ROTORCRAFT TRIM BY A NEURAL
MODEL-PREDICTIVE AUTO-PILOT
POLItecnico
di MIlano
Carlo L. Bottasso and Luca Riviello
Politecnico di Milano
Italy
31st European Rotorcraft Forum
Firenze, Italy, 13-15 September 2005
Neural Model-Predictive Auto-pilot
Outline
• Background and motivation:
• Rotorcraft trim;
• Possible solution strategies;
• Non-linear Model-Predictive (NMP) auto-pilot:
• Formulation;
• Adaptive reduced model;
• Numerical example;
• Conclusions.
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Neural Model-Predictive Auto-pilot
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:
TRIM
PROBLEM
• Given desired loads or
velocities specifying the desired
condition,
• Find resulting attitude and
constant-in-time controls.
Important remark:
• Rotorcraft systems excited by harmonic external loads;
• Periodic response of all states and loads at trim.
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Neural Model-Predictive Auto-pilot
Introduction and Motivation
Rotorcraft trim approaches:
• Periodic shooting
• Harmonic balance
• Finite elements in time
Computational cost is a
function of the number of
DOFs.
• 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).
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Neural Model-Predictive Auto-pilot
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.
Tilt-rotor whirl-flutter analysis
(about 104 degrees of freedom)
Proposed approach: use non-linear model predictive (NMP) control
technology for auto-pilot-based rotorcraft trim.
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Neural Model-Predictive Auto-pilot
Comprehensive Multibody Models
Comprehensive (multibody based) governing equations:
where:
e
x
e_ ; x
e; ȩ ; u
e ) = 0;
fe(x
e_ ; x
e) = 0;
ce(x
(dynamic & kinematic eqs.)
(constraints)
• System states : displacements/rotations, linear/angular
velocities, internal states;
• System controls
applied forces;
e
u
: e.g. actuator inputs, controlled joint rotations,
• Lagrange multipliers
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ȩ
that enforce the constraints.
Neural Model-Predictive Auto-pilot
Formulation of Rotorcraft Trim Problem
Define system outputs (problem
Z dependent):
1
ye =
T
t+T
t
e; u
e ) dt;
ge(x
•
Wind tunnel trim: components of rotor loads in fixed system;
•
Free flight: capture gross vehicle motion.
1.
Trim constraints:
¤
where
y
ye = y ¤ ;
8 t;
are desired values for the outputs;
2.
Trim conditions:
3.
Periodicity conditions:
(See Peters & Barwey 1996)
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e_ = 0; 8 t;
u
e(t + T ) = x
e(t) + ze;
x
8 t:
Neural Model-Predictive Auto-pilot
Rotorcraft Trim: Example
Wind-tunnel trim: given advance ratio, find the controls that produce
desired values of given average hub loads.
• Hub loads:
e(x
e; u
e);
g
1
e
y=
T
• Average hub loads:
• Desired average hub loads:
Z
t+T
t
y¤ :
³
´
³
´
¼
¼
µi (Ã) = µ0 + µ1s sin à ¡ i + µ1c cos à ¡ i ;
2
2
Blade pitch:
e = (µ0 ; µ1s ; µ1c )T :
Rotor controls: u
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e; u
e ) dt;
ge(x
i = 1; 2; 3; 4:
Neural Model-Predictive Auto-pilot
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):
ef = u
ei + ¢t S ¡1 G (y ¤ ¡ y)
e ;
u
where:
- Present/target outputs:
- Gain matrix:
G;
ye; y ¤ ;
- Initial/final controls:
· matrix:
- Input/output “sensitivity”
@ ye ¼ ye1 ¡ ye0 ye2 ¡ ye0
S=
@u
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¢1
;
¢2
ei ; u
ef ;
u
¡ ye ¸
yen
;:::; u
¢n
u
0
:
Neural Model-Predictive Auto-pilot
NMP Auto-pilot
Procedure:
• Predict system response using a non-linear reduced model;
• Compute controls to steer the system for a short time horizon;
• Update reduced model based on predicted-actual output errors;
• Iterate, shifting prediction forward (receding horizon control).
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Neural Model-Predictive Auto-pilot
NMP Auto-pilot
Highlights:
• Framework for guaranteeing stability of the closed-loop system;
• Superior control performance (optimal control theory);
• Constant-in-time constraints on controls explicitly enforced.
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Neural Model-Predictive Auto-pilot
NMP Auto-pilot
Model-predictive tracking problem: solution yields steering controls
Minimize cost
where
J =
Z
T
f
M (y; y ¤ ; u) dt;
T
i
M (y; y ¤ ; u) = jjy ¡ y ¤ jjS + jjujjS + jju_ jj ;
y
Subjected to:
• Reduced model equations:
where
u¤
p¤
u
u
_
_ y; u; p¤ ) = 0;
f (y;
is current estimate of model parameters.
• Initial conditions:
• Trim conditions:
• Constraints:
y(T ) = yei ;
i
_
u(t)
= 0; T < T · t · T ;
i
c
f
2
g(y; u) [g ; g
]:
min
max
Remark: constraints on controls (and states) are hard to incorporate in
other control strategies.
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.
Neural Model-Predictive Auto-pilot
Steering Problem
March forward in time multibody solver with given control inputs
computed by the tracking problem:
Solve initial value problem from current state
e_h ; x
eh ; ȩ h ; u¤ ) = 0;
fe(x
h
e_ ; x
e ) = 0;
ce(x
h
h
e(T steer )
x
0
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e0 :
=x
e0
x
u¤
as
on steering window:
Neural Model-Predictive Auto-pilot
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:
¤
_ y; u; p ) = 0;
f (y;
where the model parameters
p
ye ¼ y
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are optimized
¼ to have
when
e
u
u.
Neural Model-Predictive Auto-pilot
Reduced Model
Reduced model:
- Reference analytical model:
_ y; u) = 0;
fref (y;
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:
_ y; u) = d(y (n) ; : : : ; y; u);
fref (y;
where
d
is the unknown
¼ reference model defect that ensures
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ye
y
when
e = u.
u
Neural Model-Predictive Auto-pilot
Reduced Model Identification
Approximate
component:
d
with single-hidden-layer neural networks, one for each
di (y (n) ; : : : ; y; u) = di
NN
where
di
NN
(y (n) ; : : : ; ; u) + "i ;
(y (n) ; : : : ; y; u) = W i T ¾(V i T x + ai ) + bi ;
and
"i
= reconstruction error (universal approximator,
W i ; V i ; ai ; bi
= matrices of synaptic weights and biases;
¾(Á) = (¾(Á1 ); : : : ; ¾(ÁN ))T
n
x=
j"i j · C; 8C > 0
(y (n)T ; : : : ; y T ; uT )T
= sigmoid activation functions;
= network input.
p
The reduced model parameters
are readily identified with the
synaptic weights and
T biases of the networks:
p = (: : : ; pi ; : : :)T ;
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pi = (: : : ; W i ; V i ; ai ; bi ; : : :)T :
jk
jk
jk
jk
);
Neural Model-Predictive Auto-pilot
Numerical Example
System
Wind-tunnel trim of a four-bladed flexible
rotor:
• UH-60 rotor multibody model attached
to the ground;
• Three controls: blade collective and
longitudinal and lateral cyclic pitch angles;
• Aerodynamics: strip theory.
Reference model
Target
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Analytical blade element/momentum
theory, static flapping (performance model).
Trim for three desired average hub load
components in the inertial frame.
Neural Model-Predictive Auto-pilot
Numerical Example
Finite element based MB code
(Bauchau & Bottasso 2001).
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Neural Model-Predictive Auto-pilot
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
Error definition
Steer rotor outputs to the desired values
and evaluate controls in trim.
"(t) = kyes (t) ¡ y ¤ k
s 2
where
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y¤
s
are (scaled) target trim outputs.
Neural Model-Predictive Auto-pilot
Numerical Example
Time to trim:
T : "(t) · "max ; 8t ¸ T;
"max = 0:05:
NPMA parameters:
Activation freq.: 4/rev;
Prediction: 3 rev;
Num. of Neurons: 20;
Max. control rates: 10 deg/sec.
Classical autopilot stability
limit.
Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA.
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Neural Model-Predictive Auto-pilot
Numerical Example
Time to trim:
T : "(t) · "max ; 8t ¸ T;
"max = 0:01:
NPMA parameters:
Activation freq.: 4/rev;
Prediction: 3 rev;
Num. of Neurons: 20;
Max. control rates: 10 deg/sec.
Classical autopilot stability
limit.
Dash-dotted: auto-pilot A; Dashed: auto-pilot B; Solid: NMPA.
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Neural Model-Predictive Auto-pilot
Numerical Example
Controls: classic auto-pilot A, advance ratio 0.297.
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Neural Model-Predictive Auto-pilot
Numerical Example
Controls: NMP auto-pilot, advance ratio 0.297.
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Neural Model-Predictive Auto-pilot
Numerical Example
Outputs: classic auto-pilot A, advance ratio 0.297.
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Neural Model-Predictive Auto-pilot
Numerical Example
Outputs: NMP auto-pilot, advance ratio 0.297.
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Neural Model-Predictive Auto-pilot
Numerical Example
Outputs: classic auto-pilot A, advance ratio 0.297.
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Neural Model-Predictive Auto-pilot
Numerical Example
Outputs: NMP auto-pilot, advance ratio 0.297.
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Neural Model-Predictive Auto-pilot
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.
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Neural Model-Predictive Auto-pilot
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).
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