Adaptive Backstepping FNN Control for a Linear Synchronous Motor Drive

Adaptive Backstepping FNN Control for a Linear Synchronous Motor Drive

Chih-Hong Lin Chih-Peng Lin

Department of Electrical Engineering Department of Engineering National United University Su-Mo Enterprise Co. LTD

Miao Li, R.O.C. Taichung, R.O.C.

E-mail: [email protected] E-mail:[email protected] Abstract— The linear synchronous motor (LSM) drive

system using adaptive backstepping fuzzy neural network (ABFNN) control is investigated for the tracking of periodic reference inputs. First, the field-oriented mechanism is applied to formulate the dynamic equation of the LSM servo drive.

Then, an adaptive backstepping approach is proposed to compensate the uncertainties in the motion control system.

With the proposed adaptive backstepping control system, the mover position of the LSM drive possesses the advantages of good transient control performance and robustness to uncertainties for the tracking of periodic reference trajectories.

Moreover, to further increase the robustness of the LSM drive, a FNN uncertainty observer is proposed to estimate the required lumped uncertainty in the adaptive backstepping control system. The effectiveness of the proposed control scheme is verified by the experimental results.

Keywords: linear synchronous motor, fuzzy neural network,

adaptive backstepping control

I.

I

NTRODUCTION

The direct drive design of mechanical applications, which is based on LSM, has many advantages over its indirect counterpart: such as no backlash and less friction, high speed and high precision in long distance location, simple mechanical construction, high thrust force [1]. Therefore, the LSM is suitable for high-performance servo applications and has been used widely for the industrial robots, semiconductor manufacturing systems, and machine tools, etc. [1-3].

The backstepping design provides a systematic framework for the design of tracking and regulation strategies, suitable for a large class of state feedback linearizable nonlinear systems. The approach can be extended to handle systems with unknown parameters via adaptive backstepping [4-6]. The idea of backstepping design is to select recursively some appropriate functions of state variables as pseudo-control inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new pseudo-control design, expressed in terms of the pseudo control designs from preceding design stages.

When the procedure terminates, a feedback design for the true control input results, which achieves the original design objective by virtue of a final Lyapunov function formed by summing up the Lyapunov functions associated with each individual design stage [7]. In addition, owing to the robust control performance of adaptive backstepping control and sliding mode control, numerous combined adaptive

backstepping and sliding-mode control schemes have appeared for both linear and nonlinear systems [8–10].

Recently much research has been done on the applications of fuzzy neural network (FNN) systems, which have the advantages of both fuzzy systems and neural networks, in the control fields to deal with nonlinearities and uncertainties of the control systems [11-12]. Moreover, the FNN’s are universal approximators [13-14], which can approximate any dynamics to a prespecified accuracy by the learning process [13]. The fuzzy-neural approximator presented in Leu, Lee and Wang [15] tuned on line to approximate the unknown nonlinear dynamic systems for adaptive control. The supervisory FNN controller proposed in Lin, Hwang and Wai [16] comprised a supervisory controller which is designed to stabilize the system states around a defined bound region, and an FNN sliding-mode controller which combines the advantages of the sliding-mode control with robust characteristics and the FNN with on-line learning ability. The adaptive control schemes of nonlinear systems that incorporate the techniques of FNNs have also grown rapidly [17-19].

Due to uncertainties exist in the applications of the LSM servo drive which seriously influence the control performance, thus, adaptive backstepping controller and FNN uncertainty observer is proposed to control the mover of the LSM to track periodic references. In the proposed control scheme, an adaptive backstepping approach is proposed to compensate the uncertainties in the motion control system. Moreover, to further increase the robustness of the LSM drive, a FNN uncertainty observer is proposed to estimate the required lumped uncertainty in the adaptive backstepping control system. In addition, an on-line parameter training methodology, which is derived using the gradient descent method, is proposed to increase the learning capability of the FNN. The effectiveness of the proposed control scheme is verified by experimental results.

II. C

ONFIGURATION OF

LSM D

^RIVE

The machine model of a LSM can be described in synchronous rotating reference frame as follows [1-3]:

d e q q s

q

R i

v = + λ  + ω λ (1)

q e d d s

d

R i

v ^{= +} λ ^{ −} ω λ ⁽²⁾

where

q q q = Li

λ (3)

PM d d

d Li

λ

λ

= +

(4)

r

e P

ω

ω

=

(5)

and

v ,d vq

are the d and q axis voltages;

i ,d iq

are the d and q axis currents;

Rs

is the phase winding resistance;

q

d L

L ,

are the d and q axis inductances; ω is the angular

r

velocity of the mover; ω is the electrical angular velocity;

e

λ

PM

is the permanent magnet flux linkage;

^P

is the number of pole pairs. Moreover,

τ π

ω

r

= v

r

^/ (6)

e r

e

P ν τ f

ν = = ² (7)

where v

r

is the linear velocity; τ is the pole pitch; ν is

e

the electric linear velocity;

fe

is the electric frequency.

The developed electromagnetic power is given by [2]

( )

^{] /2}

[

3 dq d q dq e

e e

e F P i L L ii

P =

ν

=

λ

+ −

ω (8)

Thus, the electromagnetic force is

( ) ^τ

λ

π

[ ]/2

3 dq d q dq

e P i L L ii

F = + −

(9)

and the mover dynamic equation is

L r r

e

M v Dv F

F =  + + (10)

where

^M

is the total mass of the moving element system;

D

is the viscous friction and iron-loss coefficient; F is

_L

the external disturbance term.

The basic control approach of a LSM servo drive is based on field orientation [2]. The flux position in the d-q coordinates can be determined by the Hall sensors. In (4), (8) and (9), if

= 0

id

, the d-axis flux linkage λ is fixed since

d

λ is

PM

constant for a LSM, and the electromagnetic torque

Fe

is then proportional to

^iq*

which is determined by closed-loop control. The rotor flux is produced in the d-axis only, while the current vector is generated in the q-axis for the field-oriented control. Since the generated motor force is linearly proportional to the q-axis current as the d-axis rotor flux is constant in (4), the maximum force per ampere can be achieved. The resulted force equation is

τ πλ

/2 3 PMq

e i

F =

(11)

The optimal electromagnetic performance for the actuator is therefore realized by controlling the primary current distributions to lie in the q-axis, i.e.,

id =⁰

and this will yield a linear force per amp characteristic for the actuator.

The configuration of a field-oriented LSM servo drive system is shown in Fig. 1, which consists of a LSM, a ramp comparison current-controlled PWM VSI, a field-orientation mechanism, a coordinate translator, a speed control loop, a position control loop, a linear scale and Hall sensors. The flux position of the PM is detected by the output signals of the Hall sensors denoted U, V and W. Different sizes of iron disks can be mounted on the mover of LSM to change the mass of the moving element. With the implementation of field-oriented control, the LSM servo drive can be simplified to a linear control system as follows :

*q f

e

K i

F = (12)

τ λ

π / 2

3

_PM

f

P

K = (13)

D s Ms

H

_p

= 1 + )

( (14)

where K is the thrust coefficient;

_f iq^*

is the command of thrust current; s is the a Laplace's operator. The LSM used in this study is 220V 3.5A 1kW 213N type. For the convenience of the controller design, the position and speed signals in the control loop are set at 1V=0.075m and 1V=0.075m/sec. The parameters of the system are:

N/V 942 . 6 sec / kg 56 . 92

Nsec/V 2025 . 0 2.7kg

N/A, 8 . 60

=

=

=

=

=

D M K

_f

(15)

The " − " symbol represents the system parameter in the

nominal condition.

Fig. 1. Configuration of LSM Drive system.

III.

A

DAPTIVE

B

ACKSTEEPING

F

UZZY

N

EURAL

N

ETWORK

C

ONTROL

S

YSTEM

Consider a drive system with parameter variations, external load disturbance, and friction force for the actual LSM servo drive system, then

X

1

X ^

_p

= (16)

L A a a

m

A X D D U CF

A

X 

₁

= ( + Δ )

₁

+ ( + Δ ) + (17)

p

r

X

X = (18)

where X is the mover position of the LSM;

_r

X is the

_v

mover velocity of the LSM; A

_m

= − D M ; D

_a

= K

_f

M > 0 ;

M

C = − 1 ; Δ A and Δ D

_a

denote the uncertainties introduced by system parameters M and D ; U is the

_A

control input to the LSM drive system. Reformulate (17), then

H U D X A

X 

1

=

_m 1

+

_{a A}

+ (19) where H is named the lumped uncertainty and defined by

L A

a

U CF

D AX

H ≡ Δ

1

+ Δ + (20)

The lumped uncertainty H will be observed by an adaptive uncertainty observer and assumed to be a constant during the observation. The above assumption is valid in practical digital processing of the observer since the

Tb Tc

220V

60Hz Rectifier

L

C PWM

Inverter

Ramp Comparison

Current Control

Limiter

Coordinate Translator Sin/Cos Generator

and Digital

Filter and d/dt

+

−

0

*= id

*

iq

*

ia i_b^* ic^*

Ta

ia

ib

dr

W V U, , ,

LSM

∗

vr

+∑

- +∑

- Position Controller

Speed Controller

∗

dr

dr vr

e

Linear Scale

&

Hall Sensors Tb Tc

220V

60Hz Rectifier

L

C PWM

Inverter

Ramp Comparison

Current Control

Limiter

Coordinate Translator Sin/Cos Generator

and Digital

Filter and d/dt

+

−

0

*= id

*

iq

*

ia i_b^* ic^*

Ta

ia

ib

dr

W V U, , ,

LSM

∗

vr

+∑

- +∑

- Position Controller

Speed Controller

∗

dr

dr vr

e

Linear Scale

&

Hall Sensors

sampling period of the observer is short enough compared with the variation of H .

The control objective is to design an adaptive backstepping FNN control system for the output X of the

_r

system shown in (18) to track the reference trajectory

) (t

X

_m

, which is d , asymptotically. The proposed

_m

adaptive backstepping FNN control system is designed to achieve the position-tracking objective and described step by step as follows.

Step 1:

For the position-tracking objective, define the tracking error as

m

r

X

X

z

1

= − (21)

and its derivative is X

m

X

z 

1

=

1

−  (22)

Define the following stabilizing function:

X

m

z c + 

−

=

1 1

α

1

(23)

where c is positive constant. The first Lyapunov function

₁

is chosen as

12

1

2

1 z

V = (24)

Define z

₂

= X

₁

− α

₁

, then V the derivative of is

₁

12 1 2 1 1

2 1 1

1

1

z ( X X ) z ( z X ) z z c z

V  = − 

_m

= + α − 

_m

= − (25) Step 2:

The derivative of z is now expressed as

₂

1 1

2

 α

 = X −

z = A

_m

X

1

+ D

_a

U

_A

+ H + c

1

z

1

− X  

_m

( )

a A m

m

z D U H c z X

A + + + +  −  

=

2

α

1 1 1

(26) To design the backstepping control system, the lumped uncertainty H is assumed to be bounded, i.e., H ≤ H , and define the following Lyapunov function:

22 1

2

2

1 z V

V = + (27)

Using (25) and (26), the derivative of V can be derived as

₂

follows:

) ( )

,

(

_{1 2 1 2 2 1 1}² _{2 1 2}

2

z z V z z c z z z z

V  =  +  = − + + 

[ ^{z A z D U H} ]

z z

c + +

_m

+ +

_{a A}

+

−

=

_{1 1}² _{2 1}

(

₂

α

₁

) ]

[

_{1 1}

2

c z X

m

z  −  

+ (28) According to (28), a backstepping control law U is

_A

designed as follows:

)]

(

[

_{2 2 1 1 2}

1

c z z A H sgn z

D

U

_A

=

_a⁻

− − −

_m

α − ) (

_{1 1}

1 m

a

c z X

D −  +  

+

⁻

(29)

where c is positive constant. Substituting (29) into (28),

₁

the following equation can be obtained:

2 2 2

2 1 1

2 2 2 2

2 1 1

2 2 2

2 2 2

1 1 2 1 2

) (

) ( )

( ) ( )

, (

z A c z c

H H z z A c z c

H z H z z A c z c z z V

m m m

−

−

−

≤

− +

−

−

−

≤

− +

−

−

−

 =

≤ 0 (30)

Define the following term:

) , ( )

( )

( t c

₁

z

₁²

c

₂

A z

₂²

V

₂

z

₁

z

₂

W = + −

_m

≤ −  (31)

Then

∫

₀^t

W ( τ ) d τ ≤ V

₂

( z

₁

(0) , z

₂

(0)) − V

₂

( z

₁

( t ) , z

₂

( t )) (32) Since V

₂

( z

₁

(0) , z

₂

(0)) is bounded, and V

₂

( z

₁

( t ) , z

₂

( t )) is nonincreasing and bounded, then ∫ < ∞

∞

→ t

t

lim

₀

W ( τ ) d τ . Moreover, ^W ( ) ^t is bounded, then and ^W ( ) ^t is uniformly continuous [20-21]. By using Barbalat’s lemma [20-21], it can be shown that lim ( ) = 0

→∞

W t

t

. That is z and

₁

z will

₂

converge to zero as t → ∞ . Moreover,

_{r m}

t

X t = X

∞

→

( )

lim and

t

X = X 

m

∞

→ 1

lim . Therefore, the backstepping control system is asymptotically stable, even if parametric uncertainty, external load disturbance.

Step 3:

Since the lumped uncertainty H is unknown in practical application, the upper bound H is difficult to determine, therefore, a FNN uncertainty observer is proposed to adapt the value of the lumped uncertainty Hˆ . A four-layer FNN as shown in Fig. 2, which comprises the input (the i layer), membership (the j layer), rule (the k layer) and output layer (the o layer), is adopted to implement the FNN controller in this study. The signal propagation and the basic function in each layer of the FNN are introduced as follows:

Fig. 2. Structure of FNN.

Rule Layer

M embership Layer Output Layer

Input

Layer i

j k

Σ

o

3

wjk

4

wko

1

yi 2

yj 3

yk 4

yo

1

xi

Π Π Π Π

Ω Ψ

=

Ψ ^T

Hˆ( )

z1 z₁

Rule Layer

M embership Layer Output Layer

Input

Layer i

j k

Σ

o

3

wjk

4

wko

1

yi 2

yj 3

yk 4

yo

1

xi

Π Π Π Π

Ω Ψ

=

Ψ ^T

Hˆ( )

z1 z₁

Layer 1: Input Layer

For every node i in this layer, the net input and the net output are represented as

( ^{( )} ) ^{( ) , 1 , 2}

) (

, ) ( ) (

1 1

1 1

1 1

=

=

=

=

i N net N net f N y

N x N net

i i

i i

i i

(33) where x

₁¹

= and z

₁

x

¹₂

= ; N denotes the number of z 

₁

iterations.

Layer 2: Membership Layer

In this layer, each node performs a membership function.

The Gaussian function, is adopted as the membership function. For the jth node

( )

(

^net

( )

^N

) (

^{net N}

)

^{j n}

f N y

m N x

net

j j

j j

ij ij j i

, , 1 , ) ( exp ) ( )

(

, )

(

2 2

2 2

2 2 2 2

"

=

=

=

− −

=

σ

(34) where m

_ij

and

_σij

are, respectively, the mean and the standard deviation of the Gaussian function in the jth term of the ith input linguistic variable x to the node of layer 2,

ⁱ²

and n is the total number of the linguistic variables with respect to the input nodes.

Layer 3: Rule Layer

Each node k in this layer is denoted by

∏

, which multiplies the input signals and outputs the result of product. For the kth rule node

(

^{net (N)}

)

^{net (N) k l}

f ) N ( y

) N ( x w ) N ( net

k k

k k

j jk j k

, , 1 , ,

3 3

3 3

3 3 3

"

=

=

=

=∏

(35) where x represents the jth input to the node of layer 3;

³j

3

w , the weights between the membership layer and the rule

jk

layer, are assumed to be unity; ^{l =} ( ) ^{n / i}

ⁱ

is the number of rules with complete rule connection if each input node has the same linguistic variables.

Layer 4: Output Layer

The single node o in this layer is labeled with ∑ , which computes the overall output as the summation of all input signals

= ∑

k ko k

o

N w x N ,

net

⁴

( )

^{4 4}

( )

( ( ) ) ^{( )}

^{, 1}

)

( ^{4 4 4}

4 N = f net N =net N o=

y_{o o o o}

(36)

where the connecting weight

w⁴ko

is the output action strength of the oth output associated with the

k

th rule;

xk⁴

represents the

k

th input to the node of layer 4. The output of the FNN is rewritten as follows:

Ω Ψ

=

Ψ

^T

H ˆ ( )

(37)

where the error signal is the input of the FNN;

[ ^w

¹¹⁴

^w

²¹⁴

^{" " w}

^l41

]

^T

=

Ψ is the adjustable parameter vector of

the FNN; _{Ω =} [ x

₁⁴

x

₂⁴

" " x

_l⁴

]

^T

, in which x

k⁴

is determined by the selected membership function and

1

0 ≤ x

_k⁴

≤ . To develop the adaptation laws of the FNN uncertainty observer, the minimum reconstructed error Q is defined as follows:

) ( Ψ

^*

−

= H H

Q (38)

where Ψ is an optimal weight vector that achieves the

^*

minimum reconstructed error, and the absolute value of Q is assumed to be less than a small positive constant, Q (i.e.,

Q

Q ≤ ). Then, a Lyapunov candidate is chosen as ) (

) 2 (

) 1 ( ˆ 2

1

_{2 * *}

2

3

= V + Q − Q + Ψ − Ψ

^T

Ψ − Ψ

V γ λ (39)

Where γ ^and λ are positive constants; Qˆ is the estimated value of the minimum reconstructed error Q . The estimation of the reconstructed error Q is to compensate the observed error induced by the FNN uncertainty observer and to further guarantee the stable characteristic of the whole control system. Take the derivative of the Lyapunov function from (39)

[ ^{z A z D U H} ]

z z c

Q Q Q V

V

A a m

T

+ +

+ +

+

−

=

Ψ Ψ

− Ψ +

− +

=

) (

) 1 (

) ˆ ( ˆ 1

1 2 1

2 2 1 1

2 * 3

λ α

λ ^

 



Ψ Ψ

− Ψ +

− +

−

+ z

₂

[ c

₁

z 

₁

X  

_m

] 1 ( Q ˆ Q ) Q  ˆ 1 (

^*

)

^T

 λ

γ ⁽⁴⁰⁾

According to (40), an adaptive backstepping control law is proposed as follows:

ˆ ]

[

_{2 2 1 1}

ˆ

_{1 1}

1

m m

m

A

B c z z A Q H c z X

U =

⁻

− − − α − − −  +   (41) Substituting (41) into (40), the following equation can be obtained

Ψ Ψ

− Ψ +

− +

−

− +

−

−

−

=

 



T m

Q Q Q

Q z H z H z z A c z c V

) 1 (

) ˆ ( ˆ 1

ˆ ) ˆ

(

*

2 2 2 2

2 2 2

1 1 3

λ λ

Ψ Ψ

− Ψ +

− + Ψ

− Ψ +

Ψ

− +

−

−

−

−

=



_T



m

Q Q Q H

H z

H H z Q z z A c z c

) 1 (

) ˆ ( ˆ )) 1 ˆ ( ) ˆ ( (

)) ˆ ( ˆ (

) (

* 2 *

2 * 2 2

2 2 2

1 1

λ λ

Ω Ψ

− Ψ

−

−

−

−

−

−

= c

₁

z

₁²

( c

₂

A

_m

) z

₂²

z

₂

( Q ˆ Q ) z

₂

(

^*

)

^T

Ψ

Ψ

− Ψ +

−

+ 

_T



Q Q

Q ˆ ) ˆ 1 ( )

1 (

_*

λ

γ (42) The adaptation laws for Ψ and Q ˆ are designed as follows:

Ω

=

Ψ ^ λ z

₂

⁽⁴³⁾

ˆ z

2

Q  = γ

(44) Thus, (42) can be rewritten as follows:

0 ) ( )

(

_{2 2}²

12 1

3

= − c z − c − A z = W t ≤

V 

_m

(45)

By using Barbalat’s lemma [20-21], it can be shown that 0

) ( t →

W as t → ∞ . That is z and

₁

z will converge to

₂

zero as t → ∞ . As a result, the stability of the proposed

adaptive backstepping FNN control system, which is shown in Fig. 3, can be guaranteed. On the other hand, the guaranteed convergence of tracking error to be zero does not imply convergence of the estimated value of the lumped uncertainty to it real values. The persistent excitation condition [20-21] should be satisfied for the estimated value to converge to its theoretic value.

In order to train the FNN effectively, an on-line parameter training methodology [16], which is derived using the Lyapunov stability theorem and the gradient descent method, is omitted. Moreover, the update law [16] is omitted here.

Fig. 3. Adaptive backstepping FNN control system.

IV. E

XPERIMENTAL

R

ESULTS

A block diagram of the PC-based computer control system for a LSM servo drive is depicted in Fig. 1. The proposed controllers are implemented using a Pentium III computer. The current-controlled PWM VSI is implemented by the IGBT power modules with a switching frequency of 15kHz. A servo control card is installed in the control computer, which includes multi-channels of D/A and encoder interface circuits. The coordinate transformation in the field-oriented mechanism is implemented by computer.

The control gains of the proposed adaptive backstepping FNN control system are given in the following:

1

= 10

c , 35 c

₂

= , γ

=0.2

, λ = 0 . 1 ⁽⁴⁶⁾

Some experimental results are provided to demonstrate the control performance of the proposed control systems.

Two conditions are provided in the experimentation here, one being the nominal condition, another being the increasing of the mover mass to approximately 3 times the nominal value. The experimental results of the adaptive backstepping FNN control system due to periodic step and periodic sinusoidal command at the nominal case and the parameter variation case are shown in Fig. 4 and Fig. 5. The position responses of the mover at the nominal case and the parameter variation case are shown in Figs. 4(a), 4(c), 5(a) and 5(c); the associated control efforts are shown in Figs.

4(b), 4(d), 5(b) and 5(d). However, the robust control performance of the proposed adaptive backstepping FNN control system under the occurrence of parameter variations at different trajectories are obvious owing to the on-line adaptive adjustment of the FNN uncertainty observer. From

the experimental results, the control performance of the proposed the adaptive backstepping FNN control system is favorable for the tracking of periodic commands.

Fig.4. Experimental results of adaptive backstepping FNN control system due to periodical step command: (a) mover position at nominal case; (b) control effort at nominal case; (c) mover position at parameter variation case; (d) control effort at parameter variation case.

V.

C

ONCLUSIONS

The linear synchronous motor (LSM) drive system using adaptive backstepping fuzzy neural network (ABFNN) control is investigated for the tracking of periodic reference inputs. First, the field-oriented mechanism is applied to formulate the dynamic equation of the LSM servo drive.

Then, an adaptive backstepping approach is proposed to

1sec

5A Reference Model

Mover Position

8mm 0mm

0A

8mm

UA

(a)

(b)

1sec 1sec

5A Reference Model

Mover Position

8mm 0mm

0A

8mm

UA

(a)

(b)

1sec

Reference Model

Mover Position

8mm 0mm

5A 0A

8mm

UA

(c)

(d)

1sec

1sec Reference

Model

Mover Position

8mm 0mm

5A 0A

8mm

UA

(c)

(d)

1sec

1sec

+ X1

Xr

−1

Da KfF^e vr _dr

s 1

LSM Drive System FL

D Ms+

1 UA

Σ Σ Σ

Σ Σ

+ -

+ - Adaptive Backstepping FNN Control System

-

c1 ²

1 z

z α₁

-

Xm - -

c2 -

s

s2

s z¹ Am

c1 γ ^s−1Qˆ Adaptive

Law s

Fuzzy Neural Network

λ

Ω Ψ

z1 z1

Hˆ

- + - + +

X1 Xr

−1

Da KfF^e vr _dr

s 1

LSM Drive System FL

D Ms+

1 UA

ΣΣ ΣΣ Σ

ΣΣ ΣΣ

+ -

+ - Adaptive Backstepping FNN Control System

-

c1 ²

1 z

z α₁

-

Xm - -

c2 -

s

s2² s s z¹

Am

c1 γ ^s−1Qˆ Adaptive

Law s

Fuzzy Neural Network

λ

Ω Ψ

z1 z1

Hˆ

- + - +

compensate the uncertainties in the motion control system.

With the proposed adaptive backstepping control system, the mover position of the LSM drive possesses the advantages of good transient control performance and robustness to uncertainties for the tracking of periodic reference trajectories. Moreover, to further increase the robustness of the LSM drive, a FNN uncertainty observer is proposed to estimate the required lumped uncertainty in the adaptive backstepping control system. The effectiveness of the proposed control scheme is verified by the experimental results.

Fig. 5. Experimental results of adaptive backstepping FNN control system due to periodical sinusoid command: (a) mover position at nominal case; (b) control effort at nominal case; (c) mover position at parameter variation case; (d) control effort at parameter variation case.

A

CKNOWLEDGMENTS

The author would like to acknowledge the financial support of the National Science Council in Taiwan, R.O.C.

through its grant NSC 97-2221-E-239-045-MY2.

R

EFERENCES

[1] I. Boldea and S. A. Nasar, Linear Electric Actuators and Generators. London: Cambridge University Press, 1997.

[2] T. Egami, and T. Tsuchiya, “Disturbance Suppression Control with Preview Action of Linear DC Brushless Motor,” IEEE Trans.

Ind. Electronics, vol. 42, no. 5, pp. 494-500, Oct. 1995.

[3] M. Sanada, S. Morimoto, and Y. Takeda, “Interior Permanent Magnet Linear Synchronous Motor for High-Performance Drives,”

IEEE Tran. Ind. Appl., vol.33, no. 4, pp. 966-972, July/Aug. 1997.

[4] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controller for feedback linearizable system,”

IEEE Trans. Automat. Contr., vol. 36, pp. 1241-1253, 1991.

[5] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, “Adaptive nonlinear control without overparameterization,” Syst. Contr. Lett., vol. 19, pp. 177-185, 1992.

[6] M. Krstic and P. V. Kokotovic, “Adaptive nonlinear design with controller-identifier separation and swapping,” IEEE Trans.

Automat. Contr., vol. 40, pp. 426-440, 1995.

[7] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995.

[8] A. Stotsky, J. K. Hedrick, and P. P. Yip, “The use of sliding modes to simplify the backstepping control method,” Proc. of American Control Conf., pp. 1703–1708, 1997.

[9] G. Bartolini, A. Ferrara, L. Giacomini, and E. Usai, “Peoperties of a combined adaptive/second-order sliding mode control algorithm for some classes of uncertain nonlinear systems,” IEEE Trans.

Automat. Contr., vol. 45, pp. 1334–1341, 2000.

[10] F. J. Lin, P. H. Shen, and S. P. Hsu, “Adaptive backstepping sliding mode control for linear induction motor drive,” IEE Proc., Electr. Power Appl., vol. 149, pp. 184–194, 2002.

[11] C. T. Lin and C. S. G. Lee, “Neural-network-based fuzzy logic controland decision system,” IEEE Trans. Comput., vol. 40, pp.

1320–1336, Dec. 1991.

[12] Y. C. Chen and C. C. Teng, “A model reference control structure using a fuzzy neural network,” Fuzzy Sets Syst., vol. 73, no. 3, pp.

291–312, Aug. 1995.

[13] J. Zhang and A. J. Morris, “Fuzzy neural networks for nonlinear systems modelling,” Proc. IEE—Contr. Theory Applicat., vol. 142, no. 6, pp. 551–556, 1995.

[14] T. S. R. Jang and C. T. Sun, “Neural-fuzzy modeling and control,”

Proc. IEEE, vol. 83, pp. 378–405, Mar. 1995.

[15] Y. G. Leu, T. T. Lee and W. Y. Wang, “On-line turning of fuzzy-neural network for adaptive control of nonlinear dynamical systems,” IEEE Trans. Syst. Man, Cybern., vol. 27, pp. 1034-1043, 1997.

[16] F. J. Lin, W. J. Hwang, and R. J. Wai, “A supervisory fuzzy neural network control system for tracking periodic inputs,” IEEE Trans.

Fuzzy Syst.,, vol. 7, no.1, pp. 41-52, 1999.

[17] L. X. Wang, A Course in Fuzzy Systems and Control. Englewood Cliffs, NJ: Prentice-Hall, 1997.

[18] F. J. Lin, R. J. Wai, and H. P. Chen, “A PM synchronous servo motor drive with an on-line trained fuzzy neural network controller,” IEEE Transactions on Energy Conversion, vol. 13, no.

4, pp. 319-325, Dec. 1998.

[19] C. T. Lin and C. S. G. Lee, Neural Fuzzy System: A Neural-Fuzzy Synergism to Intelligent System, Englewood Cliffs, NJ:

Prentice-Hall, 1996.

[20] J. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.

[21] K. J. Astrom,andB.Wittenmark,Adaptive Control. New York, Addison-Wesley, 1995.

5A Mover Position 8mm

0mm

0A Reference Model

-8mm 8mm

UA

(a)

1sec

1sec (b)

5A Mover Position 8mm

0mm

0A Reference Model

-8mm 8mm

UA

(a)

1sec

1sec (b)

Reference Model

Mover Position 8mm

0mm

5A 1sec

1sec

0A

8mm

-8mm

UA

(d) (c) Reference

Model

Mover Position 8mm

0mm

5A 1sec

1sec

0A

8mm

-8mm

UA

(d) (c)