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Modeling dynamic systems requires to account for uncertainties arising from noises impacting the measures and/or ! the dynamics, from lack of knowledge about disturbances, and also from uncertainties on parameter values (tolerance specifications, wear processes). Some of these uncertainties, like measurement noises, can be properly modelled in stat...
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Citations
... While estimating the most likely state is sufficient for non-critical applications, critical systems require the entire set of possible states from guaranteed state estimation [1,10,11,18,22,61]. The set of possible states can be used to rigorously predict future behaviors [5,25], perform robust control [40,60], perform conformance checking [58], or apply robust fault detection [75]. We consider rigorous state estimation of linear time-invariant systems, which can also be used to observe nonlinear systems when using conservative linearization techniques, see, e.g., [8,20]. ...
Guaranteed state estimation computes the set of possible states of dynamical systems given the bounds of model uncertainties, disturbances, and noises. For the first time, we evaluate and compare a broad class of guaranteed state estimators for linear time-invariant systems regarding scalability, accuracy, and real-time applicability. In particular, we consider strip-based observers, set-propagation observers, and interval observers. The performance of most guaranteed state estimators is significantly affected by the chosen set representation. Zonotopes have emerged as a popular set representation since important operations such as linear maps and Minkowski sum can be computed exactly and efficiently. Furthermore, we provide comparisons with ellipsoids and constrained zonotopes as set representations. The comparison is conducted on various models for state estimation of autonomous vehicles.
... In 1997, Chen et al. have developped an extension of Kalman filter considering bounded uncertainties on parameters and gaussian measurement noise by using interval analysis; it is named IKF (Interval Kalman Filter) [19]. An improvement of IKF, named iIKF (improved Interval Kalman Filter) has been developped by J. Xiong during his PhD that I co-supervised with L. Travé-Massuyès [136]. In particular, the approach proposed in [19] does not provide guaranteed results because of the simplification used to avoid interval matrix inversion. ...
... The strategy of bisection is an important issue, which can influence the efficiency of an algorithm. An overview of different strategies is given in [136]. The choice of strategy is based on the algorithm requirements, considering a tradeoff between efficiency, convergence, implementation complexity and speed. ...
... The first criterion to consider is the linearity of the constraints, which defines two categories, linear CSP s and nonlinear CSP s. More information on CSP s can be found in [15] or in [136]. ...
Dans ce travail, nous nous intéressons principalement à la surveillance préventive des systèmes non linéaires à incertitudes bornées, c’est-à-dire des systèmes pour lesquels les incertitudes ne sont définies que par leur appartenance à des intervalles. Pour cela, nous nous sommes placés dans un contexte dit « ensembliste » dans lequel nous avons pu étendre deux propriétés importantes largement étudiées en contexte stochastique qui sont l’identifiabilité et la diagnosticabilité.
Au delà des définitions conceptuelles requises par ce travail, nous avons proposé des outils liés à l’algèbre différentielle permettant de vérifier ces deux propriétés.L’impact de l’identifiabilité ensembliste sur les résultats d’une estimation de paramètres a également été analysé, l’estimation de paramètres étant une des approches retenues pour la détection et l’isolation de défauts dans ce travail.
Nous avons également cherché améliorer l’estimation de paramètres en développant des critères pour la planification d’expériences, consistant ici en l’optimisation des conditions initiales, entrées et/ou période d’échantillonnage. Ces critères ont été appliqués à deux cas d’étude (pharmacocinétique et aéronautique) avec d'excellents des résultats.
Par ailleurs, nous nous sommes intéressés à la modélisation des systèmes à incertitudes « mixtes », combinant des incertitudes bornées et stochastiques, en proposant notamment une amélioration du filtre de Kalman par intervalles.
Le travail réalisé formalise des problèmes non abordés auparavant et pose des jalons pour les recherches futures. Les perspectives de travail sont nombreuses tant la thématique considérée est riche et les applications diverses.
Mots clefs : identifiabilité ensembliste, diagnosticabilité ensembliste, planification d'expériences ensembliste, estimation ensembliste, systèmes non linéaires à incertitudes bornées, incertitudes mixtes.
... In 1997, Chen et al. have developped an extension of Kalman filter considering bounded uncertainties on parameters and gaussian measurement noise by using interval analysis; it is named IKF (Interval Kalman Filter) [19]. An improvement of IKF, named iIKF (improved Interval Kalman Filter) has been developped by J. Xiong during his PhD that I co-supervised with L. Travé-Massuyès [136]. In particular, the approach proposed in [19] does not provide guaranteed results because of the simplification used to avoid interval matrix inversion. ...
... The strategy of bisection is an important issue, which can influence the efficiency of an algorithm. An overview of different strategies is given in [136]. The choice of strategy is based on the algorithm requirements, considering a tradeoff between efficiency, convergence, implementation complexity and speed. ...
... The first criterion to consider is the linearity of the constraints, which defines two categories, linear CSP s and nonlinear CSP s. More information on CSP s can be found in [15] or in [136]. ...
Dans ce travail, nous nous intéressons principalement à la surveillance préventive des systèmes non linéaires à incertitudes bornées, c’est-à-dire des systèmes pour lesquels les incertitudes ne sont définies que par leur appartenance à des intervalles. Pour cela, nous nous sommes placés dans un contexte dit « ensembliste » dans lequel nous avons pu étendre deux propriétés importantes largement étudiées en contexte stochastique qui sont l’identifiabilité et la diagnosticabilité. Au delà des définitions conceptuelles requises par ce travail, nous avons proposé des outils liés à l’algèbre différentielle permettant de vérifier ces deux propriétés.L’impact de l’identifiabilité ensembliste sur les résultats d’une estimation de paramètres a également été analysé, l’estimation de paramètres étant une des approches retenues pour la détection et l’isolation de défauts dans ce travail. Nous avons également cherché améliorer l’estimation de paramètres en développant des critères pour la planification d’expériences, consistant ici en l’optimisation des conditions initiales, entrées et/ou période d’échantillonnage. Ces critères ont été appliqués à deux cas d’étude (pharmacocinétique et aéronautique) avec d'excellents des résultats. Par ailleurs, nous nous sommes intéressés à la modélisation des systèmes à incertitudes « mixtes », combinant des incertitudes bornées et stochastiques, en proposant notamment une amélioration du filtre de Kalman par intervalles. Le travail réalisé formalise des problèmes non abordés auparavant et pose des jalons pour les recherches futures. Les perspectives de travail sont nombreuses tant la thématique considérée est riche et les applications diverses. Mots clefs : identifiabilité ensembliste, diagnosticabilité ensembliste, planification d'expériences ensembliste, estimation ensembliste, systèmes non linéaires à incertitudes bornées, incertitudes mixtes.
... The first criterion to consider is the linearity of the constraints, which defines two categories, linear CSP s and nonlinear CSP s. More information on CSP s can be found in or in Xiong (2013). ...
Background and objective:
This paper deals with the improvement of parameter estimation in terms of precision and computational time for dynamical models in a bounded error context.
Methods:
To improve parameter estimation, an optimal initial state design is proposed combined with a contractor. This contractor is based on a volumetric criterion and an original condition initializing this contractor is given. Based on a sensitivity analysis, our optimal initial state design methodology consists in searching the minimum value of a proposed criterion for the interested parameters. In our framework, the uncertainty (on measurement noise and parameters) is supposed unknown but belongs to known bounded intervals. Thus guaranteed state and sensitivity estimation have been considered. An elementary effect analysis on the number of sampling times is also implemented to achieve the fast and guaranteed parameter estimation.
Results:
The whole procedure is applied to a pharmacokinetics model and simulation results are given.
Conclusions:
The good improvement of parameter estimation in terms of computational time and precision for the case study highlights the potential of the proposed methodology.
... An MIT-rule based adaptive extended set-membership filter (MIT-ESMF) was also proposed by Wu et al. (2012), which assumes the system and measurement noise signals as unknown but bounded to better fit real applications. Xiong (2013) established another set-membership filter with an emphasis on the system model's uncertainty using interval analysis (the model uncertainty is assumed to be unknown but bounded, which is more suitable for the real case where modeling is inevitably inaccurate). Since simultaneous state and fault parameter estimation based FDD is a newly developed research area, a comparative study will be given in this paper regarding the different estimation methods to illustrate their advantages and disadvantages for actuator FDD on a UH, with emphasis on KF-UKF, MIT-UKF and MIT-ESMF. ...
Simultaneous state and parameter estimation based actuator fault detection and diagnosis (FDD) for single-rotor unmanned helicopters (UHs) is investigated in this paper. A literature review of actuator FDD for UHs is given firstly. Based on actuator healthy coefficients (AHCs), which are introduced to represent actuator faults, a combined dynamic model is established with the augmented state containing both the flight state and AHCs. Then the actuator fault detection and diagnosis problem is transformed into a general nonlinear estimation one: given control inputs and the measured flight state contaminated by measurement noises, estimate both the flight state and AHCs recursively in each time-step, which is also known as the simultaneous state and parameter estimation problem. The estimated AHCs can further be used for fault tolerant control (FTC). Based on the existing widely used nonlinear estimation methods such as the unscented Kalman filter (UKF) and the extended set-membership filter (ESMF), three kinds of adaptive schemes (KF-UKF, MIT-UKF and MIT-ESMF) are proposed by our team to improve the actuator FDD performance. A comprehensive comparative study on these different estimation methods is given in detail to illustrate their advantages and disadvantages when applied to unmanned helicopter actuator FDD.
