.. Frobenius-Perron Operator 2014 ACC Workshop on Uncertainty Analysis & Estimation Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. uq.tamu.edu
Frobenius-Perron Operator Linear Operator P t Given dynamics ẋ = F (t, x), with p(t 0, x) as the initial state density function. Evolution of density is given by P t has following properties P t is defined by p(t, x) := P t p(t 0, x). P t (λ 1 p 1 + λ 2 p 2 ) = λ 1 P t p 1 + λ 2 P t p 2 linearity P t p 0 if p 0, positivity P t p(t 0, x)µ(dx) = X p t + (pf ) = 0 X p(t 0, x)µ(dx) measure preserving Continuity equation FPK without diffusion term First order linear PDE Workshop on Uncertainty Analysis & Estimation (ACC 2014) 2 /10
First Order PDEs Method of Characteristics p p n + (pf ) = t t + pf i (t, x) x i = p n t + p F i (t, x) + p This is of the form n F i (t, x) = 0 a(t, x, p)p t + i b i (t, x, p)p xi = c(t, x, p). Lagrange-Charpit equations dt a(t, x, p) = dx i b i (t, x, p) = dp c(t, x, p) Workshop on Uncertainty Analysis & Estimation (ACC 2014) 3 /10
Characteristic Equations Lagrange-Charpit equations dt a(t, x, p) = dx i b i (t, x, p) = dp c(t, x, p) Let s be parameterization of characteristic curves Characteristic curves are given by the ODEs dt = a(t, x, p) ds dx i ds = b i(t, x, p) dp = c(t, x, p) ds Workshop on Uncertainty Analysis & Estimation (ACC 2014) 4 /10
Solution of Continuity Equation For continuity equation p n t + p F i (t, x) + p n F i (t, x) = 0 a(t, x, p) = 1, b i (t, x, p) = F i (t, x), c(t, x, p) = p Characteristic equations n F i (t, x). dt ds = 1 dx i ds = F i(t, x) dp n ds = p F i (t, x) ẋ = F (t, x) evolution of x(t) ṗ = p( F ) evolution of p along x(t) Initial Conditions x 0 Samples from p(t 0, x) p 0 = p(t 0, x 0 ) Values of p(t 0, x) at x 0 Workshop on Uncertainty Analysis & Estimation (ACC 2014) 5 /10
Parametric Uncertainty & Process Noise Given system dynamics ẋ = F (t, x, ) + n(t, ω) Expand n(t, ω) using KL expansion. New paramters: ξ := (ξ 0, ξ 0,, ξ N, ξ N ) T PDF: p ξ (ξ) Parameter PDF: p ( ) State IC PDF: p x (t 0, x) Augment state space X := x G(t, x,, ξ), with Ẋ := 0 = H(t, X) ξ 0 with p X (t 0, X) := p x (t 0, x)p ( )p ξ (ξ) and p X (t, X) := P t p X (t 0, X). Workshop on Uncertainty Analysis & Estimation (ACC 2014) 6 /10
Better Accuracy & Faster Convergence than MC 10 0 Convergence of E[x] FP MC 10 1 Convergence of E[x 2 ] FP MC 10 1 10 0 10 1 10 2 Error in E[x] 10 3 Error in E[x 2 ] 10 2 10 3 10 4 10 4 10 5 10 5 10 6 10 1 10 2 10 3 10 4 Number of samples 10 6 10 1 10 2 10 3 10 4 Number of samples (a) First moment (b) Second moment Data generated from univariate normal distribution MC: PDF from kernel density estimation FP: PDF from spline interpolation Samples generated 1000 times for a given size. Plots show average error vs sample size Requires F i(x). Workshop on Uncertainty Analysis & Estimation (ACC 2014) 7 /10
Nonlinear Example 3 DOF Vinh s Equation Models motion of spacecraft during planetary entry ḣ = V sin(γ) V = ρr0 V 2 gr0 sin(γ) 2B c vc 2 γ = ρr 0 C L V + gr 0 cos(γ) 2B c C D vc 2 ( V R 0 + h 1 V ). R 0 radius of Mars ρ atmospheric density v c escape velocity C L C lift over drag D B c ballistic coefficient h height V velocity γ flight path angle Workshop on Uncertainty Analysis & Estimation (ACC 2014) 8 /10
3DOF Vinh s Equation Gaussian initial condition uncertainty in (h, V, γ) Workshop on Uncertainty Analysis & Estimation (ACC 2014) 9 /10
Frobenius-Perron Operator Papers 1. A. Halder, R. Bhattacharya, Beyond Monte Carlo: A Computational Framework for Uncertainty Propagation in Planetary Entry, Descent and Landing, AIAA GNC 2010. 2. A. Halder, R. Bhattacharya, Dispersion Analysis in Hypersonic Flight During Planetary Entry Using Stochastic Liouville Equation, AIAA Journal of Guidance, Control, and Dynamics, 2011, 0731-5090 vol.34 no.2 (459-474) 3. P. Dutta, A. Halder, R. Bhattacharya, Uncertainty Quantification for Stochastic Nonlinear Systems using Perron-Frobenius Operator and Karhunen-Loeve Expansion. IEEE Multi-Conference on Systems and Control, Dubrovnik, Oct 2012. Workshop on Uncertainty Analysis & Estimation (ACC 2014) 10 /10