Application of the Weierstrass Theorem for Sensors Signal Modelling

Abstract

Modelling of sensor signals is a key element in implementing the procedures for estimation and tracking, sensor data fusion, fault detection and diagnosis, etc. In many cases, using traditional approaches to solve this problem is impossible due to lack of prior information about the observed process or particular sensor characteristics. Starting only from the finite rate of change of the signal at the output of real sensors, the Weierstrass theorem assumes the existence of a polynomial that approximates the signal, but does not specify how to find it. The present work attempts to solve this problem by assuming that the approximating polynomial has to retain that information in the signal which would allow its Bayesian estimation. The described approach has been successfully applied to model a humidity sensor signal, but it can be used in other cases without any problem.

Application of Polynomial Models in Bayesian Fusion of Humidity Sensors

“…between the time constants 𝑇 1 , 𝑇 2 and 𝑇 does not exceed ±5 % and the additive and multiplicative error components ±2,5 %RH, respectively. For most of these cases (1)(2)(3)(4)(5)(6)(11)(12)(13)(14)(15)(16), the prediction error is significantly larger than that for which Kalman filter work is possible (±0,01 RH%). Again using (31) for the worst cases (1 and 16), Δ𝑑 𝑘 was calculated, but this time with a different rate of change of the input signal ∆𝑢 𝑘 .…”

“…Typically, any stochastic process, and hence the one described by an ARIMA model, can be represented as the sum of two principal components [11]: a signal (trend) that describes a smooth underlying mean and a residual component, often considered as noise. Local Polynomial trend Models (PM) [12], [13] belong to this group. They have a number of unique properties that make them particularly suitable for representing different types of temporal data:…”

“…-PM describe a stochastic process with a polynomial trend. According to the Weierstrass theorem, any smooth curve (signal) defined on a finite interval can be arbitrarily well approximated by a polynomial [12]. In other words, in a short period of time for any ARIMA process, a PM with a similar trend can be found.…”

“…The description of the classical ARIMA model includes three order parameters, 𝑛 𝑝 , 𝑛 𝑑 and 𝑛 𝑞 , which respectively define the number of autoregressive terms (AR), the number of differentiations applied and the number of moving average (MA) terms in the model, the 𝑛 𝑝 + 𝑛 𝑞 coefficients associated with the AR and MA terms of the model and the variance of the innovation white noise [1]. In PM there is a parameter (𝑛 − 1) defining the order of the polynomial model where 𝑛 is the size of PM state vector and two other parameters known as the measurement noise intensity 𝑟 and the process noise intensity 𝑞 [12]. This greatly simplifies and even eliminates the need for PM identification.…”

“…In this paper, a discrete model with a local quadratic trend, also known as a second-order polynomial model ((𝑛 − 1) =2), is used to model stochastic processes [12]: (1) where 𝑘𝑘 =1, 2, ... is the discrete time, 𝜏𝜏 is the sampling period, 𝒛𝒛 𝑘 is the process state vector, 𝑪𝒛𝒛 𝑘 and 𝑦𝑦 𝑘 are the process-induced signal and its observation, 𝜻𝜻 𝑘 and 𝜂𝜂 𝑘 are white noises with zero mean and covariance matrices 5…”