Journal of China University of Science and Technology Vol.49-2011.10
Integral Fuzzy Regulation Control for a Micro Unmanned
Helicopter
微型無人操作直昇機之積分模糊調節控制
劉 建 宏
Chien-Hung Liu
中華科技大學航電系講師
Department of Avionics
China University of Science and Technology
摘 要
本文提出一個積分式模糊控制策略以解決微型無人操作直升機之輸出調節問
題。為了消除系統的偏差且保證零-偏移量之輸出調節性能，首先將系統座標轉換
到平衡點並且引入一個外加的輸出調節誤差之積分狀態，然後將此結果之增廣系
統描述成一個高木與菅野(T-S)之模糊模型。其次利用平行分佈補償(PDC)技術和直
接式李亞普諾夫(Lyapunov)法，透過解一組線性矩陣不等式(LMIs)來建立一個具輸
出調節性能之積分式狀態回饋控制律。另外所提出的控制器設計有下列之優點︰i)
能處理非線性仿射(affine)系統； ii) 在系統模型不確定的情況下具有指數穩定；iii)
擁有分段式常數輸出之調節性能。最後以微型無人操作直升機之動態模型來示範
所提之積分式模糊控制器的有效性。
關鍵詞：微型無人操作直升機、高木與菅野模糊模型、輸出調節、線性矩陣不等式、積分式模糊控
制器。
Abstract
This paper proposes an integral fuzzy control strategy to solve output regulation
problem for the micro unmanned helicopter. In order to eliminate the system's bias and
guarantee the zero-offset output regulation performance, we firstly take coordinate
translation at equilibrium point and introduce an added integral state of output
regulation error, and then the resulting augmented system is represented into a
Takagi-Sugeno fuzzy model. Next, utilizing parallel distributed compensation (PDC)
technique and direct Lyapunov method, an integral state feedback control law for output
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Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
regulation is established by solving a set of linear matrix inequalities (LMIs). Moreover,
the proposed controller design has following merits: i) capability to dealing with
nonlinear affine system; ii) giving the exponential stability in the presence of model
uncertainty; iii) possessing a piecewise constant output regulation performance. Finally,
a micro unmanned helicopter dynamic model is presented to demonstrate the validity of
the proposed integral fuzzy controller.
Keywords:
Micro unmanned helicopter, Takagi-Sugeno fuzzy model, output regulation, linear matrix
inequalities, integral fuzzy controller.
I. Introduction
The unmanned helicopter is a kind of quite flexible unmanned aerial vehicle (UAV)
not only because it has light weight, small scale, and high mobility but also it can hover
over a fixed position and vertical take-off and landing (VTOL), so that it has been a
very popular research objective in recent year. Although the helicopter has many above
stated advantages but to automatic control it is very difficult and challenging problem
due to its congenital properties with instability, nonlinearity and coupling. During the
last two decades, many studies for helicopter control [1]-[4] have been regard as a
typical application of UAV. On the other hand, fuzzy control has proved to be a useful
design technique to design controllers for nonlinear systems, where classical nonlinear
controllers are hard to design. The idea of fuzzy control of helicopters has been
investigated by a number of authors [5]-[7].
In this work, we propose an integral fuzzy control strategy to solve output regulation
problem for the micro unmanned helicopter. The study of output regulation problems
for nonlinear systems keeps attracting considerable attention due to demands from
practical dynamical processes in mechanics, economics and biology. Based on this
observation, many LMI-based fuzzy controller designs [8], [9] have been developed to
solve output regulation control problems for nonlinear systems [10]-[12]. More recently,
several well-developed approaches are proposed to design LMI-based integral controller
for the purpose of regulating the system output at piecewise constant values [13].
Moreover, in practical application, the biased nonlinear systems can not guarantee the
offset-free output regulation property using traditional fuzzy control approach in the
presence of plant parameter variations or uncertainties [14], [15]. To this end, we
proposed a unified methodology to cope with the above drawback. Once given a desired
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Journal of China University of Science and Technology Vol.49-2011.10
reference output, we augment the plant model with adding an integral state in the
forward channel of output regulation error and take coordinate translation to equilibrium
point for eliminating bias term, then the augmented system without bias term is
represented into T-S fuzzy model. Next, the integral fuzzy controller for output
regulation is realized by using PDC technique and direct Lyapunov method. Sufficient
conditions are derived for asymptotic output regulation the closed-loop controlled
system in terms of linear matrix inequalities (LMIs).
The contents of this paper are organized as follows. The micro unmanned helicopter
dynamics are introduced in Section II. In Section III, integral fuzzy output regulation
control design is presented. Numerical simulation is carried out to verify the proposed
controller in Section IV. Finally, conclusions are given in Section V.
II. Micro Unmanned Helicopter Dynamic
The micro helicopter means its wing span is under 0.5m and flying distance is 2km
at the most, which is a co-axial counter rotating aircraft system. In general, the
helicopter with coaxial counter-rotating blades has two features. One is that rotating
torque of yaw-direction of the main body can be cancelled by rotating torques between
the upper and lower rotors. The other is that a mechanical stabilizer attached above the
upper rotor has a function of keeping the upper rotor horizontally. The two features will
be considered in the dynamic model construction [5]. The dynamics of the helicopter
can be described as follows
m(u(t )  q(t ) w(t ) - r (t )v (t ))  FX (t )
m(v (t )  r (t )u(t ) - p(t ) w(t ))  FY (t )
m( w(t )  p(t )v (t ) - q(t )u(t ))  FZ (t )
p(t ) I X  q(t )r (t )( I Z - I Y )  M X (t )
(1)
q(t ) IY  r (t ) p(t )( I X - I Z )  M Y (t )
r (t ) I Z  p(t )q(t )( I Y - I X )  M Z (t )
Table I shows the parameters of the dynamic models. By considering its co-axial
counter structure, the restitutive force (generated by a mechanical stabilizer attached on
the helicopter) and gravity compensation, the dynamics can be rewritten as
129
Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
1
u( t ) r ( t ) v (t )
m
1
v ( t ) -r (t )u 
(t )
m
1
w(t )  U Z (t ),
m
1
 (t )  U (t )
IZ
X
U ( t) ,
Y
U (t ) ,
(2)
where m=0.2 and I Z =0.2857 . U X (t ) , UY (t ) , U Z (t ) and U (t ) denoted new control
input variables. We can obtain the original control inputs (to the real helicopter) from
U X (t ) , UY (t ) , U Z (t ) and U (t ) .
TABLE I. Parameters of the helicopter model
x, u
position and velocity (X-axis)
y, v
position and velocity (Y-axis)
z, w
position and velocity (Z-axis)
, p
angle and angle velocity (X-axis)
,q
angle and angle velocity (Y-axis)
,r
angle and angle velocity (Z-axis)
I X , IY , I Z
moments of inertia with respect to X, Y and Z axes
FX , FY , FZ
translational forces to X, Y and Z axes
M X , MY , M Z
rotational forces to X, Y and Z axes
m
mass
III. Integral Fuzzy Output Regulation Control
A.
Coordinate Translation and Integral Control
Consider a general nonlinear affine system as follows:
xP (t )  f ( xP (t ))  g ( xP (t ))u(t )  
yP (t )  h( xP (t ))
(3)
where x p (t )  Rn denotes the state variables; y p (t )  Rq is the output vector;
u(t )  R m is input vector; f ( x ) , g ( x ) and h( x ) are nonlinear function vectors with
appropriate dimensions and  is system bias term. Note that if the system is
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Journal of China University of Science and Technology Vol.49-2011.10
controllable and subject to a constant bias term, it may have f ( x p (t ))  0 at x p (t )  0 ,
g ( x p (t ))  0 x p (t ) and output function may have an offset such that h( x p (t ))  0 at
x p (t )  0 . Let yd  R q be a constant reference. In order to ensure zero offset output
regulation performance in the presence of plant uncertainty, we want to design an
integrator based controller such that y p (t )  yd as t   . Its principle is based on
the procedure of adding an integral action in the forward channel of tracking system to
guarantee tracking property. To this end, we introduce a new state variable to account
the resulting integrated tracking error, denoted by xe (t ) , and is computed as
xe (t )   ( yd - y p (t )) dt
(4)
Combining (3) with (4), we obtain the augmented dynamics as follows
x p (t )  f ( x p (t ))  g ( x p (t )) u(t )  
xe (t )  yd - h( x p (t ))
(5)
Let us define the regulation error as e(t )  yd - y p (t ) . Without loss of generality, there
always exists some equilibrium point for constant output reference yd in practical
industrial process control, denoted by the steady state x p (t )  Rn and the
corresponding input u (t )  R m . Then, the objective of output regulation for biased
nonlinear system will be achieved by stabilizing the system at an equilibrium state
which imply e(t )  0 . For this purpose, it follows that
f ( x p )  g ( x p )u   =0
yd  h( x p )=0.
(6)
Here (6) is assumed to have a unique solution ( x p , u ) . To eliminate the constant bias
term  from the dynamics (5), we consider the coordinate translation: let
x p (t )  x p (t )  x p and u(t )  u(t ) - u . By using the property of (6), the dynamics of the
system (5) can be written as follows:
x p (t )  f s ( x p (t ))  g s ( x p (t )) u(t )
xe (t )  hs ( x p (t ))
(7)
where f s ( x p (t ))  f ( x p (t )  x p )  g ( x p (t )  x p ) u   ; g s ( x p (t ))  g ( x p (t )  x p ) ; and
hs ( x p (t ))  yd  h ( x p (t )  x p ). Clearly, f s ( x p (t ))  0, hs ( x p (t ))  0, g s ( x p (t ))  0,
once x p (t )  0 .
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Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
B. T-S Fuzzy Modeling
From (7), the transformed augmented system can be expressed as a state-space form
with no bias terms and free-offset error signal e( k ) :
x p (t )  A( x p (t )) x p (t )  B( x p (t ))u(t )
(8)
xe (t )  C ( x p (t )) x p (t )
where A , B and C are the system matrix, the input matrix and the output matrix,
respectively. Next, we represent the system (8) into the T-S fuzzy dynamic model with
fuzzy inference rules and local analytic linear models as follows:
Model Rule i : IF z1 (t ) is F1i and … and z p (t ) is Fpi
THEN
where z (t )  [ z1 (t ),
x p (t )  Ai x p (t )  Bi u(t )
xe (t )  Ci x p (t ),
1 i  r
, z p (t )]T are known premise variables that may be functions of
the state variables; F ji ( j  1, 2,..., p ) is the fuzzy set, and r is the number of model
rules;
Ai  R nn , Bi  R nm and Ci  R qn
are system matrices of appropriate
dimensions. Using singleton fuzzifier, product inference and weighted average
defuzzifier, the fuzzy system is inferred as follows:
r
x p (t )   i ( z (t )){ Ai x p (t )  Bi u(t )}
i 1
(9)
r
xe (t )   i ( z (t )) Ci x p (t )
i 1
r
p
i 1
j 1
where i ( z (t ))  i ( z (t )) /  i ( z ( t )) with i ( z (t ))   Fji ( z j (t )) for all t ,
r
and F ji ( z j (t )) is the grade of membership of z j ( t ) in F ji . Since
 ( z (t ))  0
i 1
and i ( z (t ))  0 for i  1,2,..., r , we have
r
  ( z (t ))  1,
i 1
i
i
i ( z (t ))  0 . Denote the
augmented state vector as xa (t )  [ x p (t ) xe (t )]T . Then, the augmented system (9)
becomes as
r
xa (t )   i ( z (t )){ Ai xa (t )  Bi ua (t )}
i 1
132
(10)
Journal of China University of Science and Technology Vol.49-2011.10
where
A
Ai   i
Ci
0
 Bi 
,
B

i
C  .
0
 i
The resulting integral state feedback controller via the PDC shows as follow
r
ua (t )     j ( z ( k )) K j xa (t )
j 1
r
    j ( z ( k ))[ K pj
j 1
(11)
 x (t ) 
K ej ]  p 
 xe ( t ) 
where K pj denotes the feedback gain for x p (t ) and K ej is the gain associated with
xe (t ) . Obviously for the above stable closed-loop system, both the new augmented state
xa (t ) and the new control input ua (t ) converge to zero when t   . Accordingly,
we achieve the control objective: y p (t )  yd as t   .
C. LMI-based Integral Fuzzy Controller Design
Substituting (11) into the regulation error system (10), the closed-loop system is thus
obtained:
r
r
xa (t )   i ( z (t )) j ( z (t ))( Ai  Bi K j )xa (t )
i 1 j 1
r
(12)
r
  i ( z(t )) j ( z(t )) Gij xa (t )
i 1 j 1
where Gij  Ai  Bi K j . The fuzzy output regulation design is to determine the integral
state feedback control gains K j . In which the feedback gains K j can be determined
by an LMI-based design technique addressed in the following theorem.
Theorem 1: Let D be a diagonal positive matrix. The integral fuzzy augmented
system (10) can be exponentially stabilized via the integral fuzzy PDC controller (11) if
there exist symmetric positive definite matrices P and matrices M j such that the
following LMIs are feasible:
 Ai X  XAiT  Bi M j  M JT BiT

DX

XDT 
  0,
X 
where X  P 1 and M j  K j X
133
1  i, j  r
(13)
Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
Proof: Choose the quadratic Lyapunov function candidate is defined
V ( xa (t ))  xa (t )T Pxa (t )
where P is a symmetric positive definite matrix. Taking the derivative of V ( xa (t ))
with respect to t , we have
V ( xa (t ))  xa (t )T Pxa (t )  xa (t )T Pxa (t )
r
r
  i ( z (t )) j ( z (t )) xa (t )T (GijT P  PGij ) xa (t )
(14)
i 1 j 1
Therefore, yield V ( xa (t ))  0 once the inequality
GijT P  PGij  DPD  0
(15)
is satisfied. After pre multiplying and post multiplying X  P 1 , and denoting
M j  K j X , (15) follows that
XAiT  Ai X  M Tj BiT  Bi M j  XDX 1DX  0
(16)
for all i, j  1, 2, , r . Applying Schur's complement to (16), it yields (13).
Furthermore, according to (14) and (15), it follows that
V ( xa (t ))  - xaT (t )( DPD ) xa (t )   V ( xa (t ))
which further results in
V ( xa (t ))  V (0)e   t
with  
min ( DPD )
,
max ( P )
where min ( M ) , max ( M ) denote the minimal and maximal eigenvalue of matrix M ,
respectively. Therefore, xa (t ) 
2
V (0)   t
e
is concluded.
min ( P )
□
Therefore the LMIs, once solved, yields M j and P 1 . Then the controller gains
K j is obtained from M j P  K j . This implies that the system trajectories xa (t )  0
exponentially as t   .
For comparing the control laws, we introduce the following theorem for decay rate
controller design, which satisfying the condition V ( xa (t ))  2V ( xa (t )) [18] for all
trajectories, proposed by Tanaka et al. described as follows.
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Journal of China University of Science and Technology Vol.49-2011.10
Theorem 2: [16] The equilibrium of the integral fuzzy augmented system (10) is
asymptotically stable in the large if there exists a common positive definite matrix P
such that
Gii P  PGii  2 P  0
for all
i and
 Gij  G ji 
 G  G ji 
P  P  ij


  2 P  0
2
2




T
for i  j except the pairs (i , j ) such that i ( z (t ))  j ( z (t ))  0, t , where   0 .
Remark: In above theorem 1, we introduce the matrix D to enhance the decay rate
of the regulation error. In this form, the flexibility of assigning the decay rate is better
than the method in theorem 2 [16]. The regulation response can be significantly
improved by carefully choosing D with try and error method.
IV. Numerical Example
The aim of this numerical section is to verify the validity of the proposed integral
fuzzy output regulation controller for the micro unmanned helicopter. For comparing
the control performance, two integral fuzzy controller design methods based on decay
rate D (IFRC-DR- D ) and decay rate  (IFRC-DR-  ) will show as the following
simulation results respectively.
A.
Plant Description
For the helicopter dynamics (2), we assume that a local linear feedback control with
respect to the yaw angle  (t ) is hold. That is satisfying U (t )    (t ) , where 
is a positive value. Hence, the yaw dynamics can be stabilized by the local feedback
controller. As a result, we just only consider the remaining x ( t ) , y ( t ) and z (t )
position control. Then, the dynamics can be rewritten as
135
Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
u (t )  
v (t ) 
w( t ) 

Iz

Iz
 (t )v (t ) 
 ( t )u ( t ) 
1
U X (t ),
m
1
U Y (t ),
m
1
U Z (t ),
m
 (t )  

Iz
 (t ),
(17)
x (t )  u(t ),
y (t )  v (t ),
z (t )  w(t ).
For the integral regulation control purpose, y p (t )  yd as t   . We define the
regulation error state xe (t ) satisfies the following equation:
xe ( t )  y d  y p ( t )
(18)
where xe (t )  [ xe1 (t ) xe 2 (t ) xe 3 (t )]T , yd  [ yd 1 yd 2 yd 3 ]T and y p (t )  [ x(t ) y(t ) z(t )]T .
By solving the condition (6), we can obtain nominal equilibrium point as
x p  [0 0 0 0 yd 1 yd 2 yd 3 ]T , and U x  U y  U z  0 . Then the transformed augmented
error dynamics (8) for the helicopter system is of the following form:
xa (t )  A( x p (t )) xa (t )  BU (t )
(19)
where xa (t )  [u(t ) v (t ) w(t )  (t ) x(t ) y (t ) z (t ) xe1(t ) xe 2 (t ) xe 3 (t )]T ,


  (t )
 0
IZ


0
 I  (t )
Z

0
 0

 0
0
A 
 1
0

 0
1

0
 0
 0
0

0
 0
 0
0
0
0
0
0
0
0
0 

IZ
0
0
0
0
1
0
0
0
0
0
0
0

0 0 0


0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
,
0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
1 0 0 0 0 0 

0 1 0 0 0 0 
0 0 1 0 0 0 
0
0
136
0
1
m

0


0


B0
0
0

0
0

0
0

0
1
m
0
0
0
0
0
0
0
0

0

0

1

m
0 ,

0
0

0
0

0
0 
Journal of China University of Science and Technology Vol.49-2011.10
and U (t )  [U X (t ) UY (t ) U Z (t )]T .
For the fuzzy modeling, the premise variable can be defined as z (t )   (t ) where
 (t )  [-  ] . Fuzzy set Fi are F1 ={about - rad} and F2 ={about  rad} . Then,
the dynamical equation (19) can be exactly presented as the following two rules T-S
fuzzy model:
Plant Rule 1: IF z (t ) is F1
THEN xa (t )  A1 xa (t )  B1U (t )
Plant Rule 2: IF z (t ) is F2
THEN xa (t )  A2 xa (t )  B2U (t )
Then, the system and the corresponding input are inferred as follows:
2
xa (t )   i ( z (t )){Ai xa (t )  BU
i ( t )}
i 1
2
U (t )    j ( z (t )) K j xa (t )
j 1
According to exact modeling method [17], we assign  (t )  1d 2  2d1 with
 (t )  [d1, d 2 ] . where d1   , d 2   . Thus, we can choose the normalized
membership function for Plant Rule 1 and 2 as
1 ( z (t )) 
 ( t )  d1
d 2  d1
2 ( z (t )) 
,
d 2   (t )
d 2  d1
As a result, the system matrices in the consequent part are:

 0

 
I
 Z
 0

 0
A1  
 1

 0

 0
 0

 0
 0


0
0
0
0
0
0
0
0
0
0 
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
IZ

IZ

0 0 0


0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
,
0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
1 0 0 0 0 0 

0 11370 0 0 0 
0 0 1 0 0 0 
0
0
0
Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter

 0

 
 I
 Z
 0

 0
A2  
 1

 0

 0
 0

 0
 0

0
0
0
0
0
0
0
0
IZ

0
0 
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
IZ

0 0 0


0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
 and
0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0
1 0 0 0 0 0 

0 1 0 0 0 0 
0 0 1 0 0 0 
0
0
0
1
m

0


0


B1  B2   0
0
0

0
0

0
0

0
1
m
0
0
0
0
0
0
0
0

0

0

1

m
0 .

0
0

0
0

0
0 
B. Numerical Simulation
To proceed on the simulation, we apply the IFRC-DR- D and IFRC-DR-  to
control the unmanned helicopter with the same initial values and parameters for
comparison as follows: xa (0)=[0.5 0.5 0.5 0.5 0 0 0 0 0 0]T ,   0.0008 . Moreover,
we choose D  0.01  diag[1 1 1 1.2 1.2 1.2 1.2 0.0001 0.0001 0.0001] for
IFRC-DR- D ;   0.015 for IFRC-DR-  and let the desired output reference
yd =[0 1 2]T . For IFRC-DR- D , by using theorem 1, we obtain the corresponding
common positive definite matrix and integral control state feedback gains are given as
follows:
 1.5914
 0

 0

 0
1.3914
P
 0
 0

-0.8025
 0

 0
0
1.5914
0
0
0
1.3914
0
0
0
0
0
0
1.5914
0
0
0.9702
0
0
-0.8025
0
1.3914
0
0
-0.8025
0
1.3915
0
0
0
0
0
0
0
0
0
3.4580
0
0
-1.3913
0
1.3914
0
0
0
3.4580
0
0
-1.3913
0
1.3915
0
0
0
3.4580
0
0
0
0
0
-1.3913
0
0
1.5911
0
-0.8025
0
0
0
-1.3913
0
0
1.5911
0
-0.8025
0
0
0
-1.3913
0
0
and
138

0 

-0.8025

0 
0 
,
0 
-1.3913 

0 
0 

1.5912 
0
Journal of China University of Science and Technology Vol.49-2011.10
0
0
0 0.7647 0.0001
0
-0.2674 0.0002
0 
0.3982

K1 
0
0.3982
0
0
0
0.7647
0
0
-0.2674
0 ,


0
0.3983 0
0
0
0.7647
0
0
-0.2674 
 0
0
0
0 0.7647 0.0001
0
-0.2674 0.0002
0 
0.3982

K2 
0
0.3982
0
0
0
0.7647
0
0
-0.2674
0 .


0
0.3983 0
0
0
0.7647
0
0
-0.2674 
 0
Similarly, by using theorem 2, we obtain the corresponding common positive definite
matrix and integral control state feedback gains are given as follows:
 2.6140

0

0


0

 2.2282
P
0


0

 -1.5441

0

0


2.6140
0
0
0
2.2282
0
0
-1.5441
0 

0
2.6139
0
0
0
2.2278
0
0
-1.5441

0
0
2.3533
0
0
0
0
0
0 
0
0
0
5.7155
0
0
-2.2282
0
0 
,
2.2282
0
0
0
5.7155
0
0
-2.2282
0 
0
2.2278
0
0
0
5.7141
0
0
-2.2278

0
0
0
-2.2282
0
0
2.6140
0
0 
-1.5441
0
0
0
-2.2282
0
0
2.6140
0 

0
-1.5441
0
0
0
-2.2278
0
0
2.6139 
0
0
0
2.2282
0
0
-1.5441
0
0
and
0
0 0.7844 -0.0007
0
-0.2756 0.0005
0 
 0.3513 -0.0008

K1  0.0008 0.3513
0
0 0.0007 0.7844
0
-0.0005 -0.2756
0 ,


0
0.3512 0
0
0
0.7844
0
0
-0.2756
 0
0
0 0.7844 -0.0004
0
-0.2756 0.0003
0 
 0.3513 -0.0004

K2  0.0004 0.3513
0
0 0.0004 0.7844
0
-0.0003 -0.2756
0 ,


0
0.3512 0
0
0
0.7842
0
0
-0.2756
 0
for IFRC-DR-  .
The closed-loop control results are shows in Figs. 1 to 4. It is shown that the
proposed IFRC-DR- D obtains much better transient control performance than
IFRC-DR-  under the same initial conditions except the different decay rate. From the
simulation results, we conclude that the proposed integral fuzzy decay rate controller
design is a powerful scheme. It can regularly control the unmanned helicopter,
achieving exponential stable, shorter transient time, lower overshot and rejecting the
parameter variation.
139
Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
V. Conclusions
In this paper, an integral fuzzy control strategy to solve a way-point regulation
problem for the micro unmanned helicopter has been proposed. First, a general
nonlinear affine system can be converted into T-S fuzzy model without bias term via
coordinate translation and integral control procedure. Next, utilizing PDC technique and
direct Lyapunov method, the integral state-feedback controller which achieve the
exponential stability for output regulation is parameterized in terms of LMIs. Finally, an
unmanned helicopter is considered to demonstrate the effectiveness of the proposed
controller design methodology. From the simulation results, it has been showed that the
proposed IFRC-DR- D method not only can eliminate the bias term and guarantee
zero-offset regulation performance but also obtain much better output regulation
performance when compared to IFRC-DR-  method.
Figure 1. Control input for two different IFRC with decay rate methods.
140
Journal of China University of Science and Technology Vol.49-2011.10
Figure 2. Velocity state response for two different IFRC with decay rate methods.
Figure 3. Position state response for two different IFRC with decay rate methods.
141
Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
Figure 4. Position regulation trajectory for two different IFRC with decay rate methods
(Solid: IFRC-DR- D ; Dotted: IFRC-DR-  ).
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