Thiết kế bộ điều khiển bám đuổi thích nghi cho hệ thống phi tuyến MIMO sử dụng CMAC tt tiếng anh

Ngo Thanh Quyen Major: Automation and Control Engineering Code: 9520216 Ho Chi Minh City, March 2019 TA VAN PHUONG Dissertation Abstract Designing Adaptive Tracking Controller For Non-L

MINISTRY OF EDUCTION AND TRAINING MINISTRY OF TRANSPORT

HO CHI MINH CITY UNIVERSITY OF TRANSPORT

Instructor 1: Assoc.Prof Dang Xuan Kien

Instructor 2: Dr Ngo Thanh Quyen

Major: Automation and Control Engineering

Code: 9520216

Ho Chi Minh City, March 2019

TA VAN PHUONG Dissertation Abstract

Designing Adaptive Tracking Controller For

Non-Linear MIMO Systems Using CMAC

Abstract

Designing control system for non-linear MIMO systems has attracted many researchers for the recent decades Due to the complex characteristics, defining the dynamic model of the non-linear MIMO systems are invaluable for the practical applications Therefore, the model-based controllers cannot satisfy the desired performances

To cope with this problem, many advanced controllers have been studied and applied for the non-linear MIMO systems such as Particle Swarm Optimization (PSO), Fuzzy Logic Controller (FLC), Neural Network (NN), Fuzzy Neural Network, and so on By using these adaptive and intelligent controllers, the performances have been achieved for the practical applications However, there exists disadvantages and shortcomings that need

to be improved such as online learning problems, selection number of fuzzy rules, number of neurons and layers, the robustness of the system in the presence of disturbances, noise, and uncertainties, and so on

This thesis has proposed the CMAC, the recurrent CMAC, the redundant recurrent CMAC, and the robust recurrent cerebellar model articulation control system (RRCMACS) for the non-linear MIMO systems

to achieve the desired performances such as good tracking responses, stability, robustness, disturbances attenuation, and noise rejection The main contributions of this dissertation are presented in Chapter 2, Chapter 3, and Chapter 4 In Chapter 2, a traditional Cerebellar Model Articulation Controller is represented to show the superior properties of the CMAC to different intelligent controllers Chapter 3 presents the factors affecting the learning capability and efficiency of the CMAC and then some innovative solutions are proposed to enhance the performance and the learning effectiveness of the CMAC Besides improving the CMAC, the redundant solution is also proposed to maintain continuously the controling and supervising process A combination between the RCMAC and the robust controller to form the robust RCMAC to achieve not only good tracking response but also attenuate significantly the effects of the external disturbances, and sensor noise is shown in Chapter 4 With this combination, the stability and robustness of the control system are remarkably improved during operation Along with the main theorems, the experimental results were also provided to prove the effectiveness of the proposed solutions

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Chapter 1: INTRODUCTION OF NON-LINEAR MIMO SYSTEM

AND PROPOSED CONTROL SYSTEM

1.1 Introduction of the non-linear MIMO systems and research problems

Most of the practical systems are non-linear systems the non-linear features of the system may come from the effects of dead-zone, hysteresis, saturation, friction coefficients, cross-coupling, uncertainties, disturbances, and noises [1]-[5]

Due to the effects of the non-linear characteristics, the dynamic model of the practical systems cannot be completely obtained Therefore, they must be considered in designing the control systems From the point of view in designing control system, model-based controllers cannot achieve desired performances for the non-linear MIMO systems [6]-[15]

To cope with the the non-linear characteristics and the uncertainties of the system, Fuzzy Logic Controller (FLC) [16]-[23], Sliding Mode Controller (SMC) [24]-[30], neural network (NN) [31]-[39] have been developed for the non-linear system to achieve desired performances However, these controllers still exist shotcomings as flows

The performance of the FLC depended utterly on the selection of fuzzy sets and the number of rules However, there are not a specific method that ensure the optimal selections of fuzzy sets and rules for the controllers so far

To achieve good performance of the FLC for the practical applications, the fuzzy sets and rules were mostly selected by trial and error

For the SMC, the chattering phenomena affects long life and responses of the actuators, the selection of the boundary of uncertainties is a trade-off between the stability and the chattering phenomena

The NNs remain several shortcomings such as all weights in the structure

of the neural network are updated each learning cycle, this is unsuitable for the problems requiring real-time learning; the selection of the number of neurons and hidden layers to achieve good performances is very difficult in the practical applications

Along with the development of the NNs, Cerebellar Model Articulation Controller (CMAC) having the learning structure similar to the human brain

has been studied from the 1970s [40] The CMAC has been developing and incorporating for the complex non-linear MIMO systems because of its superior properties such as fast learning, good generation capability, and simple computation [41]-[46] The effectiveness of the CMACs rather than NNs has been proved in the practical applications [47]-[48] In the recent works, the wavelet function and recurrent technique were utilized to improve learning capability and dynamic response of the CMAC [49]-[51]

Although the above studies achieved good results in designing the controller to cope with the high non-linear MIMO systems both in simulation and experiment, however, the robustness of the system in the presence of the disturbances and noise were not fully taken into account

After studying the papers relating to the CMAC, the author proposed a new methodology to design a control system for the non-linear MIMO system basing on the Cerebellar Model Articulation Controller with the following characteristics

i) The control system is not dependence on the dynamic model Meanwhile, the stability and convergence error of the system can be obtained in case the model cannot be exactly defined

ii) The control system has dynamic response capability and avoid the local minimization during operation

iii) The control system can deal with uncertainties, disturbances, and change in parameters of the system to have good tracking response iv) The control system has redundant capability which maintains continuously the control and supervisory process

v) The control system guarantees the stability and robustness of the system in the presence of the uncertainties, disturbances, and noise

1.2 Outline of the Dissertation

This dissertation is divided into five chapters Chapter 1 mentions about the non-linear MIMO system, scientific researches relating to the non-linear system control, neccessory problems to research and proposed control system Structure of the traditional Cerebellar Model Articulation Controller (CMAC) and its applications are shown in Chapter 2 Chapter 3 points out the shortcomings of the traditional CMAC and proposes innovative solutions

to enhance the performances of the CMAC The factors affect the robustness

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and the robust CMAC designing are provided in detail in Chapter 4 Finally, the conclusion and future works are delivered in Chapter 5 Along with the representation of research problems by theory, the simulation and experimental results are also included to prove the effectiveness and merit of the proposesd control system

The organization of this dissertation is expressed in Fig 1.1 as follows

Figure 1.1: Organization of the dissertation

1.3 MIMO non-linear system including

disturbances and noise, UD x( )  F x( ) +G x u + dn x( ) ( )is lumped uncertainties, disturbances, and noise The objective of the control system synthesis is that the output signals x can not only track the desired

d

nR



x but also satisfy the robust performance in the presence

of the uncertainties, disturbances, and noise

1.4 The proposed control system

For the high-order system, the sliding error manifold was defined [2],[29]-[30] to reduce the order of variables during designing and computating the control system The sliding error manifold has the following form

Therein, exdxand  , , , n-1T

e e e e are the tracking error and error vector of the system, respectively Derivative both sides of S and combination with the dynamic equation (1.1), yields

n-1

+



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S e + Ke = x - x + K(e) = x - F (x) - G (x)u - UD(x) + K(e) (1.3)

In case the nominal functions n x n o

To cope with the drawbacks of the model-based controllers, many modern controllers have been developed such as Fuzzy Logic Controller (FLC), Sliding Mode Controller (SMC), Neural Networks (NNs), and Cerebellar Model Articulation Controller (CMAC) [17-51]

Although the above studies achieved impressive results in designing the controllers to cope with the high non-linear MIMO systems, the dynamic response and robust specifics of the system in the presence of the uncertainties were not totally mentioned

In this research, a robust recurrent cerebellar model articulation control system (RRCMACS) is proposed for the non-linear MIMO system Therein, the RCMAC is designed to imitate the ideal controller to minimize error surface and the Hrobust controller is designed to attenuate the effects of the uncertainties acting on the system to achieve the robustness performance of the system during operation

The total control system is described in (1.7) and a block diagram of the RRCMACS is depicted in Fig 1.2

RRCMACS ISM RC RCMAC

w m σ w

η , η , η , η

Robust Controller

ISM = G (x) x - F (x) - UD (x) + K(e) - ηsgn(S )

u

x

S

Figure 1.2: Block diagram of the proposed control system

The block diagram of the proposed control system includes three main control parts as follows

 The ideal sliding mode controller is used when the uncertainties are exactly known

 The RCMAC is the main controller, it is used to learn the uncertainties,

( )

UD x to minimize the error sliding surface, S With the selection of

the appropriate learning rate, the error sling surface will tend to zero

by learning capability of the RCMAC

 The robust controller uRC guarantees the robustness of the system in the presence of the uncertainties during operation

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Chapter 2: THE CEREBELLAR MODEL ARTICULATION

CONTROLLER (CMAC) 2.1 Introduction to the CMAC

Cerebellar model articulation controller (CMAC) is a neural network

model proposed by Albus [54]–[56] The CMAC with its fast learning and

good generation capability has been studied and implemented to identify and control the non-linear systems [57]-[59] Based on its superior properties, the CMAC is unnecessary to require much prior knowledge of the system Consequently, it can be considered as an intelligent controller that suits many practical non-linear systems [60]-[61] The superior properties of the CMAC

to NNs were proved in the references [62]-[64]

2.1.1 Block diagram of the proposed control system and Structure of the CMAC

The CMAC with its fast learning and good generalization capability places an important role in learning the unknown uncertainties, UD x( ) to minimize the error sliding surface The block diagram of the proposed control system is depicted in Fig 2.1 and the structure of the CMAC is shown in Fig 2.2 The controller includes input space S, association memory space A , receptive field space R , weight memory space, and output spaces O The signal propagation in the CMAC is presented as follows [48], [65]

w m σ

η , η , η

Compensator Controller

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In this research, the error-sliding surfaces are considered as input variables of input space, S= [S S S S ]1 2 3 ni Each input variable Si can be quantized into nediscrete regions corresponding to the working space The activation degrees of input variables to each layer is calculated by the Gaussian function as follows

in Fig 2.4 The blocks formed by the data overlapping of the input variables are called hypercubes Each activated element in each block or layer becomes

a firing element, thus, the weight of each block or layer can be obtained The content of thk hypercube can be defined as follows:

wherewjk is the weight of the thk hypercube corresponding to thejthoutput

2 3 4 Layer 1

Layer 2 Layer n ki

7 6

1 2 3 4 6 7

5

State (3,3)

2

S

Figure 2.4: Store data in the receptive field space of the CMAC

The output of the CMAC is the algebraic sum of the activated weights correspongding to the hypercube The output of the controller can be expressed as follows:

2.1.2 The cost function and learning rules of the CMAC

The effects of the uncertaintiesUD(x) in (1.4) is learned by the CMAC CMAC

u with approximation error ε as follows:

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The gradient descent algorithm is applied to learn to minimize the error function of the system [41], [48] The error function depends on three parameters w, m, σ The learning rules of these parameters are described by (2.7) to (2.9) The parameters of the controller are updated by (2.10)-(2.12)

i

n

T CMAC

2.2 The Estimated Boundary Compensator Controller

The most useful property of CMAC is to lean the non-linear part UD(x)

, via learning capability to minimize the sliding error surface (SES) With an appropriate learning rates selection, the SES will tend to zero as a specific period of time To guarantee the stability of the system, the controller must have the capability to maintain the SES nearby zero during operation According to sliding mode theory, a compensator controller uCC is designed for this objective as follows

1

where B is the error boundary

With the compensator controller uCCin (2.13), the stability of the system will be guaranteed in case the approximation error ε in (2.5) is bounded by

an error boundary B The selection error boundary B is a trade-off between the stability of the system and chattering phenomenon at control output As the error boundary is smaller than ε the system will be unstable, otherwise the control output will be chattering when the error boundary is too large to

ε Therefore, this parameter is estimated in this study and represented as follows

ˆ

where ˆB is the estimated value of the error boundary B

Substituting (1.7) into (1.1) yields

where uISMis represented in (1.4) excepting for UD(x)and uCMACis given

in (2.5) On the basis of some straightforward manipulations, the error equation of the system can be obtained as follows

2.3 Experiments and results

To show the effectiveness of the CMAC in reality, the controller is applied to control the pressure and water level in the tank

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The Pressure Control Model (PCM) and the Water Level Control Model (WLCM) are described in Fig 2.5 and Fig 2.6, respectively

Figure 2.5: Structure of the Tank Pressure Control Model

Figure 2.6: Structure of the Water Level Control Model

The PCM and the WLCM are non-linear and time-varying parameter systems [67]–[69] The precise dynamic equation of these systems are very difficult to define exactly In this study, the dynamic model of the PCM and the WLCM are obtained by the identification toolbox of the Matlab as bellow

where u(t)and y(t)are control signal of the inverter and the pressure or water level in the tank, respectively UD(t)is lumped uncertainties due to change in parameters, disturbances, noises, and error in linearization The dynamic equation of these systems can be rewritten in the state equation form

where F x0( )., G x0( )are nominal parameters of the system, and x is the state variable

The parameters of the CMAC are initialized as follows

a Tracking response of the pressure in the tank (Kpa)

b Tracking Error (Kpa)

c Control Effort (Volt) Figure 2.7: Experimental results of CMAC for The Pressure Control

Model due to periodic step command

The experimental results of the WLCM showed the impressive properties

of the CMAC as bellow

 Although the dynamic equation of the WLCM is obtained by the identification tool of the Matlab, the tracking responses of the system can be achieved exactly in reality

 The stability of the system can be guaranteed in the presence of the uncertainties UD(x)

2.4 Conclusion

This chapter introduces the CMAC including the structure, the cost function, and learning rules Besides, the compensator controller is combined with the CMAC to maintain the stability of the system in the presence of the uncertainties The experimental results are also provided to prove the

effectiveness of the proposed control system However, the CMAC should be improved to cope with dynamic response and local minimum problems

Chapter 3: IMPROVED CEREBELLAR MODEL ARTICULATION

CONTROLLER 3.1 Drawbacks of the traditional CMAC and proposed

improvements

The cerebellar model articulation controller (CMAC) has proposed since the 1970s by Albus [54]-[56] It has been considered as an intelligent controller which has a computational ability as the human cerebellum The structure of the CMAC is the same with a non-fully connected perceptron network It uses less memory area than other advanced controllers by overlapping data storing in the receptive field space Several advantages of the CMAC are fast learning property, good generalization capability, and ease of hardware implementation [59] Therefore, the CMAC has been used for identification and control of the complex non-linear systems [62]-[64] The CMAC, however, has several drawbacks as follows

 For the traditional CMAC, the step or triangular function was used as the activation function for input variables However, these functions

do not have differential abilities Consequently, the learning effectiveness of the controller is not good

 The structure of the CMAC was straightforward form, therefore, it was suitable to the static systems

 The controller did not design the redundant solution, consequently, it could not maintain continuously the control and supervisory process

To enhance the learning ability of the CMAC, the Gaussian and Wavelet functions were used to define the active degrees of the input variable at association memory space [70] Besides, the recurrent and redundant techniques were also studied and combined with the CMAC to improve dynamic response ability and maintaining the control and supervisory process

of the system All the proposed improvements above will be presented in

detail in the following parts

3.2 Wavelet Cerebellar Model Articulation Controller (WCMAC)

According to the Wavelet functions that is depicted in Fig 2.3 It is clear that the Wavelet has more differential ability than the step and Gaussian

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functions This characteristic is very useful in case the computation of the controller regards to derivative Consequently, the Wavelet function is combined with the CMAC to form the Wavelet CMAC (WCMAC) to improve the learning effectiveness of the controller The structure of the WCMAC is the same as the CMAC in Fig 2.2 excepting the Wavelet function was used instead of Gassian function at Association memory space The signal propagation of the WCMAC is presented as follows [48]

For the WCMAC, the Gaussian function was replaced by the Wavelet function in the association memory space Therefore, the activation degrees

of input variables to each layer is computed as follows

With the cost function is selected in the same way as the CMAC in chapter

2, the learning rules are performed by the back-propagation algorithm as bellow

s G x

(3.2)

2

i ik ik

G x

(3.3)

2

i ik ik

2 i ik 2

ik

S T

G x

(3.4)

The back-propagation algorithm was used to learn and update the parameters of the WCMAC However, the weakness of this algorithm is that

it may get stuck in local minima points [71]

To avoid getting stuck in a local minimum, the standard back-propagation algorithm is modified by adding momentum term ρ and proportional term

υ into calculating new parameters of the controller [72] The momentum term places an important role to prevent the learning algorithm from falling

in local minima and the proportional term speeds up the convergence as the activation function having a flat slope The new update rules of the WCMAC

Figure 3.1: The structure of the RCMAC

The signal propagation in the RCMAC is presented as follows [47]-[51]

3.4 The Online Learning Rules

The lumped uncertainties, UD x( ) are learned by the RCMAC uRCMAC

with an approximation error ε as follows

jk ik ri j=1 k =1 i=1

G

(3.15)

3 ik

i(k -1) 2

F is the hysteresis frictional force given as below

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where DO, MO, and KO are nominal values of D , M , and K ,

oUD(x) = F + F + f(D, F , F , K, t)

including lumped hysteresis phenomenon, external load, uncertainties, disturbances, and noises

Figure 3.2: Image of the experimental equipments of the micro motion stage The Fig 3.3 and Fig 3.4 show the experimental results of the the proportional-integral-derivative (PID) controller and the WCMAC for the position control system with sinusoidal and periodic step commands in real time

(a)

(b)

(c) Figure 3.3: Experimental results of PID and WCMAC controller with sinusoidal command (a) Tracking response (b) Error response (c) Control Signal

(a)

(b)

(c) Figure 3.4: Experimental results of PID and WCMAC controller with

periodic step command (a) Tracking response (b) Error response (c)

Control signal

The experimental results of the stage base with sinusoidal and periodic step commands in Figs 3.3 and 3.4 show that the WCMAC can deal with the uncertainties of the non-linear system to achieve desired tracking

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responses in real time Comparing to the PID controller, the WCMAC has better performances such as small convergence error, no overshoot, less chattering, and stability for a long time These results prove the effectiveness

of the WCMAC for other non-linear systems

3.5.2 Experimental results of the RCMAC

The micro-motion stage base in Fig 3.2 is reused to verify the responses

of the RCMAC To show the superior properties of the RCMAC to the CMAC, the experimental result of the CMAC was showed as well

The experimental results of the CMAC and the RCMAC due to periodic step commands are depicted in Fig 3.5

(a)

(b)

(c) Figure 3.5: Experimental results of the CMAC and the RCMAC with periodic step command (a) Tracking response (b) Error response (c) Control signal

The micro-motion stage base is a high non-linear system There exist the uncertainties and the non-linear characteristics due to cross-coupling reaction among the parts, hysteresis phenomena, and change in the parameter However, the proposed system can deal with these problems to achieved desired performances in the presence of noise during operation time Comparing to the CMAC, the RCMAC has many better performances of response time, convergence error and control signal in real time Table 1 shows a comparison of mean square error (MSE) between the CMAC (0,460 mm) and the RCMAC (0.071 mm)

These results prove the effectiveness of the RCMAC for the dynamic systems requiring real-time learning

Table 1: Comparison of mean square error (MSE)

is designed to guarantee the stability and the robustness of the system and the redundant controller aims to maintain continuous operation of the

applications

3.5.3.1 Structure and operation of the redundant system

The structure of the redundant system and a photograph of the practical experiment are shown in Fig 3.6 and Fig 3.7, respectively The redundant system includes a primary station (PS) combining with a standby station (SS)

to control the industrial applications Two of these stations aim to maintain the control and supervisory process of the system continuously in case one of them gets faults such as lost connections, power outage, controller or module

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error, and so on In particular, the redundant system uses DeviceNet Network

to control the stability pressure model and stability motor speed model The ControlNet Network is used to transfer diagnostic information between the

PS and SS to take over control as satisfying switching conditions At the beginning of the process, the earlier start station called the PS will take over control the system, the other station is in the SS mode During operation, if the PS gets the faults, the SS will take over control to maintain the control and supervisory process of the system continuously In addition, the recurrent cerebellar model articulation controller (RCMAC) is utilized to guarantee the stability and robustness of the system during operation The RCMAC was designed and downloaded to both the stations, thus the effectiveness of the control and supervisory process was the same with two stations

Figure 3.6: The structure of the redundant system

Figure 3.7: A photograph of the practical experiment

3.5.3.2 Take over control and the flowchart of the process

During operation, the control stations monitor the working conditions of each other The heartbeat parameter is used to measure the strength of the controllers As this parameter falls down under designed level, meanwhile, the controllers get faults to handle the system Therefore, the control and supervisory process will be transferred to the other station

In particular, assuming that the PP owns the system at the beginning of the operation During working, it monitors the connections to the devices in the system If there exist fault connections, it suddenly inhibits the connections to these devices, thus the SS can communicate with these devices and take over control the system In addition, the SS always monitors the status of the PP Whenever the PP has problems with connections, it immediately takes over control the system Fig 3.8 describes the process to take over control between the PP and the SS The flowchart of this process is also shown in Fig 3.9

Primary Station

Standby Station

Motor Control Inverter

Pump Control Inverter

Control Network handles changing control station

Devicenet Network Controls Device

Ethernet Communications

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Fig 3.8: Take over control between two stations

Fig 3.9: Flowchart of taking over control

Standby Station Check Heartbeat

Ow ne

rsh ip

Ow ner ship

PS takes over control

Get the fault code of output

modules

The SS own the modules Inhibit the connections of the

PS to the output modules

PS: Primary Station SS: Standby Station