In this paper, a novel control method for second order nonlinear system based on sliding mode controller, RBF neural network and adaptive control system is proposed. Firstly, a new controller is designed based on sliding mode control system.
Trang 1ADAPTIVE SLIDING MODE CONTROLLER FOR
SECOND ORDER NONLINEAR SYSTEM BASED ON RBF NEURAL NETWORK
Nguyen Dang Tien
Abstract: In this paper, a novel control method for second order nonlinear
system based on sliding mode controller, RBF neural network and adaptive control system is proposed Firstly, a new controller is designed based on sliding mode control system Then, a RBF neural network and an adaptive system are integrated
to estimate unknown parameters of the system Moreover, in order to deal with chattering phenomenon in conventional sliding controller, boundary layer method
is then applied Finally, the experimental results are presented to demonstrate the effectiveness of the proposed control method
Keywords: RBF neural network, Sliding mode controller, Adaptive controller, Chattering phenomenon,
Boundary layer method
1 PROBLEM STATEMENT
The nonlinear controller which uses linear regression method has been used extensively by researchers when the mathematic model of the object is known [1] There are many systems which have been successfully applied the adaptive controller to solve the problem of noise [2-4] However, adaptive controller is not the optimal solution for nonlinear system which is heavily affected by chattering phenomenon of noisy environment [5-7]
Neural network is one of the best solution for complicated systems which affected by noise due to the capability of learning and approximating any nonlinear formulas RBF neural is a special case of neural network with superior characteristics compare to traditional neural network such as: simple structure, fast learning algorithm and good approximation Moreover, RBF neural network
is not only able of eliminating local minimum, reduce the number of parameters
in the network but also make the initialization process become significantly easier Therefore, this network has been successfully applied in many types of controllers [8-10]
Recently, sliding mode controller is known to be the best methods for solving the problem of noise affected by surrounding environment The advantages of sliding mode control such as: stability in noisy environment, ability to respond quickly and good quality of control However, in this method, when the effect of noise is high, we have to apply a large gain in discrete control signal in order to make the system operating normally This creates the chatting phenomenon which can cause the damage for electronic equipment and make the system becomes unstable In this paper, to overcome all the above issues, we propose a new control method based on the integration of three components: adaptive controller, sliding mode controller and RBF neural network Firstly, the controller is designed based
on the model of sliding controller Then, to reduce the effect of noise, an adaptive compensator which uses RBF neural network is proposed Finally, boundary layer method is applied to solve the chattering phenomenon
Trang 22 PROBLEM ESTABLISHMENT 2.1 RBF neural network
A normal RBF neural network is constructed by three layers as follow (Fig 1)
Figure 1 RBF neural network
Input layer: Is also the input vector and can be described as follow
1 2
[ , , , ]T
N
Hidden layer: Is the second layer and has the conversion function as follow
2 2
( ) exp( ), 1,2, ,
2
i i
i
x c
b
where x is the input vector, c is the eccentricity of Gaussian distribution, i b is i
variance of Gaussian distribution and L is the number of neural in hidden layer
Output layer: Output layer of RBF neural network can be calculated by sum of
all weighted input signals
0
( )
L
i i i
y w x
2.2 Second order nonlinear system
The dynamic equation of second order nonlinear system is described as follow
( , )
x f x x mu
where f x x( , ) is an unknown nonlinear function, m is an unknown real number and u is the control signal
The dynamic equation (4) can be rewritten as follow
1 2
2 1 (x)
N
x
1
N
L
1
1
w
2
w
3
w
L
w
Input layer
Hidden layer
Output layer
Trang 3where x[ ,x x1 2]T
The aim of this paper is: designing a stable controller to make sure that the real value of nonlinear system y will be closed to the designed value y in such a d
condition that we don’t know the value of f(x) and m
2.3 Controller design
Set e t( )y t d( )y t( ) is the error vector of the controller To establish the sliding controller, firstly, we have to design the sliding surface
1
where 1diag( 11, 12, ,1n), with1i0 , i1,2, n and [ , , , ]1 2 T
n
s s s s ,
1 2
[ , , , ]T
n
e e e e It is clearly to see that s is a stable sliding surface 0 e , 0 0
e as t
According to the process of designing sliding mode controller, the control signal contains below components
( )
where K SW diag k( SW1,k SW2, ,k SWn) and k SW1,k SW2, k SWn are positive values The equivalent control signal u eq can be calculated when s 0
1 (x) y T
m
where K k p k d and T k k are positive values, p, d Ee e
It is obvious to see that the control signal contains the discrete signal (which causes the chattering phenomenon) Therefore, to overcome this problem, we propose a new controller which uses a RBF neural network to approximate f(x) ,
an adaptive controller to estimate the value of m The discrete signal is then
processed by boundary layer method
ˆ (x) ˆT (x)
where fˆ (x) is the approximate function of f(x), ˆW is adaptive weighted T
estimator, h(x) is the output of hidden layer
Adaptive parameters are calculated as follow
1
if 0 1
ˆ if 0 & ˆ 1
ˆ
if 0 &
T
(11)
Trang 4where B 0 1T
P is the positive symmetric matrix which satisfies Lyapunov equation
where Q 0 and
(13)
3 EXPERIMENTAL RESULT AND DISCUSSION 3.1 Experiment tools
To prove the effectiveness of the proposed controller, the experiment is implemented in MatLab-Simulink software
The dynamic equation of second order nonlinear system is modeled as follow
1 2
2 (x)
where x and 1 x are the location and velocity, u is the control signal, 2
f x x , m 133
The structure of RBF neural network used in this paper is 2-5-1 (two output neurons, five hidden neurons and one input neuron) This structure can be chosen based on the type of mathematic problems If the problem has many outputs and inputs (multi-output and multi-output), the number of neurons of input and output layers can be larger In addition, the approximation process will be more accurate
if the hidden layer has more neurons However, the more neurons we put, the more computational time the system needs Therefore, in a specific problem, these parameters have to be chosen so that there is a balance between the accuracy and the computational time
The input signal of the network is xx1 x2T The eccentricity of Gaussian distribution is chosen as 1 0.5 0 0.5 1 and the variance is chosen as 2
b
Q
, k p 30, k d 50, 1200,
0.0001
, mˆ (0) 120
The designed trajectory is chosen as
sin(t)
d
The initialization values of second order nonlinear system are 0, 50, 0
Trang 53.2 Discussion
Figure 2 Position and speed tracking
Figure 3 Control input
Figure 4 Estimation of f(x) and m
Trang 6The results are shown in Figs (2-4) We can clearly see that the proposed controller brings good result with small signal sticking time and low error Moreover, the control signal (see Fig 3) has eliminated the effect of chattering phenomenon and unwanted signals This proves the stability of the proposed system Fig (4) shows that RBF neural network has good approximation of function f(x) This indicates that the quality of the controller also has been improved
4 CONCLUSION
In this paper, we propose a new control method for second order nonlinear system with unknown parameters Based on the sliding mode controller, a RBF neural network is designed to estimate the effect of noise In addition, to overcome the chattering phenomenon in conventional sliding controller, a boundary layer method has been applied The experimental results have shown the effectiveness of the proposed control system
REFERENCES
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Trang 7TÓM TẮT
THIẾT KẾ BỘ ĐIỀU KHIỂN TRƯỢT THÍCH NGHI CHO HỆ PHI TUYẾN
BẬC HAI DỰA TRÊN MẠNG RBF NEURON
Trong bài báo này chúng tôi đề xuất một phương pháp điều khiển mới cho hệ phi tuyến bậc hai dựa trên bộ điều khiển trượt, mạng RBF neuron
và điều khiển thích nghi Đầu tiên, bộ điều khiển được thiết kế dựa trên phương pháp điều khiển trượt Sau đó, một mạng RBF neuron và một bộ thích nghi được thiết kế để giúp hệ thống ước lượng được các thông số chưa biết Thêm vào đó, để khắc phục tác động của hiện tượng dao động tần số cao trong bộ điều khiển trượt truyền thống, phương pháp lớp ranh giới đã được áp dụng Cuối cùng, kết quả mô phỏng đã chứng minh được
độ hiệu quả của phương pháp
Từ khóa: Mạng nơ ron RBF, Điều khiển trượt, Điều khiển thích nghi, Dao động tần số cao, Phương pháp lớp
ranh giới
Nhận bài ngày 24 tháng 3 năm 2017 Hoàn thiện ngày 04 tháng 4 năm 2017 Chấp nhận đăng ngày 05 tháng 4 năm 2017
Address: People's Police University of Technique and Logistics, Ministry of Public Security
*
Email: dangtient36@gmail.com