ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 8: State Transition Matrix, Part 2
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Homework 3 – Due Friday Sept. 18
◦ Image files okay as long as they are legible
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Lecture Quiz 2 Due Today @ 5pm
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Lecture Quiz 3 Posted by Monday Lecture
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Future Lectures
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Lecture
Lecture
Lecture
Lecture
9 – Monday 9/14 @ 9am
10 – Monday 9/14 @ 4pm
11 – Monday 9/21 @ 9am
12 – Monday 9/21 @ 4pm
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State Transition Matrix – Part 2
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Since x is linear (note lower case!) then there exists a
solution to the linear, first order system of
differential equations:
The solution is of the form:
Φ(t,ti) is the state transition matrix (STM) that maps
x(ti) to the state x(t) at time t.
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Constant!
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What is the differential equation?
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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)
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State Transition Matrix – Laplace Transform
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Laplace Transforms are useful for analysis of linear time-invariant
systems:
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electrical circuits,
harmonic oscillators,
optical devices,
mechanical systems,
even some orbit problems.
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Transformation from the time domain into the Laplace domain.
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Inverse Laplace Transform converts the system back.
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Solve the ODE
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We can solve this using “traditional” calculus:
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Solve the ODE
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Or, we can solve this using Laplace Transforms:
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Solve the ODE:
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State Transition Matrix – Analytic Approach
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Leverage the differential equation
and combine it with classic methods
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Compatible with simple equations, but not
with larger estimated state vectors or
complicated dynamics
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State Transition Matrix – Numeric Solution
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For more complicated dynamics, must
integrate X*(t) and Φ(t,t0) simultaneously in
propagator
◦ Up to n+n2 propagated states
◦ Derivative function must include the evaluation of
the A(t) matrix in addition to F(X,t)
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Use the MATLAB reshape() command to
turn matrix into a vector
◦ v = reshape( V, nrows*ncols, 1 );
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MATLAB Example…
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