Adaptive Fuzzy Dynamic Surface Control for Ball

International Journal of Fuzzy Systems, Vol. 13, No. 1, March 2011

1

Adaptive Fuzzy Dynamic Surface Control for Ball and Beam System Yeong-Hwa Chang, Wei-Shou Chan, Chia-Wen Chang, and C. W. Tao Abstract1 This paper mainly investigates the control for the ball and beam system. Combined with the conventional dynamic surface control, an adaptive fuzzy scheme is proposed for the equilibrium balance of the ball. First, with the proper defined tracking errors of state variables, an integrating of tracking errors is considered as the fuzzy input. The parameters of fuzzy controller can be optimized by using the proposed adaptive mechanism such that the control performance can be improved. For the proposed control scheme, the closed-loop stability is also analyzed by the Lyapunov theorem. Simulations are performed to evaluate the effectiveness of the proposed works. From simulation results, the proposed adaptive fuzzy dynamic surface control scheme can provide better tracking responses than the conventional dynamic surface control counterparts. Keywords: Ball and Beam System, Adaptive Fuzzy Dynamic Surface Control, Dynamic Surface Control, Underactuated System.

1. Introduction In the real world, physical systems are often under-actuated, where the degree of inputs is smaller than the degree of state space. The control of under-actuated systems is challengeable and attracts a worthy discussion. Ball and beam system, a typical under-actuated system, is a representative platform in academics due to the inherent non-linearity and instability. In the literature, the fuzzy control theory is utilized for the beam and ball system. Based on the experts’ experience, fuzzy controllers can provide satisfactory responses without the knowledge of exactly mathematical models [1]-[3]. However, the parameters of conventional fuzzy controller are required to be manually tuned to get better performance. Also, lack of Corresponding Author: C. W. Tao is with the Department of Electrical Engineering, National I-Lan University, No. 1, Sec. 1, Shen-Lung Rd., I-Lan, Taiwan, 260. E-mail: [email protected] Manuscript revised Jan. 2009; accepted Dec. 2010.

systematic procedure to analyze the stability of closed-loop system is another task under taken. Backstepping control, typically addressed in nonlinear systems, is a systematic design procedure, where the stability analysis relies on the Lyapunov stability theorem [4]-[9]. Nevertheless, as the iterative procedure going on, the formulation of the stabilizing control command will be more complex with the increasing of system order. In [10]-[15], to relieve the possible derivation burden, a so-called dynamic surface control (DSC) was proposed. Similar to the backstepping control, DSC is also an iterative design procedure. First, through the Lyapunov stability analysis, a virtual desired trajectory of each state is defined. Then a stabilizing function is determined and a low-pass filter is utilized to overcome the derivation burden mentioned in the backstepping control. As the tracking error of virtual objective function converges, the required stabilizing controller can be iteratively derived. In this paper, the last step of the conventional DSC is replaced with an adaptive fuzzy framework. The adaptive tuning mechanism can improve the tracking performance by optimizing the fuzzy parameters. The proposed adaptive fuzzy DSC (AFDSC) is utilized for a ball-and-beam system to balance the ball. The organization of this paper is as follows. In Sec. 2, the dynamic characteristics of the ball-beam system are discussed. The derivation procedures of the AFDSC are investigated in Sec. 3. The closed-loop stability of the controlled ball-and-beam system with the AFDSC is addressed in Sec. 4. In Sec. 5, simulation results and the comparisons with conventional DSC are provided. The concluding remarks are given in Sec. 6.

2. Ball and Beam System The ball and beam system is a classic dynamic system discussed in academics. It is inherently non-linear and unstable, and despite its simple nature. The ball and beam system with a central pivot point housing a servo motor is shown in Fig. 1, where balancing the ball on the beam is the main task of concern. To derive the dynamic model of the ball and beam system, the following Euler-Lagrange dynamic equation is considered.

© 2011 TFSA

d ⎡ ∂L ⎤ ∂L − =Q dt ⎢⎣ ∂q ⎥⎦ ∂q

(1)


2

where L = K − P , K is the kinetic energy, P is the potential energy, Q is the generalized force, and q is the generalized coordinate. The kinetic energy and potential energy of the ball and beam system can be determined as follows J K = 0.5(mB + B2 )r 2 + 0.5( J b + J B + mB r 2 )θb2 (2) R (3) P = mB gr sin θ b where J b is the inertia of the beam, J B is the inertia of the ball, mB is the mass of the ball, R is the radius of the ball, g is the gravity, r is the position of ball and θb is the angle of beam. The ball and beam system is an under-actuated system, where the driving torque is directly coupled on the variation of the beam angular. In (2) and (3), r and θb are two dynamic variables. Thus, from (1), the Euler-Lagrange dynamic equation of the ball-and-beam system can be represented as d ⎡ ∂L ⎤ ∂L − =0 dt ⎢⎣ ∂r ⎥⎦ ∂r d ⎡ ∂L ⎤ ∂L =τ ⎢ ⎥− dt ⎣ ∂θb ⎦ ∂θ b

eb is the back EMF voltage. According to the energy

conservation, the following energy equivalence holds (9) eb ⋅ ia = τ m ⋅ θm It is known that eb is proportional to the angular speed, e b = k eθm for some constant ke . From (8) and (9), it can be obtained that ke = kt . Since La is small enough to be neglected, (7) and (8) can be simplified as (10) e = ra ia + k eθm where ia = ke−1τ m . In Fig. 2, the gear ratio is denoted as n, and the following relationships are known θ m θm τ = = =n θ b θb τ m

By some manipulations, the torque to the ball and beam system can be derived as (12) τ = nra−1ke e − ra−1ke2θb = nra−1keu − n 2 ra−1ke2 x4 l 2

(4)

r

θb

where τ is the driving torque. Substituting (2) and (3) into (4), equivalently, we have JB ⎤ r ⎡ 0 ⎥⎡ ⎤ + ⎢(m B + R 2 ) ⎥ ⎢θ ⎥ ⎢ 0 ( J b + J B + mB r 2 )⎦ ⎣ b ⎦ ⎣ ⎡ 0 ⎢ ⎣mB rθb

2R

(5)

− mB rθb ⎤ ⎡ r ⎤ ⎡ mB g sin θb ⎤ ⎡0⎤ ⎥⎢ ⎥ + ⎢ ⎥=⎢ ⎥ mB rr ⎦ ⎣θb ⎦ ⎣mB gr cosθb ⎦ ⎣τ ⎦

Fig. 1. Illustration of the ball-and-beam system.

Let the state variables be defined as x1 = r , x2 = r , x3 = θb , and x4 = θb . Thus, the state space representation of the ball-and-beam system can be described in the following (4) (6)

+ e −

di e = La a + ra ia + eb dt dθ τ m = k t ia = J m m + Bmθm dt

(7) (8)

where θm is the angular speed, La is armature inductance, ra is the armature resistance, ia is the armature exciting current, kt is the torque constant, and

+

ia

eb −

n

Jm, Bm

τ ,θ b ,θb

τ m ,θ m ,θm

Fig. 2. Equivalent circuit of the armature-controlled DC motor.

x 4 = a (τ − 2mB x1 x2 x4 − mB gx1 cos x3 )

where a = ( J b + J B + mB x12 )−1 , b = mB (mB + J B R −2 ) −1 . It is noted that the driving torque is provided with a DC motor, shown in Fig. 2. The characteristics of the DC motor are given as

La

ra

x1 = x2

x 2 = bx1 x42 − bg sin x3 x3 = x4

(11)

Table 1. Parameters of the ball-and-beam system. mB

0.01kg

La

0.026 H

mb l

0.11kg

ke

0.058 V/rad/s

1m

ra

3.365Ω

R

0.006m

Jm

2.76×10−4 kg - m2 5.23×10−4 Nm / rad / s 15

0.058 Nm / A

−6

JB

0.144 × 10 kg - m

Jb g

9.6 × 10 −3 kg - m 2

Bm n

9.8m/s 2

kt

2

Substituting (12) into (5), we have the following equivalent state-space representation

Yeong-Hwa Chang et al.: Adaptive Fuzzy Dynamic Surface Control for Ball and Beam System

x1 = x2 x2 = bx1 x42 − bg sin x3

x3 = x4

(14) (15)

x4 = nra−1ak eu − n 2 ra−1ak e2 x4 − 2amB x1 x2 x4

(16)

− amB gx1 cos x3

τ 3 x3d (t ) + x3d (t ) = x3 , x3d (0) = x3 (0)

(13)

3. Adaptive Fuzzy Dynamic Surface Control An adaptive scheme of the dynamic surface control will be discussed in this section. Step 1: Define S1 = x1 − yd (17) where x1 is the position of ball, and yd is the expected position or command. It is desired to track yd with any initial condition. Taking the derivative of (17), we have (18) S1 = x1 − y d . Suppose x2 is a stabilizing function to be determined for (13). Choosing the Lyapunov function candidate V1 ( S1 ) = 0.5S12 , from (13) and (18), the relative derivative can be obtained as (19) V1 ( S1 ) = S1S1 = S1 ( x2 − y d ) It can be seen that (19) is negative definite if (20) x2 = x2 = −k1S1 + y d , k1 > 0 . Thus, with the designated x2 of (20), the state variable x1 can stably track the yd . To eliminate the phenomenon of oscillation, a low-pass filter is considered as follows, (21) τ 2 x2d (t ) + x2d (t ) = x2 , x2 d (0) = x2 (0) . With a proper chosen τ 2 , the smoothed x2 d (t ) can be equivalently considered as the required x2 . Step 2: Analogous to the discussion in Step 1, an error variable is defined as (22) S 2 = x2 − x2 d . From (14), the derivative of (22) can be available as (23) S 2 = x 2 − x 2 d = bx1 x42 − bg sin x3 − x 2 d . A Lyapunov function candidate is chosen as V2 ( S 2 ) = 0.5S 22 . From (23), the derivative of V2 can be derived as follows V2 ( S 2 ) = S 2 S 2 = S 2 ( Bx1 x42 − Bg sin x3 − x 2 d ) (24) It can be observed that (24) is negative definite if x3 = x3 = sin −1[(bg ) −1 (bx1 x42 − x 2 d + k 2 S 2 )], k 2 > 0 (25) Thus the state variable x2 asymptotically tracks x2 d with the designated x3 of (25). Also, to eliminate the oscillation, a low pass filter is utilized as follows

3

(26)

Step 3: The following derivation is similar to the discussion in Step 2. First, an error variable is defined as (27) S3 = x3 − x3d From (15), the derivative of (27) can be obtained as (28) S3 = x 3 − x3 d = x 4 − x3 d A Lyapunov function candidate is chosen as V3 ( S 3 ) = 0.5S 32 . From (28), the derivative can of V3 can be obtained as V3 ( S 3 ) = S 3 S3 = S 3 ( x4 − x3d ) (29) To ensure the negative definiteness of V 3 ( S 3 ) , the required x4 can be designed as x4 = x4 = x3d − k3S3 , k3 > 0 (30) It can be seen that V3 ( S3 ) is negative definite as

expected.

τ 4 x4d (t ) + x4d (t ) = x4 , x4 d (0) = x4 (0)

(31)

Step 4: Following from the Step 3, an error state is defined as (32) S 4 = x4 − x 4 d Before proceeding to the next, it will be helpful to identify the directionality of state variables. In Fig. 2, if the ball is in the right of the central pivot, x1 is positive. In addition, a positive x3 means that the beam is in the counterclockwise direction from the horizontal equilibrium state. Consequently, it can be figured out that the ball is moving to the right with a positive x2 and the beam is rotating counterclockwise with a positive x4 . Referred to (17), (22), (27), and (32), S1 , S 2 , S 3 and S 4 represent the tracking errors of ball position, ball velocity, beam angle and beam angular velocity, respectively. In order to speed the convergence rate, an integrated variable is defined as follows S = − S1 − S 2 + S 3 + S 4 (33) It is noted that the signs designated in (33) are based on the physical consistency. For example, if S1 and S 2 are negative, and S 3 and S 4 are positive, the relative S is positive. In this situation, referred to the signs defined before, negative S1 and S 2 indicate that the ball is in the left to the central pivot, and the ball is further moving away to the left. Analogously, positive S 3 and S 4 means that the beam is rotating in the counterclockwise direction. Thus a clockwise torque is required for the purpose of ball balancing, i.e. a positive S represents a counterclockwise torque. When S converges to zero, all of the subsystems guaranteed to be stable and the system states can


4

asymptotically track the respective virtual commands. In this paper, combined with the conventional DSC, an adaptive fuzzy controller is proposed, where S is considered as the input variable. Without loss of generality, the input and output membership functions are Gauss and singleton functions, respectively. The input membership functions are represented as follows ξ (S , β , C) = [ξ1(S , β1, c1),ξ2 (S , β2 , c2 ),...,ξM (S , βM , cM )]T

T ⎡ ∂ξ1 ⎡ ∂ξ1 ⎤ ~ ⎢ ∂β ⎡ ξ1 ⎤ ⎢ ∂c ⎥ ⎢ ⎥ ⎢~ ⎥ ⎢ ∂ξ ~ ⎢ ξ 2 ⎥ ⎢ ∂ξ 2 ⎥ ~ ⎢ 2 = ξ = C + ⎢ ∂β ⎢ # ⎥ ⎢ ∂c ⎥ ⎢ # # ⎢ ⎥ ⎢~ ⎥ ⎢ ∂ξ M ⎢⎣ξ M ⎥⎦ ⎢ ∂ξ M ⎥ ⎢ ⎢⎣ ∂c ⎥⎦ |c=cˆ ⎣ ∂β ~ ~ = GC + Hβ + h.o.t.

T

⎤ ⎥ ⎥ ⎥ ~ β + h.o.t. ⎥ (41) ⎥ ⎥ ⎥ ⎦ |β =βˆ

ξi ( S , β i , ci ) = exp(− β i2 ( S − ci ) 2 ) , i = 1,..., M (34) ~ where β i and ci respectively represent the width and where h.o.t. means the high-order terms, C = C − Cˆ , ~ β = β − βˆ . center of the ith input membership function, ∂ξ1 ∂ξ 2 ∂ξ M β = [ β1 , β 2 ,..., β M ]T ， C = [c1 , c2 ,..., cM ]T Let the singleton output membership function be represented as θ = [θ1,θ 2 ,...,θ M ]T . It is known that the performance of fuzzy control can be improved by tuning the parameters of membership functions. Let u fuzzy ( S , ξ ,θ ) be denoted as the fuzzy output corresponding to the fuzzy input S and membership functions ξ (S , β , C) . In this paper, an adaptive fuzzy control scheme is adopted as follows (35) u~ fuzzy = u fuzzy ( S , ξ ,θ ) − uˆ fuzzy ( S , ξˆ,θˆ) where ξˆ and θˆ are the optimized estimation parameters and uˆ fuzzy ( S , ξˆ,θˆ) is the associated defuzzified output, and

u~ fuzzy is the estimation deviation.

The estimation errors are defined as ~ ξ = ξ − ξˆ ~ θ = θ − θˆ The fuzzy IF-THEN rules are expressed as R l ：IF S is F l , THEN uˆ fuzzy is G l , l = 1,..., M

(36) (37)

(38)

Thus the defuzzified output can be obtained as M

uˆ fuzzy = ∑θîξî ( S , βî , cî ) = θˆT ξˆ( S , βˆ , Cˆ )

(39)

i =1

in which

βˆ = [ βˆ1 , βˆ2 ,..., βˆM ]T , Cˆ = [cˆ1 , cˆ2 ,..., cˆM ]T , ξˆ(S , βˆ , Cˆ ) = [ξˆ1(S , βˆ1, cˆ1),ξˆ2 (S , βˆ2 , cˆ2 ),...,ξˆM (S , βˆM , cˆM )]T θˆ = [θˆ ,θˆ ,...,θˆ ]T . 1

2

M

The proposed controller is described as u (t ) = uˆ fuzzy ( S ,θˆ, βˆ , Cˆ ) + uc (t )

(40)

where uc (t ) is the compensation controller. Taking the power series expansion, we have the following linearization equation

G =[

" ]| ˆ (42) ∂c ∂c ∂c C =C ∂ξ ∂ξ 2 ∂ξ M H =[ 1 " ]| ˆ (43) ∂β ∂β ∂β β = β ∂ξ i T ∂ξ i ∂ξ ∂ξ i =[ i " (44) ] ∂C ∂c1 ∂c2 ∂cM ∂ξ i T ∂ξ i ∂ξ ∂ξ i =[ i " ] (45) ∂β ∂β1 ∂β 2 ∂β M From (35)-(37) and (40)-(41), we have u (t ) = u fuzzy − u~ fuzzy + uc (t ) ~ ~ = θ Tξ − θ Tξ + uc (t ) ~ ~ ~ ~ = (θ + θˆ) T (ξ + ξˆ) − θ Tξ + uc (t ) (46) ~ T ˆ ˆT ~ ˆT ˆ = θ ξ + θ ξ + θ ξ + uc (t ) ~ ~ ~ = θ Tξˆ + θˆ T (GC + Hβ + h.o.t.) + θˆ Tξˆ + uc (t ) Let ε = θˆT (h.o.t.) + θˆT ξˆ . Eq. (46) can be rewritten as ~ ~ ~ u (t ) = θˆ T (GC + Hβ ) + θ Tξˆ + ε + uc (t ) (47) From (16), (32) and (47), the derivative of S 4 can be represented as ~ ~ ~ S4 = nra−1ake (θˆ T (GC + Hβ ) + θ Tξˆ + ε + uc (t )) (48) − n 2 ra−1ake2 x4 − 2amB x1 x2 x4 − amB gx1 cos x3 − x 4 d

It is denoted that η = nra−1akeε − n2 ra−1ake2 x4 − 2amB x1 x2 x4 − amB gx1 cosx3 −x 4d Equivalently, from (48), we have ~ ~ ~ S 4 = nra−1ak e (θˆ T (G C + Hβ ) + θ Tξˆ + u c (t )) + η (49) It is known that η is a uncertain term. Suppose η is bounded such that η ≤ E for a constant E . However, since E is unavailable, it is sufficient to replace E by the estimation value Eˆ , and the estimation error is defined as ~ E = E − Eˆ (50) With the following adaptive law and uc (t ) , the associated tracking errors of the closed-loop system will converge to zeros,


(52) (53) (54) (55) (56)

4. Stability Analysis In this section, the stability of the AFDSC controller ball-and-beam system will be analyzed. First, a Lyapunov function candidate is chosen as ~ ~ ~ ~ ~ ~ ~ 3 E 2 C T C β T β θ Tθ V = 0.5∑Vi + 0.5S 42 + + + + (57) 2r1

i =1

2r2

2r3

2r4

From (49), the derivative of (57) can be obtained as 1 ~ 1 ~ 1 ~ 1 ~ V = V1 + V2 + V3 + S 4 S 4 − EEˆ − C T Cˆ − β T βˆ − θ T θˆ r1 r2 r3 r4 ~ 2 2 2 T −1 ˆ = −k S − k S − k S + S {nr ak [θ (GC 2

2

3

3

4

a

e

1 ~ 1 ~ ~ + Hβ ) + θ T ξˆ + u c (t )] + η} − EEˆ − C T Cˆ r1 r2 ~

−

1 ~ T ˆ 1 ~ T ˆ β β− θ θ r3 r4

(58) Equivalently, we have ~ V = −k1 S12 − k 2 S 22 − k 3 S 32 + C T (nS 4 ra−1 ak e G Tθˆ 1 1 ~ ~ − Cˆ ) + β T (nS 4 ra−1 ak e H T θˆ − βˆ ) + θ T ⋅ r2 r3 1 1 ~ (nS 4 ra−1 ak eξˆ − θˆ) + S 4 (nra−1 ak e u c (t ) + η ) − EEˆ r4 r1 (59) Substituting (51)-(53) into (59), we have V = −k1S12 − k 2 S 22 − k3 S32 + S 4 (nra−1akeuc (t ) + η ) (60) 1 ~ ˆ −

r1

Referred to the system parameters in Table 1, the effectiveness of the proposed control scheme is evaluated. For the ball and beam system, both the conventional DSC and the proposed AFDSC method are considered. The required parameters are given as k1 = 0.5 , k 2 = 3 , k3 = 1 , τ 2 = 0.1 , τ 3 = 0.1 , τ 4 = 0.1 , r1 = 0.1 , r2 = 0.1 , r3 = 0.1 , r4 = 0.1 . Also, the parameters of input and output membership functions are given as [ β1 , β 2 , β 3 , β 4 , β5 , β 6 , β 7 ] = [1 / 8,1 / 8,1 / 8,1 / 8,1 / 8,1 / 8,1 / 8] [c1 , c2 , c3 , c4 , c5 , c6 , c7 ] = [−1,−1 / 3,−1 / 6,0,1 / 6,1 / 3,1] [θ1,θ2 ,θ3 ,θ4 ,θ5 ,θ6 ,θ7 ] = [−1,−3 / 4,−1/ 4,0,1/ 4,3 / 4,1] . Typically, the control goal of the ball and beam system is that the ball can be asymptotically balanced at the center of beam regardless of the initial conditions. In this paper, two simulations with different initial conditions are addressed, where the initial setting are x(0) = [0.3 0 0 0] and x(0) = [0.3 0 − 0.0524 0] , respectively. For the case x(0) = [0.3 0 0 0] , control response relative to the conventional DSC and AFDSC are shown in Fig. 3 and Fig. 4. It can be seen that with the proposed AFDSC the convergence rate of the ball is faster than the DSC counterpart. Similarly, with the x(0) = [0.3 0 − 0.0524 0] , the initial setting effectiveness of the proposed AFDSC can be observed from the control responses shown in Fig. 5 and Fig. 6. The control voltages of the previous simulations are shown in Fig. 7. It can be seen that the control histories are quite smooth for both the DSC and AFDSC. 0.4

=

−k1S12

− k2 S22

− k3S32

+ S4η − E S4

≤

−k1S12

− k2 S22

− k3S32

+ S 4 η − E S4

(61)

= −k1S12 − k2 S22 − k3S32 + S4 ( η − E )

From the assumption that η ≤ E , it can be concluded that V is negative definite and the closed-loop ball-and-beam system is stable with the proposed AFDSC approach.

0.2 0.1 0

EE

Furthermore, substituting (54)-(55) into (60), it leads to V = −k1S12 − k2 S22 − k3S32 − Eˆ S4 + S4η − ( E − Eˆ ) S4

DSC AFDSC

0.3

0

1

2

3

4

5 (a)

6

7

8

0.2 x 2(m/sec)

1 1

5. Simulation Results

(51)

x 1(m)

Cˆ = nr2 S 4 ra−1akeG T θˆ βˆ = nr3 S 4 ra−1ake H Tθˆ θˆ = nr4 S 4 ra−1akeξˆ Eˆ = r1 S 4 uc = − ra (nake ) −1 Eˆ sgn( S 4 ) The proposed controller is given as u (t ) = θˆT ξˆ( S , βˆ , Cˆ ) − ra (nak e ) −1 Eˆ sgn( S 4 )

5

9

10

DSC AFDSC

0 -0.2 -0.4

0

1

2

3

4

5 (b)

6

7

8

9 10 time(sec)

Fig. 3. Responses of the ball x(0) = [0.3 0 0 0] . (a) position x1 , (b) speed x 2 .


6

0.2

1 DSC AFDSC

DSC AFDSC 0.5 u(v)

x 3(rad)

0.1 0

0

-0.1 -0.2

0

1

2

3

4

5 (a)

6

7

8

9

-0.5

10

0.5

-0.5

0

1

2

3

4

5 (b)

6

7

8

DSC AFDSC

x 1(m)

0.2 0.1

0

1

2

3

4

5 (a)

6

7

8

x 2(m/sec)

0.2

9

10

DSC AFDSC

0 -0.2

0

1

2

3

4

5 (b)

6

7

8

DSC AFDSC

0.1 x 3(rad)

7

8

9

10

DSC AFDSC

0

1

2

3

4

5 (b)

6

7

8

9 10 time(sec)

In this paper, combined with DSC, an adaptive fuzzy scheme is proposed for the balance control of a ball and beam system. In the design of AFDSC, the integrated tracking error of states is considered as the fuzzy input. With the adaptive learning mechanism, the parameters of fuzzy controller can be tuned to get better control performance. The closed-loop stability of the AFDSC controlled is analyzed by the Lyapunov’s Theorem. Simulation results illustrate that the proposed AFDSC can provide better equilibrium responses than the conventional DSC counterparts.

References [1]

0.2

[2]

0 -0.1

0

1

2

3

4

5 (a)

6

7

8

9

10

[3]

1.5 DSC AFDSC

1 x 4(rad/sec)

6

Fig. 7. The control voltages. (a) x(0) = [0.3 0 0 0] , (b) x(0) = [0.3 0 − 0.0524 0] .

9 10 time(sec)

Fig. 5. Responses of the ball x(0) = [0.3 0 − 0.0524 0] . (a) position x1 , (b) speed x 2 .

0.5 0

[4]

-0.5 -1

5 (a)

6. Conclusion

0.3

-0.2

4

0.5

-0.5

9 10 time(sec)

0.4

-0.4

3

0

Fig. 4. Responses of the beam x(0) = [0.3 0 0 0] . (a) angular x 3 , (b) angular speed x 4 .

0

2

1

0

-1

1

1.5 DSC AFDSC u(v)

x 4(rad/sec)

1

0

0

1

2

3

4

5 (b)

6

7

8

9

10

Fig. 6. Responses of the beam x(0) = [0.3 0 − 0.0524 0] . (a) angular x 3 , (b) angular speed x 4 .

[5]

C. K. Chak and G. Feng, “Nonlinear system control with fuzzy logic design,” IEEE International Conference on Fuzzy Systems, pp. 1592-1597, 1994. J. Iqbal, M. A. Khan, S. Tarar, M. Khan, and Z. Sabahat, “Implementing ball balancing beam using digital image processing and fuzzy logic,” Electrical and Computer Engineering, pp. 2241-2244, 2005. Y. H. Chang, C. W. Chang, C. H. Yang, and C. W. Tao, “Swing up and balance control of planetary train type pendulum with fuzzy logic and energy compensation,” International Journal of Fuzzy System, vol. 9, no. 2, pp. 87-94, 2007. M. N. Uddin and J. Lau, “Adaptive-backstepping base design of a nonlinear position controller for an IPMSM servo drive,” Can. J. Elect. Comput. Eng., vol. 32, no. 2, pp. 97-102, 2007. F. J. Lin and C. C. Lee, “Adaptive backstepping Control for linear induction motor drive to track


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Yeong-Hwa Chang received the B.S. degree in electrical engineering from the Chung Cheng Institute of Technology, Tao-Yuan, Taiwan, the M.S. degree in control engineering from the National Chiao Tung University, Hsin-Chu, Taiwan, and the Ph.D. degree in electrical engineering from the University of Texas at Austin, U.S.A., in 1982, 1987, and 1995, respectively. He is currently an Associate Professor with the Department of Electrical Engineering, Chang Gung University, Tao-Yuan, Taiwan, R.O.C. His research interests include intelligent systems, haptics, and swarm robots.

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Wei-Shou Chan received the B.S. degree in electrical engineering from Lung Hwa University of Science and Technology, Tao-Yuan, Taiwan, R.O.C., in 2003, and the M.S. degrees in electrical engineering from Chang Gun University, Tao-Yuan, Taiwan, R.O.C., in 2006, where he is currently pursuing the Ph.D. degree in intelligent control. His current research interests are on the nonlinear control and fuzzy control. Chia-Wen Chang received the B.S. degree in electrical engineering from National I-Lan Institute of Technology, I-Lan, Taiwan, R.O.C., in 2001, and the M.S. degrees in electrical engineering from Chang Gun University, Tao-Yuan, Taiwan, R.O.C., in 2003, where he is currently pursuing the Ph.D. degree in intelligent control. His current research interests are on the intelligent control systems including fuzzy control systems and swarm intelligent optimization. C. W. Tao received the B.S. degree in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1984, and the M.S. and Ph.D. degrees in electrical engineering from New Mexico State University, Las Cruces, in 1989 and 1992, respectively. He is currently a Professor with the Department of Electrical Engineering, National I-Lan University, I-Lan, Taiwan, R.O.C.. He is an Associate Editor of the IEEE Transactions on Systems, Man, and Cybernetics. His research interests are on the fuzzy neural systems including fuzzy control systems and fuzzy neural image processing. Dr. Tao is an IEEE Senior Member.