Transactions of Nanjing University of Aeronautics and Astronautics
Dynamic Surface Control with Nonlinear Disturbance Observer for
Uncertain Flight Dynamic System
Li Fei(李飞)1* Hu Jianbo (胡剑波)1 Wu Jun(吴俊)2 Wang Jian-hao(王坚浩)3
1. Equipment Management and Safety Engineering College, Air Force Engineering University, Xi’an 710051, P. R. China；
2. Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027 , P. R. China；
3. Unit93286, People’s Liberation Army, Shenyang 110141, P. R. China.
Abstract: A new robust control method of a nonlinear flight dynamic system with aerodynamic coefficients and external
disturbance has been proposed. The proposed control system is a combination of the dynamic surface control (DSC) and the
nonlinear disturbance observer (NDO). DSC technique provides the ability to overcome the “explosion of complexity”
problem in backstepping control. NDO is adopted to observe the uncertainties in nonlinear flight dynamic system. It has
been proved that the proposed design method can guarantee uniformly ultimately boundedness of all the signals in the
closed-loop system by Lyapunov stability theorem. Finally, simulation results show that the proposed controller provides
better performance than the traditional nonlinear controller.
Keywords: nonlinear disturbance observer(NDO), dynamic surface control(DSC), uncertainties, flight control.
CLC Number: TP273
Document Code: A
crucial for a nonlinear control method to have the
Nomenclature
=aerodynamic forces about the body-fixed frame
F
Article ID:
L, M , N =aerodynamic rolling, pitching, yawing moment
ability to handle unknown disturbances. Another
important property required for a good control method
in aerospace engineering is the robustness against
I
=moment of inertia
unmodeled dynamics and variation of various
p, q, r
=roll, pitch, yaw rate about the body-fixed frame
aerodynamic derivatives[1][2].
q
=dynamic pressure
T
=thrust
common method employed in nonlinear flight
V
=velocity
control[3][4], which allows to be performed without
, 
=angle of attack, sideslip angle
state transformation. However it requires complete
Feedback linearization is probably the most
and accurate knowledge of the plant dynamics. To
 e ,  a ,  r =elevator, aileron, rudder angle
 ,
solve this issue, adaptive backstepping design methods
=roll, pith angle
is proposed[5][6]. However, the parameter estimation
1 Introduction
error is only guaranteed to be bounded and converging
aerospace
to an unknown constant value. DSC technique could
engineering technology over the past few decades,
conquer the problem of complexity in backstepping
requirements on the performance of modern flight
design resulting from the repeated differentiations by
vehicles have become more and more challenging.
introducing a first-order filtering of the synthetic input
And there have been great interests on flight dynamic
at each step[7][8]. NDO could effectively reject model
systems due to the facts that unknown disturbances
mismatches and external disturbances to compensate
widely exist in aerospace engineering, such as wind,
timely for the controller by constructing a new
friction,
systems,
dynamic system[9][10][11]. So it can improve the
environmental and electromagnetic noise etc. So it is
performance of the controller by eliminating the
With
the
fast
coupling
development
effects
from
of
other
effects of uncertainties of controlled device. NDO has
Foundation items: This work was supported by the open research project of the State Key Laboratory of Industrial Control
Technology Zhejiang University China(No. ICT1401) and Shanghai Leading Academic Discipline Project(No.J50103).
*Corresponding author: Li Fei, E-mail: [email protected]
Li Fei, et al. Dynamic Surface Control with Nonlinear Disturbance Observer…
been widely applied in robot control[10][12][13], missile
[14]
[15][16]
cos  sin 
( FT  Fx )
mV
cos 
sin  sin 
g

Fy 
Fz  (cos  sin  sin  (2)
mV
mV
V
 cos  cos  sin   sin  sin  cos  cos  )
  p sin   r cos  
control , flight control
and so on.
Accordingly, in this paper, a new robust control
design method for nonlinear flight dynamic system
with aerodynamic coefficients and external
disturbances is proposed. Firstly, we present the
nonlinear flight model with aerodynamic coefficients
and external disturbances. Secondly, the design
principles of nonlinear disturbance observer are
introduced. And the structure of the composite
controller is investigated. Then the stability of the
proposed control system is analyzed. Finally,
simulation results demonstrate that the proposed
controller could effectively resist the influence of
uncertainties and disturbances on the flight control
system.
The flight dynamic Eqs.(1–7) can be rearranged as a
2 Problem Formulation
kind
The control law design is based on the nonlinear
model of an F-16 fighter. The task of the controller is
to track the commands of ，， and  when
parameter perturbations and external disturbance exist.
The relevant equations of six-degree nonlinear model
used for the control design can be written as[17]
   p cos  tan   q  r sin  tan 
sin 
cos 

( FT  Fx ) 
Fz
mV cos 
mV cos 
g

(sin  sin   cos  cos  cos  )
V cos 
  p  tan  (q sin   r cos  )
(3)
p  I1qr  I 2 pq  I 3 L  I 4 N
(4)
q  I 5 pr  I 6 ( p 2  r 2 )  I 7 M
(5)
r   I 2 qr  I8 pq  I 4 L  I9 N
(6)
  q cos   r sin 
(7)
of
uncertain
non-linear
MIMO
system
considering the external disturbance and aerodynamic
coefficients perturbations.
 x1

 x2
 x3

 1
 2
 f1 ( x1，x3 )  g1 ( x1，x3 ) x2   1
 f 2 ( x1，x2 )  g2 ( x1 )u   2
 f 3 ( x1，x2 )
 f1 ( x1，x3 )  g1 ( x1，x3 ) x2  [h( x1 )  h( x1 )]u
 f 2 ( x1，x2 )  g2 ( x1 )u  d (t )
(8)
(1)
Where x , u , f , g , h represent the following
terms respectively:
x1   ,  ,   ,
T
x2   p, q, r  ,
T
x3   , 
T
,
u   e ,  a ,  r 
T
( s i n / c To sCx )[ qS  ( )
]  (Czc o qS
s / c o s ) ( )

1 

(
c

o
s

s
T
i

n
C

)
[
qS

(

C
)

]
qS

c
o
s


(
C
  qSs i n s i n
x
y
z )
mV 


0
( 1 / c o s ) ( s i n s i n
c o s c o s c o s )

g
 c os s i n  s i n s c sin
o 
s  sin
c o sin  cos  cos  
V


0
f1 ( x ，
1 x )
3
(
,
(9)
)
Transactions of Nanjing University of Aeronautics and Astronautics
  cos  tan 
g1 ( x1，x3 ) 
sin 

1
1
0
sin  tan 
 sin  tan  
 cos  
cos  tan  
sin 
cos 

0

Cxq ( )c 
C z ( )c

cos 
cos  q

S

cos  C y p ( )b  cos  sin  Cxq ( )c  sin  sin  C zq ( )c
4m 
0
0





cos  C yr ( )b 

0


0
sin 
cos 



Cx ( ) 
Cz (， )
0
0


cos  e
cos  e

VS 
h1 ( x1 ) 
  cos  sin  Cxe ( )  sin  sin  Cze (， ) cos  C ya (  ) cos  C yr (  ) 
2m 

0
0
0




(10)
(11)
 I1qr  I 2 pq  I 3Cl (， )qsb  I 4Cn (， )qsb 

f 2 ( x1，x2 )  
I 5 pr  I 6 ( p 2  r 2 )  I 7 Cm ( )qsc

  I 2 qr  I 8 pq  I 4 Cl (， )qsb  I 9Cn (， )qsb 
 I 3Cl ( )b  I 4 Cn ( )b
0
p
p
VS 

0
I 7 Cmq ( )c
4 
0
 I 4 Cl p ( )b  I 9Cn p ( )b

0

g2 ( x1 )  qS  I 7 Cm ( )c
e

0

f3 ( x1，x2 )  0 cos 
(12)
I 3Clr ( )b  I 4 Cnr ( )b   p 
 
 q
0
 
I 4Clr ( )b  I 9 Cnr ( )b   r 
I 3Cl ( ,  )b  I 4Cn ( ,  )b 
r
r


0
0

I 4 Cl ( ,  )b  I 9Cn ( ,  )b I 4Cl ( ,  )b  I 9Cn ( ,  )b 
a
a
r
r

I 3Cl ( ,  )b  I 4Cn ( )b
a
a
(13)
 p
 sin    q 
 r 
(14)
 1 and  2 represent the compound uncertainties in
 m， m and  m  R such that the magnitudes and
the model， f ( x ) , g ( x ) and h( x ) represent the
is the external
derivatives of f1，f 2，g1 and g2 are bounded for
all ， and   R satisfying    m and
The following assumptions are used in the design
   m . Furthermore, there exist gi 0 and gi1 such
that 0  gi 0  gi  gi1 , i  1, 2 .
aerodynamic coefficients ， d (t )
disturbance.
and analysis processes.
Assumption
1:
yc  [c，c，c ]
T
Remark 1: Note that g2 represents the input
The
desired
trajectory
angular rates p，q and r . Because the control
are bounded as
[ yc，yc，yc ]  0
(15)
where 0  R is a known positive constant and 
denotes the 2-norm of a vector or a matrix.
surfaces of the aircraft are designed to control each
axes’ angular rate of aircraft independently, the input
matrix g2 is invertible for all cases.
Assumption 2: The total velocity and dynamic
pressure are constant.
V  0，q  0
matrix of the control u to the dynamics of the
Remark
2:
Note
that
h1
represents
the
aerodynamic force component caused by the control
(16)
surface deflection. The magnitude of h1 is small
Assumption 3: There exist positive constants
compared to other aerodynamic terms in the dynamic
Li Fei, et al. Dynamic Surface Control with Nonlinear Disturbance Observer…
equation[17]. Therefore, it can be assumed that h1 is
one part of model uncertainties.
3 Nonlinear Disturbance Observer
This paper refers to the idea of [17]: the dynamic
So NDO in the inner loop of the flight dynamic
system can be designed as:
ψˆ 2   2  P2 ( x2 )

 2   L2 ( x2 )[ 2  P2 ( x2 )  f 2 ( x1 , x2 )  g2 ( x1 )u]
surface control law is designed by separating the flight
dynamics into fast and slow dynamics. So we design
NDOs in the inner and the outer loop respectively
[13][15]
:
ψˆ1   L1 ( x1 )ψˆ1  L1 ( x1 )[ x1  f1 ( x1 , x3 )  g1 ( x1 , x3 ) x2 ]
(17)
ψˆ 2   L2 ( x2 )ψˆ 2  L2 ( x2 )[ x2  f2 ( x1 , x2 )  g2 ( x1 )u] (18)
where, ψ̂1 , ψ̂2 represent the estimates of ψ1 ， ψ 2
respectively. Li ( x ), i  1, 2 is the gain matrix. From
(8), the dynamics of the above NDOs can be rewritten
as:
ψˆ1   L1 ( x1 )ψˆ1  L1 ( x1 )ψ1  L1 ( x1 )endo1
ψˆ 2   L2 ( x2 )ψˆ 2  L2 ( x2 )ψ2  L2 ( x2 )endo2
where, endoi  ψi  ψˆ i , i  1, 2 .
(28)
Remark 3: From the dynamics of NDO (19) and (20),
we can know that the output of NDOs ψ̂1 and ψ̂2
will converge to ψ1 and ψ2 via choosing appropriate
values of L1 ( x1 ) and L2 ( x2 ) .
4 Controller Design
The structure of the two-timescale controller for
flight control system is shown in Fig.1. In each
feedback loop, control laws x2c and u are designed
separately.
(19)
(20)
It is assumed that the uncertainties are slowly
time varying compared with the dynamics of NDO. So
Fig.1
ψ1  0, ψ 2  0 .
Structure of the two-timescale controller
In this paper, the dynamic surface control method
Then (19) and (20) can be rewritten as:
endo1  L1 ( x1 )endo1  0
(21)
is used to design a controller. The dynamic surface
endo 2  L2 ( x2 )endo 2  0
(22)
control method can be viewed as the two-timescale
In order to avoid the angular acceleration
approach because the fast states x2 are used as
measurement, an auxiliary nonlinear function vector is
control input for the slow states x1 intermediately.
However, this methodology considers the transient
defined to improve the NDO in the inner loop.
 2  ψˆ 2  P2 ( x2 )
(23)
responses of the fast states and, therefore, does not
require the timescale separation assumption.
where
dP2 ( x2 )
 L2 ( x2 ) x2
dt
From (23), we have:
 2  ψˆ 2  P2 ( x2 )
The controller design procedure can be given as
(24)
follows.
Step 1: define error surface S1  x1  x1c , where
(25)
x1c  yc . Its derivative is
From (20) and (24), the derivative of  2 can be
expressed as:
S1  x1  x1c
 2  L2 ( x2 )endo 2  L2 ( x2 ) x2
 L2 ( x2 )(ψ2  ψˆ 2 )  L2 ( x2 ) x2
(26)
=f1  g1 x2  ψ1  yc
(29)
 f1  g1 ( x2 c  e2 )  ψ1  yc
where, x 2c is the expected virtual control of the inner
From (8), (26) can be rewritten as:
 2   L2 ( x2 )( f 2 ( x1 , x2 )  g2 ( x1 )u)  L2 ( x2 )ψˆ 2
  L2 ( x2 )( f 2 ( x1 , x2 )  g2 ( x1 )u)  L2 ( x2 )( 2  P2 ( x2 ))
(27)
loop, e2  x2  x2c . From (17), we have
S1  f1  g1 ( x2c  e2 )  ψˆ1  endo1  yc
We choose the virtual control as:
(30)
Transactions of Nanjing University of Aeronautics and Astronautics
1

 x2c   g1 (k1 S1  f1  ψˆ1  yc )


ψˆ1   L1 ( x1 )ψˆ1  L1 ( x1 )[ x1  f1 ( x1 , x3 )  g1 ( x1 , x3 ) x2 ]
F (
constant  2  0 to obtain the filtering virtual control
is continuous.
Consider
Lyapunov
x2 c 
x2c  x2 c
2
Step 2: Define the error surface S2  x2  x2c , its
derivative is
S2  x2  x2c  f2  g2 u  ψ2  x2c
(33)
The NDO (28) is used to observe the uncertainty
ψ 2 in the inner loop:
follows:
k23 
is
a
k1  3I
L2 
1
, k2  ( g112  1) I
4
, L1 
1
I
4
,
g2
1
1
I ,  ( 11  1) such that the errors of states
4
4 2
are uniformly ultimately bounded and may be kept
arbitrary small.
 u   g21 (k2 S2  f 2  ψˆ 2  x2 c )

x2c  x2c


 x2 c 
2

 x2c   g11 (k1 S1  f1  ψˆ 1  yc )

k2  diag  k21 k22
positive number  (i.e., V   ).
Theorem 1: Suppose that nonlinear model of an
F-16 aircraft (8) with aerodynamic coefficients and
external disturbances is controlled by the proposed
controller (35). In addition, if the proposed control
system satisfies Assumptions 1~4, there exist ki , Li ,
(34)
Choose an actual control u to drive S2  0 as
(35)
Differentiating the Lyapunov candidate function
(42) and substituting (21), (22), (37), (39), and (40),
we can obtain
positive
V  S1T (k1 S1  g1 S 2  g1 y2  endo1 )
2
T
T
 S 2T (k2 S 2  endo2 )   endo,
i Li endo,i  y 2 ( 
constant.
i 1
5 Stability Analysis
 min (k1 ) S1  S1 g1 S 2  S1 g1 y2  S1 endo1
 min (k2 ) S 2
errors to represent the closed-loop system as follows:
S1  f1  g1 ( S2  x2c )  ψ1  yc
(36)
S2  k2 S2  endo2
(37)
y2  x2 c  x2 c =g11 (k1 S1  f1  ψˆ 1  yc )  x2 c (38)
Using (17) and (38), (36) can be rewritten as
follow:
(39)
Differentiating (38), we can obtain
y2
 F ( S1，S2，y2，yc，yc，yc )
τ2
1

y2
τ2
2
2
2
 S 2 endo2   min ( Li ) endo,i
2
i 1
 y2 F
(43)
Then, the boundary error is defined as:
S1  k1 S1  g1 ( S2  y2 )  endo1
y2
 F)
τ2
2
We first describe the derivative of the surface
where
(42)
Assumption 4: Assume that the Lyapunov
satisfying
S2  x2  x2c  f2  g2 u  ψˆ 2  endo2  x2c
y2  
function
candidate function (42) is bounded by any given
Therefore
where
following
1 2
T
T
V  [ ( SiT Si  endo,
i endo,i )  y2 y2 ]
2 i 1
in each step.
(32)
the
candidate
x2c , which could solve the computational complexity
 2 x2c  x2c  x2c，x2c (0)  x2c (0)
(41)
f
f
 g (k1 S1  1 x1  1 x3  ψˆ 1  yc )
x1
x3
1
1
(31)
Where, k1  diag  k11 k12 k13  , k1i  0, i  1, 2,3 .
We pass x2c through a first-order filter with time
g11
g 1
x1  1 x3 )(k1 S1  f1  ψˆ 1  yc )
x1
x3
(40)
from Assumption 1 that [ yc，yc，yc ]
T
is bounded,
thus there exists a positive constant 1
such
that F1  1 . therefore, from Assumption 3 and
applying Young’s inequality, we have
Li Fei, et al. Dynamic Surface Control with Nonlinear Disturbance Observer…
V  min (k1 ) S1
 S1
2
2
1
 endo1
4
 S1 
2
1 2
g11 S 2
4
 min (k2 ) S 2
2
2
 min ( Li ) endo,i
2

i 1
1
y2
τ2
2
2
2
 S1
 S2
 y2
2
2
2

1 2
g11 y2
4
1
 endo2
4
2
simulation on the attitude maneuver flight control
system of F-16 aircraft. The following command
2
values of d , d , and d are applied to the aircraft
in a steady-state level flight of V  200 m s ,
1
 12
4
h  4000m , FT  60kN :
(44)
k  min (k1 )  3 ,
choose
2 
1
2
1
k  min (k2 )  g112  1 ,
4

2

1

g112
1
 1 , and Li  min ( Li )  , i  1, 2 yields
4
4
2
V   ki Si
2
i 1
 Li endo,i
2
  2 y2
where ki  0 , Li  0 ,  2  0 .
Then, choose the constant 
1
 12
4
2
yc can be obtained from yd by the following
command filterer:
satisfying the
0    2 min[k1，k2， 2，L1，L2 ]
(46)
Hence, (30) can be rewritten as
(45)
The integration of (45) is multiplied by e t is:
2
1
(46)
V (t )  [V (0)  1 ]e t  12
4
4
Accordingly, the surface errors S1 and S2 , the
nonlinear disturbance observer errors endo1 and endo2 ,
the boundary layer error y2 are uniformly ultimately
bounded in the following compact set  :
  {S1 , S2 , endo1，endo2，y2
|  ( S Si  e
i 1
when
k1  3I
e
2
)  y y2  1 }
2
T
2
1
k2  ( g112  1) I
4
,
We assume that the uncertainties are time-varying.
,
L1 
(47)
1
I
4
external disturbance are given as
C j  [1  0.5sin(0.5 t )]C rj , j  x, y, z, l , m, n
2
This relation implies that V  0 when V  1 .
4
T
ndo,i ndo,i
[ yc ]i
n2
 2
，n  4，n  0.8，i  1, 2,3
[ yd ]i s  2snn  n2
The time-varying aerodynamic coefficients and
1
V   V  12
4
T
i
3 , 0  t  2s
0,0  t  2 s


10 ,2 s  t  5s
30 ,2 s  t  5s


 d  0,5s  t  10s ,  d  0 , d  0,5s  t  10s


 s  t  15s
10 ,10
30 ,10 s  t  15s
0,15s  t  20 s
0,15s  t  20s
(45)
following condition:
2
approach. The proposed controller is tested in
,
g2
1
1
L2  I ,  ( 11  1) . Besides, The compact set (47)
4
4 2
can be kept arbitrarily small by adjusting k1 , k2 , L1 ,
L2 , and  . That is, the tracking error S1 can be made
arbitrarily small.
d (t )  [0.5 1 1.5] 104 sin(t )
where Cxr , C yr , Czr , Clr , Cmr , and Cnr are the
standard aerodynamic coefficients.
The final design parameters are chosen as
k1  10 I , k2  5 I ,  2  0.05 , where I represents
the 3  3
observer gains are chosen as
L1 (， ， )  diag[2(1   2 )，
2(1   2 )，
2(1   2 )]
L2 ( p，q，r )  diag[5(1  p 2 )，
5(1  q 2 )，
5(1  r 2 )]
Simulation results are shown in Fig.2–7. Subscript
1 represents the simulation results of dynamic surface
controller with NDO, and subscript 2 represents the
simulation results of dynamic surface controller
without NDO. These figures show that under
uncertainties, the proposed NDO-enhanced dynamic
surface
6 Simulation Results
identity matrix. Nonlinear disturbance
control
method
could
signals in the closed-loop system.
application
of
the
better
performance and guarantee the boundedness of the
This section presents the simulation results from
the
exhibit
nonlinear
disturbance
observer-enhanced dynamic surface flight control
Transactions of Nanjing University of Aeronautics and Astronautics
15
30
αd
α1
α2
qd
q1
q2
20
10
q/(deg/sec)
α/deg
10
5
0
-10
0
-20
-5
0
5
10
t/sec
15
-30
0
20
Fig.2 The curves of the attacking angle tracking(  )
5
10
t/sec
15
20
Fig.6 The curves of the pitching rate tracking( q )
0.5
8
βd
0.4
β1
rd
β2
r1
r2
6
0.3
4
r/(deg/sec)
0.2
β/deg
0.1
0
2
0
-0.1
-2
-0.2
-4
-0.3
-0.4
0
5
10
t/sec
15
-6
0
20
Fig.3 The curves of the sideslip angle tracking(  )
5
10
t/sec
15
20
Fig.7 The curves of the yawing rate tracking( r )
35
Φd
Φ1
Φ2
30
7 Conclusions
25
A
Φ/deg
20
disturbance
observer-enhanced
dynamic surface flight control approach for nonlinear
15
flight dynamic system with aerodynamic coefficients
10
5
and external disturbance uncertainties has been
0
-5
0
nonlinear
proposed. This approach could solve the problem of
5
10
t/sec
15
20
Fig.4 The curves of the rolling angle tracking(  )
“explosion of complexity” caused by the traditional
backstepping
method
and
exhibited
excellent
disturbance attenuation ability and robustness against
80
pd
p1
uncertainties. All the signals in the closed-loop system
p2
60
are guaranteed to be globally uniformly ultimately
p/(deg/sec)
40
20
bounded. And the tracking error of the system is
0
proven to be converged to a small neighborhood of the
-20
origin. The dynamic surface control with NDO
-40
approach proposed in this paper may also be used in
-60
-80
0
5
10
t/sec
15
20
Fig.5 The curves of the roll rate tracking( p )
the design of the controllers for other dynamic systems
with model mismatches and external disturbances.
References
[1] Qiao B, Liu Z Y, Hu B S, et al. Adaptive angular velocity
tracking
control
of
spacecraft
with
dynamic
uncertainties[J]. Transactions of Nanjing University of
Li Fei, et al. Dynamic Surface Control with Nonlinear Disturbance Observer…
Aeronautics and Astronautics, 2014, 31, (1): 85-90.
Application, 2014, 31, (4): 431-437. (in Chinese)
[2] W u W H, Gao L, Mei D, et al. Adaptive Controller for
[10] Dong W, Gu G Y, Zhu X Y, et al. High-performance
Aircraft Attitude with Input Constraints[J]. Journal of
trajectory tracking control of a quadrotor with disturbance
Nanjing University of Aeronautics & Astronautics, 2012,
observer[J]. Sensors and Actuators A211(2014): 67-77.
44(6): 809-816. (in Chinese)
[3] Ur
R
O,
Petersen
I
[11] Wei X J, Chen N, Li W Q. Composite adaptive disturbance
R,
Fidan
B.
Feedback
observer-based for a class of nonlinear systems with
linearization-based robust nonlinear control design for
multisource
hypersonic
Adaptive Control and Signal Processing, 2013, 27:
flight
vehicles[J].
Proceedings
of
the
Institution of Mechanical Engineers, 2013, 227(1): 3-11.
disturbance[J].
International
Journal
of
199-208.
[4] Li D D, Sun X X, Dong W H, et al. Pitch control for flight
[12] Nikoobn A, Haghighi R. Lyapunov-based nonlinear
in heavy-weight airdrop based on feedback linearization
disturbance observer for serial n-link robot manipulators[J].
theory and variable-structure control[J]. Control Theory
IEEE Transactions on Industrial Electronics, 2009, 55,(2):
and Applications, 2013, 30(1): 54-60. (in Chinese)
135-153.
[5] Sonneveldt L, Chu Q P, Mulder J A, Nonlinear flight
[13] Liu X, Huang Q, Chen Y. Adaptive control with
control design using constrain adaptive backstepping[J].
disturbance observer for uncertain teleoperation systems[J].
Journal of Guidance, Control, and Dynamics, 2007, 30, (2):
Control Theory and Applications, 2012, 29, (5): 681-687.
322-336.
(in Chinese)
[6] Mei R, Wu Q X, Jiang C S. Robust adaptive backstepping
[14] Li S H, Yang J. Robust autopilot design for bank-to-turn
control for a class of uncertain nonlinear systems based on
missiles using disturbance observers[J]. IEEE Transactions
disturbance observers[J]. Science China Information
on Aerospace and Electronic Systems, 2013, 49(1):
Sciences, 2010, 53(6): 1201–1215.
558-579.
[7] Yoo S J. Adaptive track and obstacle avoidance for a class
[15] Zhang T Y, Zhou J, Guo J G. Design of predictive
mobile robots in the presence of unknown skidding and
controller for hypersonic vehicles based on disturbance
slipping[J]. IET Control Theory and Applications, 2011,
observer[J]. Acta Aeronautica et Astronautica Sinica, 2014,
5(14), pp. 1597-1608.
35(1): 215-222.
[8] Sung J Y, Park J B, Choi Y H. Adaptive dynamic surface
[16] Chen M, Zou Q Y, Jiang C S, et al. Dynamical inversion
control of flexible-joint robots using self-recurrent wavelet
flight control based on the neural network disturbance
neural networks[J]. IEEE Trans. Syst., Man, Cybern. B,
observer[J]. Control and Decision, 2008, 23(3): 283-287.
2006, 36, (6): 1342-1355.
(in Chinese)
[9] Liu C, Li Y H, Zhu X H. Adaptive sliding mode observe
[17] Lee T Y, Kim Y D. Nonlinear adaptive flight control using
for actuator fault reconstruction in nonlinear system with
backstepping and neural networks controller[J]. Journal of
mismatched
Guidance, Control, and Dynamics, 2001, 24, (4): 675-682.
uncertainties[J].
Control
Theory
and