Adaptive sliding mode controller for second order nonlinear system based on RBF neural network

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.

Nghiên cứu khoa học công nghệ ADAPTIVE 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 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 Tạp chí Nghiên cứu KH&CN quân sự, Số 48, 04 - 2017 79 Kỹ thuật điều khiển & Điện tử PROBLEM ESTABLISHMENT 2.1 RBF neural network A normal RBF neural network is constructed by three layers as follow (Fig 1) Hidden layer Input layer x1 Output layer w1 w2 w3 xN wL N L Figure RBF neural network Input layer: Is also the input vector and can be described as follow x  [ x1, x2 , , xN ]T (1) Hidden layer: Is the second layer and has the conversion function as follow  i ( x)  exp( ( x  c2i ) ), 2bi i  1,2, , L (2) where x is the input vector, ci is the eccentricity of Gaussian distribution, bi is 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 L y   wi i ( x)   (3) i 0 2.2 Second order nonlinear system The dynamic equation of second order nonlinear system is described as follow  x  f ( x, x )  mu (4) 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 x1  x2 x2  f (x)  mu y  x1 80 (5) N Đ Tiến, “Adaptive sliding mode controller for… based on RBF neural network.” Nghiên cứu khoa học công nghệ where x  [x1 , x2 ]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 yd in such a condition that we don’t know the value of f (x) and m 2.3 Controller design Set e(t )  yd (t )  y(t ) is the error vector of the controller To establish the sliding controller, firstly, we have to design the sliding surface s  e  1e (6) where 1  diag(11 ,12 , ,1n ) , with 1i  , i  1,2, n and s  [s1, s2 , , sn ]T , e  [e1, e2 , , en ]T It is clearly to see that s  is a stable sliding surface e  , e  as t   According to the process of designing sliding mode controller, the control signal contains below components   ueq  uSW  ueq  K SW sign(s) (7) where K SW  diag (kSW 1, kSW , , kSWn ) and kSW , kSW , kSWn are positive values The equivalent control signal ueq can be calculated when s  ueq   f (x)   yd  K T E (8) m   T kd  and kp , kd are positive values, E   e 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 where K  k p fˆ (x)  Wˆ T h(x) (9) where fˆ (x) is the approximate function of f (x) , Wˆ T is adaptive weighted estimator, h(x) is the output of hidden layer Adaptive parameters are calculated as follow  Wˆ   ET PBh(x)  T   E PBu  1  mˆ   E T PBu  1   if (10) E T PBu 0 if E T PBu 0 & mˆ  m (11) if E T PBu 0 & mˆ  m Tạp chí Nghiên cứu KH&CN quân sự, Số 48, 04 - 2017 81 Kỹ thuật điều khiển & Điện tử T where B   1 P is the positive symmetric matrix which satisfies Lyapunov equation T P  P  Q (12)    k p (13) where Q  and  kd   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 x1  x2 x2  f (x)  mu (14) where x1 and x2 are the location and velocity, u is the control signal, f (x)  25 x2  10 x1 , 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 x   x1 T x2  The eccentricity of Gaussian distribution is chosen as  1 0.5 0.5 1 and the variance is chosen as b2 500  The modeled parameters are Q    , k p  30 , kd  50 ,   1200 ,  500    0.0001 , mˆ (0)  120 The designed trajectory is chosen as yd  sin(t) (15) The initialization values of second order nonlinear system are  0, 50, 0 82 N Đ Tiến, “Adaptive sliding mode controller for… based on RBF neural network.” Nghiên cứu khoa học công nghệ 3.2 Discussion Figure Position and speed tracking fx Figure Control input Figure Estimation of f (x) and m Tạp chí Nghiên cứu KH&CN quân sự, Số 48, 04 - 2017 83 Kỹ thuật điều khiển & Điện tử The 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 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 [1] A Isidori, “Nonlinear Control System”, 2nd Edition, Springer, Berlin, 1989 [2] M Krstic, I Kanellakopoulos, P Kokotovic, “Nonlinear and Adaptive Control Design”, Wiley, New York, 1995 [3] R Marino, P Tomei, “Nonlinear Adaptive Design : Geometric, Adaptive, and Robust”, Prentice-Hall International (UK) Limited, London, 1995 [4] A.R Teel, R.R Kadiyala, P.V Kokotovic, S.S Sastry, “Indirect techniques for adaptive input-output linearization of non-linear systems”, Internat J Control 53 (1991) 193–222 [5] T.A Johansen, P.A Ioannou, “Robust adaptive control of minimum phase non-linear systems”, Int J Adaptive Control Signal Process 10 (1996) 61–78 [6] R Marino, P Tomei, “Robust adaptive state-feedback tracking for nonlinear systems”, IEEE Trans Automat Control 43 (1) (1998) 84–89 [7] M.M Polycarpou, P.A Ioannou, A robust adaptive nonlinear control design, Automatica 32 (1996) 423–427 [8] C K Lin.: “Nonsingular terminal sliding mode control of robot manipulator using fuzzy wavelet networks” IEEE Trans Fuzzy Syst Vol 14, No 6, 2006 [9] L Wang, T Chai, and L Zhai.: “Neural network-based terminal sliding mode control of robotic manipulators including actuator dynamics IEEE Trans” Ind Electron Vol 56, No 9, 2009 [10] Y Jiang, Q Wang, and C Dong.: “A reaching law neural network terminal sliding mode guidance law design” 2013 IEEE Region 10 Conference 84 N Đ Tiến, “Adaptive sliding mode controller for… based on RBF neural network.” Nghiên cứu khoa học cơng nghệ TĨM TẮT THIẾT KẾ BỘ ĐIỀU KHIỂN TRƯỢT THÍCH NGHI CHO HỆ PHI TUYẾN BẬC HAI DỰA TRÊN MẠNG RBF NEURON Trong báo đề xuất phương pháp điều khiển cho hệ phi tuyến bậc hai dựa điều khiển trượt, mạng RBF neuron điều khiển thích nghi Đầu tiên, điều khiển thiết kế dựa phương pháp điều khiển trượt Sau đó, mạng RBF neuron thích nghi thiết kế để giúp hệ thống ước lượng thông số chưa biết Thêm vào đó, để khắc phục tác động tượng dao động tần số cao điều khiển trượt truyền thống, phương pháp lớp ranh giới áp dụng Cuối cùng, kết mô chứng minh độ hiệu 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 ngày 24 tháng năm 2017 Hoàn thiện ngày 04 tháng năm 2017 Chấp nhận đăng ngày 05 tháng năm 2017 Address: People's Police University of Technique and Logistics, Ministry of Public Security * Email: dangtient36@gmail.com Tạp chí Nghiên cứu KH&CN quân sự, Số 48, 04 - 2017 85 ... initialization values of second order nonlinear system are  0, 50, 0 82 N Đ Tiến, Adaptive sliding mode controller for based on RBF neural network. ” Nghiên cứu khoa học công nghệ 3.2 Discussion Figure... 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,... “A reaching law neural network terminal sliding mode guidance law design” 2013 IEEE Region 10 Conference 84 N Đ Tiến, Adaptive sliding mode controller for based on RBF neural network. ” Nghiên