Skip to Main content Skip to Navigation

Stabilization of non-uniformly observable control systems and infinite-dimensional observers

Abstract : This thesis deals with two different but related topics. In the first part, we are concerned with the problem of dynamic output feedback stabilization. When only part of the state of a control system is known, a stabilizing state feedback can not be directly implemented. Hence, a common strategy to stabilize the state to some target point is to design an observer system to asymptotically estimate the state by filtering the output online, and to use as an input the stabilizing state feedback applied to the observer. This approach is known to be efficient on uniformly observable systems, that are observable for all inputs. However, it is not generic for nonlinear systems to be uniformly observable when the dimension of the output is less or equal than the dimension of the input. Hence, in the presence of observability singularities, new techniques need to be developed. In the second part, we focus on the problem of observer design for linear time-varying infinite-dimensional systems. The goal is to design a dynamical system learning the state from the output dynamics. The finite-dimensional notion of observability may be extended in several ways. In particular, one distinguishes exact and approximate observability assumptions. While exponential convergence of Luenberger observers can be proved on exactly observable systems, much less is known for approximate observability-like hypotheses, on which we focus. These observers can also be used in the context of offline reconstruction of initial data. The procedure is based on iterations of forward and backward observers, and named Back and Forth Nudging (BFN). Such methods can be applied to a batch crystallization process, where the state to be estimated is the Particle Size Distribution (PSD). Contribution 1. On Single-Input Single-Output (SISO) bilinear systems with observable target, generic perturbations of the feedback law guarantee that the inputs produced by the closed-loop system render the system observable. Contribution 2. On state-affine dissipative systems that are state feedback stabilizable, 0-detectability is a necessary and sufficient condition to the existence of a globally stabilizing dynamic output feedback. Contribution 3. On examples of nonlinear systems, we illustrate three main guidelines for output feedback stabilization at an unobservable target: - additive perturbations of the state feedback law yield new observability properties without preventing the stabilization process; - observers with dissipative error system are robust to observability singularities; - embeddings into finite or infinite-dimensional systems allow to design Luenberger observers with dissipative error systems. Contribution 4. Up to a weak detectability assumption, infinite-dimensional Luenberger observers estimate the observable part of the state in the weak topology of the state space. Strong convergence can be obtained with additional assumptions on the error system. Contribution 5. The convergence results of Contribution 4 can be adapted to the BFN context. Contribution 6. In the context of a batch crystallization process, we propose several strategies to reconstruct the PSD: - a direct approach based on a Tikhonov regularization, using the knowledge of the Chord Length Distribution (CLD); - a Kazantzis-Kravaris/Luenberger (KKL) observer, using the knowledge of temperature and solute concentration; - an infinite-dimensional Luenberger observer, based on Contributions 4 and 5, using the knowledge of the CLD.
Document type :
Complete list of metadata
Contributor : ABES STAR :  Contact
Submitted on : Tuesday, April 19, 2022 - 5:11:17 PM
Last modification on : Thursday, August 4, 2022 - 5:15:54 PM


Version validated by the jury (STAR)


  • HAL Id : tel-03247536, version 2



Lucas Brivadis. Stabilization of non-uniformly observable control systems and infinite-dimensional observers. Automatic. Université de Lyon, 2021. English. ⟨NNT : 2021LYSE1085⟩. ⟨tel-03247536v2⟩



Record views


Files downloads