AAE 666 - Final Presentation
Backstepping Based Flight Control
Asif Hossain
February 24, 2005
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Overview
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Modern Aircraft Configuration
Aircraft Dynamics
Force, Moment and Attitude Equations
Current Approaches to Flight Control Design
Backstepping Approaches to Flight Control Design
Backstepping
Backstepping Design for Flight Control
Flight Control Laws
Simulation
2
Modern Aircraft Configuration
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Aircraft Dynamics
P  ( p N p E h) T , Aircraft position expressed in an
Earth-fixed coordinate system;
V  (u v w) , The velocity vector expressed in
T
the body-axis coordinate system;
Φ  (   ) T , The Euler angles describing the
orientation of the aircraft relative to the Earth-fixed
coordinate system;
ω  ( p q r)
T
, The angular velocity of the
aircraft expressed in the body axes coordinate
system;
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Aerodynamics Forces and Moments
Body axis coordinate system
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Force Equations (Body-axes)
1
u  rv  qw  g sin   ( X  FT )
m
1
v  pw  ru  g sin  cos   Y
m
1
  qu  pv  g cos  cos   Z
w
m
Rewrite the force equations in terms of  ,  and VT
w
u
v
  arcsin
VT
  arctan
VT 
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u 2  v 2  w2
u  VT cos  cos 
v  VT sin 
w  VT sin  cos 
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Force Equations (wind-axes)
1

VT   D  FT cos  cos   mg1 
m
1
  q  ( p cos   r sin  ) tan  
( L  FT sin   mg 2 )
mVT cos 
1
  p sin   r cos  
(Y  FT cos  sin   mg 3 )
mVT
Where the contributions due to gravity are given by,
g1  g ( cos  cos  sin   sin  cos  sin   sin  cos  cos  cos  )
g 2  g (cos  cos  cos   sin  sin  )
g 3  g (cos  cos  sin   sin  cos  sin   sin  sin  cos  cos  )
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Moment and Attitude Equations
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Moment Equations:
p  (c1 r  c 2 p )q  c3 L  c 4 N
q  c5 pr  c6 ( p 2  r 2 )  c7 ( M  FT Z TP )
r  (c8 p  c 2 r )q  c 4 L  c9 N
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Attitude Equations:
  p  tan  ( q sin   r cos  )
  q cos   r sin 
q sin   r cos 
 
cos 
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Current Approaches to Flight Control Design
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Gain Scheduling


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Dynamic Inversion (feedback linearization)
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
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Divide and conquer approach is tedious since
controller must be designed for each flight envelope.
Stability is guaranteed only for low angles of attack
and low angular rates.
Cancels valuable nonlinear dynamics.
Relies on precise knowledge of the aerodynamic
coefficients
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Backstepping based flight control design
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Constructive (systematic) control design for
nonlinear systems.
Lyapunov based control design method
Avoid cancellation of “useful nonlinearities” (unlike
feedback linearization).
Stability is guaranteed for all angles of attack
(unlike gain scheduling).
Different flavors: Adaptive, robust and observer
backstepping.
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LaSalle-Yoshizawa Theory
The time-invariant system, x  f (x)
Let V (x ) be a scalar continuously differentiable function of
the state x such that
 V (x ) is positive definite
 V (x ) is radially unbounded
 Va( x )  V f ( x)  W ( x ) where W ( x) is positive semi definite.
x
Then, all solutions satisfy lim W ( x (t ))  0
t 
In addition, if W (x ) is positive definite, then the
equilibrium x  0 is Globally Asymptotically Stable (GAS).
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Control Lyapunov Function (clf)
The time-invariant system,
x  f ( x, u )
A smooth, positive definite, radially unbounded function V (x )
is called a control Lyapunov function (clf) for the system if
for all x  0 ,
V ( x)  V x ( x) f ( x, u )  0 for some u
Given a clf for the system, we can thus find a globally stabilizing control
law. In fact, the existence of a globally stabilizing control law is equivalent
to the existence of a clf, and vice versa.
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Backstepping
Consider the system
x  f ( x,  )
  u
n
Where x  R ,   R are state variables and is u  R the
control input.
Assume a virtual control law    des (x) is known such that 0
is GAS equilibrium of the system.
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Backstepping
Let, W (x ) be a clf for the subsystem x  f ( x,  ) such that
W |  des  Wx ( x) f ( x,  des ( x))  0, x  0
Then, the clf for the augmented system is given by
1
V ( x,  )  W ( x)  (   des ( x)) 2
2
Moreover, a globally stabilizing control law, satisfying
des
des
2
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V  Wx ( x) f ( x,  ( x))  (   ( x))
is given by
 des ( x)
f ( x,  )  f ( x,  des ( x)) des
u
f ( x,  )  Wx ( x)
  ( x)  
des
x
   ( x)
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Strict Feedback System
By recursive applying backstepping, globally stabilizing
control laws can be constructed for systems of the following
lower triangular form:
  f ( x,  1 )
x
1  g 1 ( x,  1 ,  2 )
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i  g i ( x,  1 , ,  i , i 1 )
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m  g m ( x,  1 , ,  m , u )
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Backstepping design for flight control
Controlled variables: General maneuvering
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Control Objectives
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Assumptions:
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Control surface deflections only produce
aerodynamic moments, and not forces.
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The speed, altitude and orientation of the aircraft
vary slowly compared to the controlled variables.
Therefore, their time derivatives can be neglected.
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Longitudinal and lateral commands are assumed
not to be applied simultaneously.
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The control surface actuator dynamics are assumed
to be fast enough to be disregarded.
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Backstepping design for flight control
The roll rate to be controlled, p s , is expressed in the stability
axes coordinate system.
T
The stability axes angular velocity, ω s  ( p s q s rs ) ,is related to
the body axes angular velocity, ω  ( p q r ) T , through the
transformation:
 s  Rsb
where
 cos  0 sin  


Rsb   0
1
0 
 - sin  0 cos  


1
T
Note that the transformation matrix Rsb satisfies Rsb
 Rsb
Introducing:
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u  (u1 u 2 u 3 ) T   s
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Aircraft Dynamics Revisited
p s  u1
  q s  p s tan  
q s  u 2
1
1
( L( )  FT sin   mg 2 )
mVT cos 
2
3
1

  rs 
(Y (  )  FT cos  sin   mg 3 )
mVT
rs  u 3
4
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Roll rate dynamics: Equation 1
Angle of attack dynamics: Equation 2-3
Sideslip dynamics: Equation 4-5
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The nonlinear control problem
The angle of attack dynamics and the sideslip dynamics can
be written as
w 1  f ( w1 , y )  w2
w 2  u
For notational convenience it is favorable to make the
origin the desired equilibrium. Let, w1   is the desired
equilibrium. x1  w1  
x 2  w2  f (, y )
 ( x1 )  f ( x1  , y )  f (, y )
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The nonlinear control problem
The dynamics become
x1  ( x1 )  x 2
x 2  u
We will use backstepping to construct a globally stabilizing
feedback control laws for the system assuming general
nonlinearity  .
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The nonlinear control problem
Assume there exists a constant,  , such that
( x1 )
  for all x1  0
x1
Then a globally stabilizing control law can be given by
u  k ( x2  ( x1 ))
where,
(( x1 )  ( x1 )) x1  0, x1  0 and 0  ( x1 )  k
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Block Diagram
The nonlinear system is globally stabilized through a cascaded control structure
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Aircraft Application
f  ( , y )   p s tan  
f  ( , y  ) 
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1
( L( )  FT sin   mg 2 )
mVT cos 
1
(Y (  )  FT cos  sin   mg 3 )
mVT
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Flight Control Laws
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Angle of attack control
u2  k ,2 (qs  k ,1 (   ref )  f ( ref , y ))
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Sideslip regulation
1
u 3  k  , 2 (rs  k  ,1  
g cos  sin  )
VT
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Stability axis roll control
u1  k ps ( psref  ps )
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Gain Selection, k p , k ,1 , k ,2 , k ,1 and k ,2
s
How should the control law parameters be selected?
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For  control, linearize the angle of attack dynamics
around a suitable operating point and then select to
achieve some desired linear closed behavior locally
around the operating point.
For  regulation, can be selected by choosing some
desired closed loop behavior using linearization of the
sideslip dynamics.
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Roll rate demand, p
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ref
s
 150 / sec
0
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Angle of attack demand, 
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ref
 15
0
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Questions
Polygonia interrogationis known as Question Mark
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