Journal of China University of Science and Technology Vol.49-2011.10
劉 建 宏
C h i i e n H u n g L i i u
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。
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
127

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.
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
128

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.
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
( ( )

( ) ( ) - ( ) ( ))

X
( )
( ( )

( ) ( ) - ( ) ( ))

Y
( )
( ( )

( ) ( ) - ( ) ( ))

Z
( )
( )
X

( ) ( )(
Z
I
Y
)

M
X
( )
( )
Y

( ) ( )(
X
I
Z
)
( )
Z

( ) ( )(
Y
I
X
)


M
M
Y
Z
( )
( )
(1)
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 u t
 r t v t
1 m
X
U ( ) , v t

- ( ) ( )
1 m



Y
U ( ) ,
1 m
U ( ),
Z
(2)
1
U

( )
I
Z where m =0.2
and I
Z
=0.2857
. U
X t , ( ) , ( ) and U

( ) denoted new control input variables. We can obtain the original control inputs (to the real helicopter) from
U
X
( ) , ( ) , ( ) and U

( ) .
TABLE I. Parameters of the helicopter model x u y v z w

, p

, q

, r
I
X
, I
Y
, I
Z
X
,
Y
,
Z
M
X
, M
Y
, M
Z m position and velocity (X-axis) position and velocity (Y-axis) position and velocity (Z-axis) angle and angle velocity (X-axis) angle and angle velocity (Y-axis) angle and angle velocity (Z-axis) moments of inertia with respect to X, Y and Z axes translational forces to X, Y and Z axes rotational forces to X, Y and Z axes mass
A.
Coordinate Translation and Integral Control
Consider a general nonlinear affine system as follows:
P
( )

(
P
( ))

(
P
( )) ( ) y t
P
 h x t
P
( ))
 
(3) where x t p
( )

R n
denotes the state variables; y t

R is the output vector; p
( ) q
( )

R m
is input vector; ( ) , ( ) and ( ) are nonlinear function vectors with appropriate dimensions and

is system bias term. Note that if the system is
130

Journal of China University of Science and Technology Vol.49-2011.10 controllable and subject to a constant bias term, it may have ( p
( ))

0 at x t p
( )

0 g x t p p
( )

0

0
. Let
 x t p y d

R q
and output function may have an offset such that ( p
( ))

0 at
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 t p
 y d
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 x t e
( ) , and is computed as x t e
 
( y d
y t p dt (4)
Combining (3) with (4), we obtain the augmented dynamics as follows x t p
 f x t p
 g x t p u t
  e
( )
 y d
- ( p
( ))
(5)
Let us define the regulation error as ( )
 y d
y t p
( ) . Without loss of generality, there always exists some equilibrium point for constant output reference y d
in practical industrial process control, denoted by the steady state x t

R n
and the p
( ) corresponding input ( )

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 ( )

0 . For this purpose, it follows that f x p
)
 g x u p
) y d

( p
)=0.
 
=0
(6)
Here (6) is assumed to have a unique solution ( x p
, ) . To eliminate the constant bias term

from the dynamics (5), we consider the coordinate translation: let p
( )
 p
( )
 x p
and ( )

( ) u . By using the property of (6), the dynamics of the system (5) can be written as follows: x t
 f x t
 g x t u t
( )
 s
( p
( ))
(7) where f x t h x t s
( p
( )) s p
 y d

( p
( )
 x p once x t p
( )

0 .

( p
( )

).
x p
)

( p
( )
 x p
) u
 
; s
( p
( ))

( p
( )
 x p
)
Clearly, f x t s
( p
( ))

0, h x t s
( p
( ))

0, g x t s p
; and

0,
131

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 ( ) : p
( )

( p
( )) p
( )

( p
( )) ( )
(8) x t e

C x t x t p p
( ) 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 z t is
1
( ) F
1 i
and … and z t p
( ) is F pi
THEN p
( )
 i p
( )
 i
( ) e
( )
 i p
( ), 1 i r where z t
 z t , ( )]
T
are known premise variables that may be functions of the state variables; F ji
( j

1, 2,..., ) is the fuzzy set, and r is the number of model rules; A i

R , B i

R and C i

R are system matrices of appropriate dimensions. Using singleton fuzzifier, product inference and weighted average defuzzifier, the fuzzy system is inferred as follows: e p

 r  i

1
 i
( ( )){ i p
( )
 i
( )}
(9) r  i

1
 i z t C x t i p
( ) where
 i and F ji
( ( )) j
  i i r 

1
 i
with
 i

is the grade of membership of z t j
( ) in p  j

1
F z t ji
( ( )) j
for all t ,
F ji
. Since i r 

1
 i
( ( ))

0 and
 i
( ( ))

0 for i

1 , 2 ,..., r , we have i r 

1
 i
( ( ))

1 ,
 i
( ( ))

0 . Denote the augmented state vector as x t a

[ x t p x t e
T
. Then, the augmented system (9) becomes as a
 i r 

1
 i
( ( )){ A x t i a
( )
 i a
( )} (10)
132

Journal of China University of Science and Technology Vol.49-2011.10 where A i



A
C i i
0
0


, B i

B i
  .
The resulting integral state feedback controller via the PDC shows as follow a
  j r 

1
 j j a
( )
  j r 

1
 j
( ( ))[ K pj
K ej
]

 p e


(11) where K pj
denotes the feedback gain for x t p
( ) and K ej
is the gain associated with e
( ) . Obviously for the above stable closed-loop system, both the new augmented state a
( ) and the new control input ( ) converge to zero when t
 
. Accordingly, we achieve the control objective: y t p
( )
 y d
as t
 
.
C. LMI-based Integral Fuzzy Controller Design
Substituting (11) into the regulation error system (10), the closed-loop system is thus obtained: a
 i r r 

1 j

1
 i z t
 j
( ( ))( A i
 i j
) ( )
(12)
 r r  i

1 j

1
 i z t
 j z t G x t where G ij

A i

B 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:


A X

XA i
T 
B M j
 T T
M B i
XD
T
DX

X



0, 1

,
 r (13) where X

P

1
and M j

K X j
133

Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
Proof: Choose the quadratic Lyapunov function candidate is defined
 a
( )
T a
( ) where P is a symmetric positive definite matrix. Taking the derivative of V x t a with respect to t , we have
( a
( ))
 a
( )
T
( )
 a
( )
T
V x t x t Px t x t ( ) a a
 i r r 

1 j

1
 i z t
 j z t x t
T T
G P
 ij
) a
( )
(14)
Therefore, yield ( ( ))

0 once the inequality
G P ij

PG ij

DPD

0 (15) is satisfied. After pre multiplying and post multiplying
M j

K X , (15) follows that
X

P

1
, and denoting
XA i
T 
A X i

M B j i
T 
B M i j

XDX DX

0 (16) for all ,

1, 2, , r . Applying Schur's complement to (16), it yields (13).
Furthermore, according to (14) and (15), it follows that
V x t a

-
T x t DPD x t a
) a
( )
  
V x t a
( )) which further results in where
 min
( M ) ,
( a
( ))

V (0) e
  t
with
 
 min
(
 max
DPD )
,
 max
( M ) denote the minimal and maximal eigenvalue of matrix M , respectively. Therefore, a
2 
V
 min
(0) e
  t
is concluded. □
Therefore the LMIs, once solved, yields M j
and P

1
. Then the controller gains
K j
is obtained from M P

K j
. This implies that the system trajectories x t

0 exponentially as t
 
.
For comparing the control laws, we introduce the following theorem for decay rate controller design, which satisfying the condition ( ( ))
 
2

( ( )) [18] for all trajectories, proposed by Tanaka et al. described as follows.
134

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
G P ii

PG ii

2

P

0 for all i and


G ij

G ji
2
 T
P

P


G ij

G ji
2



2

P

0 for i
 j except the pairs ( , ) such that
 i
( ( ))
 j
( ( ))

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.
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-DRD ) 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

( ) is hold. That is satisfying U

( )
  
, 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 ( ) , ( ) and ( ) position control. Then, the dynamics can be rewritten as
135

Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
 

I
  z
I
  z

1 m
U ( ),
X

1 m
Y


1 m
U ( ),
Z
 


I z
( )

( ),
( )

( ),
(17)
( )

( ).
For the integral regulation control purpose, y t p
( )
 y d
as t
 
. We define the regulation error state x t e
satisfies the following equation: e
( )
 y d
 p
( ) (18) where x t e

[ x e 1 t x e 2 t x e 3 t
T
, y d

[ y d 1 y d 2 y d 3
]
T
and y t p
 x t y t z t
T
. x p

By solving the condition (6), we can obtain nominal equilibrium point as
[0 0 0 0 y d 1 y d 2 y d 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:
( )

( p
( )) a
( )

( ) (19)
A


















 0
I
 
Z
0
0
1
0
0
0
0
0 where x t a
 u t v t w t
 t x t y t z t x e 1 t x e 2 t x e 3 t
T
,

I
 
Z
0
0
0
0
1
0
0
0
0
( ) 0
0
0
0
0
0
0

0
0
0
0
0
0
I

Z
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 0 0 0
1 0 0 0 0 0


0

1 0 0 0 0
0 0 0 0

1 0 0 0
















, B
















1 m
0
0
0
1 m
0
0
0
1 m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0


















,
136

Journal of China University of Science and Technology Vol.49-2011.10 and ( )

[ U
X
( )
Y
( )
Z
( )]
T
.
For the fuzzy modeling, the premise variable can be defined as ( )
 
( ) where

( )
  
. Fuzzy set F i
are F
1

and F
2
. Then, the dynamical equation (19) can be exactly presented as the following two rules T-S fuzzy model:
Plant Rule 1: IF ( ) is F
1
THEN a
( )
 a
( )

( )
Plant Rule 2: IF ( ) is F
2
THEN a
( )
 a
( )

( )
Then, the system and the corresponding input are inferred as follows: a

2  i

1
 i
( ( )){ i a
( )
 i
( )}
  j
2 

1
 j j a
( )
According to exact modeling method [17], we assign
  
1 d
2
 
with
 t
 d
1 d
2
] . where d
1
  
, d
2
 
. Thus, we can choose the normalized membership function for Plant Rule 1 and 2 as

1


( )
 d
1 d
2
 d
1
,

2
 d
 
2 d
2
 d
1
As a result, the system matrices in the consequent part are:
A
1


















 0

I
Z
0
0
1
0
0
0
0
0


I
Z
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
I

Z
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0

1 0 0 0 0 0
0 0 0 0

1
137
0 0 0 0
0 0 0 0 0

1 0 0 0


















,

Integral Fuzzy Regulation Control for an Micro Unmanned Helicopter
A
2




















0

0
0
1
0
0
0
0
I
Z
0

I
Z
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
I

Z
0 0 0 0 0 0

0 0 0 0 0 0




1 0 0 0 0 0 0 0 0


0
0
0
1
0
0
0
0
0

0
0
1 0 0 0 0 0



0
1
0
0
0
0
0
0
0
0


0 0 0 0 0

1 0 0 0


and B
1

B
2

















1 m
0
0
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-DRD and IFRC-DR-

to control the unmanned helicopter with the same initial values and parameters for comparison as follows: x a
(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-DRD ;
 
0.015
for IFRC-DR-

and let the desired output reference y =[0 1 2] d
T
. For IFRC-DRD , by using theorem 1, we obtain the corresponding common positive definite matrix and integral control state feedback gains are given as follows:
P





1.5914
0


0
0

1.3914






0


0
-0.8025
0
0 and
0
1.5914
0
0
0
1.3914
0
0
-0.8025
0
0
0
1.5914
0
0
0
1.3915
0
0
-0.8025
0
0
0
0
0
0
0
0
0.9702
0
1.3914
0
0
0
3.4580
0
0
-1.3913
0
0
0
1.3914
0
0
0
3.4580
0
0
-1.3913
0
0
0
1.3915
0
0
0
3.4580
0
0
-1.3913
-0.8025
0
0
0
-1.3913
0
0
1.5911
0
0
0
-0.8025
0
0
0
-1.3913
0
0
1.5911
0
0
0
-0.8025




0
0
0
-1.3913
0
0










1.5912
,
138

Journal of China University of Science and Technology Vol.49-2011.10
K
1
 



0.3982
0
0
0
0.3982
0
0
0
0.3983 0
0 0.7647
0.0001
0 0 0.7647
0 0
0
0
0.7647
-0.2674
0.0002
0 -0.2674
0 0
0

0

,
-0.2674


K
2
 0.3982
 

0
0
0
0.3982
0
0
0
0 0.7647
0.0001
0
0.3983 0
0
0
0.7647
0
0
0
0.7647
-0.2674
0.0002
0
0
-0.2674
0
0 
0

.
-0.2674


Similarly, by using theorem 2, we obtain the corresponding common positive definite matrix and integral control state feedback gains are given as follows:
P













2.6140
0


0
0

2.2282
0
0
-1.5441
0
0 and
0
2.6140
0
0
0
2.2282
0
0
-1.5441
0
0
0
2.6139
0
0
0
2.2278
0
0
-1.5441
0
0
0
0
0
0
0
2.3533
0
0
2.2282
0
0
0
5.7155
0
0
-2.2282
0
0
0
2.2282
0
0
0
5.7155
0
0
-2.2282
0
0
0
2.2278
0
0
0
5.7141
0
0
-2.2278
-1.5441
0
0
0
-2.2282
0
0
2.61
40
0
0
0
-1.5441
0
0
0
-2.2282
0
0
2.6140
0
0
0
-1.5441
0
0
0
-2.2278
0
0














,
2.6139
K
1
 



0.3513 -0.0008
0.0008
0.3513
0 0
0
0
0.3512
0
0 0.7844
-0.0007
0 0.0007
0.7844
0 0
0
0
0.7844
-0.2756
0.0005
-0.0005 -0.2756
0 0
0 
0

,
-0.2756


K
2
 


0.3513 -0.0004

0.0004
0.3513
0 0 for IFRC-DR-

.
0
0
0.3512
0
0 0.7844
-0.0004
0 0.0004
0.7844
0 0
0
0
0.7842
-0.2756
0.0003
-0.0003 -0.2756
0 0
0

0

,
-0.2756


The closed-loop control results are shows in Figs. 1 to 4. It is shown that the proposed IFRC-DRD 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
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-DRD 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-DRD ; Dotted: IFRC-DR-

).
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