Robust Adaptive Fuzzy Controller Design for Wing Rock Syetems
Chih-Min Lin*
*
Chun-Fei Hsu
Department of Electrical Engineering, Yuan-Ze University,
Chung-Li, Tao-Yuan, 320, Taiwan, Republic of China
e-mail: cml@ee.yzu.edu.tw
Abstract: In this study, a robust adaptive fuzzy control (RAFC) system is proposed. The
RAFC system is comprised of a computation controller and a robust controller. The
computation controller containing a neural-network-estimator is the principal controller and
the robust controller is designed to achieve H  tracking performance. An on-line tuning
method is derived to tune the parameters of the neural network for estimating the controlled
system dynamic function. To investigate the effectiveness of the RAFC, the design
methodology is applied to control a wing rock system. The phenomenon of the wing rock
system is manifested by a limit cycle oscillation predominantly in rolling motion for an
aircraft operating at subsonic speeds and high angles of attack. Simulation results demonstrate
that the proposed RAFC system can achieve favorable tracking performances for the wing
rock system.
Key-Words: Fuzzy control, Robust control, Adaptive law, Neural network, Limit cycle, Wing
rock system.
control proposed in [1] has combined the
H  attenuation technique in the fuzzy
controller design. In recently developed
adaptive fuzzy control schemes, only the
consequence parts of fuzzy rules are tuned.
To increase the learning ability, some
researches used a radial basis function (RBF)
neural network to tune the premise and
consequence parts of the fuzzy rules
simultaneously [5, 9].
This paper develops a robust adaptive
fuzzy control (RAFC) system to achieve H 
tracking performance for the wing rock
system. Both the H  control design and the
adaptive fuzzy control approach have been
employed together to design the new control
system. By the adaptive fuzzy control
approach, this study uses a RBF neural
network to approximate the system dynamic
function. All the neural network parameters
are tuned based on the Lyapunov function,
thus the stability of the system can be
guaranteed. By the H  control design
technique, the approximation error can
attenuate to a desired small level. Finally,
simulation results demonstrate that the
developed RAFC system can achieve H 
1 Introduction
Many high-performance aircraft often
require for operating at subsonic speeds and
high angles of attack. They may become
unstable or enter into a self-induced limit
cycle oscillation, mainly rolling motion
known as wing rock [2,3]. This wing rock is a
concern because it may have adverse effects
on maneuverability, tracking accuracy, and
operational safety. A series of papers have
considered the control of the wing rock
system based on adaptive control technique
[6]. In the adaptive techniques, the
knowledge of the structure of the
aerodynamic functions is required; however,
the structure is difficult to obtain.
The application of fuzzy set theory to
control problems has been the focus of
numerous studies [4]. The fuzzy control is a
model free design method; however, it has
not been view as a rigorous approach due to
the lack of formal synthesis techniques that
can guarantee the system stability. To tackle
this problem, some researches have been
directed by the use of the Lyapunov synthesis
approach to construct a stable adaptive fuzzy
control [1, 8, 10]. The adaptive fuzzy
1
where  denotes an approximation error,
w * and Φ* are the optimal parameter
vectors of w and Φ , respectively, and m*
and s * are the optimal parameter vectors of
m and s , respectively. The optimal
weighting vectors w * , m* and s * that are
needed to best approximate a given nonlinear
function  are difficult to determine. An
estimated RBF approximator is defined as
ˆ (x, m
ˆ , sˆ)
ˆ TΦ
(9)
yˆ  w
where ŵ and Φ̂ are the estimated vectors
of w * and Φ* , respectively, and m̂ and ŝ
are the estimated vectors of m* and s * ,
respectively. Define the estimated error ~y
as
robust tracking performance for the wing
rock system with unknown system dynamic.
2 Description of Radial
Function Neural Network
Basis
A RBF neural network maps according to
l
y   wk  k ( xi  mik ,  ik )
(1)
k 1
where x i , i  1,2,.., n and y contain the
input variables and the output variable of the
RBF neural network, respectively; wk
represents the connective weight between the
hidden layer and the output layer;  k
represents the firing weight of the k-th neuron
in the hidden layer; and m ik and  ik are the
center and width of the activation function,
respectively. The firing weight can be
represented as
 k  e net
k
~
y    yˆ  y *  yˆ  
~ ~T~
~ TΦ
ˆ w
ˆ TΦ w
(10)
w
Φ 
~
*
*
~
ˆ . In the
where w  w  wˆ and Φ  Φ  Φ
following, some tuning laws will be
developed to on-line tune the parameters of
the RBF approximator to achieve favorable
estimation.
The
Taylor
expansion
linearization technique is employed to
transform the nonlinear function into a
partially linear form, i.e. [9]
(2)
where
n
netk  
i 1
[ xi  mik ] 2
 ik2
(3)
or
n
netk   sik2 [ xi  mik ] 2
i 1
(4)
  
  1 
~   1
 s 

m
1
  
 ~    2 
~  2  

 2
~
Φ
  m  |m mˆ m   s  |s sˆ ~s  H
  
  
~    
  l 
  l    l 
 m 
 s 
in which sik  1  ik is the inverse radius of
the radial basis function. For simplicity of
discussion, define the vectors m and s
collecting all parameters of the hidden layer
as
m  [m11  mn1 m12  mn 2  m1l  mnl ]T
(5)
T
s  [s11  sn1 s12  sn 2  s1l  snl ] . (6)
Then, the output of the RBF neural network
can be represented in a vector form as
(7)
y(x, m, s, w)  wT Φ(x, m, s)
T
where x  [ x1 x2 ... xn ] , w  [w1 w2 ... wl ]T
and Φ  [1  2 ... l ]T . It has been proven
that there exists an RBF approximator of (7)
such that it can uniformly approximate a
nonlinear even time-varying function  . By
the universal approximation theorem, there
exists an optimal RBF approximator y *
such that [8]
(11)
or
~
~  B T ~s  H
Φ  AT m
~  m*  m
ˆ , ~s  s *  sˆ ,
where m
vector
higher-order
 k
 l 
   2
B 1

|s sˆ , and

m
s 
 s s
and
 k
s
are defined as


 k
 k
  k 
0

00 (13)
 m   0( k


 1)l m1k mnk ( l  k )l 
T


 k  k
  k 
0

00 . (14)
 s   0( k
1)l s
(
l

k
)

l
s nk


1k


T
  y * (x, m* , s * , w * )  
 w *T Φ* (x, m* , s * )  
of
 l 
   2
A 1

|m mˆ
m 
 m m
(12)
H is a
terms,
,
Substituting (12) into (10), gives
~
~  B T ~s  H)  w
~ TΦ
~ TΦ
~y  w
ˆ w
ˆ T (AT m

(8)
2
~ T Aw
~ TΦ
ˆ m
ˆ  ~s T Bw
ˆ 
w
(15)
T
T ~
T
~
ˆ A m  m Aw
ˆ
where
and
w
T
T~
T
~
ˆ
ˆ
w B s  s Bw are used since they are
 (degree/sec)
﹡
scales; and the sum of matching error
~
~ TΦ
ˆ TH  w
 w
.
﹡
﹡: initial conditions
3 Problem Statement of Wing Rock
Systems
 (degree )
Fig. 1. Phase-plane portraits
uncontrolled wing rock system.
The nonlinear motion equation for an
80-deg slender delta wing has been developed
by Elzebda et al. as [2]
(16)
(t )  ( pU 2 Sb / 2I xx )Cl (t )  u(t )
where  (t ) is the roll angle, an over-dot
denotes a derivative with respect to time,
u (t ) is the control effort which represents the
aileron deflection, p is the density of air,
U  is the freestream velocity, S is the wing
reference area, b is the chord, I xx is the
mass moment of inertia, and C l (t ) is the roll
moment coefficient. The roll-moment
coefficient is written as [2]
C l  c0  c1  c 2  c3    c 4    c5 3 .
The control problem is to find a control
law so that the output  can track the
command trajectory  m , where the tracking
error vector is defined as
e  [e(t ) e (t )]T  [ m   m  ]T . (19)
Assume that the parameters of the wing rock
system in (18) are well known, there exits an
ideal controller as [7]
(20)
u id   f ( , )  m  k 2 e  k1e .
Substituting (20) into (18), gives the
following equation
e  k 2 e  k1 e  0 .
(21)
If k1 and k 2 are chosen to correspond to
the coefficients of a Hurwitz polynomial, it
implies lim e(t )  0 . However, the system
The aerodynamic parameters c i , i  0,1,...,5
have been provided that they are nonlinear
functions for different angles of attack [2].
Substituting (17) into (16), system (16) can
be rewritten as
(18)
(t )  f ( , )  u(t )
f ( , )  b0  b1  b2  b3  
where
and the parameters
the
4 Robust Adaptive Fuzzy Control of
Wing Rock Systems
(17)
 b4    b5 3
of
t 
dynamic function f ( , ) in (20) is a
nonlinear time-varying function and it can’t
be exactly obtained, so uid can not be
implemented. An RBF approximator will be
used to observe the unknown system dynamic
function
f ( , ) . By the universal
approximation theorem, there exists an
optimal RBF neural network such that [8]
(22)
f ( , )  f * (θ* , t )  
*
*
* * T
where θ  [w m s ] is the optimal weight
vector of the RBF neural network and 
denotes the approximation error. However,
the optimal weight vector is difficult to
determine. Define the weight estimated error
~
vector θ as
~
θ  θ *  θˆ
(23)
T
ˆ
ˆ sˆ ] is the estimated vector
where θ  [wˆ m
of the optimal vector θ* . While  appears,
bi ,
i  0,1,...,5 are given by bi  ( pU2 Sb / 2I xx )ci .
To observe the dynamic behavior, the
open-loop system time response with
u (t )  0 was simulated for two initial
conditions: a small initial condition
(  (0)  6 deg , (0)  3 deg/ sec ) and a large
initial
condition
(  (0)  30 deg
,
(0)  10 deg/ sec ). The phase-plane plot is
shown in Fig. 1. For the small initial
condition a limit cycle oscillation exists, and
for the large initial condition the roll angle is
divergent.
3
the following H  tracking performance is
requested [1]

T
0
system can be guaranteed.
~   eT PB Φ
ˆ
ˆ  w
w
1
m
~   e T PB Aw
ˆ  m
ˆ
m
2
m
T
~
ˆ
sˆ   s   3 e PB m Bw
T
~
1~
eT Qe dt  eT (0)Pe(0)  θ T (0)θ(0)   2   2 dt

0
(24)
where T  [0, ] and   L2 [0, T ] . The
Q  Q T and P  PT are given positive
weighting matrices,  is a design gain, and
 is a prescribed attenuation level. If the
system starts with initial conditions e(0)  0
uR 
~
sup
 L2 [ 0 ,T ]

0

e T Qe dt
T
0
 2 dt

PA m  A Tm P  Q 
(25)
2

PB m B Tm P 
1

2
PB m B Tm P  0
(35)
if and only if
where the L2 -gain from  to the tracking
error e must be equal to or less than  .
For achieving a favorable tracking
performance and an arbitrarily small
attenuation level simultaneously, a RAFC
system is shown in Fig. 2 with the controller
u RF  u CP  u R
(26)
in which the computation controller is chosen
as
u CP   fˆNN  m  k 2 e  k1 e
(27)
with the dynamic approximator chosen as
ˆ (e, m
ˆ , sˆ, w
ˆ , sˆ ) .
ˆ)w
ˆ TΦ
fˆNN (e, m
(28)
The estimated vectors ŵ , m̂ and ŝ are the
optimal vectors of w * , m* and s * ,
respectively. By substituting (26) into (18)
and using (20), the tracking error dynamic
equation can be obtained as follows:
e  A m e  B m ( f  fˆNN  u R )
(29)
 0
1
(34)
B Tm Pe

where  1 ,  2 and  3 are the learning rates
with positive constants,  is a positive
weighting factor, and positive matrix P  PT
is the solution of the following Riccati-like
equation
and θ (0)  0 , then the H  performance in
(24) can be rewritten as
T
(31)
(32)
(33)
2


1
2
 0 or 2 2   .
■
Proof: Refer to [1], [5] and [9].
Therefore, for a prescribed  in H 
tracking control, in order to guarantee the
solvability of H  tracking performance, the
weight  on control law u R of (34) must
satisfy the above inequality. Then, the H 
tracking performance in (24) can be achieved
for a prescribed attenuation level  .
5 Simulation Results
The parameters bi for the aerodynamic
coefficients in (18) are given in [6]. Solve the
Riccati-like equation (35) with  2   , it is
obtained that
 1 0 
1.7625 0.7812 
Q
and P  
 .

0.7812 0.8088 
 0  1
1 
where A m  
, and B m  0 1T .

  k1  k 2 
Using (15), equation (29) can be rewritten as
~ T Aw
~ TΦ
ˆ m
ˆ  ~s T Bw
ˆ    uR ) .
e  A m e  B m (w
(30)
Therefore, the following theorem can be
stated.
Theorem 1: Consider a wing rock system
represented by (18), if the robust adaptive
fuzzy control system is designed as (26), in
which the adaptation laws of the system
dynamic function approximator are designed
as in (31) ~ (33), and the robust controller is
designed as in (34), then the stability of the
(36)
It should be emphasized that the derivation of
RAFC system does not need to use the
aerodynamic parameters and the structure of
the aerodynamic functions. The system
parameters are used only for simulations. To
investigate the effectiveness of the developed
control system, two initial conditions are
simulated. The RBF neural network with 7
hidden layer neurons in rule layer is utilized
to approach the wing rock system dynamics.
The learning rates are selected as k1  0.64 ,
k 2  1.6 , 1   2   3  20 . To attenuate to a
small level via H  tracking design
4
technique, the simulation results of the RAFC
system with   0.1 for small and large
initial conditions are shown in Fig. 3. The
state responses are shown in Figs. 3(a) and
3(b); and the associated control efforts are
shown in Figs. 3(b) and 3(d) for small and
large initial conditions, respectively. From
these simulation results, it can be seen that
robust tracking performance can be achieved
without any knowledge of system dynamic
functions.
u (degree/sec2 )
control effort
Time (sec)
(d)
Fig. 3. The time responses of robust adaptive
fuzzy control wing rock system with   0.1 .
6 Conclusions
This paper developed a robust fuzzy
adaptive control (RAFC) system to attenuate
the effects of the dynamic function
approximation error on the tracking
performance using H  tracking technique.
The RAFC system is comprised of a
computation controller and a robust controller.
The computation controller including a
neural-network-estimator is used to mimic the
system dynamic function and the robust
controller is used to attenuate the effects of
the approximation error. Finally, the
developed RAFC system is applied to control
a wing rock system to demonstrate its
effectiveness.
Fig. 2. Block diagram of robust Adaptive fuzzy
control wing rock system.
 (degree)
Acknowledgment
This work was supported by the National
Science Council of the Republic of China
under Grant NSC 90-2213-E-155-016.
reference
trajectory
RAFC response
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6