Localization of Invariable Sparse Errors in Dynamic Systems

Understanding the dynamics of complex systems is a central task in many different areas ranging from biology via epidemics to economics and engineering. Unexpected behaviour of dynamic systems or even system failure is sometimes difficult to comprehend. Such a data-mismatch can be caused by endogenous model errors including misspecified interactions and inaccurate parameter values. These are often difficult to distinguish from unmodelled process influencing the real system like unknown inputs or faults. Localizing the root cause of these errors or faults and reconstructing their dynamics is only possible if the measured outputs of the system are sufficiently informative. Here, we present criteria for the measurements required to localize the position of error sources in large dynamic networks. We assume that faults or errors occur at a limited number of positions in the network. This invariable sparsity differs from previous sparsity definitions for inputs to dynamic systems. We provide an exact criterion for the recovery of invariable sparse inputs to nonlinear systems and formulate an optimization criterion for invariable sparse input reconstruction. For linear systems we can provide exact error bounds for this reconstruction method.