Set-based state estimation and fault diagnosis using constrained zonotopes and applications

This doctoral thesis develops new methods for set-based state estimation and active fault diagnosis (AFD) of (i) nonlinear discrete-time systems, (ii) discrete-time nonlinear systems whose trajectories satisfy nonlinear equality constraints (called invariants), (iii) linear descriptor systems, and (iv) joint state and parameter estimation of nonlinear descriptor systems. Set-based estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. To address this issue, the present doctoral thesis proposes new methods for efficiently propagating constrained zonotopes (CZs) through nonlinear mappings. Besides, this thesis improves the standard prediction-update framework for systems with invariants using new algorithms for refining CZs based on nonlinear constraints. In addition, this thesis introduces a new approach for set-based AFD of a class of nonlinear discrete-time systems. An affine parametrization of the reachable sets is obtained for the design of an optimal input for set-based AFD. In addition, this thesis presents new methods based on CZs for set-valued state estimation and AFD of linear descriptor systems. Linear static constraints on the state variables can be directly incorporated into CZs. Moreover, this thesis proposes a new representation for unbounded sets based on zonotopes, which allows to develop methods for state estimation and AFD also of unstable linear descriptor systems, without the knowledge of an enclosure of all the trajectories of the system. This thesis also develops a new method for set-based joint state and parameter estimation of nonlinear descriptor systems using CZs in a unified framework. Lastly, this manuscript applies the proposed set-based state estimation and AFD methods using CZs to unmanned aerial vehicles, water distribution networks, and a lithium-ion cell.Comment: My PhD Thesis from Federal University of Minas Gerais, Brazil. Most of the research work has already been published in DOIs 10.1109/CDC.2018.8618678, 10.23919/ECC.2018.8550353, 10.1016/j.automatica.2019.108614, 10.1016/j.ifacol.2020.12.2484, 10.1016/j.ifacol.2021.08.308, 10.1016/j.automatica.2021.109638, 10.1109/TCST.2021.3130534, 10.1016/j.automatica.2022.11042

Classification Algorithms based on Generalized Polynomial Chaos

Classification is one of the most important tasks in process system engineering. Since most of the classification algorithms are generally based on mathematical models, they inseparably involve the quantification and propagation of model uncertainty onto the variables used for classification. Such uncertainty may originate from either a lack of knowledge of the underlying process or from the intrinsic time varying phenomena such as unmeasured disturbances and noise. Often, model uncertainty has been modeled in a probabilistic way and Monte Carlo (MC) type sampling methods have been the method of choice for quantifying the effects of uncertainty. However, MC methods may be computationally prohibitive especially for nonlinear complex systems and systems involving many variables. Alternatively, stochastic spectral methods such as the generalized polynomial chaos (gPC) expansion have emerged as a promising technique that can be used for uncertainty quantification and propagation. Such methods can approximate the stochastic variables by a truncated gPC series where the coefficients of these series can be calculated by Galerkin projection with the mathematical models describing the process. Following these steps, the gPC expansion based methods can converge much faster to a solution than MC type sampling based methods. Using the gPC based uncertainty quantification and propagation method, this current project focuses on the following three problems: (i) fault detection and diagnosis (FDD) in the presence of stochastic faults entering the system; (ii) simultaneous optimal tuning of a FDD algorithm and a feedback controller to enhance the detectability of faults while mitigating the closed loop process variability; (iii) classification of apoptotic cells versus normal cells using morphological features identified from a stochastic image segmentation algorithm in combination with machine learning techniques. The algorithms developed in this work are shown to be highly efficient in terms of computational time, improved fault diagnosis and accurate classification of apoptotic versus normal cells