MINISTRY OF EDUCATION AND TRAINING MINISTRY OF TRANSPORT HO CHI MINH CITY UNIVERSITY OF TRANSPORT PHAM THANH TUNG RESEARCH AND DEVELOPMENT OF OPTIMAL NEURAL NETWORKS’ STRUCTURE APPLIED IN NONLINEAR SYSTEM CONTROL Major: Automation and Control Engineering Code: 9520216 SUMMARY OF THE THESIS TP.HCM – 2019 The work was completed at Ho Chi Minh City University of Transport Instructor 1: Assoc Prof Dr Đong Van Huong Instructor 2: Assoc Prof Dr Nguyen Chi Ngon Independent reviewer 1: Assoc Prof Dr Hoang Duc Tuan Independent reviewer 2: Dr Hoang Minh Tri Reviewer 1: Reviewer 2: Reviewer 3: The dissertation will be protected before the Dissertation Marking Council meets at Ho Chi Minh City Transport University At …… hours …… day …… month …… year 2019 The thesis can be found at the library - National Library of Vietnam - Library of Ho Chi Minh City University of Transport CHAPTER 1: INTRODUCTION 1.1 Problem Sliding mode control (SMC) is an effective approach to control of nonlinear systems with the remarkable characteristic is the robust stability with disturbance or variable model parameters of the system [6, 21] and quick response [44] However, sliding mode control signal exists in the chattering phenomenon of the phase trajectory around the sliding surface [56, 98] To improve control quality, studies [3, 34, 58, 61] proposed an adaptive sliding controller; [51] adaptive sliding mode control with neural network; [100] backstepping sliding mode control; [17] adaptive backstepping sliding mode control; [56, 107] adaptive integral sliding backstepping control The simulation results show that proposed methods eliminate chattering, improved sustainability, less errors and faster convergence In the above methods, neural networks emerge as adaptive controllers, contributing to improving the control quality of the sliding controller However, the difficulty in training RBF networks is to select the appropriate number of hidden layer’s neurons, centers, thresholds, and connection weights [60, 67, 104, 109] In addition, neural network training algorithms should also be considered to enhance network performance, in which Gradient Descent algorithm [10, 24, 29, 51] is often used However, this algorithm is limited such as: slow convergence speed, easy to fall into the local minimum and the ability to search the whole world inefficient [22, 37] This study proposes using genetic algorithms to optimize the structure of neural networks After optimization, neural networks are trained online by Quasi-Newton algorithm due to the feedback from output signals of the plant The neural networks are considered as an adaptive controller The proposed controller is used to control the nonlinear system The simulation results are done on MATLAB / SIMULINK 1.2 Limitations of the thesis The thesis focuses on researching MIMO nonlinear system - Omnidirectional mobile robot (OMR) and developing optimal algorithm of neural network structure - RBF (Radial Basic Function) to control trajectory tracking of the plant to improve control quality 1.3 The objective of the thesis 1.3.1 Overall objectives Optimizing the structure of the RBF neural network to control the trajectory tracking of nonlinear system – Omni-directional mobile robot, to improve the quality of system control 1.3.2 Detail objectives - Sliding mode control is designed to control the trajectory tracking of nonlinear system - The neural network online training algorithm is built to approximate the nonlinear functions in the sliding control law - The adaptive sliding mode control with neural networks is simulated to control nonlinear object to improve the control quality of the system - Suitable neural networks structure is evaluated and selected in nonlinear system control - Genetic algorithms are studied to optimize neural networks’ structure - The adaptive sliding controller with optimized neural network by genetic algorithm is simulated to control nonlinear system - The proposed algorithm is applied in Omni-directional mobile robot trajectory tracking control in the nominal state; in the presence of noise and changes in the parameters of the object 1.4 Research methods, approaches - Researching documents: collecting; analyze, synthesize documents, identify advantages for scientific basis for the thesis, and improvements in the existing in those documents - Experimental mathematical model of nonlinear system (Omnidirectional mobile robot) on MATLAB / SIMULINK - Information processing: observe the system's response and adjust the parameters of the controller (if any) so as to meet the control quality criteria 1.5 Object and scope of the study - Research subjects: MIMO nonlinear system is described by state equation - Research scope: Math description method for MIMO nonlinear system (Omni-directional mobile robot); adaptie control; neural network and genetic algorithm 1.6 Scientific and practical significance 1.6.1 Scientific significance The study proposes an optimal algorithm for artificial neural network structure to control the nonlinear system trajectory tracking, the responses of the system converge to references without steady-state error and less affected by noise 1.6.2 Practical significance The optimal algorithm is verified the practical application to control nonlinear systems proposed by soft tools 1.7 The contributions of the thesis 1.7.1 Theory - Nonlinear matrix online estimate algorithm is built by Quasi-Newton algorithm due to the feedback from output signals of the system - Online optimal algorithm for neural network structure is built by genetic algorithm 1.7.2 Practice - The adaptive sliding mode control with neural network is simulated to control nonlinear system using Quasi-Newton algorithm to achieve good quality criteria - The qualities of the sliding mode control with Quasi-Newton algorithm are improved compared to traditional Gradient Descent algorithm - The optimal results of neural network structure using genetic algorithm are applied to control nonlinear systems to achieve more quality targets than the randomly generated neural network structure 1.8 The structure of thesis Chapter is introduction; Chapter presents the adaptive sliding mode control with neural network; the effect of the Quasi-Newton algorithm in nonlinear system trajectory tracking control is assessed as Chapter 3; Chapter presents the optimal method of neural network structure using genetic algorithms and Chapter is the results, conclusions and recommendations CHAPTER 2: THE ADAPTIVE SLIDING MODE CONTROL WITH NEURAL NETWORKS 2.1 Introduction This chapter presents a method to design the adaptive sliding mode control using the neural network (RBF: Radial Basic Function) to control MIMO nonlinear system trajectory tracking (Omni-directional mobile robot) The sliding mode control is designed to ensure the actual trajectory of the robot reaching a reference trajectory The RBF neural network is considered as an adaptive controller that is trained online by Gradient Descent algorithm The simulation results are compared with the traditional sliding mode control law through the achieved quality criteria of the system 2.2 The objective - Nonlinear object control system structure is constructed using adaptive sliding controller - Sliding mode controller is designed to control nonlinear system - The Radial Basic Functions are used to estimate online nonlinear functions in the sliding mode control law - The proposed algorithm is applied in Omni-directional mobile robot trajectory tracking control 2.3 Modeling nonlinear objects (Omni-directional mobile robot) Omni-directional mobile robot has been known to perform by the development of special wheels [47] or movement mechanisms [47, 82] It is assumed that the absolute coordinate system Ow - XwYw is fixed on the plane and the moving coordinate system Om – XmYm is fixed on the c.g for the mobile robot as shown in Fig 2.1 Fig 2.1 Model of an Omni-directional mobile robot [47] Dynamic equation of the robot as (2.1): −a2d   xw   b1  xw   a1  y  = a  a1   yw  + b1  w  d w   0 a3  w   b2 = AW β + BW U + D f b1 b1 b2 2b1 cos    u1   D fx    2b1 sin   u2  +  D fy  b2   u3   D f   (2.1) where D f =  D fx D fy  a1  AW =  a2   T D f   are unknown system disturbances −a2 a1 0  b1   ; BW = b1  b2 a3  U = u1 a2 = − a2 = u2 b1 b1 b2 2b1 cos   2b1 sin   ; b2  u3  T 3I w ;  = − sin  − cos  ;  = sin  − cos  (3I w + 2Mr )  = cos  − sin  ;  = − cos  − sin  2.4 Design of the sliding mode control law for the Omni-directional mobile robot 2.4.1 Design of the sliding mode control law for the robot Sliding mode controller is presented to ensure the actual trajectory of the robot reaching a reference trajectory Sliding mode control law is defined as (2.2) [108]:  (  ) U = − BW −1 AW  +  d + ke +  sign ( S ) −1 W (2.2) * Prove that the B matrix is invertible We have: b1 − sin  − cos   BW =  b1 cos  − sin   b2  ( ( ) b ( sin  − cos ) ) b ( − cos  − sin  ) 1 b2 determinant of BW as (2.4): det ( BW ) = 3b12b2  kr krL where: b1 = ; b2 = (3I w + 2Mr ) (3I w + I v r ) 2b1 cos    2b1 sin   (2.3) b2   (2.4) Contains parameters of the robot such as the radius of the wheel (r); the moment inertia of the wheel around the driving shaft (Iv); the mass of the robot (M); Moment inertia of the wheel (Iw); Distance between any assembly and the center gravity of the robot (L) −1 This shows that the BW matrix is reversible and shows that the BW matrix exists Thus, there is a control rule (2.2) for robot Fig 2.2 The structure of sliding mode control for robot To prove the stability of the control law we need to prove that the sliding mode surfaces converge to according to Lyapunov Lyapunov function is defined as (2.5): T S S 0 Get the derivative of V, we have: V = S T − AW  −  sign ( S ) V= ( ( ) (2.5) ) = − S T  +  sign ( S )  2.4.2 The parameters of the system and simulation results Sliding mode controller is simulated to control an Omni-directional mobile robot to ensure the following contents: (1) selection of a sliding surface so as to achieve the desired system behavior, when the control system reaches the sliding surface; and (2) selection of a control law such that the existence of sliding mode can be guaranteed The parameters of the system and controller are respectively given in Table 2.1 and 2.2 Table 2.1 The parameters of the mobile robot [47] Notation Meaning Value Unit Iv Moment inertia of the mobile robot 11.25 kgm M Mass of the robot 9.4 kg L Distance between any assembly and the center gravity of the robot 0.178 m k Driving gain factor 0.448 c Viscous friction factor of the wheel 0.1889 kgm2 / s I Moment inertia of the wheel 0.02108 kgm r Radius of the wheel 0.0245 m 10 The simulation diagram of the QN-ASMC-RBF in MATLAB /SIMULINK to control Omni-directional mobile robot with simulation parameters of the robot still used as in Chapter ➢ Case 1: the input is unit step function Fig 3.2 Reponses xw, yw and w Fig 3.3 Errors xw, yw and w of of the QN-ASMC-RBF with the QN-ASMC-RBF with Step Step Table 3.1 The quality criteria of the QN-ASMC-RBF with Step Reponses QN-ASMC-RBF POT (%) exl (mm) tqđ (s) xw 1.5 0.2 yw 0.19 0.2 w 0.05 0.18 Table 3.2 The error performances between GD-ASMC-RBF and QNASMC-RBF Reponses QN-ASMC-RBF GD-ASMC-RBF ADD MSE RMSE ADD MSE RMSE xw 9.8×10-4 1.9×10-6 0.0014 2.2×10-4 9.4×10-8 3.0×10-4 yw 7.1×10-4 1.0×10-6 1.0×10-3 2.6×10-4 1.5×10-7 3.7×10-4 w 2.0×10-5 8.3×10-10 2.9×10-5 9.2×10-5 1.6×10-8 1.3×10-4 21 ➢ Case 2: the input is unit step function when the noise impacts at the output Fig 3.4 Reponses xw, yw Fig 3.5 Errors xw, yw and and w of the QN-ASMC- w of the QN-ASMC-RBF RBF with Step when the with Step when the noise noise impacts at the output impacts at the output The advantage of this controller is the convergence speed of BFGS method is faster than Gradient Descent method; the quality criteria of the adaptive sliding control with neural network are trained online by QuasiNewton algorithm with BFGS smaller than Gradient Descent algorithm in the same network structure; control rules adapts according to the operating conditions of the robot However, the parameters of neural networks, such as the number of hidden layers, thresholds and initialization weights are randomly selected This problem will be solved in Chapter to ensure the network structure is optimal 3.3.3 Investigate the influence of the RBF neural networks parameters on the control quality of the adaptive sliding mode control The author investigates the influence of the neural networks structure on the control quality of the adaptive sliding mode control with the number of hidden layers is 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19 The weight update result and the performances of this algorithm are presented in Table 3.4 and 3.5 22 CHAPTER 4: OPTIMIZING THE STRUCTURE OF NEURAL NETWORK USING GENETIC ALGORITHM 4.1 Introduction In this Chapter 4, the study optimizes the structure and the parameters of the RBF neural network by genetic algorithm First, genetic algorithm is used to determine the number of neurons in the hidden layer of the RBF neural network; then this result is used to find the best values of the centers, widths and initial weights After optimizing, the radial basis function neural network is online trained by Quasi - Newton algorithm in trajectory tracking control of the omni-directional mobile robot 4.2 Optimize the structure and the parameters of the neural network using genetic algorithm The overall diagram of the optimizing the structure and the parameters of neural networks using genetic algorithm is shown in Fig 4.1: Fig 4.1 Diagram of GA-ASMC-RBF Control Parameters for the complete GA algorithm are shown in Table 4.1 ➢ Using the network training error and the number of hidden neurons to determine the RBF neural networks’ corresponding fitness of the chromosomes Suppose the training error is 𝐸, the number of neurons in the hidden layer is 𝑠, and upper limit of the number of neurons in the hidden layer is 𝑠max 23 - Condition:  s  smax  −6  E  25  10 (m ) (4.1) - The fitness 𝐹 of GA is defined by (4.7) [90, 104]: s F =C−E ( m) smax (4.2) where, C (m2) is a constant selected through experimentation, smax is the maximum number of neurons in the hidden layer, we choose smax = 50 ➢ In each generation of GA, we use the formula (4.3) as the target function for the Quasi-Newtonian algorithm to update the parameters of neural networks [75, 104] (4.3) E = ( d −  w )2 Where, E is the Mean Square Error, βd is the desired input signal and βw is the output of the object Table 4.1 Parameters of the GA algorithm Optimal structure Optimal parameters the RBF network the RBF network Generation number 15 15 Population size 20 30 0.7 0.7 0.01 – 0.001 0.01 – 0.001 Parameters Frequency of hybridization Probability of mutation Selective Objective function Roulette s F =C−E smax 24 Roulette E = ( d −  w )2 The results of the RBF neural network using GA are shown in Fig 4.2 4.3 Fig 4.2 Diagram indicate the Fig 4.3 Diagram indicate the value of the objective function in value of the objective function in structure optimal RBF network weights optimal RBF network The number of hidden layer neurons and parameters of RBF neurons after optimization by genetic algorithms are presented in Table 4.2 Table 4.2 Structure and parameters of the RBF network using the GA algorithm Notation Value b 10.6022 1.76 −11.01  12.41 −0.47 −1.73 −12.08 9.89  −4.39 1.19 −13.67 −2.88 9.83 10.35 −13.91   −1.65 12.30 7.07 −9.59 3.02 −3.92 −12.82     6.25 5.15 −2.06 −2.69 −10.11 −2.20 3.98  13.58 4.00 −0.88 −3.90 −1.98 4.42 −1.46  4.81 0.77 −1.24 0.90 −1.78 1.58 −2.41 cij T1 T2 T3 −3.92 0.87 3.57 −2.82 −1.86 −1.48 −2.24 −2.00 −2.45 2.38 −1.16 −4.50 4.18 1.77 j From Table 4.2, it is observed that the genetic algorithm has determined the specific values of the hidden layer neuron, center, 25 threshold, and initial weight of the RBF network These values will use to training online the RBF network in the adaptive sliding mode control in Chapter The optimal results of the hidden layer neuron in this thesis is less than the results of studies [60, 75, 90, 104] Specifically, as Table 4.3: Table 4.3 Compare optimal results of the hidden layer neuron Studies The hidden layer neuron Min Gan et al [60] - 2012 31 Shifei Ding et al [90] -2012 32 Weikuan Jia et al [104] - 2014 28 P.S Mishra [75]- 2018 42 Thesis 4.3 Trajectory tracking control of the Omni-directional mobile robot with the RBF neural network is optimized by Genetic algorithm (GAASMC-RBF) The structure of the controller is described as Fig 4.4 Fig 4.4 The structure of the GA-ASMC-RBF controller 26 Fig 4.5 Detailed controller structure ➢ Case 1: the input is unit step function Fig 4.6 Responses xw, yw w Fig 4.7 Errors xw, yw w of of GA-ASMC-RBF with Sep GA-ASMC-RBF with Step Table 4.4 The quality criteria of GA-ASMC-RBF controller with Step Responses GA-ASMC-RBF POT (%) exl (mm) tqđ (s) xw 0.79 0.19 yw 0.19 0.17 w 1.21 2.3 0.10 27 Table 4.5 The error performances between GA-ASMC-RBF and QNASMC-RBF Responses QN-ASMC-RBF ADD MSE -4 xw 9.8×10 yw 7.1×10-4 w 2.0×10-5 1.9×10 GA-ASMC-RBF RMSE -6 ADD MSE -4 5.7×10 RMSE -8 2.4×10-4 0.0014 1.9×10 1.0×10-6 1.0×10-3 1.0×10-4 2.2×10-8 1.5×10-4 8.3×10-10 2.9×10-5 4.7×10-5 4.4×10-9 6.7×10-5 ➢ Case 2: the input is unit step function when the noise impacts at the output Fig 4.8 Responses of GA- Fig 4.9 Errors of GA- ASMC-RBF controller with ASMC-RBF controller with Step when the noise impacts Step when the noise impacts at the output at the output ➢ Case 3: the input is unit step function when the inertial moment of the robot decreases by 25% and 50% Table 4.6 The quality criteria of the GA-ASMC-RBF controller with Step when the inertial moment of the robot decreases by 25% and 50% The inertial moment of the The inertial moment of the robot decreases by 25% robot decreases by 50% Responses POT (%) exl (mm) tqđ (s) POT (%) exl (mm) tqđ (s) xw 0.74 1.0 0.19 0.34 7.8 0.19 yw 0.94 0.17 0.93 5.9 0.17 w 0.37 0.10 1.2 0.10 28 Fig 4.10 Response xw of GA- Fig 4.11 Response yw of GA- ASMC-RBF controller with Step the ASMC-RBF controller with inertial moment of the robot Step the inertial moment of the decreases by 25% and 50% robot decreases by 25% and 50% ➢ Case 4: the input is unit step function when the viscosity coefficient of the robot increases by 25% and 50% Fig 4.12 Response xw of the Fig 4.13 Response yw of the GA-ASMC-RBF controller GA-ASMC-RBF controller with with Step when the viscosity Step when the viscosity coefficient of the robot coefficient of the robot increases by 25% and 50% increases by 25% and 50% 29 Table 4.7 The quality criteria of the GA-ASMC-RBF controller with Step when the viscosity coefficient of the robot increases by 25% and 50% Responses The viscosity coefficient of The viscosity coefficient of the the robot increases by robot increases by 50% 25% POT (%) exl (mm) tqđ (s) POT (%) exl (mm) tqđ (s) xw 0.71 2.9 0.19 0.69 7.8 0.19 yw 0.28 6.1 0.17 0.95 5.9 0.17 w 0.34 0.10 0.31 1.2 0.10 ➢ Case 5: other inputs Fig 4.14 The simulation results with Circular trajectory of the GAASMC-RBF controller Fig 4.15 The simulation results with Astroid trajectory of the GAASMC-RBF controller 30 Fig 4.16 The simulation results with Rose trajectory of the GAASMC-RBF controller Fig 4.17 The simulation results with Eight curve trajectory of the GA-ASMC-RBF controller 31 CHAPTER 5: RESULTS, CONCLUSIONS AND RECOMMENDATIONS 5.1 Results After the implementation period, the thesis “research and development of optimal neural networks’ structure applied in nonlinear system control” has achieved the following results: - The study has proposed the adaptive sliding mode controller with neural networks which are trained online with Gradient Descent algorithm for Omni-directional mobile robot The simulation results are compared with traditional SMC controller, indicating that the control quality and noise resistance of GD-ASMC-RBF controller is better At the same time, the study has also verified the response of the GD-ASMC-RBF controller with different trajectories and the simulation results still show the superiority of this controller compared to the traditional SMC controller - The study has proposed using Quasi-Newton algorithm to training online the neural network which is applied to control the Omni-directional mobile robot The simulation results compared with the GD-ASMC-RBF controller show that the control quality and noise resistance of the QNASMC-RBF controller is better At the same time, the study used a manual method to investigate the influence of neural network structure on the control quality of the proposed controller - The study has applied the genetic algorithm to optimize the structure of the neural network The results of the RBF neural network using GA converge across 15 generations, the most convergent point being 0.0715 (for the number of neurons hidden layer optimization) and 0.0085 (for weight optimization) The number of neurons hidden layer and and the parameters of the neural network after optimization by genetic algorithm is presented in Table 4.2 These results are used in the GA-ASMC-RBF 32 controller to control the Omni-directional mobile robot trajectory The simulation results in MATLAB / SIMULINK show that the effectiveness of GA-ASMC-RBF controller compared to QN-ASMC-RBF controller with the steady-state error is 2.3 (mm); the overshoot is 0.07 (%) 5.2 Conclusions The thesis has implemented the following contents: - Sliding mode control is designed to control the trajectory tracking of nonlinear system - The neural network online training algorithm is built to approximate the nonlinear functions in the sliding control law - The adaptive sliding mode control with neural networks is simulated to control nonlinear object to improve the control quality of the system - Suitable neural networks structure is evaluated and selected in nonlinear system control - Genetic algorithms are studied to optimize neural network’s structure - The adaptive sliding controller with optimized neural network by genetic algorithm is simulated to control nonlinear system - The proposed algorithm is applied in Omni-directional mobile robot trajectory tracking control in the nominal state; in the presence of noise and changes in the parameters of the object 5.3 Recommendations Developed as DSP-based controller, microcontroller or IPC controller for nonlinear systems 33 LIST OF WORKS OF AUTHOR Journals [1] Pham Thanh Tung, Đong Van Huong, Chi-Ngon Nguyen (2016), “Identification of MIMO nonlinear system using radial basis function neural networks”, Journal of Transportation Science and Technology Ho Chi Minh City University of Transport, No 20-8/2016, pp 31-36 [2] Dinh Tu Nguyen, Chi Cuong Tran, Hoang Dang Le, Thanh Tung Pham, Chi-Ngon Nguyen (2018), “Training the RBF neural network – based adaptive sliding mode control by BFGS algorithm for Omni-directional mobile robot”, International Journal of Mechanical Engineering and Robotics Research, Vol 7, No 4, pp 367-373 [3] Thanh-Tung Pham, Chi-Ngon Nguyen, Ngo-Phong Nguyen, Huong Dong Van, Kieu-Mai Le Thi, Hoang-Tam Vo (2018) “Adaptive sliding mode control with RBF neural networks for Omni-directional mobile robot”, Journal of Technical Education Science Ho Chi Minh City University of Technology and Education, No.49, pp 80-87 [4] Pham Thanh Tung, Nguyen Đinh Tu, Le Thi Kieu Mai, Nguyen Hua Duy Khang, Đong Van Huong, Chi-Ngon Nguyen (2018), “Performance evaluation of the Quasi-Newton algorithm in adaptive sliding mode control using radial basis function neural networks”, Can Tho University Journal of Science, Vol 54, No 7A, pp 27-34 [5] Van Huong Dong, Thanh Tung Pham, Kieu Mai Le Thi, Chi-Ngon Nguyen and Chi Cuong Tran (2018), “Radial Basis Function Newral Network and Genetic Algorithm in Trajectory Tracking Control of The Omni-Directional Mobile Robot”, International Journal of Mechanical Engineering & Technology (IJMET) - Scopus Indexed, Volume:9, Issue:11, pp 670-683 Conferences [6] Nguyen Đinh Tu, Tran Chi Cuong, Le Hoang Đang, Pham Thanh Tung, Chi-Ngon Nguyen (2016), “Modeling and Control of Three Wheeled 34 Omni - Directional Mobile Robot”, The 8th Vietnam Conference on Mechtronics, pp 517-523 [7] Le Hoang Đang, Nguyen Đinh Tu, Tran Chi Cuong, Pham Thanh Tung, Chi-Ngon Nguyen (2016), “Control of 3-Wheels Omni-Directional Mobile Robot using Self-Tuning RBF-PD Controller”, The 8th Vietnam Conference on Mechtronics, pp 744-749 [8] Tung Thanh Pham, Đang Hoang Le, Chi-Ngon Nguyen, Tu Đinh Nguyen, Cuong Chi Tran (2017), “Optimizing the Structure of RBF Neural Network-Based Controller for Omni-directional Mobile Robot Control”, International Conference on Systems Science and Engineering, pp 313-318 [9] Tung Thanh Pham, Dong Van Huong, Chi-Ngon Nguyen, Thanh Le Minh (2017), “Online Training the Radial Basis Function Neural Network Based on Quasi-Newton Algorithm for Omni-Directional Mobile Robot Control”, Recent Advances in Electrical Engineering and Related Sciences: Theory and Application, Vol 465, Springer, pp 607-616 [10] Tung Thanh Pham, Dong Van Huong, Chi-Ngon Nguyen, Thanh Le Minh (2017), “Comparison of SMC and RBF-SMC on mobile robot control system”, Asia Maritime and Fisheries Universities Forum, pp 325339 [11] Pham Thanh Tung, Đong Van Huong, Chi-Ngon Nguyen, Le Thi Kieu Mai (2018), “The optimization the parameters of radial basis function using genetic algorithm in omni-directional mobile robot control”, The 4th Scientific and Technological Conference – Ho Chi Minh City University of Transport, pp 263-272 Measurement, Control and Automation [12] Le Hoang Đang, Nguyen Đinh Tu, Tran Chi Cuong, Pham Thanh Tung, Chi-Ngon Nguyen (2016), “Experiment on an Omni-Directional Mobile Robot using RBF-PD Supervisory Controller”, Special Issue on Measurement, Control and Automation, No 17, pp 51-55 35 ... neuron in this thesis is less than the results of studies [60, 75, 90, 104] Specifically, as Table 4.3: Table 4.3 Compare optimal results of the hidden layer neuron Studies The hidden layer neuron... Frequency of hybridization Probability of mutation Selective Objective function Roulette s F =C−E smax 24 Roulette E = ( d −  w )2 The results of the RBF neural network using GA are shown in Fig... structure optimal RBF network weights optimal RBF network The number of hidden layer neurons and parameters of RBF neurons after optimization by genetic algorithms are presented in Table 4.2 Table 4.2