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# A Computational Framework to Robustness Analysis and Gain Tuning

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1. A Computational Framework toRobustness Analysis and Gain Tuning Luis G. Crespo National Institute of Aerospace Sean Kenny, Dan Giesy Dynamic Systems and Control Branch, NASA LaRC AEM, University of Minnesota, April 22, 2011

2. Outline • Robustness Analysis • Control Tuning • Adaptive Control Example • Conclusions • Bonus

3. Robustness Analysis: Framework AOA [deg] CG location Time [s] Time delay Parameter Space Reference model, nominal controller, adaptive controller

4. Robustness Analysis: Framework AOA [deg] CG location Time [s] Time delay AOA [deg] Parameter Space Time [s] Command, nominal controller, adaptive controller

5. Robustness Analysis: Framework AOA [deg] Adaptive CG location Nominal Time [s] Time delay AOA [deg] Parameter Space Time [s] Command, nominal controller, adaptive controller

6. Robustness Analysis: Motivation AOA [deg] Adaptive CG location Nominal Time [s] Time delay AOA [deg] Parameter Space Time [s] • There may be parameter realizations where any of the controllers outperforms the other one

7. Robustness Analysis: Framework • Plant: Consider the dynamical system where is the state, is the reference command, is an unknown parameter and is the control input

8. Robustness Analysis: Framework • Plant: Consider the dynamical system where is the state, is the reference command, is an unknown parameter and is the control input • Control design: Find the controller with gains such that the closed-loop requirements are satisfied. Lets describe the satisfaction of the requirements as • Safe domain • Failure domain

9. Robustness Analysis: Framework • Plant: Consider the dynamical system where is the state, is the reference command, is an unknown parameter and is the control input • Control design: Find the controller with gains such that the closed-loop requirements are satisfied. Lets describe the satisfaction of the requirements as • Safe domain • Failure domain • Robustness analysis: Study how depends on

10. Robustness Analysis: Framework • Plant: Consider the dynamical system where is the state, is the reference command, is an unknown parameter and is the control input • Control design: Find the controller with gains such that the closed-loop requirements are satisfied. Lets describe the satisfaction of the requirements as • Safe domain • Failure domain • Robustness analysis: Study how depends on by sizing subsets of the safe domain • Control verification:Arbitrary structures for , , and

11. Robustness Analysis: Framework S2 S1 • PSM, CPV, Sampling vs. optimization

12. Outline • Robustness Analysis • Control Tuning • Adaptive Control Example • Conclusions • Bonus

13. Control Tuning: Framework • Plant: Consider the dynamical system where is the state, is the reference command, is an unknown parameter and is the control input • Control design: Find the controller with gains such that the closed-loop requirements are satisfied. Lets describe the satisfaction of the requirements as • Safe domain • Failure domain • Robustness analysis: Study how depends on by sizing subsets of the safe domain • Control tuning:Search for the controller gain that makes sufficiently large. State of the practice.

14. Control Tuning: Framework S2 S1 • Conflictive objectives, robustly optimal controller

15. Outline • Robustness Analysis • Control Tuning • Adaptive Control Example • Conclusions

16. GTM Example: The Plant • Dynamically scaled flight test article • High-fidelity mathematical model having non-linear aerodynamics, avionics, engine and sensor dynamics, atmospheric model, telemetry effects, etc. (278 states) • Control inputs: • Commands:

17. GTM Example: The Architecture Nominal Controller Plant Pilot - + × Adaptive Controller Adaptive law Reference model • Nonlinear controller: integrator of x2, dependence on IC, transient resp. • Reference model dynamics: attainability • Triggers of adaptation: parametric uncertainty, nonlinearities, time delay • Switching adaptation on and off 19

18. GTM Example: The Architecture ACTS ADAPTIVE CONTROLLER Failures Adaptive law Reference model Damages Pilot Time delay Saturation modification Anti-wind up modification Uncertainties Dead zone Projection algorithm Generic Transport Model NOMINAL CONTROLLER Longitudinal controller Sensor dynamics Aero- dynamics Lateral/dir controller Equations of motion Actuator dynamics Auto- throttle Anti-wind up Avionics Telemetry 20

19. GTM Example: The Uncertainties • Actuator failure • Additional time delay , • Scaling command , • CG longitudinal displacement , • Aerodynamic uncertainties • Pitch stiffness (via inner elevators) • Roll damping (via flaps) • Yaw damping (via lower rudder) Loss of effectiveness Locked-in-place

20. GTM Example: The Requirements • Structural loading (1) • Command tracking (2,3,4,5) • Reliable flight envelope (6) • Riding/Handling quality (7) • Reference tracking (8,9) The evaluation of the 9 performance functions composing requires a simulation of the closed-loop system for a fixed set of commands

21. GTM Example: The Tasks • Determine the merits of adaptation by comparing a model reference adaptive controller with a non-adaptive, flight-validated controller • Design, analyze and tune both the non-adaptive and adaptive controllers • Determine the benefits and drawbacks of adaptation from a safety perspective

22. Non-adaptive Baseline Controller Pitch damping uncertainty Roll damping uncertainty • This controller was approved by the pilot after extensive real time simulation

23. Non-adaptive Tuned Controller Pitch damping uncertainty Roll damping uncertainty • This controller tolerates 48% more roll damping uncertainty than the baseline for > min

24. Non-adaptive Tuned Controller • Pilot-approved in the real time simulator • Flight tested in the NASA GTM Test Article • This non-adaptive controller exhibited as much robustness to pitch stiffness and roll damping uncertainty than DFMRAC and L1 adaptive controllers in both the high-fidelity simulation and flight tests

25. Tuning of An Adaptive Controller • Nonlinear dependence of the adaptive gains on the error and the state obscures causality • Ad-hoc and random search-based strategies for setting ranges and rates of adaptation • Good adaptive rates for some uncertainties may be too large/small for others • Lack of stability margins increases the risk of over-tuning

26. Tuning of An Adaptive Controller Γ 4Γ Pitch damping uncertainty Pitch damping uncertainty Roll damping uncertainty Roll damping uncertainty 8Γ 12Γ Pitch damping uncertainty Pitch damping uncertainty • Sudden transition to instability Risk of over-tuning

27. Tuning of An Adaptive Controller Γ 4Γ Pitch damping uncertainty Pitch damping uncertainty Roll damping uncertainty Roll damping uncertainty 8Γ 12Γ Pitch damping uncertainty Pitch damping uncertainty •  Using P[F] as a robustness metric: geometry of F is not important, Γtrend

28. Tuning of An Adaptive Controller Γ 4Γ Pitch damping uncertainty Pitch damping uncertainty Roll damping uncertainty Roll damping uncertainty 8Γ 12Γ Pitch damping uncertainty Pitch damping uncertainty •  Using PSM as a robustness metric: geometry of F is important, Γtrend

29. Adaptive Tuned Controller: Benefits GAIN Maximal set of cada, tuned Pitch damping uncertainty Roll damping uncertainty • This controller tolerates up to 22% more roll damping uncertainty than the baseline for > min

31. Adaptive Tuned Controller: Risks g(ΛRD, cnominal), g(ΛRD, cadaptive) Yaw rate performance requirement Locked in place surface Loss of control effectiveness 33

32. Adaptive Tuned Controller: Risks g(ΛRD, cnominal), g(ΛRD, cadaptive) Locked in place surface Loss of control effectiveness Instability 34

33. Conclusions • Framework • Scope: arbitrary plant models, control structures and requirements • Implementation:standard optimization algorithms • Analysis: weakly sensitive to the uncertainty model • Caveats: curse of dimensionality, convergence to global optima • Outcomes • Deterministic and probabilistic robustness metrics • Critical combination of uncertain parameters • Identification of strengths, weaknesses and trade-offs • Means to justify/reject additional complexity based on its merits • Unifying framework for control tuning and control verification • Supports V&V and certification of control systems

34. A Computational Framework toRobustness Analysis and Gain Tuning Luis G. Crespo National Institute of Aerospace Sean Kenny, Dan Giesy Dynamic Systems and Control Branch, NASA LaRC AEM, University of Minnesota, April 22, 2011

35. A New Paradigm in Uncertainty Analysis Luis G. Crespo National Institute of Aerospace Sean Kenny, Dan Giesy Dynamic Systems and Control Branch, NASA LaRC Cesar Munoz, Anthony Narkawicz Safety Critical Avionics Systems Branch, NASA LaRC

36. State of the Practice • Are the requirements satisfied?

37. State of the Practice • Uncertainty model

38. State of the Practice • Failure probability

39. State of the Practice x x x x x x x x x x x • Monte Carlo Analysis

40. State of the Practice: The Good x x x x x x x x x x x • The failure probability is a meaningful reliability measure • Easy to calculate

41. State of the Practice: The Bad x x x x x x x x x x x • It may be computationally expensive to evaluate (e.g., 99.9%)

42. State of the Practice: The Ugly x x x x x x x x x x x • Fails to describe some of the desired attributes of the system • Inherently linked to the uncertainty model: liability • Refinement of uncertainty models makes previous effort obsolete (e.g. Ares)

43. New Paradigm • Fresh start…

44. New Paradigm • Let and be inner and outer approximations to the failure domain • The probability of subsets of and can be calculated exactly

45. New Paradigm • What can be said about the failure probability?

46. New Paradigm • We can give exact probability bounds

47. New Paradigm • Imagine the approximations and approaching the failure domain

48. New Paradigm • Consequently, the failure probability bounds become tighter