Skip to Main content Skip to Navigation

Non linear Unknown Input Observability: Basic Properties and the Case of a Single Unknown Input

Agostino Martinelli 1
1 E-MOTION - Geometry and Probability for Motion and Action
Inria Grenoble - Rhône-Alpes, LIG - Laboratoire d'Informatique de Grenoble
Abstract : This paper investigates an open problem in control theory (the unknown input observability in the non linear case) and provides its solution in the case of a single unknown input. Additionally, it provides sufficient conditions in the case of multiple unknown inputs. The first contribution is an extension of the observability rank criterion to deal with these systems driven by several unknown inputs. This extension is obtained by a suitable state augmentation and is called the extended observability rank criterion. In the case of a single unknown input, the paper provides a complete answer to the problem of state observability (second and main contribution). In this case, it is provided an analytical method to directly obtain the entire observable codistribution. As in the standard case of only known inputs, the observable codistribution is obtained recursively by computing Lie derivatives along the vector fields that characterize the state dynamics. On the other hand, in correspondence of the unknown input, the corresponding vector field must be replaced by dividing the original one by the first order Lie derivative of the considered output along the original vector field. Additionally, the observable codistribution also includes the span of the gradients of the Lie derivatives of the outputs along a new set of vector fields. This set is obtained iteratively, starting by computing all the Lie brackets of the vector fields that correspond to the known inputs with the vector field that corresponds to the unknown input and by rescaling the obtained vectors by dividing by the previously mentioned first order Lie derivative. In practice, as in the standard case of only known inputs, the entire observable codistribution is obtained by a very simple recursive algorithm. Hence, the overall method to obtain all the observability properties is very simple. On the other hand, the analytic derivations required to prove that this codistribution fully characterizes the state observability properties are very complex. Finally, it is shown that the recursive algorithm converges in a finite number of steps and the criterion to establish that the convergence has been reached is provided (third contribution). Also this proof is based on several tricky and very complex analytical steps. The proposed approach is used to derive the observability properties of several systems, starting from simple ones. The last application is a very complex unknown input observability problem. Specifically, we derive the observability properties for the visual-inertial structure from motion problem in the case when part of the inertial inputs are missing.
Complete list of metadatas
Contributor : Agostino Martinelli <>
Submitted on : Thursday, January 8, 2015 - 8:39:40 PM
Last modification on : Tuesday, February 9, 2021 - 3:14:06 PM
Long-term archiving on: : Thursday, April 9, 2015 - 10:51:23 AM


Files produced by the author(s)


  • HAL Id : hal-01071314, version 2


Agostino Martinelli. Non linear Unknown Input Observability: Basic Properties and the Case of a Single Unknown Input. [Research Report] RR-8608, INRIA Grenoble. 2014. ⟨hal-01071314v2⟩



Record views


Files downloads