Aeroengine Nonlinear Sliding Mode Control based on
Artificial Bee Colony Algorithm
LU Bin-bin, XIAO Ling-fei
(Jiangsu Province Key Laboratory of Aerospace Power System，College of Energy and Power，
Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)
Abstract: For a class of aeroengine nonlinear systems, a novel nonlinear sliding
mode controller design method based on Artificial Bee Colony (ABC) algorithm is
proposed. In view of the strong nonlinearity and uncertainty of aeroengines, sliding
mode control strategy is applied to design controller for the aeroengine. On basis of
exact linearization approach, the nonlinear sliding mode controller is obtained
conveniently. By using Artificial Bee Colony algorithm, the parameters in the
designed controller can be tuned to achieve optimal performance, and the closed-loop
system has satisfying dynamic performance and high steady accuracy as a result.
Simulation on an aeroengine verifies the effectiveness of the presented method.
Keywords: aeroengine; nonlinear control; sliding mode control; Artificial Bee
Colony (ABC) algorithm;
1 Introduction
Aeroengines are a kind of strong nonlinear systems of complicated thermodynamic,
while their working conditions are very poor generally. In the actual aeroengine
systems, there exist the nonlinear unknown disturbances and nonlinear parameter
perturbations, so the system needs to be controlled in order to reach good dynamic
performance and robustness
[1][2]
. The design of controller is an essential problem in
the field of aeroengines. There are several common control methods, such as PID
control[3], LQR control [4], H∞ control [5], adaptive control[6], robust control [7], etc. The
general linear control method cannot meet the requirement of performance in large
deviation of aeroengine due to its complicated thermodynamic process and nonlinear
This work is supported by the fundamental research funds for the central universities, Jiangsu
Province Key Laboratory of Aerospace Power System (NJ20140022), the Open Research Project
of the State Key Laboratory of Industrial Control Technology, Zhejiang University ,China
(No.ICT1505), and Postdoctoral Science Foundation of China (2015M571749).
Corresponding author: Lingfei Xiao, e-mail: [email protected]
factors. Hence, research on the aeroengine nonlinear control is necessary[8][9].
The designed controller based on approximate linearization cannot guarantee the
system stability in large initial deviation, and this will have a negative effect on
aeroengines. While the exact linearization theory based on the differential geometry
can exactly solve the control problem of nonlinear system in large deviation under
certain conditions
[10]
. Sliding mode control (SMC), also known as variable structure
control (VSC) or sliding mode variable structure control (SMVSC), purposely
changes the control structure according to the current system state, then forces the
system's trajectories to reach the sliding mode in finite time
[11][12][13]
. SMC has
invariability when the parameters change and the disturbances exist, so it possesses
great research value. ABC algorithm is an optimization algorithm on basis of the
intelligent behavior of honey bee swarm
[14]
. This algorithm has advantages of
easiness, simple calculation, good optimization performance and strong robustness, so
it can be used to tune the parameters of the nonlinear controller of aeroengines to
obtain the optimal performance.
In recent years, many advanced design methods of nonlinear control system have
been applied to aeroengines, such as nonlinear predictive control
back-stepping control
[16]
[15]
, nonlinear
, intelligent control [17][18][19], nonlinear control based on a
generalized Gronwall-Bellman lemma
[20]
, etc. At present, ABC algorithm has been
used in aerospace domain. In Ref. [21], ABC algorithm is used for parametric
optimization of spacecraft attitude tracking controller. In Ref. [22], ABC algorithm is
used for research on path planning of unmanned air vehicle. For a class of affine
nonlinear mathematical model of aeroengines, this paper designs a nonlinear sliding
mode controller based on ABC algorithm.
The remainder of this paper is organized as follows: In Section 2, the theoretical
preliminaries are described. The aeroengine nonlinear sliding mode control law and
the optimization procedure of ABC algorithm are derived in Section 3. Then
simulation on an aeroengine nonlinear model with interference at two initial
conditions is given in Section 4. Finally, Section 5 concludes this paper.
2 Preliminaries
2.1 Exact Linearization Theory
First, only an affine nonlinear system
x  f ( x)  B( x)u  d (t )
(1)
can be exactly linearized, where x is a n-dimension state vector, f ( x) is a smooth
n-dimension vector function and f  [ f1 ,..., f n ]T , B ( x ) is a n  m function matrix
and B  [b1 ,..., bm ] , u is an m-dimension control vector, d (t ) is an n-dimension
interference vector and d (t ) satisfies the matching condition
rank (B, d )  rank (B)
(2)
Exact linearization theory is based on the differential geometry. Next, some
definitions are clarified as follows:
1) Lie derivative and Lie bracket
Lie derivative of two functions, q and f can be defined as
L f q  (q ) f
(3)
where q is a continuous and smooth scalar function, q  q( x) , f is a smooth
n-dimension vector function, and f  [ f1 ( x),..., f n ( x)]T , q is Jacobian matrix
of q , and L0f q  q ， Lif q  L f ( Lif1q)  (Lif1q) f ， i  2 . Besides, Lie bracket
of q and f can be defined as ad f b  [ f , b]  (b) f  (f )b ， ad 0f b  b ，
ad if b  [f, ad if1b] ， i  1 .
2) Involutiveness
A collection of linearly independent vector functions {G1 ,
, Gm } on
  R n is involutive , if it satisfies
[Gi , G j ]  span{G1 ,
, Gm } (i, j  1,
, m)
(4)
Combining the conditions for exact linearization of nonlinear system in [10],
this paper analyses sufficient and necessary conditions for exact linearization of
the
affine
nonlinear
system
(1)
as
follows
(defining
a
Gi (x)  {adkf b j : 0  k  i,1  j  m,0  i  n 1} ):
1)
rank ( B, d )  rank ( B) ,namely the matching condition of interference;
set
2)
when 0  i  n1 , the rank of Gi ( x) does not change with x ;
3)
when 0  i  n 2 ,each pair of Gi (x) s are involutive;
4)
rank (Gn1 (x))  n .
2.2 Artificial Bee Colony Algorithm
ABC algorithm is a random search algorithm based on group cooperation. It solves
the optimization of target problems by imitating the honey process of bee colony.
ABC algorithm model is systematically proposed by a Turkish academic Karaboga D
in 2005, and used to solve the optimal value of the objective multivariate function. It
can be found that bees have high efficiency in honey process and show an intelligent
behavior. In the honey process, the scouts in the bee colony are in charge of searching
the food sources, and become the employed bees after find a food source. The
employed bees fly to the waiting area, and share the information with the onlooker
bees by their own dance movements. The range of the dance movements reflects the
quality of the food sources. Onlooker makes decisions to choose a food source by
comparing the dance movements. The efficiency of gathering honey can be greatly
increased by the mechanism that food sources of high quality can attract more
onlookers. In ABC algorithm, a solution of optimization problem corresponds to a
food source, it means that the higher quality the food source has, the better the
solution will be. The solution of problem is optimized by the artificial bee colony,
which will search better solutions.
Main feature of ABC algorithm is that special information of problems is not
necessary, instead, this algorithm only needs to compare the advantages and
disadvantages of the problems, and enables each individual to have local
optimization-oriented operation, then finds the global optimum value in the colony.
ABC algorithm model contains three elements of food source, employed bees, and
unemployed bees. And the unemployed bees can be divided into onlookers and scouts.
In this paper, employed bees account for half of the colony size. ABC algorithm
flowchart is shown in Figure 1.
Start
Initialization Phase
Employed Bee Phase
Onlooker Phase
Scouter Phase
Cycle<Maxcycles？
yes
no
End
Figure 1:
Flow chart of Artificial Bee Colony algorithm
ABC algorithm process includes initialization phase, employed bee phase, onlooker
phase, and scouter phase.
In the initialization phase, scouts initialize the solution vectors by equation
X mi  lower (i)  rand  upper (i)  lower (i) 
(5)
where, rand is a random variable and rand  [0,1] , m is an integer between 1 and
half of the colony size, i is an integer between 1 and the dimension of solution
vector, lower (i ) and upper (i ) are the lower limit and upper limit of
X mi
respectively.
In the employed phase, the employed bees find the previous food source or solution
vector by memory, and conduct neighborhood search by equation
Vmj  X mj  mj ( X mj  X kj )
(6)
where, k is a random variable and k  m , mj is a random variable, mj  [a, a]
and a values in 1 in this paper, j is a random integer, j 1, Dim and Dim is
the number of parameters or the dimension of solution vector. If Vmj exceeds the
value range, then it is substituted by the nearest limit value. Next, the greedy selection
mechanism is used to choose better solution vector, that is, if the new solution vector
is better than the old one, the solution vector is updated.
In the onlooker phase, the probability of an employed bee being selected is
calculated by equation
fit ( X )
Pm  Colonysize /2 m
 fit ( X i )
(7)
i 1
The onlookers conduct neighborhood search by equation (6) after selecting a food
source.
In the scouter phase, if the solution vector of a employed bee has not been updated
more than the pre-set limit times in this cycle, then the employed bee transforms into
a scouter again, and initializes the solution vector by equation (5).
The optimal solution is obtained while ABC algorithm execute the loops for the
pre-set max times. ABC algorithm possesses advantages of less control parameters,
easiness, simple calculation, good optimization performance and strong robustness, so
it can self-tune the parameters of aeroengine nonlinear controller to get a optimal
performance.
3 ABC based Aeroengine Nonlinear Sliding Mode controller
design
There are lots of aeroengines that can be described by the affine nonlinear model
(1), as shown in [9], [16], [20], etc. For example, the aeroengine nonlinear model
given in Ref. [20] is
 4.1476
x
 0.2975
8.7379
y  3.3033
 2.1940
1.4108 
 12 x12  x22   0.2491
x



u
2
2
3.1244 
0.2336

1.7
x

x



1
2

3.8025  x
2.5749 
0
(8)
where u is the increment of fuel flow and u  W f , x  [ x1 , x2 ]T , and x1 and x2 are
intermediate variables, y  [ PCN 2R P56 / P25 P16 / P56]T , and each component
of x and y is a normalized relative increment and and saves the increment sign
 . In y , PCN 2R indicates percent corrected fan speed, P56 indicates
high-pressure turbine exit pressure, P25 indicates compressor inlet pressure, and
P16 indicates bypass duct pressure. Let
 12 x12  x22 
 4.1476 1.4108 
f ( x)  
x



2
2
 0.2975 3.1244 
 1.7 x1  x2 
 12 x12  x22  4.1476 x1  1.4108 x2 


2
2
 1.7 x1  x2  0.2975 x1  3.1244 x2 
(9)
and
 0.2491
B

0.2336
(10)
x  f ( x)  Bu
(11)
then
On this basis, add an interference d  Dg (t ) into it, where g (t ) is a scalar
function. So the aeroengine nonlinear system with interference can be represented as
x  f ( x)  Bu  d (t )
(12)
just like (1).
In the following, a novel nonlinear sliding mode controller design method for
aeroengines which are in the form of (1), will be given based on ABC algorithm.
First of all, a smooth m-dimension vector function h is introduced, where
h  [h1 ,...hm ]T . The derivative of hi ( 1  i  m ) along system (1) is represented as
m
hi  hi ( f  Bu  d )  L f hi   Lb j hiu j Ld hi
(13)
j 1
When Lb j hi  0 , by the matching condition(2), it can be concluded that Ld hi  0 , and
Ld ( Lkf hi )  0 as long as Lb j ( Lkf hi )  0 . At this point, the second derivative of hi
along system (1) is expressed as
m
hi  L2f hi   Lb j ( L f hi )u j Ld ( L f hi )
(14)
j 1
When there exists a minimum integer ri which satisfies
Lb j ( Lkf hi )  0 1  i, j  m; 0  k  ri  1
(15)
Lb j ( Lrfi 1hi )  0
(16)
and j satisfies
then system (1) have relative orders {r1 ,..., rm } and total relative order r  r1  ...  rm .
For each hi ,
hi( k )  Lkf hi

m
 ( ri )
ri
h

L
h

Lb j ( Lrfi 1hi )u j Ld ( Lrfi 1hi )

f i
 i
j 1

(17)
The coefficient matrix of input vector u in (17), namely the decoupling matrix,
can be represented as
 Lb1 ( Lr1f 1h1 )

E ( x)  
 L ( Lrm 1h )
 b1 f m
Assuming
E ( x)
is
reversible
Lbm ( Lr1f 1h1 ) 


rm 1
Lbm ( L f hm ) 

(18)
P(x)  [ Lr1f h1 ,
，
, Lrfm hm ]T
and
, Ld ( Lrfm 1hm )]T , the non-linear sate transformation
Q( x)  [ Ld ( Lr1f 1h1 ),
 zij  Ti k (x)  Ljf hi , 0  i  m, 0  j  ri  1

 zk  Tk (x) r  1  k  n
(19)
and the input transformation
u  E 1 ( x)[v  P(x)  Q(x)]
(20)
are applied, then system (1) can be represented as
 x  Ax  Bv

   (x,  )   (x,  )v
x  [z10 ,
where,
v  [v1 ,
, vm ]T
,
, z1r1 1 ,
zm0 ,
, zmrm 1 ]T
A  diag ( A1 ,
, Am )
  ( zr 1 ,
,
,
0
0
m

vi   Lbk ( Lrfi 1h1 ) uk  Lrfi hi  Ld ( Lrfi 1hi ) , Ai  

k 1
0
0
(21)
B  diag (b1 ,
1
0
0 1
0 0
0 0
，
, zn ) T
, bm )
,
and
0
0

0
0
 

, bi   
,

 
1
0
1 
0  r r
ri 1
i i
1 i  m .
Evidently, x and  are an n-dimension state vector and an n-r dimension state
vector respectively. Therefore, only when r  n ,  will not exist, and then system (1)
can be represented as
x  Ax  Bv
(22)
If sufficient and necessary conditions for exact linearization of (1) can be satisfied,
by reference to section 2.1, then the existence of h and the equation r  n will be
satisfied.
Aimed at the system(22), this paper designs a nonlinear sliding mode controller and
optimizes the parameters by the ABC algorithm. Obviously, system (22) can be
partitioned into m subsystems as follows:
zi  Ai zi  bi vi (i  1,
where, zi  [ zi0 ,
(23)
, m)
, ziri 1 ]T . For each subsystem, the switching function of sliding
mode control is set as
ri  2
si  Ci zi   cij zij  ziri 1 (i  1,
(24)
, m)
j 0
where, Ci  [ci0 ,
, ciri  2 ,1] and each cij is a design parameter which ensures that the
polynomial p ri 1  ciri  2 p ri  2 
 ci1 p  ci0 satisfies Hurwitz stability ( p is Laplace
operator). Thus, the system is asymptotic stable after reaching each sliding surface
si  0 .
By the "reaching law approach", the reaching law for each subsystem can be
designed as
si  ki si   i sgn(si ) (i  1,
(25)
, m)
where,  i  0 and ki  0 . By Lyapunov theory, a lyapunov function is defined as
Vi 
1 2
si , then Vi  si si , namely
2
Vi  ki si2   i sgn(si )  si
 ki si2   i si
(i  1,
(26)
, m)
Obviously, Vi  0 , hence the system can reach the sliding surface si  0 in finite
time. Because
si  Ci zi  Ci Ai zi  Cibi vi (i  1,
(27)
, m)
Ci bi  1 , so Ci bi is reversible, thus
vi  (Cibi )1[Ci Ai zi  ki si   i sgn(si )] (i  1,
, m)
(28)
Plugging (28) into (20), the designed nonlinear sliding mode controller can be
represented as
  (C1b1 )1[C1 A1 z1  k1s1  1 sgn(s1 )] 





u  E 1 ( x)  
  P(x)  Q(x) 


1

  (Cmbm ) [Cm Am zm  km sm   m sgn( sm )]

(29)
In order to eliminate the chattering phenomenon in sliding mode control, this paper
uses a quasi-sliding mode method, namely adopting the saturation function sat ( s ) to
replace the sign function sgn( s) . The saturation function sat ( s ) can be represented
as
s
1

sat ( s)  (1/ ) s | s | 
1
s  

(30)
where,  is the boundary layer thickness. In this paper,  values in 0.001 to avoid
the degradation of the anti-jamming capability due to large boundary layer thickness.
Then the designed controller will be
  (C1b1 )1[C1 A1 z1  k1s1  1sat (s1 )] 





u  E 1 ( x)  
  P( x)  Q( x) 


1

  (Cmbm ) [Cm Am zm  km sm   m sat (sm )]

(31)
Next, this paper uses ABC algorithm to optimize the controller parameters to obtain
optimal performance. In (31), the design parameters are ci0 ,
k i (  i  0 & ki  0 , i  1,
, ciri  2 ,  i and
, m ).
So the solution vector of the designed controller in ABC algorithm is an n  m
dimensional vector c10 ,
, c1r1  2 , 1 , k1 ,
, cm0 ,
, cmrm  2 ,  m , km  . Then the objective
performance function for ABC algorithm is designed based on the systems’ state
responses as follows:

J    w1 e1 (t ) 
0
where, e1 (t ),
 wn en (t ) dt
en (t ) are respectively the errors of x1 ,
the weights and meets w1 
xn at time t . w1 ,
(32)
wn are
 wn  1 .
The steps for parameter optimization of aeroengine nonlinear sliding mode
controller based on ABC algorithm are shown below:
Step1: Set n  m dimensional solution vectors corresponding to the n  m design
parameters ( c10 ,
, c1r1  2 , 1 , k1 ,
, cm0 ,
, cmrm  2 ,  m , km ) of the designed aeroengine
nonlinear sliding mode controller, and initialize all the vectors by equation (5);
Step2: Calculate the objective functions (32) of all the solution vectors, then set the
minimum function value and the corresponding optimum vector as the global
minimum function value and global optimum vector;
Step3: Start the cycles, and setting the numbers of the no-updated times for
employed bees as 0;
Step4: The employed bees execute neighborhood search by equation (6) to find new
solution vectors, and choose better vectors through the greedy selection
mechanism.
Step5: If some employed bee don’t find a better solution vector, then the number of
the no-updated times for it will be the old value plus 1;
Step6: Calculate the probabilities of all the employed bees being selected by
equation (7), and the onlookers execute random selection, the employed bees
which have larger probability are more likely to be selected;
Step7: The onlookers execute neighborhood search ,choose better vectors through
the greedy selection mechanism, and calculate the numbers of the no-updated
times with method in Step5;
Step8: Confirm the minimum function value in this cycle, if it is less than the value
in the last cycle, then set it as the global minimum function value, and set the
corresponding solution vector as the global optimum vector;
Step9: If the number of the no-updated times for some employed bee is larger than
the pre-set limit times, then this employed bee turns into a scouter, and initialize
the solution vector by equation (5);
Step10: If the cycle times do not achieve the pre-set maximum cycle times, go to
Step2, otherwise, the cycle ends with the optimum controller parameters got.
4 Simulation
Aimed at the aeroengine system (12), supposing D  B , the interference matching
condition (2) will be satisfied. In the simulation, g (t )  50sin(10 t ) is set. As for
system (12), m  1 and B  [b] . Then
  0.2491 
G0 (x)  {b}   

 0.2336 
 5.806 x1  0.4672 x2  0.7391  
  0.2491
G1 ( x)  {b, ad f b}   
, 


0.4672 x2  0.8225 x1  0.6579 
 0.2336
(33)
Obviously, sufficient and necessary conditions for exact linearization of (12) can be
satisfied, by reference to Section 2.1.
Then, the smooth m-dimension vector function h( x) is introduced, and
Lb h  0
Lb ( L f h)  0
(34)
such that
h
b0
x
(35)
Let
h
x
(36)
 (x)b  0
(37)
 ( x) 
where  (x) is a 1  2 function matrix, so
If h( x) is smooth sufficiently, then h( x) has second partial derivatives and they
are equal, namely
2h
2h
=
x1x2 x2x1
(38)
1 ( x)  2 ( x)

x2
x1
(39)
then
If  ( x)  [1 ( x), 2 ( x)] exists and satisfies (37) and (39), then h( x) can be solved
using integration method by plugging  (x) into (36). This paper gets a solution
of h( x) , that is
h  0.02728448x12  0.029257805x22  0.05650784x1 x2
(40)
then
L f h  0.75089085x13  0.57861754x12 x2  0.25787492x12
 0.1110768x1 x22  0.48575962x1 x2  0.00200777x23  0.10310491x22
(41)
Lb ( L f h)  0.40975643x12  0.33183025 x1 x2
 0.011286438x1  0.028276523x22  0.16567587x2
(42)
L2f h  28.015720351x14  13.50915992x13 x2  17.1383344x13
 4.15397205x12 x22  15.51057289x12 x2  2.42288998x12
 0.93508148x1 x23  0.55187985x1 x22  4.32987154x1 x2
(43)
 0.10505348x24  0.4174377213x23  0.04102770x22
Ld ( L f h)   g (t )(0.40975643x12  0.33183025 x1 x2
 0.011286438x1  0.028276523x22  0.16567587x2 )
(44)
By the method proposed in Section 3, E ( x)  Lb (L f h) , E ( x) is reversible，
P( x)  L2f h , Q( x)  Ld ( L f h) . Then the nonlinear state transformation
x  [h, L f h]T
(45)
v  Lb (L f h) u  L2f h  Ld ( L f h)
(46)
and the input transformation
are applied. So the original system x  f ( x)  Bu  d (t ) is transformed into
x  Ax  Bv
(47)
0 1 
0 
where, A  
and B    .

1 
0 0 
In (47), x1  x 2 , thus the switching control function can be designed as
s( x)  Cx  cx1  x2
(48)
where, C  [c,1] and c is a design parameter which ensures that p  c satisfies
Hurwitz stability ( p is Laplace operator). By Hurwitz criterion, the root of p  c  0
should possess a negative real part, namely c  0 . Plunging x  [h, L f h]T into
s  s ( x ) , then
s( x)  s( x) |x [ h , L
f
h ]T
 ch( x)  L f h( x)
(49)
By Section 3, the reaching law is designed as
s  ks   sat ( s )
(50)
Because
s
s
x  C ( Ax  Bv)  CAx  CBv
x
(51)
and CB  1 which means CB is reversible, so
v  (CB) 1 (CAx  ks   sat ( s))
(52)
Plugging (52) into (46), the designed nonlinear sliding mode controller can be
represented as
u  [ Lb ( L f h)]1{(CB) 1 (CAx  ks   sat ( s))  [ L2f h  Ld ( L f h)]} (53)
Then, ABC algorithm is used to optimize the controller parameters to obtain
optimal performance. By Section 3, the objective performance function of ABC
algorithm is

J    w1 e1 (t )  w2 e2 (t ) dt
0
(54)
where, e1 (t ) and e2 (t ) are respectively the errors of x1 and x2 at time t . In
8.7379 0


system(8), y  3.3033 3.8025  x , so x1 has greater impact than
 2.1940 2.5749 
x2 on average,
hence this paper set w1  0.9 & w2  0.1 . In the simulation, the ranges of the design
parameters are defined around experiential values，which are listed in Table 1 and
parameter settings of ABC algorithm are shown in Table 2.
Table 1. Range of the design parameters
Parameters
Minimum
Maximum
c
0.5
5
k
3
30

10
60
Table 2.
Parameter settings of ABC algorithm
Colony size
10
Maximal cycle times
50
Maximal no-updated times
10
Then the ABC-based aeroengine nonlinear sliding mode control algorithm is firstly
simulated at a same initial condition ( x(0)  [0.5  1]T ) in Ref. [20] to be compared
with the controller designed by Gronwall-Bellman Lemma Approach. At
x(0)  [0.5  1]T , Figure 2 shows the convergence of the global optimal value in
ABC algorithm, obviously, the ABC algorithm converges very fast. In the end, the
global optimal value is 0.0107377 and the global optimal solution is
[c, k ,  ]  [3.5652,14.9458,50.6788] . Figure 3 and Figure 4 show the response of the
designed controller using ABC-based aeroengine nonlinear sliding mode control
algorithm. Because the sliding surface expressed in the original state vector x
( x  [ x1 , x2 ]T ) is s ( x)  ch( x)  L f h( x)  0 and the analytic solutions of this function
is not unique, this paper uses the sliding surface after state transformation
s( x)  cx1  x2  0 and response of x1 and x2 to draw phase diagram (see (1) in
Figure 3). The phase diagram and the s response (see (2) in Figure 3) show that the
system can reach the sliding surface in a short time. The u response shows that the
designed controller can resist disturbance well (see (3) in Figure 3). The state
response (see (4) in Figure 3) and the output response (see Figure 4) show that the
ABC-based aeroengine nonlinear sliding mode control algorithm has faster responses
than the Gronwall-Bellman Lemma Approach in Ref. [20] (see Figure 5 and Figure 6).
0.0135
BestJ
0.013
BestJ
0.0125
0.012
0.0115
0.011
0.0105
Figure 2:
0
5
10
15
20
25
Cycles
30
35
40
45
50
J functions of Artificial Bee Colony algorithm at x(0)  [0.5  1]T
0.4
0.3
~
x(0) =[-0.0507,0.3899]
T
phase locus
sliding surface
0.3
s response
0.25
0.2
0.2
~
x
s
2
0.15
0.1
0.1
0.05
0
0
-0.1
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
-0.05
0.01
0
0.5
1
Time(s)
~
x
1
1.5
2
(2)
(1)
600
0.4
ΔW f response
500
0.2
Disturbance
0
300
-0.2
u
x(t)
400
200
-0.4
100
-0.6
0
-0.8
-100
0
0.5
1
Time(s)
1.5
2
-1
x 1 with ABC-SMC
x 2 with ABC-SMC
0
0.5
1
Time(s)
(3)
1.5
2
(4)
Figure 3: Response at x(0)  [0.5  1]T using ABC-based aeroengine nonlinear sliding
mode control algorithm.
Output response with ABC-SMC
0
0.5
0
-0.5
-0.5
-1
0
-0.5
-1.5
-1
-1
-2
-2
P16/P56
-1.5
P56/P25
PCN2R
-1.5
-2.5
-3
-2
-2.5
-3.5
-2.5
-3
-4
-3.5
-3
-4.5
-4
-5
-3.5
-4.5
0
1
Time(s)
2
0
1
Time(s)
2
0
1
Time(s)
2
Figure 4: Output response at x(0)  [0.5  1]T using ABC-based aeroengine nonlinear
sliding mode control algorithm.
Figure 5: State response at x(0)  [0.5 1]T using Gronwall-Bellman Lemma
Approach in Ref. [20].
Figure 6: Output response at x(0)  [0.5 1]T using Gronwall-Bellman Lemma
Approach in Ref. [20].
In order to verify the adaptive ability in large deviation of the designed controller, a
simulation at another initial condition x(0)  [0.3 0.5]T is given. Finally, the global
optimal value is 0.0049289 and the global optimal solution is
[c, k ,  ]  [3.5064,14.8431, 42.04709] . The results show that the ABC-based
aeroengine nonlinear sliding mode control algorithm has a satisfying control effect as
well.
-3
5.6
x 10
BestJ
5.5
5.4
BestJ
5.3
5.2
5.1
5
4.9
4.8
Figure 7:
0
5
10
15
20
25
Cycles
30
35
40
45
50
J functions of Artificial Bee Colony algorithm at x(0)  [0.3 0.5]T
-3
0.07
15
~
x(0) =[-0.0133,0.0611]
0.06
T
s response
phase locus
sliding surface
0.05
x 10
10
~
x
s
2
0.04
5
0.03
0.02
0
0.01
0
-0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002
-5
0
0
0.5
~
x
1
1
Time(s)
1.5
2
(2)
(1)
200
0.6
x 1 with ABC-SMC
0.5
x 2 with ABC-SMC
0
0.4
-200
x(t)
u
ΔW f response
Disturbance
0.3
0.2
-400
0.1
-600
-800
0
0
0.5
1
Time(s)
1.5
2
-0.1
0
0.5
(3)
1
Time(s)
1.5
2
(4)
Figure 8: Response at x(0)  [0.3 0.5]T using ABC-based aeroengine nonlinear
sliding mode control algorithm.
Output response with ABC-SMC
3
1.8
2.5
2.5
1.6
2
1.4
2
1
P16/P56
P56/P25
PCN2R
1.2
1.5
1.5
1
1
0.8
0.6
0.5
0.4
0.5
0
0.2
-0.5
0
1
Time(s)
2
0
1
Time(s)
2
0
1
Time(s)
2
Figure 9: Output response at x(0)  [0.3 0.5]T using ABC-based aeroengine
nonlinear sliding mode control algorithm.
5 Conclusions
In this paper, a novel design approach of aeroengine nonlinear sliding mode
controller based on Artificial Bee Colony (ABC) algorithm is presented. The designed
nonlinear controller fully utilizes the advantages of sliding mode control strategy and
the optimization capacity of Artificial Bee Colony algorithm. The ABC based
aeroengine nonlinear sliding mode controller can implement optimal control
performance for the aeroengine. Simulation results show that the system has quick
responses, good anti-interference ability and strong robustness, and can obtain optimal
performance due to the designed control algorithm.
References
[1] Wang, C., Li, D., Li, Z., & Jiang, X. (2009). Optimization of Controllers for Gas
Turbine Based on Probabilistic Robustness. Journal of Engineering for Gas
Turbines and Power, 131(5), 054502.
[2] Brown, H., & Elgin, J. A. (1985). Aircraft engine control mode analysis.Journal
of Engineering for Gas Turbines and Power, 107(4), 838-844.
[3] Liang B B, Jiang J, Zhen Z Y, et al. Improved Shuffled Frog Leaping Algorithm
Optimizing Integral Separated PID Control for Unmanned Hypersonic Vehicle[J].
Transactions of Nanjing University of Aeronautics and Astronautics, 2015, 32(1):
110-114.
[4] LIU, J. X., LI, Y. H., CHEN, Y. G., & WANG, C. (2004). Aeroengine LQR
Control Based on Fuzzy-Neural Networks [J]. Journal of Aerospace Power, 6,
019.
[5] Härefors, M. (1997). Application of h∞ robust control to the RM12 jet
engine. Control Engineering Practice, 5(9), 1189-1201.
[6] Pu H Z, Zhen Z Y, Xia M. Flight Control System of Unmanned Aerial Vehicle[J].
Transactions of Nanjing University of Aeronautics and Astronautics, 2015, 32(1):
1-8.
[7] Frederick, D. K., Garg, S., & Adibhatla, S. (2000). Turbofan engine control
design using robust multivariable control technologies. Control Systems
Technology, IEEE Transactions on, 8(6), 961-970.
[8] Zhen Z Y, Pu H Z, Chen Q, et all. Nonlinear Intelligent Flight Control for
Quadrotor Unmanned Helicopter[J]. Transactions of Nanjing University of
Aeronautics and Astronautics, 2015, 32(1): 29-34.
[9] Improving control allocation accuracy for nonlinear aircraft dynamics[M].
Defense Technical Information Center, 2002.
[10] Blaquiere A. Nonlinear system analysis[M]. Elsevier, 2012.
[11] Benelghali, S., El Hachemi Benbouzid, M., Charpentier, J. F., Ahmed-Ali, T., &
Munteanu, I. (2011). Experimental validation of a marine current turbine
simulator: Application to a permanent magnet synchronous generator-based
system second-order sliding mode control. Industrial Electronics, IEEE
Transactions on, 58(1), 118-126.
[12] Panda, S., & Bandyopadhyay, B. (2008). Sliding mode control of gas turbines
using multirate-output feedback. Journal of Engineering for Gas Turbines and
Power, 130(3), 034501.
[13] Wu, L., & Ho, D. W. (2010). Sliding mode control of singular stochastic hybrid
systems. Automatica, 46(4), 779-783.
[14] Karaboga, D., Gorkemli, B., Ozturk, C., & Karaboga, N. (2014). A
comprehensive
survey:
artificial
bee
colony
(ABC)
applications.Artificial Intelligence Review, 42(1), 21-57.
algorithm
and
[15] Brunell, B. J., Bitmead, R. R., & Connolly, A. J. (2002, December). Nonlinear
model predictive control of an aircraft gas turbine engine. In Decision and
Control, 2002, Proceedings of the 41st IEEE Conference on (Vol. 4, pp.
4649-4651). IEEE.
[16] ZHANG, H. B., SUN, J. G., & SUN, F. C. (2007). Applications of nonlinear
backstepping control strategy in multivariable control of aeroengine [J].Journal of
Aerospace Power, 7, 027.
[17] Yao, Y. L., & Sun, J. G. (2008). Aeroengine Direct Thrust Control Based on
Neural Network Inverse Control. J. Propul. Technol, 29(2), 249-252.
[18] Cai, K., Yu, K., & Lv, B. (2008, June). Multi-variable neural network adaptive
control for aeroengine. In Intelligent Control and Automation, 2008. WCICA
2008. 7th World Congress on (pp. 8687-8692). IEEE.
[19] Luo, G. Q., Liu, B., & Song, D. Y. (2011). Hybrid particle swarm optimization in
solving aero-engine nonlinear mathematical model. Gas Turbine Exp Res,24(2),
5-8.
[20] Wang J, Ye Z, Hu Z. Nonlinear Control of Aircraft Engines Using a Generalized
Gronwall-Bellman Lemma Approach[J]. Journal of Engineering for Gas Turbines
and Power, 2012, 134(9): 094502.
[21] Zhong S, Dong Y F, Sarosh A. Artificial Bee Colony Algorithm for Parametric
Optimization of Spacecraft Attitude Tracking Controller[M]//Foundations and
Practical Applications of Cognitive Systems and Information Processing.
Springer Berlin Heidelberg, 2014: 501-510.
[22] HU, Z. H., & ZHAO, M. (2010). Research on path planning of UAV based on
ABC algorithm. Transducer and Microsystem Technologies, 3, 012.
[23]
基于人工蜂群算法的航空发动机非线性滑模控制
卢彬彬，肖玲斐
(南京航空航天大学 能源与动力学院 江苏省航空动力系统重点实验室南京 210016)
摘要：针对一类航空发动机非线性系统，提出了一种新颖的、基于人工蜂群算法的航
空发动机非线性滑模控制器设计方法。针对航空发动机的强非线性和不确定性，采用滑模控
制策略设计控制器。基于精确线性化理论，得到非线性的滑模变结构控制器。通过人工蜂群
算法整定非线性滑模控制器中的设计参数，以得到最优的控制性能。对某型航空发动机进行
仿真验证，结果表明，闭环系统有满意的动态性能以及高度的稳定性，验证了所提方法的有
效性。
关键词： 航空发动机；非线性控制； 滑模控制；人工蜂群算法