A state estimation approach based on stochastic expansions
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A state estimation approach based on stochastic expansions R. H. Lopez1 · J. E. Souza Cursi2 · A. G. Carlon1
Received: 26 August 2016 / Revised: 13 June 2017 / Accepted: 14 September 2017 / Published online: 11 October 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract This paper presents a new approach for state estimation problems. It is based on the representation of random variables using stochastic functions. Its main idea is to expand the state variables in terms of the noise variables of the system, and then estimate the unnoisy value of the state variables by taking the mean value of the stochastic expansion. Moreover, it was shown that in some situations, the proposed approach may be adapted to the determination of the probability distribution of the state noise. For the determination of the coefficients of the expansions, we present three approaches: moment matching (MM), collocation (COL) and variational (VAR). In the numerical analysis section, three examples are analyzed including a discrete linear system, the Influenza in a boarding school and the state estimation problem in the Hodgkin–Huxley’s model. In all these examples, the proposed approach was able to estimate the values of the state variables with precision, i.e., with very low RMS values. Keywords Uncertainty quantification · State estimation · Polynomial chaos Mathematics Subject Classification 60G15 Gaussian processes · 60G35 Signal detection and filtering · 60H07 Stochastic calculus of variations and the Malliavin calculus · 60H10 Stochastic ordinary differential equations · 60H30 Applications of stochastic analysis
Communicated by Dr. Jose Alberto Cuminato.
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A. G. Carlon [email protected] R. H. Lopez [email protected] J. E. Souza Cursi [email protected]
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Department of Civil Engineering, Center for Optimization and Reliability in Engineering (CORE), UFSC, Rua Joao Pio Duarte, s/n, Florianópolis, SC, Brazil
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Institut National des Sciences Appliques (INSA) de Rouen, 76801 Saint Etienne du Rouvray CEDEX, France
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R. H. Lopez et al.
(to PDE, etc.) · 60H35 Computational methods for stochastic equations · 65C30 Stochastic differential and integral equations
1 Introduction State estimation of discrete/continuous time systems is a fundamental problem in engineering sciences. Indeed, many models of physical systems use mathematical descriptions involving a finite number of variables, usually referred to as state variables, collected in a state vector s. The evolution of the system is modeled as changes in the vector s and may be described either by a function of time t −→ s(t) (continuous time), or a sequence of discrete values s(0) , s(1) , s(2) , . . ., where s( p) corresponds to time t ( p) (discrete time). Evolution may be caused by internal changes or external actions u defined by a function t −→ u(t) (continuous time) or discrete values u(0) , u(1) , u(2) , . . ., where u( p) corresponds to time t ( p) (discrete time). Continuous time models usually describe
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