1. POLI di MI tecnicolano Carlo L. BottassoLuca Riviello 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

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

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

4. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Rotorcraft trim approaches Rotorcraft trim approaches: Periodic shooting Periodic shooting Harmonic balance Harmonic balance Finite elements in time Finite elements in time Auto-pilot 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). Introduction and Motivation Computational cost is a function of the number of DOFs.

5. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO High-fidelity comprehensive aeroelastic models: non-linear MB dynamics Based on non-linear MB dynamics formulations; complex aerodynamic modelsCFD. Coupled with complex aerodynamic models or CFD. Need for efficient trim procedures. Current auto-pilots: unstable Are unsuitable for unstable systems; stability Offer no guarantee of stability; limit cycle Often find limit cycle solutions. Proposed approachnon-linear model predictive (NMP) control Proposed approach: use non-linear model predictive (NMP) control technology for auto-pilot-based rotorcraft trim. Introduction and Motivation Tilt-rotor whirl-flutter analysis (about 10 4 degrees of freedom)

6. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Comprehensive Multibody Models Comprehensive (multibody based) governing equations: (dynamic & kinematic eqs.) (constraints) where: System states : System states : displacements/rotations, linear/angular velocities, internal states; System controls : e System controls : e.g. actuator inputs, controlled joint rotations, applied forces; Lagrange multipliers Lagrange multipliers that enforce the constraints. e u e ¸ e f ( _ e x ; e x ; e ¸ ; e u ) = 0 ; e c ( _ e x ; e x ) = 0 ; e x

7. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Formulation of Rotorcraft Trim Problem system outputs Define system outputs (problem dependent): Wind tunnel trim: components of rotor loads in fixed system; Free flight: capture gross vehicle motion. 1.Trim constraints 1.Trim constraints: where are desired values for the outputs; 2.Trim conditions 2.Trim conditions: 3.Periodicity conditions 3.Periodicity conditions: (See Peters & Barwey 1996) y ¤ e y = y ¤ ; 8 t ; _ e u = 0 ; 8 t ; e x ( t + T ) = e x ( t ) + e z ; 8 t : e y = 1 T Z t + T t e g ( e x ; e u ) d t ;

8. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Rotorcraft Trim: Example Wind-tunnel trim: average hub loads. 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: e u = ( µ 0 ; µ 1 s ; µ 1 c ) T : µ i ( Ã ) = µ 0 + µ 1 s s i n ³ Ã ¡ ¼ 2 i ´ + µ 1 c cos ³ Ã ¡ ¼ 2 i ´ ; i = 1 ; 2 ; 3 ; 4 : e g ( e x ; e u ) ; e y = 1 T Z t + T t e g ( e x ; e u ) d t ; y ¤ :

9. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Trim Solution Strategies: Auto-pilot Procedure 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. proportional auto-pilot A possible proportional auto-pilot control law (in discrete form): where: - Present/target- Initial/final - Present/target outputs: - Initial/final controls: - Gain - Gain matrix: - Input/output - Input/output “sensitivity” matrix: e u f = e u i + ¢ t S ¡ 1 G ( y ¤ ¡ e y ) ; e y ; y ¤ ; e u i ; e u f ; G ; S = @ e y @ u ¼ · e y 1 ¡ e y 0 ¢ 1 ; e y 2 ¡ e y 0 ¢ 2 ;:::; e y n u ¡ e y 0 ¢ n u ¸ :

10. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO NMP Auto-pilot Procedure Procedure: Predict non-linear reduced model Predict system response using a non-linear reduced model; steer Compute controls to steer the system for a short time horizon; Update Update reduced model based on predicted-actual output errors; Iterate Iterate, shifting prediction forward (receding horizon control).

11. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO NMP Auto-pilot Highlights Highlights: guaranteeing stability Framework for guaranteeing stability of the closed-loop system; performance Superior control performance (optimal control theory); constraints Constant-in-time constraints on controls explicitly enforced.

12. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Model-predictive tracking problem 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: Remark: constraints on controls (and states) are hard to incorporate in other control strategies. NMP Auto-pilot u ¤ J = Z T f T i M ( y ; y ¤ ; u ) d t ; M ( y ; y ¤ ; u ) = jj y ¡ y ¤ jj S y + jj u jj S u + jj _ u jj _ u ; y ( T i ) = e y i ; g ( y ; u ) 2 [ g m i n ; g max ] : p ¤ f ( _ y ; y ; u ; p ¤ ) = 0 ; _ u ( t ) = 0 ; T i < T c · t · T f ;

13. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Solution of Tracking Problem Optimal Control Problem Optimal Control Governing Eqs. Discretize NLP Problem Numerical solution Direct Indirect Indirect approach Indirect approach: Need to derive optimal control governing equations; Need to provide initial guesses for co-states; For state inequality constraints, need to define a priori constrained and unconstrained sub-arcs. Direct approach Direct approach: all above drawbacks are avoided.

14. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO March forward in timegiven control inputs March forward in time multibody solver with given control inputs as computed by the tracking problem: initial value problem Solve initial value problem from current state on steering window: Steering Problem u ¤ e f ( _ e x h ; e x h ; e ¸ h ; u ¤ h ) = 0 ; e c ( _ e x h ; e x h ) = 0 ; e x ( T s t eer 0 ) = e x 0 : e x 0

15. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Adaptive NMP Auto-pilot Stability 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 Proposed solution: Identify adaptive parametric reduced model Identify adaptive parametric reduced model to control the approximation error and converge to exact trim solution: optimized where the model parameters are optimized to have when f ( _ y ; y ; u ; p ¤ ) = 0 ; p e y ¼ y e u ¼ u.

16. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Reduced Model Reduced model: - Reference analytical 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 - Augmented reduced model: unknowndefect where is the unknown reference model defect that ensures when f re f ( _ y ; y ; u ) = 0 ; d e y ¼ y f re f ( _ y ; y ; u ) = d ( y ( n ) ;:::; y ; u ) ; e u = u.

17. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Reduced Model Identification single-hidden-layer neural networks 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. reduced model parameters The reduced model parameters are readily identified with the synaptic weights and biases of the networks: d i ( y ( n ) ;:::; y ; u ) = d i NN ( y ( n ) ;:::;; u ) + " i ; d W i ; V i ; a i ; b i ¾ ( Á ) = ( ¾ ( Á 1 ) ;:::; ¾ ( Á N n )) T x = ( y ( n ) T ;:::; y T ; u T ) T p = ( :::; p i T ;::: ) T ; p i = ( :::; W i j k ; V i j k ; a i j k ; b i j k ;::: ) T : p " i j " i j · C ; 8 C > 0 d i NN ( y ( n ) ;:::; y ; u ) = W i T ¾ ( V i T x + a i ) + b i ;

18. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Target average hub load components inertial frame Trim for three desired average hub load components in the inertial frame. System Wind-tunnel trim of a four-bladed flexible rotor: UH-60 rotor multibody model attached to the ground; collective longitudinallateral cyclic Three controls: blade collective and longitudinal and lateral cyclic pitch angles; A strip theory Aerodynamics: strip theory. Analytical blade element/momentum theory, static flapping (performance model). Reference model

19. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Finite element based MB code Finite element based MB code (Bauchau & Bottasso 2001). Numerical Example

20. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Steer rotor outputs to the desired values and evaluate controls in trim.Goal estimate the forces (ouputs) trim straight level flight 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. Error definition where are (scaled) target trim outputs. Methodology " ( t ) = k e y s ( t ) ¡ y ¤ s k 2 y ¤ s

21. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Time to trim Time to trim: Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA. Classical auto- pilot stability limit. T :" ( t ) · " max ; 8 t ¸ T ; " max = 0 : 05 : NPMA parameters NPMA parameters: Activation freq.: Activation freq.: 4/rev; Prediction: Prediction: 3 rev; Num. of Neurons: Num. of Neurons: 20; Max. control rates: Max. control rates: 10 deg/sec.

22. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Time to trim Time to trim: T :" ( t ) · " max ; 8 t ¸ T ; " max = 0 : 01 : Classical auto- pilot stability limit. Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA. NPMA parameters NPMA parameters: Activation freq.: Activation freq.: 4/rev; Prediction: Prediction: 3 rev; Num. of Neurons: Num. of Neurons: 20; Max. control rates: Max. control rates: 10 deg/sec.

23. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Controlsclassic auto-pilot A, Controls: classic auto-pilot A, advance ratio 0.297.

24. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example ControlsNMP auto-pilot, Controls: NMP auto-pilot, advance ratio 0.297.

25. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Outputsclassic auto-pilot A, Outputs: classic auto-pilot A, advance ratio 0.297.

26. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example OutputsNMP auto-pilot, Outputs: NMP auto-pilot, advance ratio 0.297.

27. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example Outputs: classic auto-pilot A, Outputs: classic auto-pilot A, advance ratio 0.297.

28. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example OutputsNMP auto-pilot, Outputs: NMP auto-pilot, advance ratio 0.297.

29. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Numerical Example reduced model defect Time history of norm of reduced model defect: Increasing advance ratio f re f ( _ y ; y ; u ) = d ( y ( n ) ;:::; y ; u ) ;

30. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Conclusions new formulation for the auto-pilot arbitrarily complex rotorcraft models A new formulation for the auto-pilot approach was proposed, applicable to arbitrarily complex rotorcraft models; superior performance stability Non-linear model predictive approach implies superior performance and leads to provable stability; constant-in-time constraints The solution specifically accounts for the presence of constant-in-time constraints on controls (trim conditions): no limit cycles; Model adaptivitylearning Model adaptivity and learning reduce the need of tuning to different flight conditions and different models; maneuvering flight Extension to maneuvering flight: paper #29, Session C4, Flight Mechanics, Tue. 5:00-5:30.

31. Neural Model-Predictive Auto-pilot POLITECNICO di MILANO Acknowledgements US Army Research Office 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).