Modeling and Control of FixedWing Aircraft in Longitudinal Flight

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Modeling and Control of Fixed-Wing Aircraft in
Longitudinal Flight
Allison Ryan Qualifying Exam Presentation
November 14, 2006
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Outline

• Modeling and control of fixed-wing aircraft
• Model derivation
• Sliding mode control
• Future research plans

Goal (I) Derive a physically intuitive model
for longitudinal flight and design a robust
non-linear controller Goal (II) Introduce a
direction of research that is motivated by my
recent work
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Aircraft Modeling and Control

• Motivation
• Derive a model that enhances physical intuition
• Investigate a control method that could be
  extended from longitudinal to 6 degree of freedom
  motion
• Combine a simplified model with robust control
  in place of a very high fidelity model
• Applications
• Fly by wire control for piloted aircraft
• Autopilot for unmanned aircraft

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Notation for Aircraft Dynamics
Reference Control of spacecraft and aircraft
Arthur E. Bryson, Jr, 1994
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Longitudinal Aerodynamic Forces
?
Reference Foundations of aerodynamics Kuethe
and Chow, 1998
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Longitudinal Non-linear Model

• No equilibrium point for uncontrolled system.
  Each control (ut, ue) defines a trim condition.
• 3 Aerodynamic constants will be estimated from
  trim condition
• this will introduce parametric modeling error

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Sliding Mode Control General Formulation

• Express control objective s.t. tracking error
  converges to 0 when s 0 and control inputs
  appear in d/dt(s)
• Select a desired function d/dt(s) such that
• Select control inputs to satisfy (2) for all
  allowable models
• We now have a Lyapunov function for s dynamics
• s converges to zero due to stability from
  Lyapunovs direct method, giving

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Smoothed Formulation
When si0, sliding condition leads to
infinite-frequency switching across si0
surface Instead, set so s converges to
within boundary layer ? around 0. Control
objective Result si converges to boundary
layer
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Modeling Error and Need for Robustness
Dominant contributions to model error
Aerodynamic parameter estimation
(Lift) (Elevator lift) (Drag)

• Equations above represent a family of models for
  s dynamics
• The sliding condition must be satisfied for any
  model in family

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Final Sliding Mode Controller
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Sliding Mode Controller Design Parameters
? desired convergence rate of s dynamics ?
boundary layer for convergence of s near zero ?
convergence rate of error dynamics after s
converges
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Effects of Smoothing and Actuator Saturation

• Effects of smoothing with model error
• In smoothed formulation, s does not converge to
  zero, leading to steady state error
• Effect is controlled by the choice of ?

Effects of actuator saturation The sliding mode
controller does not account for actuator
saturation. It may decrease performance.
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Effects of Parameter Estimation Error
Similar performance More aggressive
actuation
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Conclusions from Control Design

• A distributed-parameter system is represented by
  a reduced order model with estimated parameters
• Sliding mode control explicitly deals with model
  uncertainty and provides performance by
  aggressive actuation
• Actuator saturation is modeled but not accounted
  for in the control, which influences performance
  by slowing convergence of s-dynamics

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Research Interest Mobility Control for Sensor
Networks
Control the motion of a team of mobile agents to
acquire desired information based on - vehicle
motion models - sensor models - communication
models - prior knowledge
Application Unmanned aerial vehicle sensing
missions - agents have significant onboard
processing ability - motion may have
non-holonomic constraint - communication is
range- and bandwidth-limited
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Related Areas

• Distributed control of stationary sensor networks
• Optimize power use on very large number of nodes
• Emphasis on inference nodes tend not to have
  actuators in the usual sense, so control is
  limited
• Simultaneous localization and mapping
• Mobilize agents to gain information and maintain
  communication
• Update joint distribution (map) with sequential
  observations and use as basis for mobility
  control

Smart Lighting (Agogino et. al), Structural
Monitoring (Stojadinovic et. al)
Centibots (Ortiz et. al), ACFR (Durrant-Whyte et.
al)
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Interesting Problem Aspects

• Tight coupling of control with estimation
• Tight coupling of control with communication
• Information value theory may address both of
  these express value of sensor placement and
  value of communication in terms of expected
  information gain

Mobile sensor network
Classical control
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Applicable Theory

• Distributed inference algorithms
• expectation maximization (Cooperative tracking)
• sum-product algorithm
• Distributed Kalman Filter
• Methods for multi-agent cooperation
• Auction-type algorithms (Berkeley UAV flight
  demo)
• Distributed model predictive control
• Information value theory
• Model how control effects information gain

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Potential Applications

• Chemical plume source location
• Water or oil exploration

Idea Position sensor where we expect to make
useful observations
A priori model
Sensor data
Estimate
Motion
Control
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Additional Slides to Follow

• More details of aircraft model derivation
• Previous work UAV sensing missions
• Applicable theory

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Longitudinal Motion
Assumptions on kinetics, forces, and moments
Resulting balance equations
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Balance Laws
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Derivation of Aerodynamic Forces

• Thin airfoil
• Small angle of attack
• Incompressible non-viscous fluid
• Within 10 of experiment for thin airfoil up to
  a 12
• (my model absorbs all constants including density
  into CL)

For a flapped wing (elevator) lift change is
linear in flap deflection
Reference Foundations of aerodynamics Kuethe
and Chow, 1998
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Estimation of Aerodynamic Coefficients
Engine calculations 7.5 Hp engine, assume 30
efficiency. PFV 100 throttle 84 N Trim
condition d/dt(u) d/dt(w) q 0 Unknowns
CL, CE, CD Results CD 0.12 CL 6.55 CE
1.24
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Linearization by Tayler Expansion at Trim
Condition
Local stability of equilibrium point depends on
eigenvalues of A matrix via Lyapunovs indirect
method
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Results of Linearization

• Linearized about trim condition from Sig Rascal
  flight data
• x0 19.98 0.86 0.08 0T u0 55 0.03T
• Linearized model is asymptotically stable at trim
  condition
• Linearization at trim condition results in short
  mode and
• phugoid mode standard for fixed-wing aircraft

Poles in complex plane
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Previous Work UAV Sensing Missions

• UAV-assisted search and rescue (hardware in the
  loop simulation)
• Control UAV formation as slaves to human-piloted
  helicopter in USCG search pattern
• Proceedings of the IEEE Conference on Decision
  and Control, 2005
• Distributed task allocation for UAV teams (flight
  tested)
• Task destination points provided by user
• Efficiently route vehicles in a robust and
  distributed manner with limited communication
• Proceedings of the AIAA Conference on Guidance
  Navigation and Control, 2006
• Distributed target tracking using EM algorithm
  (simulated)
• Associate noisy measurements using EM algorithm
• Distribute calculations using message passing on
  tree structure
• Allocate and control UAVs based on tracking
  estimates

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Expectation Maximization

• An iterative maximum likelihood estimate for
  partly hidden models
• Equivalent to coordinate ascent optimization
• Application data association for multiple
  target tracking

Mixture models

• Distribution ? consists of individual
  distributions ?i each with marginal probability
  ?I all unknown
• Given observations x(1)x(m) estimate labels
  z(1)z(m) and distribution ?

Expectation Calculate P(z(k) j) for each
observation x(k) and each distribution ?j given
estimated ? Maximization Update ? to maximize
likelihood of observations, given P(z(k) j)
from expectation
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Decentralized Kalman Filter

• Mathematically equivalent to standard KF
• Requires state estimate and variance from each
  node distributed to each other node
• Efficient when size of state is smaller than size
  of combined observations

Local prediction
Local observations
Local update
State and variance terms
Communicate
Complete update
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Information gain
Expected information gain by observing A Reflects
the amount of dependency between X and A

• Expected value decision making for sensing
  missions
• Choose utility function for sensing mission
• Relate utility function to information gain
• Control sensors to maximize utility function via
  information gain
• Example move sensor to increase expected accuracy