Outline

1. ROTORCRAFT TRIM BY A NEURAL MODEL-PREDICTIVE AUTO-PILOTCarlo L. Bottasso and Luca RivielloPolitecnico di Milano, Italy62nd American Helicopter SocietyAnnual Forum and Technology Display

2. Outline • Background and motivation: • Rotorcraft trim; • Possible solution strategies; • Non-linear Model-Predictive (NMP) auto-pilot: • Formulation; • Adaptive reduced model; • Numerical example; • Conclusions.

3. Introduction and Motivation 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.

4. Introduction and Motivation • Rotorcraft trim approaches: • Periodic shooting • Harmonic balance • Finite elements in time • Auto-pilot: • Adjust control settings to “fly” the system to the trimmed solution (Peters, Kim & Chen, 1984)(Peters, Chouchane & Fulton, 1990); • Very efficient even for large vehicle models (cost does not depend on the number of DOFs). Computational cost is a function of the number of DOFs.

5. Introduction and Motivation • High-fidelity comprehensive aeroelastic models: • Based on non-linear MB dynamics formulations; • Coupled with complex aerodynamic models or CFD. • Need for efficient trim procedures. • Current auto-pilots: • Are unsuitable for unstable systems; • Offer no guarantee of stability; • Often find limit cycle solutions. • Proposed approach: use a non-linear adaptive control technique for auto-pilot-based rotorcraft trim. Tilt-rotor whirl-flutter analysis (about 104 degrees of freedom)

6. e e e ¸ x u e e _ ( ) e e e f ¸ 0 x x u = ; ; ; ; _ ( ) e e e 0 c x x = ; ; Comprehensive Multibody Models • Comprehensive (multibody based) governing equations: • (dynamic & kinematic eqs.) • (constraints) • where: • System states :displacements/rotations, linear/angular velocities, internal states; • System controls :e.g. actuator inputs, controlled joint rotations, applied forces; • Lagrange multipliersthat enforce the constraints.

7. ¤ 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)

8. ¤ y : T + t T Z 1 ( ( ) ) e e e e µ µ µ u g x u ; = 0 1 1 ( ) e e e e d s c ³ ´ ³ ´ t ¼ ¼ ; ; ; : y g x u ; = ( ) µ Ã µ µ Ã µ Ã i i i i 1 2 3 4 ; + ¡ + ¡ T s n c o s = = i 0 1 1 s c ; ; ; ; : t 2 2 Rotorcraft Trim: Example Wind-tunnel trim: given advance ratio, find the controls that produce desired values of given average hub loads. • Hub loads: • Average hub loads: • Desired average hub loads: Blade pitch: Rotor controls:

9. e e e G ¤ u y y u ; ; ; f i ; ; · ¸ e e e e e e e @ ¡ ¡ ¡ 1 ¡ y y y y y y y ( ) e e e ¤ S G ¢ 1 0 2 0 0 t + ¡ n u u y y = S f i u ¼ ; = ; ; : : : ; : @ ¢ ¢ ¢ u 1 2 n u Trim Solution Strategies: Auto-pilot • General auto-pilot procedure: • Controls are promoted to dynamic variables; • Error on trim constraints is measured; • A suitable control law is designed to converge to the trim solution. • A possible (classical) proportional auto-pilot strategy (in discrete form) is • where: • Present/targetoutputs: - Initial/finalcontrols: • Gain matrix: • Input/output“sensitivity” matrix:

10. NMP Auto-pilot • Proposed auto-pilot strategy: • 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).

11. NMP Auto-pilot • Highlights: • Framework forguaranteeing stabilityof the nonlinear closed-loop system; • Superior controlperformance (optimal control theory); • Constant-in-timeconstraintson controls explicitly enforced.

12. T Z ( ( ) ) e f ¤ ¤ ¤ T _ f 0 y p u y y u y p ; = = i i ( ) ¤ ; ; ; ; d J M t y y u = ( ( ) ) j j j j j j j j j j j j ¤ ¤ ; ; ; _ _ M T T T 0 · · t t ¡ + + < y y u u y y u u ; = = S S S f i T ; ; ; ; c _ y u u i NMP Auto-pilot • Model-predictive tracking problem: solution yields steering controls . • Minimize cost • where • Subjected to: • Reduced model equations: • where is current estimate of model parameters. • Initial conditions: • Trim conditions: • Remark: constraints on controls (and states) are hard to incorporate in other control strategies.

13. ( ) e e f ¤ _ 0 ¼ ¼ y u p y y u y u p = ; . ; ; ; Adaptive NMP Auto-pilot • Stability: guaranteed for infinite prediction horizon and reduced model identical to the plant. • Approximations: • Finite prediction horizon to lower computational cost; • Reduced model only approximates plant response. • Proposed solution: • Identify adaptive parametric reduced model to control the approximation error and converge to exact trim solution: • where the model parameters areoptimizedto have • when

14. e e d ¼ y u y u = . ( ) ( ) ( ( ) ) n f f d _ _ 0 y y u y y u y y u ; = = f f r e r e ; ; ; ; ; : : : ; ; ; Reduced Model • Reduced model: • Reference analytical model: • Reference model is problem dependent. • E.g.: wind tunnel trim  classical performance rotor model based on blade element theory with uniform inflow (Prouty 1990). • Augmented reduced model: • where is the unknown reference model defect that ensures • when

15. ( ) T T T T i i j j ( ) » d 8 C C n 0 · > p y y u " " = ; ; : : : ; ; T T T ( ( ) ) ( ) T T i i i i i i i i i i i i i i ( ( ( ) ) ) ( ( ( ) ) ) n n n d d d » b b W V W V + + + p y p y y y u u p p y ¾ u p a a " = = = = k k k k N N N N j j j j : : : ; ; ; ; : : : : : : : : : ; ; ; ; ; ; : : : ; ; : : : ; ; ; ; ; ; ; ; : ; : : : Reduced Model Identification Approximate with single-hidden-layer neural networks, one for each component: where and = reconstruction error (universal approximator, ); = network input. The reduced model parameters are readily identified with the synaptic weights and biases of the networks:

16. 1 T T ¯ ¯ J ¡ + T p p p z e ´ ´ = = k k k k ( ) ( ) 1 + d d J = ¯ = ¯ ¡ ¡ ; k k p p z z = ; 2 d ¡ e z = k k k : ( ) d d y u = ; ; ( ) d z y u p = N N ; ; : Model adaption: steepest descent strategy Cost function for adaption: Steepest descent update:

17. e ¤ u x i T T T ¡ = f i p NMP Auto-pilot Summary: at every controller activation the algorithm is Prediction horizon Computed optimal control history Initial state prediction

18. e e e ¤ u x x y i ; T T T T T T ¡ ¡ < = f i i p c p NMP Auto-pilot Summary: at every controller activation the algorithm is Prediction horizon Computed optimal control history Initial state prediction Steering (MB integration fwd in time) Control horizon System response

19. e e e ¤ u x x y y i ; T T T T T T ¡ ¡ < = f i i p c p NMP Auto-pilot Summary: at every controller activation the algorithm is Prediction horizon Computed optimal control history Initial state prediction Predicted response model adaption Steering (MB integration fwd in time) Control horizon System response

20. Numerical Example System • Wind-tunnel trim of a four-bladed flexible rotor: • UH-60 rotor multibody modelattached to the ground; • Three controls: bladecollectiveand longitudinal and lateral cyclicpitch angles; • Aerodynamics:strip theory. Reference model Analytical blade element/momentum theory, static flapping (performance model). Target Trim for three desired average hub load componentsin theinertial frame.

21. ( ) k ( ) k e ¤ ¤ t t ¡ y y y " = s 2 s s Numerical Example Methodology • Given rotorcraft advance ratio (flight speed/tip speed) and weight,estimate the forces (ouputs)necessary to trim in straight level flight. • Then: • Initialize the controls to small values; • Activate the auto-pilot. Goal Steer rotor outputs to the desired values and evaluate controls in trim. Error definition where are (scaled) target trim outputs.

22. ( ) m a x m a x 8 T T 0 0 1 · ¸ t t : " " " = ; ; : : Numerical Example Time to trim: NPMA parameters: Activation freq.:4/rev; Prediction:3 rev; Num. of Neurons:20; Max. control rates:10 deg/sec. Classical auto-pilot stability limit. Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA.

23. Numerical Example Controls: classic auto-pilot A, advance ratio 0.297.

24. Numerical Example Controls: NMP auto-pilot, advance ratio 0.297.

25. ( ) m a x m a x 8 T T 0 0 1 · ¸ t t : " " " = ; ; : : Numerical Example Time to trim: NPMA parameters: Activation freq.:4/rev; Prediction:3 rev; Num. of Neurons:20; Max. control rates:10 deg/sec. Solid: NMPA, no continuation; Dashed: NMPA, continuation.

26. P 1 ¡ ^ ^ K ¢ _ ¢ 0 T T ¡ ¢ + p p p p p e K P H V H P H = = = k k k k k k + 1 1 + + = ; ; ; k k k 2 ; ( ( ) ) d d d d ^ H ¢ ¢ + + ( ) z y u y p p u p v v P I K H P = = = k k k k k k N N N N ¡ 2 2 = ; ; ; ; : : ; k k k 1 + ; d ¡ T e z = [ ] k k k V E : v v = 2 2 2 : Model adaption:Extended Kalman (EK) approach Model for parameter estimation: Discretization and linearization: Enforcing optimality we get Kalman gain matrix Difference Riccati eq. for , prediction error covariance matrix and the EK parameter estimate is

27. Model adaption: comparison Outputs: NMP auto-pilot, advance ratio 0.3. Steepest descent EK approach

28. K + T ¯ p p e = k k k k 1 + + ; p p z e ´ = k k k 1 + = ¯ : k p ( ) P I K H P ¡ = k k k 1 + ; T T 1 ¡ ( ) K P H V H P H + = k k k 2 : Model adaption: comparison Parameters: NMP auto-pilot, advance ratio 0.3. Steepest descent EK approach

29. K + T ¯ p p e = k k k k 1 + + ; p p z e ´ = k k k 1 + = ¯ : k p ( ) P I K H P ¡ = k k k 1 + ; T T 1 ¡ ( ) K P H V H P H + = k k k 2 : Model adaption: comparison Parameters: NMP auto-pilot, advance ratio 0.3. Steepest descent EK approach

30. Conclusions • A new formulation for the auto-pilot approach was proposed, applicable to arbitrarily complex rotorcraft models; • Non-linear model predictive approach implies superior performance and leads to provable stability; • The solution specifically accounts for the presence of constant-in-time constraints on controls (trim conditions): no limit cycles; • Model adaptivity and learning reduce the need of tuning to different flight conditions and different models.

31. Acknowledgements This work is supported in part by the US Army Research Office, through contract no. 99010 with the Georgia Institute of Technology, and a sub-contract with the Politecnico di Milano (Dr. Gary Anderson, technical monitor).