Observability of nonlinear systems with unmeasured inputs

This paper presents a geometric algorithm to investigate the theoretical observability of nonlinear systems with partially measured inputs and outputs. The algorithm is based on Lie algebra and applies to systems whose state and measurement equations are analytical and affine in all inputs. It investigates whether the system satisfies a necessary observability condition that is named the Observability Rank Condition for systems with Direct Feedthrough (ORC-DF). The presented algorithm allows to assess the observability of the dynamic system states, the identifiability of constant-in-time parameters, and the ability to track unmeasured inputs, which is referred to as system invertibility. It is also shown how the developed methodology can be extended to investigate the observability of non-smooth systems that can be broken into different smooth branches, often encountered in mechanical applications related to sliding and damage. Possible applications are illustrated with several examples from structural engineering