Observability of nonlinear systems with unmeasured inputs

Abstract

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.

Lie Symmetries, Observability and Model Transformation of Nonlinear Systems With Unknown Inputs

“…On the contrary, the parameters k 1 , k 2 , m, β and γ are unidentifiable and the unmeasured excitation F is unobservable. The observability results suggested by the symmetries are in agreement with the results output from the observability algorithm ORC-DF [11].…”

“…A :ṙ = v 1 − βv 1 r 2 − γv 1 r 2 , when v 1 > 0 and r > 0 (22) B :ṙ = v 1 + βv 1 r 2 − γv 1 r 2 , when v 1 < 0 and r > 0 C :ṙ = v 1 + βv 1 r 2 − γv 1 r 2 , when v 1 > 0 and r < 0 D :ṙ = v 1 − βv 1 r 2 − γv 1 r 2 , when v 1 < 0 and r < 0 For the sake of brevity, only Lie symmetries of branch A are studied, while the detailed discussions on the observability properties of the complete non-smooth system can be referred in the works [1,11]. For branch A, Algorithm I gives a 2-parameter group of Lie symmetries that is a combination of a group of translations and a group of scalings:…”

“…Recent developments of novel identification methods such as [6,7,8,9] allow for estimating the unmeasured and hence unknown inputs of a system reliably as well as its state variables and parameters. To meet the demand of observability analysis when invoking such identification methods, the Observability Rank Condition (ORC) [3] was extended to alleviate the constraint that all inputs need to be measured [10,11]. Recently the authors achieved efficient automated observability testing of large linear systems with unknown parameters [12], relaxing the significant computational constraints of the default implementations of the ORC.…”

Full observability and estimation of unknown inputs, states and parameters of nonlinear biological models

“…However, this would come at the expense of a loss of generality, and our approach is currently meant to be applicable to any analytic nonlinear system. A promising step in this direction is the algorithm by Maes et al [ 55 ], which has been recently proposed for mechanical systems that are affine in all inputs.…”

Tracking of inputs, states and parameters of linear structural dynamic systems

“…This finding is in line with what was found by Maes et al [14] for the case of joint input-state estimation. Furthermore, the authors have recently developed an analytical method to investigate the observability of nonlinear systems in presence of unmeasured inputs, which can also be applied to the case of joint input-state-parameter estimation [15].…”