EML 6229

Introduction to Random Dynamical Systems

Mrinal KumarAssistant Prof., MAE

http://www.mae.ufl.edu/~mrinalkumar

Syllabus…

Uncertainty: A Fundamental Challenge

✣Nature is far too complex for engineers

✣ Understanding nature: via math:: analysis via observation:: instruments

Bothimperfect!!

✣Outcome: we must live with uncertainty, aka stochasticity

Specifically, we must make decisions while limited by stochastic information!

Uncertainty: Examples

Weather prediction

Hazardous event management Catastrophic event decision making

Apophis collision probability in 2029: 2.7% (2004 estimate)

Understanding turbulence:affordable transportation

2010 Iceland eruption map

due to practical limitations, e.g. ⇾ model not good enough, ⇾ not enough measurements available, ⇾ neglected effects

Characterization of Uncertainty

✣ Given that uncertainty is unavoidable, how do we best capture it in engineering/scientific/financial/economic/sociological systems?

Uncertainty Quantification (UQ)

UncertaintyTypes

Epistemic:

Aleatory: due to fundamental limitations, e.g.⇾ accuracy of instruments⇾ computational limits⇾ nonrepeatability of experiments

Epistemic uncertainty is reducible to aleatory uncertainty in an ideal world

Uncertainty in System Models

System (physics)…. to be modeled

States… entities that identify/characterize/quant

ify the system

Accurate math model(no uncertainty)…but too complex!!

Uncertainty in System Models

A much simpler, reduced order model, but with uncertainty

noise….

Also need: initial conditions: almost always come from measurements

Uncertainty in Measurements

✣ There is always aleatory measurement uncertainty: unavoidable and irreducible

✣ In today’s world, epistemic measurement uncertainty is also dominant (essentially too much information to track, and too few resources)

Example: consider the so-called potentially hazardous asteroids

Uncertainty in Measurements

✣ There is always aleatory measurement uncertainty: unavoidable and irreducible

✣ In today’s world, epistemic measurement uncertainty is also dominant (essentially too much information to track, and too few resources)

Example: or something closer to home: our space debris

View of debris in LEO Expanded view of debris to include HEO

plus….active satellites!!

limited resources….

Uncertainty Propagation

✣ When measurements cannot be made (due to lack of allocation), only way to quantify uncertainty is to propagate (forecast) it through use the best known models