EML 6229 Introduction to Random Dynamical Systems Mrinal Kumar Assistant Prof., MAE

Syllabus…

Uncertainty: A Fundamental Challenge ✣ Nature is far too complex for engineers ✣ Understanding nature: via math:: analysis via observation:: instruments Both imperfect!! ✣ 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) Uncertainty Types 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