Optimal Filtering for Polynomial Systems

Abstract

Let (Ω,F,P) be a complete probability space with an increasing right-continuous family of σ-algebras F _t ,t ≥ t₀, and let (W₁(t), F _t , t ≥ t₀) and (W₂(t), F _t , t ≥ t₀) be independent Wiener processes. The F _t -measurable random process (x(t),y(t)) is described by a nonlinear differential equation with polynomial drift for the system state

$$ dx(t) = f(x,t) dt + b(t)dW_{1}(t), ~~ x(t_0)=x_{0}, ~~(1.1) $$

and a linear differential equation for the observation process

$$ dy(t) = (A_{0}(t)+A(t)x(t))dt+ B(t) dW_{2}(t). ~~ (1.2) $$

Here, x(t) ∈ Rⁿ is the state vector and y(t) ∈ Rⁿ is the linear observation vector, such that the matrix A(t) ∈ R^n×n is invertible. The initial condition $x_0\in R^{n}$ is a Gaussian vector such that x₀, W₁(t), and W₂(t) are independent. It is assumed that B(t)B^T(t) is a positive definite matrix. All coefficients in (1.1)–(1.2) are deterministic functions of time of appropriate dimensions.