SLIDING MODE CONTROL USING RADIAL BASIS FUNCTION NEURAL NETWORKS Tran Quang Thuan, Lecturer, PTIT-HCM, Duong Hoai Nghia, Lecturer, HCM City University of Technology, and Dong Si Thien C
Trang 1SLIDING MODE CONTROL USING RADIAL BASIS FUNCTION
NEURAL NETWORKS
Tran Quang Thuan, Lecturer, PTIT-HCM, Duong Hoai Nghia, Lecturer, HCM City University of
Technology, and Dong Si Thien Chau, Lecturer, Ton Duc Thang University
Abstract This paper considers the problem sliding
mode control of nonlinear dynamical system using
radial basis function neural networks This paper
presents sliding mode control (SMC) approach,
nonlinear system identification using radial basis
function (RBF) neural networks approach The
application of the nonlinear system identification
results to design sliding mode control law for a
nonlinear dynamic plant will be discussed in this paper
Simulation results are presented
I INTRODUCTION
Earliest notion of sliding mode control strategy
was constructed on a second order system in the
late 1960s by Emelyanov [5] Since then, the
theory has greatly been improved by researchers
To design a sliding mode control system, we must
have exactly model of the plant To the concrete,
designers not always have exactly model of the
plant To decide this problem, we propose to
identify model of the control plant base on RBF
neural network In comparison with traditional
nonlinear identification technique, the designers
which use this method don’t need to determine the
structure of the model The application of the
model to design sliding mode control law for a
nonlinear dynamic plant will be discussed in this
paper
The remainder of the paper is organized as
follows The second section presents nonlinear
system identification using radial basis function
neural networks SMC using RBF neural network
are presented in the third section The following
section describes the dynamic model of the plant
and the result of the simulations References
constitute the last part of the paper
II NONLINEAR SYSTEM IDENTIFICATION
USING RADIAL BASIS FUNCTION NEURAL
NETWORK
Consider a nonlinear system:
)
(
T n k k n k
k
X =[ , , − +1, , , − +1] (II.2)
where, yk is the output, ukis the input, nuand
n are integers In this paper, we consider the
problem identifying the system (II.1) from input-output data using a RBF model:
2
/ 1
1
n
i
+
=
where, yˆk is the output of the model, Xk is defined as (II.2), n is the number of RBF functions, θi are linear weights, Zi and σi
respectivery, the centers (reference points) and the scaling factor of the RBF [3], [4]
For identification purpose, we rewrite (II.3) as:
k T k k
1=Φ
where, θˆk =[θ1,θ2, ,θn]Tis the linear weights vector at time k, and
, ,
T
is the regressive vector
To identify the linear weights, we can use the stochastic approximation approach [8]:
] ˆ [
ˆ ˆ
1 1
T k k k k k
with
k T k k
k k
F
Φ Φ +
=
α
Here, αk is a stochastic approximation parameter which satisfies:
∞
=
∑∞
=1 k k
1
lim sup( k k )
−
∞
<
∑∞
=1 k
p k
k
δ is the Tikhonov parameter which satisfies the following conditions:
+
∞
→
→ 0
k k
δ , δ δk k+1> , 1 δk> 0 (II.9) III SMC USING RBF NEURAL NETWORK Let a nonlinear system be defined as
=
Trang 2Here, X =[x x& x(n−1)]T is state
vector, y is the output, u is control signal, n is
number of the state variables The
functions f = f X( ), g=g X( ) are not exactly
known, but the extent of the imprecision on f, g
are bounded by known:
max
f ≤ ≤ , 0<gmin ≤g ≤gmax (III.2)
Consider two of sliding mode control problems:
- Tracking control: the control problem is to get
feedback control law u = u(r, X) so that y→ r with
r is a reference signal
- Regulation control: the control problem is to get
feedback control law u = u(X) so that y → 0 when
t→ ∞
A time varying surface S(t) is defined in the state
space R(n) by equating the variable S(X;t), defined
below, to zero
n
S =x − +a − x − + +a x +a x (III.3)
with a0, a1,…, an-2 are constans which the
specificity function of (III.3):
0
2
2
−
−
a p a p
a
n
n
(III.4) has all roots which real of them are positive
In this paper, regulation control is presented
Consider a nonlinear system:
=
&
We can choose sliding surface:
) ( 1
x
with x1 which satisfies the following conditions:
) ( 1
has root x1→0 when t→∞
and
1
We can choose:
1 1
1 )
τ
We rewrite (III.7) as:
0
1
1
x
τ
This equation has root 1= − t τ →0
Ae
0
/ 1
2=−x =−Ae− t τ →
Sliding surface:
1
τ
We choose control law:
2
( )
so that
where, k is positive constant
IV SIMULATION
A Introducion about plant to control
In this study, a couple double pendulum systems, which are illustrated in Fig.1, is used to elaborate performance of the method discussed The differential equations characterizing the behavior
of the system are given in (IV.1), in which the angular positions x1 and x2, the angular velocities
x3 and x4 define the state vector The control inputs, which are denoted by u1 and u2, are provided to the relevant pendulum by the base servomotors The model introduced in this section has been studied by Spooner and Passino [1], Efe, Kaynak, Yu and Wilamowski [2], Efe [7] The parameters of the plant are given in Table 1, which states that as b<a, the two pendulums each other in the upright position
1
2
( )
( )
1
1
J
J
=
=
=
=
&
&
&
&
(IV.1)
Fig.1: Physical structure of the double pendulum system
x 1 , u 1 x 2 , u 2
Trang 31
2
TABLE 1: THE PARAMETERS OF THE PLANT
The pendulum end mass 1 M 1 2 kg
The pendulum end mass 2 M 2 2.5 kg
The moment of inertia 1 J 1 0.5 kg.m2
The moment of inertia 2 J 2 0.625 kg.m 2
The spring constant of the
connecting sping
k s 100 N/m The natural length of the spring a 0.5 m
The distance between the
pendulum hinges
The pendulum height r 0.5 m
Gravitational accelecetion g 9.81 m/s2
B Nonlinear System Identification
To design sliding control law for couple double
pendulum system, we must identify equations
f1(X) and f2(X) in (IV.1) using RBF networks
which their structure were choose such as Fig.4
Location of references of RBF model is given in
Fig.5 The input and output data are given in Fig
2 and Fig.3
The scaling factors σi of the RBF network are
ln(.5)
ln(.5)
i
i
+
+
(IV.2)
where x1m, x2m, x3m, x4m are the maximum value
of inputs
With 90,000 data, αk = 0.0001, δk = 1/(k+2), the identification results are given in Fig.6 At the end
of the identification, we have:
1
ˆ
θ =[4.9148 5.0498 4.9023 5.1254]T
2
ˆ
θ =[-3.9742 -4.0372 -3.991 -3.9928]T
Fig.2: The input
Fig.3: The output
Fig.5: Location of reference points of RBF model
x 1
x 4
-x 1m
x 4m
-x 4m
x 1m
x 3
-x 2m
x 3m
-x 3m
x 2m
x 2
RBF Model
x 1
x4
ˆ 1 f
RBF Model
x 2
x 3
ˆ 2 f
Fig.4: Structure of RBF network
Trang 4Fig.8b: Signals ẏ and ẏ of ideal SMC system using RBFNN
C SMC using RBFNN
At the end of the identification, we have f X ˆ ( )1
and f X Cˆ ( )2 ontrol laws:
1
1
τ
2
1
τ
With k1>sup(f1− fˆ1) (IV.5)
2 sup( 2 ˆ2)
The simulation results are given in Fig.7,
Fig.8, Fig.9 and Fig.10 The parameters of
control system are time-constants τ1 = τ2 =
0.2 sec, positive constants k1 = k2 = 15 and initial
conditions [x1 x2 x3 x4]T = [-π/4 π/7 0 0]T Under
these conditions, phase orbit motions depicted in
Fig 10 are obtained The simulation results of
SMC system using RBFNN and ideal SMC
system, which use exactly f1(X) and f2(X), are
analogous
Fig.6: The result of identification
Fig.7a: Signals y1 and y 2 of ideal SMC system
Fig.7b: Signals y1 and y 2 of SMC system using RBFNN
Fig.8a: Signals ẏ1 and ẏ 2 of ideal SMC system
Trang 5V REFERENCES
Preriodicals:
[1] Jeffrey T.Spooner and M Passino, “Decentralized Adaptive Control of Nonlinear Systems Using Radial Basis Neural Networks”, IEEE Transactions on Automatic Control, vol.44, No.11, pp.2050-2057, 1999
[2] M.Önder Efe, Okayay Kaynak, Xinghuo Yu, Bogdan M.Wilamowski, “Sliding Mode Control of Nonlinear Systems Using Gaussian Radial Basis Function Neural Networks”, IEEE Transactions on Neural Networks,
pp.474-479, 2001
[3] Hui Peng, Tohru Ozaki, “A Parameter Optimization Methode for Radial Basis Function Type Models”, IEEE Transactions on Neural Networks, Vol.14, No.2 (2003), pp.432-438
[4] Chu Kiong Loo, Mandava Rajeswari, M.V.C Rao,
“Novel Direct and Self – Regulating Approaches to Determine Optimum Growing Multi-Expert Network Structure”, IEEE Transactions on Neural Networks , Vol.15, No.6 (2004), pp.1378-1395
Books:
[5] Emelyanov, S.V “Variable structure control sytems”, Moscow, Nauka, 1967
[6] H Khalil, “Nonlinear Systems”, 2002
Papers fom Conference Proceedings [7] M.Önder Efe, “Variable Structure Systems Theory in Training of Radial Basis Function Neurocontrollers – Part II: Applications”, NIMIA-SC2001 – 2001 NATO Advanced Study Institude on Neural Networks for Instrumentation, Measurement, and Related Industrial Applications: Study Cases Crema , Italy, 9-20 October 2001
[8] Dong Si Thien Chau and Duong Hoai Nghia, “Nonlinear System Identification Using Radial Basis Neural Networks”,
in Proc International Symposium on Electrical & Electronics Engineering 2005 – HCM City, Vietnam
Fig.9a: Sliding surfaces of ideal SMC system
Fig.10a: Phase space of ideal SMC system
S 1 =0
S 1 >0
S 1 <0
S 2 >0
S2<0
S 2 =0 Fig.9b: Sliding surfaces of SMC system using RBFNN
Fig.10b: Sliding surfaces of ideal SMC system using RBFNN