Download presentation

Presentation is loading. Please wait.

Published byCristian Foster Modified over 2 years ago

1
University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 5: State Deviations and Fundamentals of Linear Algebra

2
University of Colorado Boulder Homework 2– Due September 12 Make-up Lecture ◦ Today @ 3pm, here 2

3
University of Colorado Boulder Effects of State Deviations Linear Algebra (Appendix B) 3

4
University of Colorado Boulder 4 Quantifying Effects of Orbit State Deviations

5
University of Colorado Boulder Quantification of such effects is fundamental to the OD methods discussed in this course! 5 Time

6
University of Colorado Boulder Let’s think about the effects of small variations in coordinates, and how these impact future states. 6 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Example: Propagating a state in the presence of NO forces

7
University of Colorado Boulder What happens if we perturb the value of x0? 7 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0)

8
University of Colorado Boulder What happens if we perturb the value of x0? 8 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf+Δx, yf, zf, vxf, vyf, vzf)

9
University of Colorado Boulder What happens if we perturb the position? 9 Initial State: (x0, y0, z0, vx0, vy0, vz0) Force model: 0 Initial State: (x0+Δx, y0+Δy, z0+Δz, vx0, vy0, vz0) Final State: (xf+Δx, yf+Δy, zf+Δz, vxf, vyf, vzf)

10
University of Colorado Boulder What happens if we perturb the value of vx0? 10 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Initial State: (x0, y0, z0, vx0-Δvx, vy0, vz0)

11
University of Colorado Boulder What happens if we perturb the value of vx0? 11 Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: 0 Final State: (xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf) Initial State: (x0, y0, z0, vx0+Δvx, vy0, vz0)

12
University of Colorado Boulder What happens if we perturb the position and velocity? 12 Force model: 0

13
University of Colorado Boulder We could have arrived at this easily enough from the equations of motion. 13 Force model: 0

14
University of Colorado Boulder This becomes more challenging with nonlinear dynamics 14 Force model: two-body

15
University of Colorado Boulder This becomes more challenging with nonlinear dynamics 15 Force model: two-body Initial State: (x0, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.

16
University of Colorado Boulder This becomes more challenging with nonlinear dynamics 16 Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body

17
University of Colorado Boulder 17 Select Topics in Linear Algebra

18
University of Colorado Boulder Matrix A is comprised of elements a i,j The matrix transpose swaps the indices 18

19
University of Colorado Boulder Matrix inverse A -1 is the matrix such that For the inverse to exist, A must be square We will treat vectors as n×1 matrices 19

20
University of Colorado Boulder 20

21
University of Colorado Boulder 21

22
University of Colorado Boulder If we have a 2x2, nonsingular matrix: 22 Asking you to invert a full 2x2 matrix on an exam is fair game!

23
University of Colorado Boulder The square matrix determinant, |A|, describes if a solution to a linear system exists: 23 It also describes the change in area/volume/etc. due to a linear operation:

24
University of Colorado Boulder 24

25
University of Colorado Boulder A set of vectors are linearly independent if none of them can be expressed as a linear combination of other vectors in the set ◦ In other words, no scalars α i exist such that for some vector v j in the set {v i }, i=1,…,n, 25

26
University of Colorado Boulder The matrix column rank is the number of linearly independent columns of a matrix The matrix row rank is the number of linearly independent rows of a matrix rank(A) = min( col. rank of A, row rank of A) 26

27
University of Colorado Boulder 27

28
University of Colorado Boulder 28

29
University of Colorado Boulder When differentiating a scalar function w.r.t. a vector: 29

30
University of Colorado Boulder When differentiating a function with vector output w.r.t. a vector: 30

31
University of Colorado Boulder If A and B are n×1 vectors that are functions of X: 31

32
University of Colorado Boulder The n×n matrix A is positive definite if and only if: 32 The n×n matrix A is positive semi-definite if and only if:

33
University of Colorado Boulder The point x is a minimum if 33 and is positive definite.

34
University of Colorado Boulder Given the n×n matrix A, there are n eigenvalues λ and vectors X≠0 where 34

35
University of Colorado Boulder Other identities/definitions in Appendix B of the book ◦ Matrix Trace ◦ Maximum/Minimum Properties ◦ Matrix Inversion Theorems Review the appendix and make sure you understand the material 35

Similar presentations

OK

2. Review of Matrix Algebra Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Technical.

2. Review of Matrix Algebra Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Technical.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on quality of higher education in india Ppt on as 14 amalgamation define Ppt on polynomials for class 8 Using ppt on ipad Ppt on coal petroleum and natural gas Ppt on cross-sectional study limitations Ppt on action research in education Ppt on types of rocks Ppt on event driven programming vs procedural programming Ppt on corporate etiquettes of life