Certain Thoughts on Uncertainty Analysis for Dynamical Systems

Transcription

1 Certain Thoughts on Uncertainty Analysis for Dynamical Systems!"#$$%&'(#)*+&!"#$%"&$%'()$%(*+&$,$-$+.(",/(0,&122341,&(5$#$67(8'.&1-.(!"#$%%&'()*+,-.+/01'&2+,304 5,#')67, ,:!'-(:'&4;4<,)2*#':,4=-.(-,,)(-.4

2 Iceland Volcano Eruption Motivation The eruption at Eyjafjallajokull, Iceland, had wreaked havoc on European aviation since ash emissions began on 14 April 2010.! London VAAC (Volcanic Ash Advisory Center) began issuing code Red VAAs on 16 April 2010, which was rapidly followed by complete shutdown of British airspace.! Over the next hours and days, airspace over all of Europe was shut down. Losses were estimated at over $200 million/day (over $1 billion in total), with almost 7 million stranded passengers.!

3 Iceland Volcano Eruption Motivation

4 Space Object Tracking Motivation! In July, 1994, the comet Shoemaker-Levy 9 collided with Jupiter at over 133,000 mph.!! Asteroid 2004 MN4 is predicted to make a very close encounter in April of 2029, followed by other approaches in 2034 and 2036.! The uncertainty of the 2029 approach ( 1 Earth radii) is quite large compared to the predicted miss distance of 5.6 Earth radii.! This close encounter is not presently considered a high risk for collision, however the subsequent encounters have larger uncertainties and cannot be accurately predicted due to the nonlinearity of the dynamics and the large uncertainty associated with the 2029 approach.! 4

5 Space Object Tracking Motivation Feb unintentional collision " between Russia s Cosmos 2251 " satellite and a US Iridium satellite! Resulted in about 500 pieces! of debris" The debris had to be accurately tracked in order to analyze threat to other satellites in similar orbits.! The increasing amount of debris and inactive resident space objects (RSOs), particularly those with high area-to-mass ratios, in both the low-earth-orbit (LEO) and geosynchronous-earth orbit (GEO) regime pose a threat to active RSOs and must be accurately tracked for threat analysis.!

6 Space Object Tracking Motivation we live in a bustling neighborhood!

7 Introduction Dynamical Systems Requirements for simulation of dynamical systems: " Knowledge of Governing Dynamics (Equations of motion) " Knowledge of Initial Conditions (Where to start the solution) Different initial states lead to different trajectories.

8 Uncertainty Characterization Types of Uncertainty '%,-.+/0-&1(2$3$#0+*&45"+0,#/&67,3&8,#0#","/90:$&;<#+:(-/=& '%,-.+/0-&1(2$3$#-$&45"+0,#/&67,3&1(/-3$%$90:$&;<#+:(-/=& 1(/-3$%$90:$&:$+/"3$:$#%&:,;$*& A,;$*&>#-$3%+(#%<B&-.+3+-%$3(C$;& ;(/%3(D"0,#/. I<D3(;&J$.+K(,3&L&A,;$*&>#-$3%+(#%<&M&!+3+:$%3(-&>#-$3%+(#%<&

9 Data Assimilation Sources of Uncertainty Physical Process Observation Data Sensor Model Errors Sensor Noise & Quantization Errors Sensor Failure Best Estimates Mathematical Model Numerical Solution Modeling Errors Initial/Boundary Condition Uncertainty Input Uncertainty Numerical Algorithm Error

10 Uncertainty Propagation Conventional Methods Approximate Solution to exact problem: Multiple-model estimation method, Unscented Kalman Filter (UKF), Monte Carlo (MC) methods. Exact solution to approximate problem: Extended Kalman Filter (EKF), Gaussian closure, Equivalent Linearization, and Stochastic Averaging.

11 Uncertainty Characterization Uncertainty Propagation N R& Uncertainty Propagation captures the spatio-temporal evolution of uncertainty as constrained by the system dynamics and (if available) sensor observations U (& N P& S6N P TU P =& S6N Q TU Q =& S6N O TU O =& NQ& N R T% R P T% P Q T% Q O T% O =&

12 Main Idea Gaussian Sum Approximation " With sufficient number of Gaussian components, any pdf can be approximated as closely as desired. FPKE Linearized propagation True pdf + + = Component 1 Component 2 Component 3 Anderson, B. D. and Moore, J. B., Optimal Filtering, Prentice-Hall, 1979, Approximation

13 Uncertainty Characterization Classic Gaussian Sum Mixture = " Assumption: covariances are small enough such that the linearizations become representative for the dynamics around the means " Easily violated: large initial uncertainty, strong nonlinearities, computational constraints Anderson, B. D. and Moore, J. B., Optimal Filtering, Prentice-Hall, 1979,

14 Uncertainty Characterization Adaptive Gaussian Sum Mixture ! " Update I: Continuous-time dynamical systems Updates the weights by constraining the Gaussian sum approximation to satisfy the Fokker-Planck equation " Update II: Discrete-time nonlinear systems Weights to minimize the integral square difference between the true forecast pdf and its Gaussian sum approximation Terejanu, Singla, et al. IEEE Transactions on Automatic Control, 2011

15 Uncertainty Propagation Exact Problem The Fokker-Planck-Kolmogorov equation (FPKE) provides the exact description of the uncertainty propagation problem under white-noise excitation " Exact solution given by the FPKE " Where

16 Solving Fokker-Planck Equation Inherent Difficulties in Numerically Solving the FPKE Following issues make the FPKE a difficult numerical problem: Positivity of the pdf: Normality of the pdf: No unique domain for numerical implementation: The actual domain of the FPKE is. But we need a finite domain for numerical implementation. Dimensionality: FPKE contains the partial derivatives w.r.t. all the states. e.g. 2-D planar motion requires discretization of 4 dimensions.

17 Solving FPKE Gaussian Sum Approximation! Let us assume that underlying pdf can be approximated by a finite sum of Gaussian pdfs.! Question is how to find unknown parameters of this Gaussian Sum Mixture?

18 Solving FPKE Gaussian Sum Approximation EKF UKF Now, update the weights of Gaussian Sum Mixture such that FPKE error is minimized.

19 Solving FPKE Gaussian Sum Approximation " Pdf approximated by a finite sum of Gaussian densities " Propagation of Gaussian component moments " Update the weights of Gaussian Sum Mixture such that FPKE error is minimized. " Compute FPKE error

20 Solving FPKE Gaussian Sum Approximation

21 Solving FPKE Gaussian Sum Approximation

22 Solving FPKE Gaussian Sum Approximation " New weights are obtained by minimizing " Where, " Integrals involving Gaussian pdfs over volume V which can be computed exactly for polynomial nonlinearity and in general can be approximated by the Gaussian Quadrature method or unscented transformation.

23 Uncertainty Propagation Discrete Dynamical Systems! The Gaussian sum weights can be obtained by minimizing the following integral square difference between the true pdf and its approximation in the least square sense:

24 Uncertainty Propagation Discrete Dynamical Systems! Making use of the Chapman-Kolmogorov Equation:

25 Uncertainty Propagation Discrete Dynamical Systems! Finally,

26 Illustrating Key Ideas Duffing Oscillator " 16 Gaussian components " Coordinates of the means linearly spaced [-5,5] " Equal covariance matrices: eye(2) " Initial weights are randomly assigned a value between 0 & 1. " Duffing oscillator " Propagation: 100 sec " Stationary pdf: Terejanu, Singla, et al. AIAA JGCD, 2008

27 Example Duffing Oscillator True Stationary PDF Final PDF (without weight update) FPKE Feedback CKE Feedback Terejanu, Singla, et al. AIAA JGCD, 2008

28 Example Vanderpol Oscillator " 100 Gaussian components " Coordinates of the means linearly spaced [-4,4] " Equal covariance matrices: eye(2) " Initial weights are assumed to be equal. " Energy dependent damping oscillator " Propagation: 200 sec " Stationary pdf: Terejanu, Singla, et al. AIAA JGCD, 2008

29 Example Vanderpol Oscillator True Stationary PDF Final PDF (without weight update) Final PDF (with FPKE Feedback) Final PDF (with CKE Feedback) Terejanu, Singla, et al. AIAA JGCD, 2008

30 Uncertainty Characterization Adaptive Gaussian Sum Mixture " Lorenz system [Lorenz31] " Performance Measures Terejanu, Singla, et al. IEEE Transactions on Automatic Control, 2011.

31 Uncertainty Characterization Adaptive Gaussian Sum Mixture Terejanu, Singla, et al. IEEE Transactions on Automatic Control, 2011.

32 Simulation Results HAMR Object in LEO Orbit! Consider the dynamics of an object in low-earth-orbit (LEO) with nonconservative atmospheric drag forces!! Initial Condition:!

33 Simulation Results HAMR Object in LEO Orbit 50,000 Monte Carlo Samples Histogram! Adaptive Gaussian Sum Mixture! Uncertainty After 0.5 Orbits! Giza, Singla, & Jah AIAA GNC, 2009.

34 Simulation Results HAMR Object in LEO Orbit 50,000 Monte Carlo Samples Histogram! Adaptive Gaussian Sum Mixture! Uncertainty After 1 Orbits! Giza, Singla, & Jah AIAA GNC, 2009.

35 Simulation Results HAMR Object in LEO Orbit 50,000 Monte Carlo Samples Histogram! Adaptive Gaussian Sum Mixture! Uncertainty After 1.5 Orbits! Giza, Singla, & Jah AIAA GNC, 2009.

36 Simulation Results HAMR Object in LEO Orbit 50,000 Monte Carlo Samples Histogram! Adaptive Gaussian Sum Mixture! Uncertainty After 2 Orbits! Giza, Singla, & Jah AIAA GNC, 2009.

37 Simulation Results HAMR Object in LEO Orbit Giza, Singla, & Jah AIAA GNC, 2009.

38 Data Association Information Theory Based Metric Kullback-Leibler Divergence: is defined as the ratio of the prediction error obtained with an assumed (incorrect) spectral density to the one obtained with the correct spectral density:! Giza, Singla, et al. AIAA GNC, 2010.

39 Simulation Results HAMR Object in LEO Orbit Giza, Singla, et al. AIAA GNC, 2010.

40 Simulation Results HAMR Object in LEO Orbit Giza, Singla, et al. AIAA GNC, 2010.

41 Simulation Results HAMR Object in LEO Orbit Giza, Singla, et al. AIAA GNC, 2010.

42 even by improving the propagation method, chances are that we will still get a poor approximation to the forecast probability distribution where it matters. e.g. in the tails of the distribution. Because we still have a finite representation.

43 Helping Decision Maker Conventional Methodology action NO evacuation Decision Maker 10,000 residents Uncertainty Propagation (Approximations) params

44 Helping Decision Maker New Approach!! EVACUATE action Decision Maker 10,000 residents 10,000 residents Interaction Level Uncertainty Propagation (Approximations) 0.0 params Terejanu, Singla, et al. Advances in Information Fusion, Dec

45 CBRN Incident Decision Centric Approach Reference Truth (10,000 MC Runs) Gaussian Approximation Gaussian Sum Approximation Hazard Maps: Probability of chemical concentrations being more than Small black ellipse represents the initial source uncertainty and red ellipse represents a populated area of interest. The black arrows represent the nominal windfield. Terejanu, Singla, et al. Advances in Information Fusion, Dec

46 Helping Decision Maker New Approach!! Interaction Level between decision maker and the prediction module incorporate contextual information held by the decision maker Progressive Selection of Gaussian Components to supplement the initial uncertainty with new Gaussian components that are sensitive to the loss function at the decision time. The cost of the overall improvement is an increase in the number of Gaussian components The new probability density function addresses the region of interest and provides a better approximation overall, if any probability density mass is moving naturally towards the region of interest. Significantly enhanced accuracy within the decision maker s region of interest. Terejanu, Singla, et al. Advances in Information Fusion, Dec

47 Uncertainty Characterization Polynomial Chaos " Originally used by Norbert Wiener in 1938, to describe the members of the span of Hermite polynomial functionals of standard Gaussian random variables. " Generalized (Xiu, 2002) to efficiently use the orthogonal polynomials from the Askey-scheme to model various probability distributions. " Applications in modeling parametric uncertainty. " A dynamical system, represented by a set of Difference Equations with stochastic parameters, can be transformed into a deterministic system of equations in the coefficients of a series expansion, using this approach. " Used to approximate the stochastic system states in terms of a finitedimensional series expansion in the infinite-dimensional stochastic space. " The completeness of the space allows for the accurate representation of any random variable

48 Iceland Volcano Eruption Uncertainty Analysis Meteosat-9 SEVIRI data products compared with model output for a 9 4 run implementation of PCQ Clenshaw-Curtis sampling of the input space.! Uncertain Variables: Vent Velocity, Vent Radius, Mean Particle Size, Particle Size Sigma.!

49 Parametric Uncertainty Uncertainty Marriage

50 Parametric Uncertainty Uncertainty Marriage

51 Parametric Uncertainty Uncertainty Marriage '%,-.+/0-&1(2$3$#0+*&45"+0,#/&67,3&8,#0#","/90:$&;<#+:(-/=& Initial Condition + Stochastic Forcing Kolmogorov Equation Parametric Uncertainty

52 Parametric Uncertainty Uncertainty Marriage Uncertainty in parameter vector:

53 Parametric Uncertainty Uncertainty Marriage

54 Concluding Remarks What have been done " Two update schemes for the forecast weights are presented in order to obtain a better Gaussian sum approximation to the forecast pdf. " Continuous Dynamical Systems: minimize the FPKE error. " Discrete Dynamical Systems: minimize the integral square difference between the true pdf and its approximation. " Both methods result in a convex quadratic minimization problems. " guaranteed to have a unique solution. " Progressive selection of centers. " Uncertainty Marriage: gpc + AGMM

55 <0627'6,34B'?(.'C2-4;4D0(3'-:,482)4>EF4 " A Central Goal: Develop nonlinear control & estimation algorithms enabling unmanned vehicle motion in uncertain environment. V,K$*&W*,D+*9X,-+*&I($3+3-.<Y& Z&#,K$*&:$%.,;,*,)<&%,&+"%,:+%$&%.$& $[-($#%&@3,;"-0,#&,7&%.$&%,@,)3+@.(-& -,#%,"3&:+@/&D<&/0%-.(#)&%,)$%.$3&K+3(,"/& *,-+*&%$33+(#&:+@/Y& 1$/()#&,7&W!'9*$//&*,-+*(C+0,#&+*),3(%.:/& $#+D*(#)&+"%,:+%$;&#+K()+0,#&,7&K$.(-*$&(#& "#F#,E#&$#K(3,#:$#%Y& 1$/()#&,7&,@0:+*&-,#%3,*&+*),3(%.:/&"#;$3& /$#/,3<&-,#/%3+(#%/&\&"#-$3%+(#0$/Y&

56 Multi-Resolution Modeling Radiation Therapy " Novel adaptive algorithms to precisely predict the motion of tumor & Internal Organs " Correlation of real-time imagery data from external & internal markers. Collaborators: Prof. Singh (MAE), Dr. Podgorsoak (Rosewell Park)

57 Thank you Students & Sponsors " Gabriel Terejanu " Uma Konda " Kumar Vishwajeet " Cheng Jin " Reza Madankan " Jemin George " Daniel Giza

58 Thank you Adriaan Fokker Max Planck Andrey Kolmogorov

59