Filtering Theory for a Weakly Coloured Noise Process
- PDF / 1,863,737 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 107 Downloads / 189 Views
Filtering Theory for a Weakly Coloured Noise Process Shaival H. Nagarsheth1 · Dhruvi S. Bhatt1 · Shambhu N. Sharma1
© Foundation for Scientific Research and Technological Innovation 2020
Abstract The problem of analyzing the Itô stochastic differential system and its filtering has received attention. The classical approach to accomplish filtering for the Itô SDE is the Kushner equation. In contrast to the classical filtering approach, this paper presents filtering for ( ) ( )the stochastic differential system affected by weakly coloured noise, i.e., ẋ t = f xt + g xt 𝜉t , where the input process 𝜉t is a weakly coloured noise process. As a special case, the process 𝜉t can be regarded as the Ornstein–Uhlenbeck (OU) process, i.e., d𝜉t = −𝛼𝜉t dt + 𝛽dBt , where(𝛼 > ) 0. More ( ) precisely, the filtering model of this paper can be cast as ẋ t = f xt + g xt 𝜉t , t
zt = ∫ h(x𝜏 )d𝜏 + 𝜂t , t0
where h(xt ) is the measurement non-linearity and 𝜂 = {𝜂t , t0 ≤ t < ∞} is the Brownian motion process. The former expression describes the structure of a noisy dynamical system, and the latter is the observation equation. The novelties of this paper are two (1) the extension of the filtering theory for the Itô stochastic differential system to the filtering theory for the ‘weakly coloured noise-driven’ stochastic differential system (2) the theory of this paper is based on a pioneering contribution of Ruslan Stratonovich involving the perturbation-theoretic approach to noisy dynamical systems in combination with the notion of the ‘filtering density’ evolution. The stochastic evolution of condition moment is derived by utilizing the filtering density evolution equation. A scalar Duffing system driven by the OU process is employed to test the effectiveness of the filtering theory of the paper. Numerical simulations involving four different sets of initial conditions and system parameters are utilized to examine the efficacy of the filtering algorithm of this paper. Keywords Non-markovian stochastic system · Ornstein–Uhlenbeck process · Brownian motion process · Classical filtering equations · Kushner equation · Filtering density · Stochastic differential equation * Shaival H. Nagarsheth [email protected] Dhruvi S. Bhatt [email protected] Shambhu N. Sharma [email protected] 1
Electrical Engineering Department, Sardar Vallabhbhai National Institute of Technology, Surat, India
13
Vol.:(0123456789)
Differential Equations and Dynamical Systems
Introduction The Stochastic Differential Equation (SDE) formalism has found remarkable success in diverse field, e.g., adaptive control, satellite trajectory estimations, helicopter rotor, stochastic networks, mathematical finance, neuronal activity, protein kinematics [1, 2]. The Itô stochastic differential equation is widely used in analyzing randomly perturbed dynamical systems. Rigorous mathematical treatments about stochastic differential equations can be found in [3, 4]. In 1945, Kiyoshi Itô developed the rigorous mathematical framework for the Brownian motion proce
Data Loading...
