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## A Computational Framework to Robustness Analysis and Gain Tuning

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**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**Outline**• Robustness Analysis • Control Tuning • Adaptive Control Example • Conclusions • Bonus**Robustness Analysis: Framework**AOA [deg] CG location Time [s] Time delay Parameter Space Reference model, nominal controller, adaptive controller**Robustness Analysis: Framework**AOA [deg] CG location Time [s] Time delay AOA [deg] Parameter Space Time [s] Command, nominal controller, adaptive controller**Robustness Analysis: Framework**AOA [deg] Adaptive CG location Nominal Time [s] Time delay AOA [deg] Parameter Space Time [s] Command, nominal controller, adaptive controller**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**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**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: 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**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**Robustness Analysis: Framework**S2 S1 • PSM, CPV, Sampling vs. optimization**Outline**• Robustness Analysis • Control Tuning • Adaptive Control Example • Conclusions • Bonus**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.**Control Tuning: Framework**S2 S1 • Conflictive objectives, robustly optimal controller**Outline**• Robustness Analysis • Control Tuning • Adaptive Control Example • Conclusions**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:**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**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**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**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**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**Non-adaptive Baseline Controller**Pitch damping uncertainty Roll damping uncertainty • This controller was approved by the pilot after extensive real time simulation**Non-adaptive Tuned Controller**Pitch damping uncertainty Roll damping uncertainty • This controller tolerates 48% more roll damping uncertainty than the baseline for > min**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**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**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**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**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**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**Adaptive Tuned Controller: Risks**g(ΛRD, cnominal), g(ΛRD, cadaptive) Yaw rate performance requirement Locked in place surface Loss of control effectiveness 33**Adaptive Tuned Controller: Risks**g(ΛRD, cnominal), g(ΛRD, cadaptive) Locked in place surface Loss of control effectiveness Instability 34**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**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**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**State of the Practice**• Are the requirements satisfied?**State of the Practice**• Uncertainty model**State of the Practice**• Failure probability**State of the Practice**x x x x x x x x x x x • Monte Carlo Analysis**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**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%)**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)**New Paradigm**• Fresh start…**New Paradigm**• Let and be inner and outer approximations to the failure domain • The probability of subsets of and can be calculated exactly**New Paradigm**• What can be said about the failure probability?**New Paradigm**• We can give exact probability bounds**New Paradigm**• Imagine the approximations and approaching the failure domain**New Paradigm**• Consequently, the failure probability bounds become tighter