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

1

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

2

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.

3

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.

4

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.

5

- 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 1 is introduction; Chapter 2 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 4 presents the optimal method of neural network structure using

genetic algorithms and Chapter 5 is the results, conclusions and

recommendations.

6

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.

7

Fig. 2.1. Model of an Omni-directional mobile robot [47]

Dynamic equation of the robot as (2.1):

−a2d 0 xw b1 1

xw a1

y = a

a1

0 yw + b1 3

w 2 d

w 0

0

a3 w b2

= AW β + BW U + D f

b1 2

b1 4

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

0

T

D f are unknown system disturbances.

−a2

a1

0

0

b1 1

0 ; BW = b1 3

b2

a3

U = u1

a2 = 1 − a2 =

u2

b1 2

b1 4

b2

2b1 cos

2b1 sin ;

b2

u3

T

3I w

; 1 = − 3 sin − cos ; 2 = 3 sin − cos

(3I w + 2Mr 2 )

3 = 3 cos − sin ; 4 = − 3 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]:

8

(

)

U = − BW −1 AW + d + ke + sign ( S )

−1

W

(2.2)

* Prove that the B matrix is invertible

We have:

b1 − 3 sin − cos

BW = b1 3 cos − sin

b2

(

(

) b ( 3 sin − cos )

) b ( − 3 cos − sin )

1

1

b2

determinant of BW as (2.4): det ( BW ) = 6 3b12b2 0

kr

krL

where: b1 =

; b2 =

2

(3I w + 2Mr )

(3I w + I v r 2 )

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 0 according to Lyapunov. Lyapunov

function is defined as (2.5):

9

1 T

S S 0

2

Get the derivative of V, we have:

V = S T − AW − sign ( S )

V=

(

(

)

(2.5)

)

= − S T + sign ( S ) 0

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 2

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 2

r

Radius of the wheel

0.0245

m

10

Table 2.2. The parameters of the sliding mode control

Meaning

Sliding

parameters

Notation

Value

x 0 0

0 y 0

0 0

20 0 0

0 20 0

0 0 20

kx

0

0

25 0 0

0 25 0

0 0 58

0

ky

0

0

0

k

➢ Case 1: the input is unit step function

Fig. 2.3. Responses xw , yw ,w

Fig. 2.4. Errors xw , yw ,w and

and xd , yd ,d with Step

xd , yd ,d with Step

Table 2.3. The quality criteria with unit step function

The quality criteria

Response

POT (%) exl (mm)

tqđ (s)

xw

0.20

0.22

0.45

yw

0.22

0.024

0.45

w

0.15

0.15

1.0

11

➢ Case 2: the input is unit step function when the mass of the robot

increases 25% and 50%

Fig. 2.5. Response xw when the

Fig. 2.6. Response yw when

mass of the robot increases

the mass of the robot

25% and 50%

increases 25% and 50%

➢ Case 3: the input is unit step function when the noise impacts at the

output

Fig. 2.7. Responses xw, yw and w

Fig. 2.8. Errors xw, yw và w when

when the noise impacts at the

the noise impacts at the output

output

The simulation results show that sliding mode controller has well

controlled robot converge to reach the reference trajectory. However, the

response of the SMC has yet to give optimal results with the achieved

quality criteria. To improve control quality, this study proposes using the

radial basic function neural network to estimate nonlinear function in

SMC law. The RBF is trained online by Gradient Descent algorithm and

12

considered as an adaptive controller due to the feedback from output

signals of the plant.

2.5. Design of the adaptive sliding mode control with neural networks

for the robot

2.5.1. The adaptive sliding mode control using RBF neural networks

In this part, the study uses Gradient Descent algorithm to estimate

online AW matrix in (2.2) SMC law. This matrix 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).

The algorithm of RBF neural network as (2.6) [41]:

X −c

i

ij

hij = exp −

2

2bij

2

i =1,3; j =1,5

(2.6)

The general block diagram of the radial basis function neural network

is used to approximate the function fi in AW matrix is given in Fig. 2.9.

Fig. 2.9. Block diagram of the radial basis function neural network

where:

X1 e (1) e (1) d (1)

X i = X 2 = e ( 2 ) e ( 2 ) d ( 2 )

X 3 e ( 3) e ( 3) d ( 3)

hij

= hi1 hi 2 hi 3

i =1,2,3

13

d (1) d (1)

d ( 2 ) d ( 2 )

d ( 3) d ( 3)

hi 4

hi 5

(2.7)

(2.8)

wij

i =1,2,3

= wi1

wi 2

wi 3

wi 4

wi 5

(2.9)

The output fi as (2.10): fi = wijT hij

(2.10)

The approximation matrix AW can be rewritten in the following form:

w1jT h1j − w2jT d h2j

0

T

T

ˆ

(2.11)

AW = w2j h2j

w1j h1j

0

T

0

0

w3j h3j

Now, the adaptive sliding mode control using the RBF neural network

with Gradient Descent algorithm (GD-ASMC-RBF) is presented as (2.12):

(2.12)

U = − B −1 Aˆ + ( + ke ) + sign ( S )

W

W

d

When the actual trajectory of the robot deviates from the reference

trajectory due to the impact of actual conditions such as road surface

friction, changing inertial torque, .... the error changes. At that time, the

RBF network will be automatically updated with Gradient Descent

algorithm, which AˆW will also change so that the error reaches the

minimum value. We say the control law (2.12) adapts to the robot's

operating conditions

To prove the stability of the control law we need to prove that the

sliding mode surfaces converge to 0 according to Lyapunov. Lyapunov

function is defined as (2.13):

1 T

S S 0

2

Get the derivative of V, we have:

V = S T − AˆW − sign ( S ) = S T − − sign ( S )

V=

(

)

(

(

(2.13)

)

)

= − S T + sign ( S ) 0

2.5.2. The parameters of the system and simulation results

- Sliding mode controller is defined to guarantee state trajectory not

only converges to the sliding surface in finite time but also remains on the

sliding surface even when the system parameters are changed.

14

- The Gradient Descent algorithm effects in online approximation of

nonlinear matrix of the SMC law.

- The adaptive sliding mode control is stable when the actual

trajectory of the robot deviates from the reference trajectory due to the

impact of actual conditions or by changing the parameters of the plant.

The parameters of the radial basis function neural network are shown

in Table 2.4:

Table 2.4. The parameters of the radial basis function neural network

Notation

Meaning

Value

Momentum factor

0.05

Learning factor

0.5

0.5 0.5

cij

Center vector for the neural

network

−1

−1

−1

−1

−1

−0.5 0 0.5 1

−0.5 0 0.5 1

−0.5 0 0.5 1

−0.5 0 0.5 1

−0.5 0 0.5 1

bj

Certain threshold of the

neural network

1

Parameter vector

0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5

W

V

Z

T

1 1 1 1

T

i

Total inputs

5

j

Total neurons in hidden

layers

5

15

➢ Case 1: the input is unit step function

Fig. 2.10. Responses GD-

Fig. 2.11. Errors GD-ASMC-

ASMC-RBF and SMC

RBF and SMC controller with

controller with Step

Step

Table 2.5. The quality criteria of the GD-ASMC-RBF controller with

unit step function

Response

The quality criteria

POT (%)

exl (mm)

tqđ (s)

xw

0.15

0

0.3

yw

0.62

0.55

0.3

w

0.80

0

0.4

Table 2.6. The error performances between GD-ASMC-RBF and SMC

Response

SMC

GD-ASMC-RBF

ADD

MSE

RMSE

ADD

MSE

RMSE

xw

1.4×10-4

3.9×10-8

2.0×10-4

4.9×10-5

4.9×10-9

7.0×10-5

yw

1.1×10-4

2.5×10-8

1.6×10-4

8.6×10-5

1.5×10-8

1.2×10-4

w

2.3×10-5

1.1×10-9

3.3×10-5

4.2×10-5

3.6×10-9

6.0×10-5

16

➢ Case 2: the input is unit step function when the noise impacts at the

output

Fig. 2.12. Responses of the

Fig. 2.13. Errors of the GD-

GD-ASMC-RBF and SMC

ASMC-RBF and SMC with

with unit step function

unit step function when the

when the noise impacts at

noise impacts at the output

the output

The simulation results of the GD-ASMC-RBF controller for Omnidirectional mobile robot have good results, the response of the robot

converge to reach the trajectory without steady-state error even

disturbances of the system. This is the remarkable feature of the sliding

mode controller as well as the Gradient Descent algorithm in the online

approximation of the Aw matrix in SMC law. This shows the feasibility of

GD-ASMC-RBF controller in nonlinear system control

17

CHAPTER 3: EVALUATING THE EFFECTIVENESS OF THE

QUASI-NEWTON ALGORITHM IN NONLINEAR SYSTEM

TRAJECTORY TRACKING CONTROL

3.1. Problem

In chapter 2, the author is used the RBF neural network to estimate

nonlinear functions in the sliding mode control law which is calculated

based on Lyapunov stability theory of adaptive sliding mode controller

based on RBF neural network. The simulation results with

MATLAB/SIMULINK show that the proposed algorithm is efficient, the

response of the robot converge to reach the trajectory without steady-state

error is about, and the rise time is about 0.3 ± 0.001(s) . However,

Gradient Descent 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]. In this chapter 3, research proposes using a

Quasi-Newton algorithm to train online the RBF neural network, at the

same time the research investigates the influence of the parameters of

neural network (number of hidden) on the control quality of the adaptive

sliding mode control.

3.2. The object

- The RBF neural networks are trained online using Quasi-Newton

algorithm with initial parameters are randomly generated

- The proposed algorithm is applied in Omni-directional mobile robot

trajectory tracking control.

- the research investigates the influence of number of hidden layers on

the control quality of an adaptive sliding mode control using radial basis

function neural networks.

3.3. Evaluate the performance of the Quasi-Newton algorithm in

nonlinear system trajectory tracking control

18

3.3.1. The Quasi-Newton algorithm

This algorithm onsists of 4 steps with k iteration

Step 1: Initialize search direction by (3.1)

d k = − H k−1 gk .

(3.1)

Step 2 : Search in the direction d k to find the step length k .

f ( wk + k d k ) = min ( f ( wk + k d k ) ) .

0

E ( wk + k d k ) E ( wk ) + k gkT d k

g ( wk + k d k ) − gkT d k

T

(3.2)

The weight update process is presented as (3.3).

wk +1 = wk + k d k

(3.3)

Step 3 : The Hessian matrix update process is presented as (3.4).

g g T H T H

(3.4)

H k +1 = H k + k T k − k T k k k

gk k

k H k k

Step 4 : heck convergence with some standards. Typically, this

standard is defined as a direction of the objective function to ensure that

it is less than a given value to stop iteration as (3.5)

gkT gk .

(3.5)

The iteration begins when k = 0 , the initialization process is random,

the Hessian matrix H 0 is a symmetric matrix and is a positive

determination matrix with initial value H 0 = I .

3.3.2. Training online the RBF neural network by the Quasi-Newton

algorithm applied in the adaptive sliding mode control Omni-directional

mobile robot.

In this part, the study uses the Quasi-Newton algorithm to approximate

AW matrix in SMC law (2.2) instead of Gradient Descent.

The algorithm of the RBF neural network as (2.6).

The RBF neural network’s structure is used to approximate the fi

component in AW matrix as Fig. 2.9 in Chapter 2.

19

Where Xi and hij as (2.7) and (2.8).

And t ij i =1,2,3 = ti1 ti 2 ti 3 ti 4 ti 5

(3.6)

The output fi as (3.7): fi = tijT hij

(3.7)

Hence, the approximation matrix AW can be rewritten as (3.8):

t1jT h1j −t 2jT d h2j

0

T

T

ˆ

(3.8)

AW = t 2j h2j

t1j h1j

0

T

0

0

t3j h3j

The adaptive sliding mode control law using the RBF neural network

which is trained online by the Quasi-Newton algorithm (QN-ASMCRBF) as (3.9):

(

)

U = − BW −1 AˆW + d + ke + sign ( S )

(3.9)

The result of online training of RBF neural network with QuasiNewton algorithm (BSGS) presented as Fig. 3.1 with MSE (Mean Square

Error) are smaller than Gradient Descent algorithm. This has great

significance in improving controller response, especially in controller

online training

Fig. 3.1. Compare MSE of Quasi-Newton (BSGS) and Gradient

Descent

20

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 2.

➢ 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

0

1.5

0.2

yw

0.19

0

0.2

w

0.05

0

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 4 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

2

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].

1

(4.3)

E = ( d − w )2

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

1

E = ( d − w )2

2

The results of the RBF neural network using GA are shown in Fig. 4.2

và 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

7

From Table 4.2, it is observed that the genetic algorithm has

determined the specific values of the hidden layer neuron, center,

25

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

1

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

2

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.

3

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.

4

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.

5

- 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 1 is introduction; Chapter 2 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 4 presents the optimal method of neural network structure using

genetic algorithms and Chapter 5 is the results, conclusions and

recommendations.

6

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.

7

Fig. 2.1. Model of an Omni-directional mobile robot [47]

Dynamic equation of the robot as (2.1):

−a2d 0 xw b1 1

xw a1

y = a

a1

0 yw + b1 3

w 2 d

w 0

0

a3 w b2

= AW β + BW U + D f

b1 2

b1 4

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

0

T

D f are unknown system disturbances.

−a2

a1

0

0

b1 1

0 ; BW = b1 3

b2

a3

U = u1

a2 = 1 − a2 =

u2

b1 2

b1 4

b2

2b1 cos

2b1 sin ;

b2

u3

T

3I w

; 1 = − 3 sin − cos ; 2 = 3 sin − cos

(3I w + 2Mr 2 )

3 = 3 cos − sin ; 4 = − 3 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]:

8

(

)

U = − BW −1 AW + d + ke + sign ( S )

−1

W

(2.2)

* Prove that the B matrix is invertible

We have:

b1 − 3 sin − cos

BW = b1 3 cos − sin

b2

(

(

) b ( 3 sin − cos )

) b ( − 3 cos − sin )

1

1

b2

determinant of BW as (2.4): det ( BW ) = 6 3b12b2 0

kr

krL

where: b1 =

; b2 =

2

(3I w + 2Mr )

(3I w + I v r 2 )

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 0 according to Lyapunov. Lyapunov

function is defined as (2.5):

9

1 T

S S 0

2

Get the derivative of V, we have:

V = S T − AW − sign ( S )

V=

(

(

)

(2.5)

)

= − S T + sign ( S ) 0

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 2

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 2

r

Radius of the wheel

0.0245

m

10

Table 2.2. The parameters of the sliding mode control

Meaning

Sliding

parameters

Notation

Value

x 0 0

0 y 0

0 0

20 0 0

0 20 0

0 0 20

kx

0

0

25 0 0

0 25 0

0 0 58

0

ky

0

0

0

k

➢ Case 1: the input is unit step function

Fig. 2.3. Responses xw , yw ,w

Fig. 2.4. Errors xw , yw ,w and

and xd , yd ,d with Step

xd , yd ,d with Step

Table 2.3. The quality criteria with unit step function

The quality criteria

Response

POT (%) exl (mm)

tqđ (s)

xw

0.20

0.22

0.45

yw

0.22

0.024

0.45

w

0.15

0.15

1.0

11

➢ Case 2: the input is unit step function when the mass of the robot

increases 25% and 50%

Fig. 2.5. Response xw when the

Fig. 2.6. Response yw when

mass of the robot increases

the mass of the robot

25% and 50%

increases 25% and 50%

➢ Case 3: the input is unit step function when the noise impacts at the

output

Fig. 2.7. Responses xw, yw and w

Fig. 2.8. Errors xw, yw và w when

when the noise impacts at the

the noise impacts at the output

output

The simulation results show that sliding mode controller has well

controlled robot converge to reach the reference trajectory. However, the

response of the SMC has yet to give optimal results with the achieved

quality criteria. To improve control quality, this study proposes using the

radial basic function neural network to estimate nonlinear function in

SMC law. The RBF is trained online by Gradient Descent algorithm and

12

considered as an adaptive controller due to the feedback from output

signals of the plant.

2.5. Design of the adaptive sliding mode control with neural networks

for the robot

2.5.1. The adaptive sliding mode control using RBF neural networks

In this part, the study uses Gradient Descent algorithm to estimate

online AW matrix in (2.2) SMC law. This matrix 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).

The algorithm of RBF neural network as (2.6) [41]:

X −c

i

ij

hij = exp −

2

2bij

2

i =1,3; j =1,5

(2.6)

The general block diagram of the radial basis function neural network

is used to approximate the function fi in AW matrix is given in Fig. 2.9.

Fig. 2.9. Block diagram of the radial basis function neural network

where:

X1 e (1) e (1) d (1)

X i = X 2 = e ( 2 ) e ( 2 ) d ( 2 )

X 3 e ( 3) e ( 3) d ( 3)

hij

= hi1 hi 2 hi 3

i =1,2,3

13

d (1) d (1)

d ( 2 ) d ( 2 )

d ( 3) d ( 3)

hi 4

hi 5

(2.7)

(2.8)

wij

i =1,2,3

= wi1

wi 2

wi 3

wi 4

wi 5

(2.9)

The output fi as (2.10): fi = wijT hij

(2.10)

The approximation matrix AW can be rewritten in the following form:

w1jT h1j − w2jT d h2j

0

T

T

ˆ

(2.11)

AW = w2j h2j

w1j h1j

0

T

0

0

w3j h3j

Now, the adaptive sliding mode control using the RBF neural network

with Gradient Descent algorithm (GD-ASMC-RBF) is presented as (2.12):

(2.12)

U = − B −1 Aˆ + ( + ke ) + sign ( S )

W

W

d

When the actual trajectory of the robot deviates from the reference

trajectory due to the impact of actual conditions such as road surface

friction, changing inertial torque, .... the error changes. At that time, the

RBF network will be automatically updated with Gradient Descent

algorithm, which AˆW will also change so that the error reaches the

minimum value. We say the control law (2.12) adapts to the robot's

operating conditions

To prove the stability of the control law we need to prove that the

sliding mode surfaces converge to 0 according to Lyapunov. Lyapunov

function is defined as (2.13):

1 T

S S 0

2

Get the derivative of V, we have:

V = S T − AˆW − sign ( S ) = S T − − sign ( S )

V=

(

)

(

(

(2.13)

)

)

= − S T + sign ( S ) 0

2.5.2. The parameters of the system and simulation results

- Sliding mode controller is defined to guarantee state trajectory not

only converges to the sliding surface in finite time but also remains on the

sliding surface even when the system parameters are changed.

14

- The Gradient Descent algorithm effects in online approximation of

nonlinear matrix of the SMC law.

- The adaptive sliding mode control is stable when the actual

trajectory of the robot deviates from the reference trajectory due to the

impact of actual conditions or by changing the parameters of the plant.

The parameters of the radial basis function neural network are shown

in Table 2.4:

Table 2.4. The parameters of the radial basis function neural network

Notation

Meaning

Value

Momentum factor

0.05

Learning factor

0.5

0.5 0.5

cij

Center vector for the neural

network

−1

−1

−1

−1

−1

−0.5 0 0.5 1

−0.5 0 0.5 1

−0.5 0 0.5 1

−0.5 0 0.5 1

−0.5 0 0.5 1

bj

Certain threshold of the

neural network

1

Parameter vector

0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5

W

V

Z

T

1 1 1 1

T

i

Total inputs

5

j

Total neurons in hidden

layers

5

15

➢ Case 1: the input is unit step function

Fig. 2.10. Responses GD-

Fig. 2.11. Errors GD-ASMC-

ASMC-RBF and SMC

RBF and SMC controller with

controller with Step

Step

Table 2.5. The quality criteria of the GD-ASMC-RBF controller with

unit step function

Response

The quality criteria

POT (%)

exl (mm)

tqđ (s)

xw

0.15

0

0.3

yw

0.62

0.55

0.3

w

0.80

0

0.4

Table 2.6. The error performances between GD-ASMC-RBF and SMC

Response

SMC

GD-ASMC-RBF

ADD

MSE

RMSE

ADD

MSE

RMSE

xw

1.4×10-4

3.9×10-8

2.0×10-4

4.9×10-5

4.9×10-9

7.0×10-5

yw

1.1×10-4

2.5×10-8

1.6×10-4

8.6×10-5

1.5×10-8

1.2×10-4

w

2.3×10-5

1.1×10-9

3.3×10-5

4.2×10-5

3.6×10-9

6.0×10-5

16

➢ Case 2: the input is unit step function when the noise impacts at the

output

Fig. 2.12. Responses of the

Fig. 2.13. Errors of the GD-

GD-ASMC-RBF and SMC

ASMC-RBF and SMC with

with unit step function

unit step function when the

when the noise impacts at

noise impacts at the output

the output

The simulation results of the GD-ASMC-RBF controller for Omnidirectional mobile robot have good results, the response of the robot

converge to reach the trajectory without steady-state error even

disturbances of the system. This is the remarkable feature of the sliding

mode controller as well as the Gradient Descent algorithm in the online

approximation of the Aw matrix in SMC law. This shows the feasibility of

GD-ASMC-RBF controller in nonlinear system control

17

CHAPTER 3: EVALUATING THE EFFECTIVENESS OF THE

QUASI-NEWTON ALGORITHM IN NONLINEAR SYSTEM

TRAJECTORY TRACKING CONTROL

3.1. Problem

In chapter 2, the author is used the RBF neural network to estimate

nonlinear functions in the sliding mode control law which is calculated

based on Lyapunov stability theory of adaptive sliding mode controller

based on RBF neural network. The simulation results with

MATLAB/SIMULINK show that the proposed algorithm is efficient, the

response of the robot converge to reach the trajectory without steady-state

error is about, and the rise time is about 0.3 ± 0.001(s) . However,

Gradient Descent 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]. In this chapter 3, research proposes using a

Quasi-Newton algorithm to train online the RBF neural network, at the

same time the research investigates the influence of the parameters of

neural network (number of hidden) on the control quality of the adaptive

sliding mode control.

3.2. The object

- The RBF neural networks are trained online using Quasi-Newton

algorithm with initial parameters are randomly generated

- The proposed algorithm is applied in Omni-directional mobile robot

trajectory tracking control.

- the research investigates the influence of number of hidden layers on

the control quality of an adaptive sliding mode control using radial basis

function neural networks.

3.3. Evaluate the performance of the Quasi-Newton algorithm in

nonlinear system trajectory tracking control

18

3.3.1. The Quasi-Newton algorithm

This algorithm onsists of 4 steps with k iteration

Step 1: Initialize search direction by (3.1)

d k = − H k−1 gk .

(3.1)

Step 2 : Search in the direction d k to find the step length k .

f ( wk + k d k ) = min ( f ( wk + k d k ) ) .

0

E ( wk + k d k ) E ( wk ) + k gkT d k

g ( wk + k d k ) − gkT d k

T

(3.2)

The weight update process is presented as (3.3).

wk +1 = wk + k d k

(3.3)

Step 3 : The Hessian matrix update process is presented as (3.4).

g g T H T H

(3.4)

H k +1 = H k + k T k − k T k k k

gk k

k H k k

Step 4 : heck convergence with some standards. Typically, this

standard is defined as a direction of the objective function to ensure that

it is less than a given value to stop iteration as (3.5)

gkT gk .

(3.5)

The iteration begins when k = 0 , the initialization process is random,

the Hessian matrix H 0 is a symmetric matrix and is a positive

determination matrix with initial value H 0 = I .

3.3.2. Training online the RBF neural network by the Quasi-Newton

algorithm applied in the adaptive sliding mode control Omni-directional

mobile robot.

In this part, the study uses the Quasi-Newton algorithm to approximate

AW matrix in SMC law (2.2) instead of Gradient Descent.

The algorithm of the RBF neural network as (2.6).

The RBF neural network’s structure is used to approximate the fi

component in AW matrix as Fig. 2.9 in Chapter 2.

19

Where Xi and hij as (2.7) and (2.8).

And t ij i =1,2,3 = ti1 ti 2 ti 3 ti 4 ti 5

(3.6)

The output fi as (3.7): fi = tijT hij

(3.7)

Hence, the approximation matrix AW can be rewritten as (3.8):

t1jT h1j −t 2jT d h2j

0

T

T

ˆ

(3.8)

AW = t 2j h2j

t1j h1j

0

T

0

0

t3j h3j

The adaptive sliding mode control law using the RBF neural network

which is trained online by the Quasi-Newton algorithm (QN-ASMCRBF) as (3.9):

(

)

U = − BW −1 AˆW + d + ke + sign ( S )

(3.9)

The result of online training of RBF neural network with QuasiNewton algorithm (BSGS) presented as Fig. 3.1 with MSE (Mean Square

Error) are smaller than Gradient Descent algorithm. This has great

significance in improving controller response, especially in controller

online training

Fig. 3.1. Compare MSE of Quasi-Newton (BSGS) and Gradient

Descent

20

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 2.

➢ 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

0

1.5

0.2

yw

0.19

0

0.2

w

0.05

0

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 4 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

2

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].

1

(4.3)

E = ( d − w )2

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

1

E = ( d − w )2

2

The results of the RBF neural network using GA are shown in Fig. 4.2

và 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

7

From Table 4.2, it is observed that the genetic algorithm has

determined the specific values of the hidden layer neuron, center,

25

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