1 / 29

Outline

ROTORCRAFT TRIM BY A NEURAL MODEL-PREDICTIVE AUTO-PILOT Carlo L. Bottasso and Luca Riviello Politecnico di Milano Italy 31st European Rotorcraft Forum Firenze, Italy, 13-15 September 2005. Outline. Background and motivation: Rotorcraft trim; Possible solution strategies;

kuper
Download Presentation

Outline

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. ROTORCRAFT TRIM BY A NEURAL MODEL-PREDICTIVE AUTO-PILOTCarlo L. Bottasso and Luca RivielloPolitecnico di MilanoItaly31st European Rotorcraft Forum Firenze, Italy, 13-15 September 2005

  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 non-linear model predictive (NMP) control technology 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 • 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 proportional auto-pilot control law (in discrete form): • where: • Present/targetoutputs: - Initial/finalcontrols: • Gain matrix: • Input/output“sensitivity” matrix:

  10. NMP Auto-pilot • 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).

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

  12. T Z ( ( ( ) ) [ ) ] e f ¤ ¤ ¤ T _ f 0 2 u p g y y y u y u y p g g ; = = i i i ( ) ¤ ; ; ; ; ; ; : d J M m a x m n 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 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: • Constraints: • Remark: constraints on controls (and states) are hard to incorporate in other control strategies.

  13. e ¤ x u 0 e e _ ( ) e e ¤ f ¸ 0 x x u = h h h h ; ; ; ; _ ( ) e e e 0 c x x = h h ; ; t ( ) e e s e e r T x x = 0 0 : Steering Problem March forward in time multibody solver with given control inputs as computed by the tracking problem: Solve initial value problem from current state on steering window:

  14. ( ) 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

  15. 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

  16. ( ) T T T T T i i i i i i j ( j ) ( ( ( ) ( ) ) ) d Á 8 Á b Á W V C C n 0 · > x p ¾ y a y u " " ¾ ¾ = = N 1 ; ; ; ; ; : : : ; ; : : : ; ; n 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 p y y y y u u p y ¾ x u 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, ); = matrices of synaptic weights and biases; = sigmoid activation functions; = network input. The reduced model parameters are readily identified with the synaptic weights and biases of the networks:

  17. 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.

  18. Numerical Example Finite element based MB code (Bauchau & Bottasso 2001).

  19. ( ) 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.

  20. ( ) m a x m a x 8 T T 0 0 5 · ¸ 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.

  21. ( ) 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.

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

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

  24. Numerical Example Outputs: classic auto-pilot A, advance ratio 0.297.

  25. Numerical Example Outputs: NMP auto-pilot, advance ratio 0.297.

  26. Numerical Example Outputs: classic auto-pilot A, advance ratio 0.297.

  27. Numerical Example Outputs: NMP auto-pilot, advance ratio 0.297.

  28. 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; • Extension to maneuvering flight: paper #29, Session C4, Flight Mechanics, Tue. 5:00-5:30.

  29. 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).

More Related