Acknowledgements

Numerical Simulation of Aero-Servo-Elastic Problems, with Application to Wind Turbines and Rotary Wing VehiclesCarlo L. BottassoPolitecnico di MilanoCOMPDYN 2007 Rethymno, Crete, Greece, June 13-15, 2007

Acknowledgements Work in collaboration with: A. Croce, D. Leonello, L. Riviello, B. Savini; Work supported by: AgustaWestland, US Army Research Office, Leitner S.p.A.

Outline • Multidisciplinary FEM-based multibody modeling; • Active control of complex aero-servo-elastic models: three challenging applications; • Adaptive identification of reduced models; • Examples; • Conclusions and outlook.

Multidisciplinary FEM Multibody Modeling Rotorcraft aeroelastic model: finite element multibody code (Bauchau & Bottasso 2001). Highlights: arbitrarily complex topologies, non-linearly stable energy decaying schemes. Aerodynamic models: lifting lines; inflow models (Pitt-Peters, Peters-He); free-wake; CFD. Other models: controllers; actuators (hydraulic, engine, etc.); sensors; unilateral contact.

Multidisciplinary FEM Multibody Modeling

Some Trends and Needs • Integration of complex multi-physics models with active controls (e.g., add virtual pilots to vehicle models, systematically explore operational envelope boundaries, etc.); • Model reduction (e.g., for model-based control); • Model identification from experimental data; • Real-time performance (e.g., for pilot-in-the-loop applications).

Active Control of Complex Aero-Servo-Elastic Models • Three challenging applications : • Simulation of maneuvers (Maneuvering Multibody Dynamics, MMBD); • Finding periodic solutions (“trimming”); • Control of wind turbine generators.

Maneuvering Multibody Dynamics • Limiting factors (maximum loads, vibrations, noise, etc.) are experienced in the maneuvering regime and at the performance envelope boundaries. • It is virtually impossible to guess the controls that will produce a complex maneuver of long duration, guaranteeing to stay within the performance envelope boundaries. Example: Cat-A continued take-off. TDP Two model predictive problems: trajectory planning & tracking.

Maneuvering Multibody Dynamics

Example: Cat-A Continued TO

MMBD: Model Predictive Planning & Tracking Updated reference trajectory 1. Maneuver planning problem (reduced model) Reference trajectory 2. Tracking problem (reduced model) Fly the comprehensive model along the reference trajectory and, at the same time, update the reduced model (learning). 5. Re-plan with updated reduced model 3. Steering problem (comprehensive model) 4. Reduced model update Prediction window Prediction window Tracking cost Tracking cost Prediction window Tracking cost Reference trajectory Prediction error Prediction error Steering window Steering window Prediction error Trajectory flown by comprehensive model Predictive solutions Steering window

Trimming Trim: control settings, attitude and cargo disposition for a desired steady (flight) condition. Performance, loads, noise, handling qualities, stability, etc. depend strongly on the trim condition. • Procedure: • Given desired loads or velocities specifying the desired condition, • Find resulting attitude and constant-in-time controls. TRIM PROBLEM • Important remark: • Rotorcraft systems excited by harmonic external loads; • Periodic response of all states and loads at trim.

¤ y T + t _ Z 1 ( ) ( ) e e e e e ¤ 8 8 8 T 0 t t t t t + + x y u y x z ; = = = ( ) e e e e d t ; ; ; ; : y g x u ; = ; T t Formulation of Rotorcraft Trim Problem • Define system outputs (problem dependent): • Wind tunnel trim: components of rotor loads in fixed system; • Free flight: capture gross vehicle motion. • Trim constraints: • where are desired values for the outputs; • Trim conditions: • Periodicity conditions: • (See Peters & Barwey 1996)

Model Predictive Trimming • Procedure: • Predictsystem response using a non-linear reduced model; • Compute controls tosteerthe system for a short time horizon; • Update reduced model based on predicted-actual output errors; • Iterate, shifting prediction forward (receding horizon control).

Control of Wind Turbine Generators • Goals: • Regulate wind turbine by adjusting blade pitch (and possibly generator torque) to react against wind turbulence and gusts. • Minimize fatigue damage and maximize power output.

Reduced Model Identification System Identification Comprehensive multibody-based model: many dofs, captures fine scale solution details. This procedure is common to all three previous problems. Reduced model: few dofs, captures gross to-be-controlled response.

Reduced Model Identification • Goal: • Develop reduced model capable of predicting the behavior of the plant with minimum error (same outputs when subjected to same inputs); • Reduced model must be self-adaptive (capable of learning) to adjust to varying operating conditions, and to react to disturbances. Prediction (tracking) window Prediction error to be minimized Steering window Predictive solutions

e e e e e e ¸ ¼ y u u x y y u = , e ( ( ) ) e e f h _ 0 y y y u p x = = ; ; ; : ; e e _ ( ) e e e f ¸ 0 x x u = ; ; ; ; _ ( ) e e e 0 c x x = ; ; Reduced Model Identification • Comprehensive (multibody based) governing equations: • where are the states, the controls, the Lagrange multipliers. • Define outputs that capture the to-be-controlled outputs: • Find reduced parametric model • such that when • i.e. • the reduced model captures the gross motion of the comprehensive one (plant).

u u y y ( ) f _ 0 y y u ; = f r e ; ; Reduced Model Formulation • Neural augmented reference model: • Reference (problem dependent) analytical model: • For example, in this work: • - Rotorcraft problems: 2D rigid body model, actuator-disk rotor (blade element theory + uniform inflow). • = CG position & velocity, pitch & pitch rate, rotor speed; • = main & tail rotor collective, lateral & longitudinal cyclics, available power. • - Wind turbine problems: actuator-disk rotor + springs to model tower flexibility. • = rotor speed, tower tip position & velocity; • = blade pitch, generator torque.

e e e e d d 6 y u u y y y u u = = = = . . ( ) ( ) ( ) n f d _ y y u y y u = f r e ; ; ; : : : ; ; ; Reduced Model Formulation • Remark: reference model will not, in general, ensure adequate predictions, i.e. • when • Augmented reference model • where is the unknown reference model defect that ensures • when • Hence, if we knew , we would have perfect prediction capabilities.

Reduced Model Formulation • Approach: • - Approximate the unknown defect using a parametric function (neural network); • - Adjust the function parameters to ensure good approximation of the defect (hence, good predictions). • Reasons for using a reference model: • - Reasonable predictions even before any learning has taken place (otherwise would need extensive pre-training); • - Easier and faster adaption: the defect is typically a small quantity, if the reference model is well chosen.

Results Rotorcraft maneuver problem: pitch rate for multibody, reference, and neural-augmented reference with same prescribed inputs. Red: reference model Black: multibody model Blue: reference model +neural network Short transient = fast adaption Good prediction, even for changing flight condition.

Results Rotorcraft trim problem: rotor thrust for multibody, reference, and neural-augmented reference with same prescribed inputs. Black: multibody model Blue: reference model +neural network Red: reference model

i d " i Results Rotorcraft trim problem: defect and remaining reconstruction error after adaption. Red: defect Blue: remaining reconstruction error

Results Wind turbine control problem: tower-tip velocity for multibody, reference, and neural-augmented reference with same prescribed inputs. Fast adaption Red: reference model Black: multibody model Blue: reference model +neural network Good prediction, even with turbulent wind.

i d " i Results Wind turbine control problem: defect and remaining reconstruction error after adaption. Red: defect Blue: remaining reconstruction error

Application: Rotorcraft Minimum Time Obstacle Avoidance • Optimal Control Problem (with unknown internal event at T1) • Cost function: • Constraints and bounds: • - Initial trimmed conditions at 30 m/s • - Power limitations

Minimum Time Obstacle Avoidance

Minimum Time Obstacle Avoidance Effect of reduced model adaption: Blue: planned trajectory Red: tracked trajectory Trajectories at 1st iteration Trajectories at 4th iteration

Minimum Time Obstacle Avoidance Effect of reduced model adaption: Red: tracked trajectory Blue: planned trajectory Pitch vs. time at 4th iteration Pitch vs. time at 1st iteration

Conclusions • Observations: • Computational procedures now blend traditionally separate disciplines, e.g. aero-servo-elasticity with flight mechanics; • Mathematical models of vehicles are becoming so complex that there is a trend to use methods for analyzing experimental data (e.g. stability analysis, system identification, etc.); • Outlook: • These trends will continue (virtual lab); • Real-time simulation; • Human behavior models.

Acknowledgements