(ROBUST)CONTROLTHEORYFUNDAMENTALS PabloA.Parrilo-MITEECS6.S979TopicsinDeployableMLFall2019

AroundRobustness… •  RobustStatistics(1960s--)•  RobustControl(1980s--)•  RobustOptimization(1990s--)•  RobustEconomics(2000s--)•  RobustMachineLearning(2015s--)

Historyrepeatsitself,firstastragedy,secondasfarce.

KarlMarx

Thislecture

• Whatis“ControlTheory”?• Whatis“RobustControl”?•  (Some)insightsandresults•  Afewtechnicalnuggets• Modelsofuncertainty,andassessmentofrobustness• ManyquestionsthatIdon’tknowtheanswerto

DynamicsandControlTwokeynotions:Dynamicalsystems:•  Systemsthatevolveover“time”Control/Decisions:•  Valuesofsomeofthevariablescanchosenbyus.

Manydifferentflavors…•  Continuous/DiscreteTime•  Differential/DifferenceEquations•  Differential/AlgebraicEquations(DAEs)•  StochasticDifferentialEquations(SDEs)•  MarkovDecisionProcesses(MDPs)•  DiscreteEventSystems(DES)•  (more…)Also:•  Single/MultipleDecisionMakers(e.g.,games)•  “Classical”vs.“Adaptive”vs.“Robust”

Aroughclassification

OneDecisionMaker

SeveralDecisionMakers

Static(notime/sequential)

Optimization GametheoryRobustOptimization

Dynamic(sequentialdecisions)

OptimalControl DynamicGamesRobustControl

Keynotions

Analysis:Givenaclosed-loopsystem(G,K)andspecifications,determineifthespecificationsaresatisfied.Synthesis:Givenanopen-loopsystemGandspecifications,findacontrollerKsuchthatthespecificationsaresatisfied.

But…

Mathematical/Computationalmodels(regardlessofdetail)arealwaysmerelyapproximationstothe“real”phenomenon.Robustness:Simultaneouslymodelthesystem,andtheuncertaintyassociatedwithitE.g.,unknownparameters,uncertaindynamics,environment…

Optimalitydoesnotsuffice J.C.Doyle(1978),IEEETransactionsinAutomaticControl

State-of-the-artdesignmethods(optimalquadraticcontrol,LQG)Arbitrarilysmallperturbationscandestabilizesystem!

ModelsystemanduncertaintyManydescriptionsofuncertainty:

•  Set-based:unknownelementsinagivenset(worst-case)

•  Probabilistic:Setofallowableperturbations,plusadistribution

Whatcanweefficientlydo?Models/theoremsaregreat,butwhentherubbermeetstheroad,whatreallymattersiswhatyoucancomputeeffectively(andthismaymeandifferentthingstodifferentpeople)Formulatingvs.solvingOptimalControl:Hamilton-Jacobi-Bellman(HJB)equationNonlinearfiltering:Kushner,ZakaiequationsDynamicProgramming:“inprinciple”cansolveanywell-definedgame,but..

(Ontheboard…)Linearsystems–whydolikethemsomuch,andwhatwecandowiththemGoingnonlinear–analysisisgreat,synthesisnotsomuchUncertainty–IQCmodelsBottomline:incertaincases,reliable,algorithmicanalysis/designmethods

AFewLessons“Optimality”doesnotguaranteerobustness–infact,mayhurtitValuablestructuralinsights(e.g.,controllability/observability)Cleverreformulationscanhelpalot,evenifintheendwestillresorttocomputationalmethodsValuabletoolkittoexploitstructure

Certifyingperformance

•  Semidefiniterelaxationsarecentralintoday’srobustcontrol/optimization

• Why?Becausewedon’tknowmanywaysofrigorouslycertifyingpropertiesoffunctionsinmanyvariables.

•  Speculative:linksbetweenrobustnessandverifiability

BillionDollarquestion

WhatwillbetheepistemologyofdeployableML?Howwillweconvinceourselvesthatthings“work”?Whatwillthismean?Theorems?Extensivesimulations?Distinctionbetweencat/no-catvs.safety-critical?(e.g.,90000dailyflightsinUS.99.9%successrate->90dailyplanecrashes)Howtobridgeacrossallthesedifferentcultures?