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1 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Data-Driven Behavioral Formulation of the Adaptive Feedback Control Problem Michael G. Safonov University of Southern California

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2 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB An Achilles heel of modern system theory has been the habit of ‘proof by assumption’ Theorists typically give insufficient attention to the possibility of future observations which may be at odds with assumptions.

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3 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Controller Adaptation should be intelligent. Intelligence is consistency between FACTS, DECISIONS, & GOALS. FACTS assumptions, beliefs, models or data DECISIONS controller, estimator GOALS cost functions, objectives

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4 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB FACTS But, there are two kinds of FACTS DATA 1. Observed DATA (data/evidence) BELIEFS 2. Prior BELIEFS (assumptions/models/axioms) DATABELIEFS Science gives DATA precedence over BELIEFS : BELIEFSDATA BELIEFS inconsistent with DATA are rejected.

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5 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB “data-driven” Galileo: open-eyed “data-driven” Reality is what we observe. And two ways to learn & adapt: Observation vs. Introspection vs. ‘Curve-Fitting’ Plato: introspective “assumption driven” Reality is an ideal, observable only through noisy sensors. ‘Probabilistic Estimation’ MODELS approximate Observed DataData approximates Unobserved TRUTH

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6 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Example: Linear Regression CURVE FIT: (Galilean) Given data (y i,u i ), i=1,2,... (y 1,u 1 ) (y 2,u 2 ) (y 3,u 3 ) (y 4,u 4 ) (y 5,u 5 ) (y 6,u 6 ) (y 7,u 7 ) height 2 { BAYESIAN ESTIMATE (Platonic): Given data (y i,u i ), i=1,2,... ‘noise’ v=N(0, ) ),ba (y |x,maxprob subject to prior beliefs

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7 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB The data driven Galilean curve fitter will remain open minded: He will wait to look and see how his model fits the data. Both the Bayesian probabilist and the Galilean curve-fitter use the same formula to estimate model parameters (a,b), but a naive Bayesian may have some (rather unrealistic) expectations for his model. (y 1,u 1 ) (y 2,u 2 ) (y 3,u 3 ) (y 4,u 4 ) (y 5,u 5 ) (y 6,u 6 ) (y 7,u 7 ) height 2 { The assumption driven Bayesian ‘knows’ a priori that 2/3 of his future data must eventually lie in his predicted 2 confidence bound, and 1/3 outside

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8 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB And, Galilean vs Bayesian : two kinds of adaptive control “Data Driven” goals are modest: –no guaranteed predictions of future stability –just consistency of goals, decisions and data no troublesome assumptions, parsimonious formulation : –DATA –GOALS –DECISIONS no approximation “Assumption Driven” goals are ambitious: –guaranteed future stability –Cost & estimation convergence many troublesome assumptions –“the ‘true’ plant is in a given model set”, –“noise independent identically distributed” –“bounds on parameters, probabilities” –…, “linear time-invariant, minimum- phase plant, order < N” Remarkably, some leading “assumption driven” control theorists have held that observed DATA inconsistent with ASSUMPTIONS should be ignored (cf. M. Gevers et al., "Model Validation in Closed-Loop", ACC, San Diego, 1999)

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9 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Data Driven Adaptive Control candidate controllers FALSIFIED COMPUTER SIEVE LEARNING FEEDBACK LOOPS K Unfalsified Controllers K given M. G. Safonov. In Control Using Logic-Based Switching, Spring-Verlag, GOALS ACTIVE CONTROLLER SELECTOR DATA DECISIONS

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10 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB The Behavioral Approach to Adaptive Control Data Driven: Let the data speak... –Don’t let modeling beliefs trump observation Unfalsify (validate) models and/or controllers against hard criteria: –Choose criteria expressible directly in terms of observed data (sensor outputs, actuator inputs) –Avoid criteria that that rely on “noise model” and other prior beliefs

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11 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Unfalsification Algorithm Brugarolas & Safonov, CCA/CACSD ‘99 UNFALSIFIED ADAPTIVE CONTROL: The ability of each candidate controller to meet the performance goal is treated as a hypothesis to be tested directly against evolving real-time measurement data. The controller need not be in the loop to test the hypothesis. UnfalsificationAlgorithmUnfalsificationAlgorithm

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12 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Plant Data:at time t=0, (u,y)=(1,1) Candidate K’s: u=Ke real gain Goal:|e(t)/r(t)| K is unfalsified if |1/(1+K)| < 0.1 => unfalsified K’s: K>9 or K<-11 Trivial Example Unknown Plant K gain r uye - +

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13 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB DATA-DRIVEN Behavioral Problem Formulation Observations operator maps input-output signals to measurement signals. Truncated Space results from applying the observations operator to a signal space. Typically is the experimental observation time sampling operator, so returns values of z(t) only for past time intervals over which experimental observations of z(t) have been recorded. is a projection operator.

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14 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB BACKGROUND: Willems’ Behavioral System Theory

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15 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB BEHAVIORAL ADAPTIVE CONTROL

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16 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Application: Behavioral MRAC Behavioral Representation of Standard Class of Candidate MRAC Controllers

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17 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Behavioral MRAC (cont.)

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18 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Main Result: Solution of Behavioral MRAC Problem A matrix pencil e-value computation gives optimal.

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19 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Data Driven Behavioral MRAC Computation

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20 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Noisy Plant Simulation Example

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21 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Data Driven Behavioral MRAC Simulation Results

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22 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Discussion: Data Driven Behavioral MRAC Unstable plant stabilized in real time Computation effort does not grow with No need for controller parameter gridding Exponential forgetting factor in cost – no controller ‘gain windup’ – behaves like classical - modification

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23 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Other Examples: Unfalsified PID Control ————— Jun & Safonov, CCA/CACSD ‘99 Example: Adaptive PID Goal: Unfalsified adaptive control loop stabilizes in real-time Unstable Plant 30 Candidate PID Controllers: K I =[2, 50, 100] K D = [.5,.6] K P = [5, 10, 25, 80, 110]

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24 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Benchmark Simulation Jun & Safonov, CCA/CACSD ‘99 control input u(t) proportional gain k P (t) integral gain k I (t) derivative gain k D (t), plant output y(t) Evolution of unfalsified setTime responses Tsao & Safonov, IEEE Trans, AC-42, 1997.

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25 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Another Example: Missile Learns control gains Adapts quickly to compensate for damage & failures Superior performance Specified target response bound Actual response Commanded response Brugarolas, Fromion & Safonov, ACC ’98 Unfalsified adaptive missile autopilot: discovers stabilizing control gains as it flies, nearly instantaneously maintains precise sure-footed control Brugarolas, Fromion and Safonov, ACC98

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26 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Other People’s Successes with Unfalsified Control Emmanuel Collins et al. (Weigh Belt Feeder adaptive PID tuning, CDC99) Kosut (Semiconductor Mfg. Process run-to- run tuning, CDC98) Woodley, How & Kosut (ECP Torsional disk control, adaptive tuning, ACC99 … maybe others soon?

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27 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Conclusion We have analytic tools for controlling assumed models DATA-DRIVEN analytic tools needed to reliably close adaptive feedback loops with experimentally observed data A SOLUTION: DATA-DRIVEN Unfalsified Adaptive

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28 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Acknowledgement Bob Kosut’s mid-1980’s work on time-domain model validation and identification for control played a key role in laying the foundations of this work, as did later contributions of Jim Krause, Pramod Khargonekar, Carl Nett, Kamashwar Poolla, Roy Smith and many others who have advanced the use of validation methods in control-oriented identification. Tom Mitchell’s early 1980’s “candidate elimination algorithm’’ for machine learning is closely related to the unfalsified control methods presented here. And of course, none of this would have been possible without the superb graduate education that I received at MIT so many years ago under the guidance of first Jan Willems and later Michael Athans.

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29 Michael G. Safonov 30 July - 2 August 2001 AFOSR Dynamics and Control Meeting, WPAFB Selected References

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