University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 8: State Transition Matrix, Part 2

University of Colorado Boulder  Homework 3 – Due Friday Sept. 18 ◦ Image files okay as long as they are legible  Lecture Quiz 2 Due 5pm  Lecture Quiz 3 Posted by Monday Lecture  Future Lectures ◦ Lecture 9 – Monday 9am ◦ Lecture 10 – Monday 4pm ◦ Lecture 11 – Monday 9am ◦ Lecture 12 – Monday 4pm 2

University of Colorado Boulder 3 State Transition Matrix – Part 2

University of Colorado Boulder  Since x is linear (note lower case!) then there exists a solution to the linear, first order system of differential equations: 4  The solution is of the form:  Φ(t,t i ) is the state transition matrix (STM) that maps x(t i ) to the state x(t) at time t.

University of Colorado Boulder  What is the differential equation? 5 Constant!

University of Colorado Boulder  There are four methods to generate the STM: ◦ Solve from the direct Taylor expansion ◦ If A is constant, use the Laplace Transform or eigenvector/value analysis ◦ Analytically integrate the differential equation directly ◦ Numerically integrate the equations (ode45) 6

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University of Colorado Boulder 9 State Transition Matrix – Laplace Transform

University of Colorado Boulder 10  Laplace Transforms are useful for analysis of linear time-invariant systems: ◦ electrical circuits, ◦ harmonic oscillators, ◦ optical devices, ◦ mechanical systems, ◦ even some orbit problems.  Transformation from the time domain into the Laplace domain.  Inverse Laplace Transform converts the system back.

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University of Colorado Boulder  Solve the ODE  We can solve this using “traditional” calculus: 12

University of Colorado Boulder  Solve the ODE  Or, we can solve this using Laplace Transforms: 13

University of Colorado Boulder  Solve the ODE: 14

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University of Colorado Boulder 17 State Transition Matrix – Analytic Approach

University of Colorado Boulder  Leverage the differential equation 18 and combine it with classic methods  Compatible with simple equations, but not with larger estimated state vectors or complicated dynamics

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University of Colorado Boulder 25 State Transition Matrix – Numeric Solution

University of Colorado Boulder  For more complicated dynamics, must integrate X * (t) and Φ(t,t 0 ) simultaneously in propagator ◦ Up to n+n 2 propagated states ◦ Derivative function must include the evaluation of the A(t) matrix in addition to F(X,t) 26

University of Colorado Boulder  Use the MATLAB reshape() command to turn matrix into a vector ◦ v = reshape( V, nrows*ncols, 1 );  MATLAB Example… 27