Điều khiển trượt dùng mạng nơron hàm cơ sở xuyên tâm

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 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

problem identifying the system (II.1) from inputoutput data using a RBF model:
n

− X −Z
yˆ k +1 = ∑ θi e k i

2

/σi

(II.3)

i =1

where, yˆ k is the output of the model, X k is
defined as (II.2), n is the number of RBF
functions, θ i 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:

yˆ k +1 = ΦTkθˆk
(II.4)
T
where, θˆk = [θ1 ,θ 2 ,...,θ n ] is the linear weights
vector at time k, and

Φ k = e


2

− X k − Z1 / σ1

,..., e

− X k − Zn

2

/σ n




T

(II.5)

is the regressive vector.
To identify the linear weights, we can use the
stochastic approximation approach [8]:

θˆk +1 = θˆk + Fk +1Φ k [ yk +1 − ΦTkθˆk ]
with

Fk +1 =

αk
δ k + ΦTk Φ k

(II.6)

(II.7)

Here, α k is a stochastic approximation parameter
which satisfies:
∞

∑α

k

=∞,

∑α

p
k

< ∞ , p≥2 .

k =1
∞

lim sup(α k−1 − α k−−11 ) < ∞ , and

k →∞

(II.8)

k =1

δ k is the Tikhonov parameter which satisfies the
following conditions:
δ k → 0 + , δ k δ k +1 > 1 , δ k > 0

(II.9)

k →∞

III. SMC USING RBF NEURAL NETWORK
Consider a nonlinear system:

y k +1 = f ( X k )

(II.1)

X k = [ yk ,..., yk − n y +1 , uk ,..., uk − nu +1 ]T

(II.2)

where, yk is the output, uk is the input, nu and

n y are integers. In this paper, we consider the

Let a nonlinear system be defined as

 x ( n ) = f ( X ) + g ( X ).u

y = x

(III.1)


Here, 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:
f min ≤ f ≤ f max , 0 < g min ≤ g ≤ g max
(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.
S = x ( n −1) + an − 2 x ( n − 2) + ... + a1 x (1) + a0 x (III.3)
with a0, a1,…, an-2 are constans which the
specificity function of (III.3):
p n −1 + an − 2 p n − 2 + ... + a1 p + a0 = 0
(III.4)
has all roots which real of them are positive.
In this paper, regulation control is presented.
Consider a nonlinear system:

 x&1 = x2

 x&2 = f ( x) + g ( x).u

u=

−1  1


x2 + f ( X ) + k .sign( S ) 
g ( X ) τ


(III.13)

so that

S& = −k .sign( S )

(III.14)

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 bother in the upright position.

(III.5)

We can choose sliding surface:

S = x2 − ϕ ( x1 )

(III.6)
with x1 which satisfies the following conditions:
(III.7)
x&1 = ϕ ( x1 )

has root x1 → 0 when t → ∞

ϕ ( x1 )

and

x1 = 0

=0

x1, u1

x2, u2

(III.8)

We can choose:

1

ϕ ( x1 ) = − x1
τ

(III.9)

Fig.1: Physical structure of the double pendulum system

We rewrite (III.7) as:

x&1 +

1

τ

x1 = 0

(III.10)

This equation has root x1 = Ae − t τ → 0 and

x2 = − x1 = − Ae − t / τ → 0 when t → ∞ .
Sliding surface:

S=

1

τ

x1 + x2

1
1
S& = x&1 + x&2 = x2 + f ( X ) + g ( X ).u

τ

τ

We choose control law:

(III.11)
(III.12)

 x&1 = x3
&
 x2 = x4

1
 x&3 = f1 ( X ) + u1
J1


 x& = f ( X ) + 1 u
2
2
 4

J2

 y1 = x1
y = x
2
 2

(IV.1)


with
 M gr k r 2 
k r2
k r
1
− s  sin( x1 ) + s sin( x4 ) + s ( a − b )
 J1

4 J1 
4 J1
2 J1


f1 ( X ) = 

 M gr ks r 2 
k r2
k r
2
−

 sin( x2 ) + s sin( x3 ) − s ( a − b )
 J2

4J 2 
4J 2
2J 2

TABLE 1: THE PARAMETERS OF THE PLANT

f2 ( X ) = 

The pendulum end mass 1
The pendulum end mass 2
The moment of inertia 1
The moment of inertia 2
The spring constant of the
connecting sping
The natural length of the spring
The distance between the
pendulum hinges
The pendulum height
Gravitational accelecetion

M1
M2
J1
J2
ks

2 kg

2.5 kg
0.5 kg.m2
0.625 kg.m2
100 N/m

a
b

0.5 m
0.4 m

r
g

0.5 m
9.81 m/s2

Fig.3: The output

x1
x4

B. Nonlinear System Identification

RBF Model

fˆ1

RBF Model

fˆ2

x2

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.

x3

Fig.4: Structure of RBF network
x4

x3

x4m

x3m
x1
x1m

-x1m
-x4m

x2
-x2m

x2m
-x3m

Fig.5: Location of reference points of RBF model

The scaling factors σi of the RBF network are
x2 + x2
σ 1i = σ 1 = − 1m 4 m ,
i = 1, N
ln(.5)
(IV.2)
x22m + x32m
,
i = 1, N
σ 2i = σ 2 = −
ln(.5)
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:

θˆ1 =[4.9148 5.0498 4.9023 5.1254]T
θˆ =[-3.9742 -4.0372 -3.991 -3.9928]T
2

Fig.2: The input

Fig.7a: Signals y1 and y2 of ideal SMC system

Fig.6: The result of identification

Fig.7b: Signals y1 and y2 of SMC system using RBFNN

C. SMC using RBFNN

At the end of the identification, we have fˆ1 ( X )

and fˆ2 ( X ) . Control laws:

1

u1 = − J1  x3 + fˆ1 ( X ) + k1.sign( S1 ) 
τ 1

1

u2 = − J 2  x4 + fˆ2 ( X ) + k2 .sign( S 2 ) 
τ 2

With k1 > sup( f1 − fˆ1 )
k > sup( f − fˆ )
2

2

2

(IV.3)
(IV.4)
(IV.5)
(IV.6)

Fig.8a: Signals ẏ1 and ẏ2 of ideal SMC system

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.8b: Signals ẏ1 and ẏ2 of ideal SMC system using RBFNN


V. REFERENCES
Preriodicals:

Fig.9a: Sliding surfaces of ideal SMC system

[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.474479, 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

Fig.9b: Sliding surfaces of SMC system using RBFNN
[7] M.Önder Efe, “Variable Structure Systems Theory in
Training of Radial Basis Function Neurocontrollers – Part
II: Applications”, NIMIA-SC2001 – 2001 NATO Advanced

S1=0
S1>0
S1<0

S2>0

S2<0
S2=0

Fig.10a: Phase space of ideal SMC system

Fig.10b: Sliding surfaces of ideal SMC system using RBFNN

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.