A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

A set-membership state estimation algorithm based on DC programming

This paper presents a new approach to guaranteed state estimation for nonlinear discrete-time systems with a bounded description of noise and parameters. The sets of states that are consistent with the evolution of the system, the measured outputs and bounded noise and parameters are represented by zonotopes. DC programming and intersection operations are used to obtain a tight bound. An example is given to illustrate the proposed algorithm.Ministerio de Ciencia y Tecnología DPI2006-15476-C02-01Ministerio de Ciencia y Tecnología DPI2007-66718-C04-01

A unified framework for solving a general class of conditional and robust set-membership estimation problems

In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the computation of optimal conditional and robust projection estimates in a nonlinear estimation setting where the operator relating the data and the parameter to be estimated is assumed to be a generic multivariate polynomial function and the uncertainties affecting the data are assumed to belong to semialgebraic sets. By noticing that the computation of both the conditional and the robust projection optimal estimators requires the solution to min-max optimization problems that share the same structure, we propose a unified two-stage approach based on semidefinite-relaxation techniques for solving such estimation problems. The key idea of the proposed procedure is to recognize that the optimal functional of the inner optimization problems can be approximated to any desired precision by a multivariate polynomial function by suitably exploiting recently proposed results in the field of parametric optimization. Two simulation examples are reported to show the effectiveness of the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic Control (2014

ГАРАНТИРОВАННОЕ ОЦЕНИВАНИЕ СОСТОЯНИЯ ДИНАМИЧЕСКИХ СИСТЕМ, ВОЗМУЩЕНИЙ И ПОМЕХ В УСЛОВИЯХ НЕПОЛНОТЫ ИНФОРМАЦИИ

The paper considers the problem of set-valued dynamic systems state estimation under conditions of uncertainty, when the sets of disturbances and noises possible values are known and statistical information about them is absent or cannot be obtained. An algorithm for feasible set polyhedral approximation is described, when the sets of possible values of disturbances and noises are polyhedra. The algorithm is based on the implicit description of the information set with linear equations and inequalities systems and solving a number of linear programming problems. Methods for increasing the estimation accuracy by taking into account additional information about disturbances and noises models are considered. Set-valued estimation of the dynamical system state vector is described when the disturbances are given as a system of functions with unknown coefficients. In this case, due to the use of information that the coefficients are constant, the dynamic system state estimates are more accurate than in the case when the disturbances are known up to a set of possible values. A numerical example is presented to demonstrate the algorithm performance. Aim. The aim of the research is to develop dynamic system state, disturbance and noises set-valued estimation algorithms. Research methods. Methods of optimization theory, filtering, linear algebra, MATLAB software package were used in the work. Results. Dynamic system state estimation algorithm was described. The algorithm takes into account additional information about disturbances and noises models. A method of feasible set polyhedral approximation is described, which makes it possible to obtain a set-valued estimate of a state vector, a vector of disturbances and noises, and an evolution of reachable sets. It can be used in the adaptive estimation and control algorithms development. The algorithm for set-valued estimation of the system state vector and coefficients in the disturbance decomposition as a system of given functions is developed. Conclusion. An algorithm for feasible set polyhedral approximation was described.The numerical example was performed and the analysis of the estimateswas presented.Рассматривается задача гарантированного оценивания состояния динамических систем в условиях неопределенности, когда известны только множества возможных значений возмущений и помех, а статистическая информация о них отсутствует или не может быть получена. Описан алгоритм полиэдральной аппроксимации информационных множеств, когда множества возможных значений возмущений и помех являются многогранниками. Алгоритм основан на неявном описании информационного множества системами линейных уравнений и неравенств и решении ряда задач линейного программирования. Рассмотрены методы повышения точности оценивания с помощью учета дополнительной информации о характере возмущений и помех. Описано гарантированное оценивание вектора состояния динамической системы, когда возмущения заданы в виде системы функций с неизвестными коэффициентами. В этом случае за счёт использования информации о том, что коэффициенты разложения являются постоянными, оценка вектора состояния получается точнее, чем в случае, когда возмущение известно с точностью до множества возможных значений. Приведен численный пример, демонстрирующий работу алгоритма. Целью исследования является разработка методов гарантированного оценивания состояния, возмущений и помех. Методы исследования. В работе использовались методы теории оптимизации, фильтрации, линейной алгебры, пакет прикладных программ MATLAB. Результаты. Описан метод гарантированного оценивания вектора состояния динамической системы с учётом дополнительной информации о характере возмущений. Описан метод полиэдральной аппроксимации информационных множеств, позволяющий получать гарантированную оценку вектора состояния, вектора возмущений и помех, а также множества прогнозов, что может быть использовано при разработке адаптивных алгоритмов оценивания и управления. Разработан алгоритм гарантированного оценивания вектора состояния системы и коэффициентов в разложении возмущения по системе заданных функций. Заключение. Приведен алгоритм полиэдральной аппроксимации информационных множеств, численный пример и анализ полученных оценок

Robust Adaptive Model Predictive Control: Performance and Parameter Estimation

For systems with uncertain linear models, bounded additive disturbances and state and control constraints, a robust model predictive control algorithm incorporating online model adaptation is proposed. Sets of model parameters are identified online and employed in a robust tube MPC strategy with a nominal cost. The algorithm is shown to be recursively feasible and input-to-state stable. Computational tractability is ensured by using polytopic sets of fixed complexity to bound parameter sets and predicted states. Convex conditions for persistence of excitation are derived and are related to probabilistic rates of convergence and asymptotic bounds on parameter set estimates. We discuss how to balance conflicting requirements on control signals for achieving good tracking performance and parameter set estimate accuracy. Conditions for convergence of the estimated parameter set are discussed for the case of fixed complexity parameter set estimates, inexact disturbance bounds and noisy measurements

Robust identification of non-linear greenhouse model using evolutionary algorithms

[EN] This paper presents the non-linear modelling, based oil first principle equations, for a climatic model of a greenhouse and the estimation of the feasible parameter set (FPS) when the identification error is bounded simultaneously by several norms. The robust identification problem is transformed into a multimodal optimization problem with an infinite number of global minima that constitute the FPS. For the optimization task, a special evolutionary algorithm (epsilon-GA) is presented, which characterizes the FPS by means of a discrete set of models that are well distributed along the FPS. A procedure for determining the norm bounds, such that FPS not equal 0, is (c) 2007 Elsevier Ltd. All rights reserved.Partially supported by MEC (Spanish government) and FEDER funds: projects DPI2005-07835 and DPI2004- 8383-C03-02.Herrero Durá, JM.; Blasco, X.; Martínez Iranzo, MA.; Ramos Fernández, C.; Sanchís Saez, J. (2008). Robust identification of non-linear greenhouse model using evolutionary algorithms. Control Engineering Practice. 16(5):515-530. https://doi.org/10.1016/j.conengprac.2007.06.001S51553016

Set-valued estimation of switching linear system: an application to an automotive throttle valve

This paper introduces a polyhedral approximation algorithm for set-valued estimation of switching linear systems. The algorithm generates set-valued estimates for any possible sequence of switching parameters, under the assumption that the system has unknown but bounded disturbances and measurement noises. Our algorithm has practical implications; namely, set-valued estimates were generated for the position and electrical current of a real-time automotive electronic throttle valve, and the corresponding experimental data demonstrate the practical benefits of our approach.Postprint (author's final draft

A Bayesian approach to robust identification: application to fault detection

In the Control Engineering field, the so-called Robust Identification techniques deal with the problem of obtaining not only a nominal model of the plant, but also an estimate of the uncertainty associated to the nominal model. Such model of uncertainty is typically characterized as a region in the parameter space or as an uncertainty band around the frequency response of the nominal model. Uncertainty models have been widely used in the design of robust controllers and, recently, their use in model-based fault detection procedures is increasing. In this later case, consistency between new measurements and the uncertainty region is checked. When an inconsistency is found, the existence of a fault is decided. There exist two main approaches to the modeling of model uncertainty: the deterministic/worst case methods and the stochastic/probabilistic methods. At present, there are a number of different methods, e.g., model error modeling, set-membership identification and non-stationary stochastic embedding. In this dissertation we summarize the main procedures and illustrate their results by means of several examples of the literature. As contribution we propose a Bayesian methodology to solve the robust identification problem. The approach is highly unifying since many robust identification techniques can be interpreted as particular cases of the Bayesian framework. Also, the methodology can deal with non-linear structures such as the ones derived from the use of observers. The obtained Bayesian uncertainty models are used to detect faults in a quadruple-tank process and in a three-bladed wind turbine