ASEN 5070: Statistical Orbit Determination I
Fall 2014
Professor Brandon A. Jones
Lecture 7: Linearization and the State Transition
Matrix
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Homework 2 – Due September 12
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Lecture Quiz – Due Friday @ 5pm
◦ Full credit for doing it
◦ We will discuss the answers/results in class on
Monday
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Linearization – How we do it? (wrap-up)
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State Transition Matrix (STM)
◦ Derivation
◦ Solution Methods
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Linearization – Why do we need it? (review)
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How do we estimate X ?
How do we estimate the errors εi?
How do we account for force and observation model
errors?
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This is the “normal form” of the least squares estimator
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We assumed that the state-observation relationship was
linear, but the orbit determination problems is nonlinear
◦ We will linearize the formulation of the problem
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Linearization – How do we do it? (continued)
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Computed, not measured values!
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Linearization – State Transition Matrix
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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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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 – Alternative Derivation
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Expand X(t) in a Taylor series about X*(t):
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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 Integration
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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 Demo…
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