Observability of nonlinear discrete-time systems with unmeasured inputs

The observability analysis of a dynamic system for a given sensor setup offers useful insights on the ability to estimate its states and parameters. Recent works have further extended the concept of observability to systems under unmeasured inputs, while efficient algorithms have allowed the study of realistic dynamic systems which typically involve multiple degrees of freedom and parameters to be identified. Observability algorithms however typically rely on the assumption that the measurements are obtained continuously in time. While the results obtained from this assumption are still useful, some of the observations from the observability analysis, e.g., the derivatives of the measurements needed for the system to become observable, are hard to link to system identification algorithms. Such algorithms use measurements that have been logged at some finite sampling frequency, while the inputs are often represented using a discretization in time. In this work, the concept of (state, parameter and input) observability under the presence of unmeasured inputs is extended to discrete-time systems. This allows accounting for the measurements and inputs being sampled at some finite frequency. The outcomes of the observability analysis are extended to include the number of measurements needed for a state, parameter or input to become observable. The work will discuss how this property is especially important in the case of unmeasured inputs. An efficient algorithm to calculate the observability of discrete systems under the presence of unmeasured inputs is introduced. An efficient numerical implementation of the algorithm is introduced using stabilization of the results across different numerical precisions. The algorithm is then used for dynamic systems which are discretized by methods of different order, and the effect of this process on the observability properties of the system is discussed. Suitable examples demonstrate how the observability of discrete-time systems with unmeasured inputs can guide significant decisions in terms of sensor placement and the modelling of the system.