c can be uniquely characterized from given measurements [41,43].…”

Searching for Errors in Models of Complex Dynamic Systems

“…In nearly all cases we have to deal with the presence of unknown structural model errors . We call these model errors structural , for they can lie in the functional form or in the very network topological structure of a system and can not be fixed by adjusting the parameters, compare (Engelhardt et al, 2016 , 2017 ; Kahl et al, 2019 ; Villaverde et al, 2019 ).…”

“…Algorithms for recovering the errors of a given model (1) from data implicitly assume that the root cause for an error can uniquely be identified (Kolodziej and Mook, 2005 ; Engelhardt et al, 2016 , 2017 ; Villaverde et al, 2019 ; Newmiwaka et al, 2020 ). However, this uniqueness assumption is often violated because the output function c does not provide sufficient information (Engelhardt et al, 2016 ; Kahl et al, 2019 ) about the root cause for the observed discrepancy y − y data between model and data. Based on the representation of model errors as unknown inputs (Kolodziej and Mook, 2005 ; Engelhardt et al, 2016 ) to the system (1), it was possible to relate the problem of error reconstruction to the problem of system invertibility (Kahl et al, 2019 ).…”

“…However, this uniqueness assumption is often violated because the output function c does not provide sufficient information (Engelhardt et al, 2016 ; Kahl et al, 2019 ) about the root cause for the observed discrepancy y − y data between model and data. Based on the representation of model errors as unknown inputs (Kolodziej and Mook, 2005 ; Engelhardt et al, 2016 ) to the system (1), it was possible to relate the problem of error reconstruction to the problem of system invertibility (Kahl et al, 2019 ). This enabled us to utilize algebraic and graph theoretic results (Fliess, 1988 ; Wey, 1998 ) to derive conditions on the system (1), which guarantee the unique reconstruction of model errors (Kahl et al, 2019 , 2020 ).…”

“…Based on the representation of model errors as unknown inputs (Kolodziej and Mook, 2005 ; Engelhardt et al, 2016 ) to the system (1), it was possible to relate the problem of error reconstruction to the problem of system invertibility (Kahl et al, 2019 ). This enabled us to utilize algebraic and graph theoretic results (Fliess, 1988 ; Wey, 1998 ) to derive conditions on the system (1), which guarantee the unique reconstruction of model errors (Kahl et al, 2019 , 2020 ). The analysis of biological systems revealed that the unique reconstruction of model errors is often difficult, unless the measurements [i.e., c in (1)] are carefully selected (Kahl et al, 2019 ).…”