Adaptive self-constructing fuzzy neural network controller for hardware implementation of an inverted pendulum system

Applied Soft Computing 11 (2011) 3962–3975

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Adaptive self-constructing fuzzy neural network controller for hardware implementation of an inverted pendulum system Hung-Ching Lu a,∗ , Ming-Hung Chang a , Cheng-Hung Tsai b a b

Department of Electrical Engineering, Tatung University, 40 ChungShan North Road, 3rd Sec, Taipei, Taiwan Department of Electrical Engineering, China University of Science and Technology, Taipei, Taiwan

a r t i c l e

i n f o

Article history: Received 14 February 2010 Received in revised form 28 November 2010 Accepted 14 February 2011 Available online 21 March 2011 Keywords: Self-constructing fuzzy neural network Rule elimination Rule generation Adaptive control Mahalanobis distance Linear induction motor Inverted pendulum

a b s t r a c t A tracking control of a real inverted pendulum system is implemented in this paper via an adaptive self-constructing fuzzy neural network (ASCFNN) controller. The linear induction motor (LIM) has many excellent performances, such as the silence, high-speed operation and high-starting thrust force, fewer losses and size of motion devices. Therefore, the experiment is implemented by integrating the LIM and an inverted pendulum (IP) system. The ASCFNN controller is composed of an ASCFNN identifier, a computation controller and a robust controller. The ASCFNN identifier is used to estimate parameters of the real IP system and the computational controller is used to sum up the outputs of the ASCFNN identifier. In order to compensate the uncertainties of the system parameters and achieve robust stability of the considered system, the robust controller is adopted. Furthermore, the structure and parameter learning are designed in the ASCFNN identifier to achieve favorable approximation performance. The Mahalanobis distance (M-distance) method in the structure learning is also employed to determine if the fuzzy rules are generated/eliminated or not. Concurrently, the adaptive laws are derived based on the sense of Lyapunov so that the stability of the system can be guaranteed. Finally, the simulation and the actual experiment are implemented to verify the effectiveness of the proposed ASCFNN controller. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In the past decades, various intelligent control researches have been investigated in controlling the non-linear system. These researches adopt fuzzy logic, neural networks (NNs), fuzzy neural networks (FNNs), and so on. The fuzzy logic has the advantage that the solution of the problem can be imagined by human impressions so that human experience can be used in the design of the controller. The NNs can also arbitrary approximate linear or non-linear systems so that the NNs are adopted as the powerful control scheme for the complex non-linear systems [1–3]. Recently, the fuzzy neural networks (FNNs) integrate the advantages of fuzzy reasoning in handling uncertain information [4,5] and the advantages of the NNs in learning from processes [6,7]. The FNNs possess the merits of low-level learning and computational ability of the NNs, and the high-level human knowledge representation of fuzzy theory [8]. Therefore, many researches of FNNs have been investigated by applying in the control field to deal with non-linearities and uncertainties of the control systems [9–12]. Generally, learning for most traditional FNN applications is only parameter learning based on

∗ Corresponding author. Fax: +886 2 25941371. E-mail address: [email protected] (H.-C. Lu). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.02.025

supervised backpropagation algorithm, in which the parameters of the membership functions and the associated weights are adjusted, and the structure of FNN is determined and fixed by trial-and-error in advance [13–15]. Though the literatures adopt the ability of online parameter learning and the fixed structure FNN, the proposed control performance are usually acceptable [5], but they could not avoid using large numbers of nodes in the hidden layers for complex controlled plant. To solve the problem of structure determination, several FNN researches consist of structure and parameter learning phases have been proposed in [16–18]. Traditionally, the two learning phases are executed sequentially in [17], i.e., the structure learning is employed to determine the numbers of fuzzy rule first, and then parameter learning is utilized to adjust the parameters of each rule (e.g. membership functions and the connected weights). However, the feasibility of sequential approach [17] is only off-line operated, and usually requires a lot of time. To overcome the problem and to achieve the purpose of fast learning, much interest has been focused on how to automatically generate the optimal numbers of neuron. Recently, several approaches of the self-constructing FNN (SCFNN) have been developed by many researchers in [19–21]. Park et al. [22] proposed a self-structuring NN control method, which can create new neurons on the hidden layer to decrease the approximated error; unfortunately, the proposed approach can-

H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

Nomenclature uF ,uG Gaussian function m mean of Gaussian function  standard deviation of Gaussian function ␾ output of layer 3 w link weight yF, yG output of layer 4 emin , e˙ min threshold of structure learning DM M-distance ka ,kd threshold of M-distance M total rule number ı uncertainty bound  learning rate ε lumped uncertainty  uncertainty term k sample period t times Suffix item N nominal term i the ith input of ASCFNN p input number of ASCFNN F estimated value F G estimated value G j jth rule of the uF l lth rule of the uG m rule number of the uF n rule number of the uG Superscript item new generated new rule * optimal value ˆ estimated value ∼ estimated error

not avoid the structure of NN growing unboundedly [8]. To solve the problem, in this paper, Mahalanobis distance (M-distance) method [23] is employed to determine the numbers of fuzzy rule are generated/eliminated or not. Then the rule machine can directly and immediately generate suitable rules (or eliminate insignificant rules) during structure learning phase. For the traditional selfconstructing approaches, a new rule is generated when a new input signal is too far away from the current clusters and an existing rule is canceled when the fuzzy rule is insignificant [24]. Concurrently, the parameters of the existed rules are updated by adaptive laws in the parameter learning phase. The adaptive laws are derived based on the sense of Lyapunov so that the stability of the system can be guaranteed. Therefore, the adaptive laws are employed to obtain the optimum parameters in this paper. The inverted pendulum (IP) problem is one of the most popular problems and has been extensively studied in the control field. It provides many challenging problems to control design for the researchers and engineers. The system, which has the characteristics of high order, non-linear, unstable, nonminimum phase and underactuated, can be approximated as the control of biped robot [25], the pitch dynamics of a rocket, and Pan & Tilt. It is also used in the investigation of wheeled motion [26], and balancing mechanisms [27]. Therefore, these challenges made the IP system a classic tool in the control laboratories, and many researches are discussed such as the problem of non-linear, robustness, stabilization and tracing of the system [28–30]. In this paper, the adaptive SCFNN (ASCFNN) controller is proposed for an IP system. The effectiveness of the proposed ASCFNN

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controller is demonstrated by simulating an IP system. The proposed algorithm is performed its superior tracing ability and can approximate the unknown parameters of non-linear system. Finally, the integrated architecture of LIM and IP is implemented to verify the effectiveness of the proposed ASCFNN controller. 2. Problem formulation Consider the nth uncertain non-linear dynamic system in the following form: x(n) = f (x) + g(x)u(t) + d(t),

(1)

T ˙ · ·x(n−1) ] [x x·

where x = [x1 x2 · · ·xn ]T = ∈ Rn is the state vector of the system, which is assumed to be available for measurement, f(x) and g(x) are unknown but bounded real continuous functions, u(t) ∈ R is represented as the input of the system, and d(t) is an uncertain external perturbation which is bounded. Assume that the nominal model of the uncertain non-linear system (1) can be represented as x(n) = fN (x) + gN (x)u(t),

(2)

where fN (x) and gN (x) are the nominal terms of f(x) and g(x), respectively. If the uncertainty includes the external perturbation in the system, the entire uncertainty of the system (1) can be reformulated as x(n)

= (fN (x) + f (x)) + (gN (x) + g(x))u(t) + d(t) = fN (x) + gN (x)u(t) + ε(t),

(3)

where f(x) and g(x) are the estimated errors; ε(t) ≡ f(x) + g(x)u(t) + d(t) is the lumped uncertainty, including the estimated of system and external disturbance, and  errors  assume that ε(t) <  is bounded, where  is a positive constant. In summary, the f(x) and g(x) of the nth uncertain non-linear dynamic system are time-varying unknown parameters and d(t) is a time-varying uncertain external perturbation in practical applications. For a tracking control, unknown parameter and uncertain external perturbation causes the system to be unstable. To overcome the problem and to achieve accurate tracking performance, the proposed ASCFNN controller will be designed in this paper. 3. Sliding-mode control Consider the tracking problem between the state vector x and reference trajectory vector xd , the first step is to select a sliding surface in the SMC. Then, design the control law such that the system trajectory is forced toward the sliding surface and stay on it. The sliding surface in the error space is chosen as



s = e(n−1) + kn e(n−2) + . . . + k2 e + k1

t

e() d,

(4)

0

where e ≡ x − xd is the tracking error and the tracking error T

˙ · ·e(n−1) ] = [e1 e2 · · ·en ]T ; vector is represented as E = [e e· K ≡ [k1 k2 · · · kn ]T is chosen such that all roots of the polynomial sn + kn sn−1 + . . . + k1 = 0 are located in the open left half plane. ˙ According to equivalent control theorem s(x)| u=ueq = 0, the sliding surface can be assumed that s(x) = 0. Then, the equivalent controller ueq is designed as ueq = gN (x)−1 (−fN (x) + h(t)), xdn

(5)

T

where h(t) = − K E. According to [34], the ideal SMC can be expressed as usmc = ueq + uht ,

(6)

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H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

and the hitting controller uht is designed as

the Gaussian function. In this paper, the membership functions u

uht = gN (x)−1 [−sgn(s)],

(7)

and uGl are defined as i



where sgn() is a sign function. Substituting (5)–(7) into (3) yields s˙ = −sgn(s) + ε

(8)

In order to show that the controller (6) via SMC stabilizes the system (3), then, a Lyapunov function candidate can be chosen in the following form: V (s(t)) =

1 2 s . 2

(9)

Differentiating (9) with respect to time and using (8), Eq. (10) can be obtained. V˙ (s(t))

˙ = s(t)s(t) = −s(t)sgn(s) + s(t)ε ≤ |s(t)||ε| − |s(t)| = −|s(t)|( − |ε|) ≤ 0.

u

j i

F

= exp

 uGl = exp

−

i



ji2 (xi − mli )2

,

j = 1, . . . , m,

(12)

,

l = 1, . . . n,

(13)



li2

where mji , mli ,  ji and  li are the means and the standard deviations of the Gaussian function in the jth term and lth term associated with ith input variable. Layer 3: Each node in this layer denoted by , represents the precondition part of one fuzzy rule which multiplies the incoming signals from layer 2 and produces the result of the product. The outputs of the jth and lth rule nodes are represented as the following

(10) Fj =

Thus, (6) can guarantee that control system is asymptotically stable in the sense of Lyapunov function. However, the system nominal terms fN (x) and gN (x) may be unknown or perturbed in practical application, (6) cannot be precisely obtained, and the upper bound of uncertainty  is difficult to determine under the practical consideration. To overcome the problem, an ASCFNN identifier is designed to estimate the parameters of system dynamics. The robust controller is not only utilized to reduce the influence of approximation error between the real non-linear system and an approximate ASCFNN controller dynamics but also developed to estimate the uncertainty bound. Therefore, the ASCFNN controller is proposed to deal with the problems of non-linear system with unknown parameters, and will be described in the next section.

−

(xi − mji )2

j i

F



p 

u

Gl =

j i

F

i=1

p

(xi − mji )2

= exp −



p 

i=1

p

(xi − mli )2

uGl = exp − i

i=1

ji2

i=1



,

(14)

 (15)

li2

Layer 4: It is seen that this layer acts as a defuzzifier. The node in this layer is labeled by , which computes the overall output as the summation of all input variables. yF = wF T F =

m

wFj Fj ,

(16)

j=1

yG = wG T G =

n

wGl Gl ,

(17)

l=1

4. Controller design There are three principal components in the proposed adaptive ASCFNN controller: ASCFNN identifier, computation controller, and a robust controller. The main part of the ASCFNN identifier is the Gaussian membership function base, which consists of j number of rules to estimate the controlled system dynamics. The computation controller including an ASCFNN identifier is the principal controller and a robust controller is designed to compensate the difference between the ASCFNN identifier and controlled system dynamics.

where yF and yG are the output of the ASCFNN identifier, the link weight wFj is the output action strength associated with the jth rule, and link weight wGl is the output action strength associated with the lth rule. In this paper, the outputs of the ASCFNN identifier are used to estimate the unknown parameters F(x) and G(x) of the IP system so that a two-output of the ASCFNN identifier can be expressed in the following vector notation. yF = wF T F = wF1 F1 + wF2 F2 + · · · + wFm Fm ,

(18)

yG = wG T G = wG1 G1 + wG2 G2 + · · · + wGn Gn , 4.1. Design of the ASCFNN identifier

where ⎡ ⎤ ˚F = [␾F1 ␾⎡F2 · · · ␾⎤Fm

The fuzzy logic rule used in the ASCFNN identifier has the following form: Rj :

j

j

j

IF x1 is A1 and x2 is A2 and . . . xp is Ap ,

THEN y = bj

(11)

where Rj is the jth rule, xn and y denote the input and output varij

ables, respectively; Ai is the linguistic term of the precondition part with membership function u j , bj is the constant consequent part, A

i

and p is the number of the input variables. The operation of each layer in the ASCFNN identifier is described as follows. Layer 1: Each node in this layer is an input node, just transmits the input variables xi (i = 1, . . ., p). These nodes only pass the input signal to the next layer. In this paper, the input variables are given as x1 = x − xd = e (the position error) and x2 = e˙ (the derivative of position error). Layer 2: Each node performs the behavior of a membership function. The membership functions are the linguistic terms with

]T ,

(19) ]T ,

˚G = [␾G1 ␾G2 · · · ␾Gn wF = wF1 wG1 ⎢ wF2 ⎥ ⎢ wG2 ⎥ ⎢ . ⎥, and wG = ⎢ . ⎥. ⎣ .. ⎦ ⎣ .. ⎦ wFm wGn Therefore, the outputs yF and yG of the ASCFNN identifier are ˆ respectively. defined as yF = Fˆ and yG = G, For a lot of published literatures [8,18–20], the structure of NN may grow unboundedly. To solve the problem of structure determination, an on-line generation and elimination structure learning algorithm is proposed. The structure learning is to determine whether to generate some new nodes or eliminate some old (existing) nodes in layer 2 and 3. Therefore, the structure learning is necessary if emin ≤ |e| ˙ where emin and e˙ min are the preset positive constants. or e˙ min ≤ |e|, In another aspect, if a new data falls within the local region, the structure learning will not be executed. Then, parameters of the existing rules will be updated.

H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

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In the estimated parameter F(x) of structure learning, each rule of the membership function is expressed in (14). It can be regarded as a regularized function of the Mahalanobis distance (M-distance) [23], i.e.,

Otherwise, if Dmax,G > ka,G is satisfied, then the total number of membership functions MG (k) will become

2 (j)), Fj = exp(−DM,F

Consider a non-linear dynamic system with unknown parameters f(x) and g(x). Firstly, assume that the system nominal terms fN (x) and gN (x) are unknown in advance, then the system dynamics (3) can be reformulated as

where



DM,F ( j) =

(x1 − mj1 )2 2 j1

(20)

+

(x2 − mj2 )2 2 j2

+ ··· +

(xp − mjp )2 2 jp

(21)

is given as the M-distance. According to the M-distance, the criterion of generating a new fuzzy rule for new incoming data is described as follows. Firstly, it is to find the minimum degree Dmin,F ,



Dmin,F = arg



min (DM,F (j))

.

(22)

1≤j≤m

If Dmin,F > kd,F is satisfied, then a new rule is generated, where kd,F is a preset positive constant. If Dmin,F > kd,F is satisfied, the existed fuzzy rules of ASCFNN identifier are not sufficient to perform a conversion from a crisp point into fuzzy rules such that a new rule should not be generated. On the other hand, the new generated rule will be influenced by the variation of kd,F . In the generated new rule, the mean and the standard deviation of the new membership function are assigned with the preset values as follows:

MG (k + 1) = MG (k) − 1.

(33)

x(n) = fN,U (x) + gN (x)u(t),

(34)

where fN,U (x, t) = fN (x) + ε(t). (n)

Substituting (34) into (8) and using h(t) = xd − K T E, it yields s˙ = gN(x)(F(x, t) + u(t) + G(x, t)h(t)),

(35)

t) = gN(x)−1 fN,U (x)

t) = − gN(x)−1

and G(x, are assumed where F(x, to be unknown functions that will be identified by the proposed ASCFNN identifier. By the universal approximation theorem [35], the neural network can be treated as a useful tool for function approximation. In this paper, the optimal ASCFNN identifier is used to approximate the parameters of system dynamic, such that ∗

F = F ∗ (w∗F , m∗F , ␴∗F ) + εF = w∗T F F + εF , G=G

∗

(w∗G , m∗G , ␴∗G ) + εG

=

∗ w∗T G ˚G

(36)

+ εG ,

(37) ∗ G ,

= xi , mnew F

(23)

Fnew

= pre,F ,

(24)

wFnew

= wpre,F ,

(25)

T ˆ ˆ ˆ F, ␴ ˆ F ˆ F, m ˆ F) = w F(X| w F,

(38)

T ˆ ˆ ˆ G, m ˆ G, ␴ ˆ G ˆ G) = w G(X| w G,

(39)

MF (k + 1) = MF (k) + 1,

(26)

where MF (k) is the total number of the existing rules at the sampling period k. Next, the structure learning is considered to determine whether or not to eliminate the existing fuzzy rules. Therefore, the maximum degree Dmax,F is found as Dmax,F = arg( max (DM,F (j))).

(27)

1≤j≤m

If Dmix,F > ka,F is satisfied, where ka,F is a preset positive constant, it shows that the existing fuzzy rule is not important such that the jth fuzzy rule, the associated membership functions, and weights could be deleted. Then the total number of membership functions MF (k) become MF (k + 1) = MF (k) − 1.

ˆ F,  ˆ G, m ˆ G,  ˆ F, m ˆ G, ␴ ˆ F, w ˆ F , and ␴ ˆ G are denoted as the estiwhere w ∗ ∗ mated values of w∗F , w∗G , F , G , m∗F , m∗G , ␴∗F , and ␴∗G , respectively. ˜ can be defined as Then, the estimated errors F˜ and G T ˜ T ˆ T ˜ ˆ F ˆ F ˆ F F˜ = F − Fˆ = w F+w F+w F + εF ,

˜ =G−G ˆ = G

= xi , mnew G

(29)

Gnew = pre,G ,

(30)

wGnew = wpre,G ,

(31)

T ˜ ˆ G w G

T ˆ ˆ G +w G

(40)

+ εG ,

(41) ∗ F

ˆ F , and  ˜G= −

G

In the following discussion, it is necessary to attain favorable estimation of system parameters by using a linearization technique. Thus a Taylor expansion-linearization technique is adopted to transform the Gaussian function into a partially linear form [36], ˜ F and  ˜ G in Taylor series can be obtained at the the expansion of  same time.

⎡ ˚˜ ⎤ F1

˜F=⎢  ⎣

⎡ +

⎢ ⎢ ⎢ ⎢ ⎣

˜ F2 ˚ . . . ˜ Fm ˚

∂˚F1 ∂␴F ∂˚F2 ∂␴F . . . ∂˚Fm ∂␴F

⎡

⎢ ⎥=⎢ ⎦ ⎢ ⎢ ⎣ ⎤

∂˚F1 ∂mF ∂˚F2 ∂mF . . . ∂˚Fm ∂mF

⎤ ⎥ ⎥ ⎥ |mF =mˆ F m˜ F ⎥ ⎦

⎥ ⎥ ⎥ |F =ˆ F ␴˜ F + OF , ⎥ ⎦

where  pre,G and wpre,G are preset constants. Then, the total number of membership functions MG (k) is increased as

or

MG (k + 1) = MG (k) + 1.

˜ F + BF T ␴ ˜ F + OF , F = AF T m

(32)

T ˜ ˆ G +w G

˜F= ˆ F = w∗F − w ˆ F, w ˆ G = w∗G − w ˆ G,  where w ∗ ˆ  − G .

(28)

Likewise, as noted previously, in the estimated parameter G(x) of structure learning, each rule of the membership function is expressed in (15), if Dmin,G > kd,G is satisfied, the generated new rule, the mean and the standard deviation of the new membership function are assigned with preset values and given as follows.

w∗G ,

∗ F ,

m∗F , where εF , εG are the approximation error, ∗ ∗ ∗ mG , ␴F , and ␴G are the optimal parameters of wf , wG , F , G , mF , mG ␴F , and ␴G , respectively. For the approximated dynamic parameters, the proposed ASCFNN identifier is designed as

where  pre,F and wpre,F are preset constants. Then, the new membership function is adopted and the total number of membership functions MF (k) is increased as

w∗F ,

(42)

(43)

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H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

Fig. 1. Structure of ASCFNN identifier.

⎡ ˜ ⎤

⎡

G1 ⎢ ˜ G2  ⎢ ⎥=⎢ ˜G=  . ⎣ . ⎦ ⎢ ⎢ . ⎣

˜ Gm 

∂˚G1 ∂mG ∂˚G2 ∂mG . . . ∂˚Gm ∂mG

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ |mG =mˆ G m˜ G + ⎢ ⎥ ⎢ ⎦ ⎣

∂˚G1 ∂␴G ∂˚G2 ∂␴G . . . ∂˚Gm ∂␴G

⎤



⎥ ⎥ ⎥ |G =ˆ G ␴˜ G + OG , ⎥ ⎦

(44)



˜ G = AG T m ˜ G + BG T ␴ ˜ G + OG , 

(45)

where the matrices AF , BF , AG , and BG are selected as AF = [(∂˚F1 /∂mF )· · ·(∂˚Fm /∂mF )]|mF =mˆ F , BF = [(∂˚F1 /∂␴F )· · ·(∂˚Fm /∂␴F )]|F =ˆ F , AG = and BG = [(∂˚G1 /∂mG )· · ·(∂˚Gn /∂mG )]|mG =mˆ G [(∂˚G1 /∂␴G )· · ·(∂˚Gn /∂␴G )]|G =ˆ G , respectively. OF , OG are ˜ F = m∗F − m ˆ F, denoted as the vector of higher order terms. m ∗ ∗ ∗ ˆ G, ␴ ˜ G = mG − m ˜ F = ␴F − ␴ ˆ F, ␴ ˜ G = ␴G − ␴ ˆ G ; Since m∗F , m∗G , m ␴∗F , and ␴∗G are optimal parameter vectors of mF , mG , ␴F , and ˆ F, m ˆ G, ␴ ˆ F , and ␴ ˆ G are estimation vectors ␴G , respectively; m of m∗F , m∗G , ␴∗F , and ␴∗G , respectively; and ∂˚Fj /∂mF , ∂˚Gl /∂mG , ∂˚Fj /∂␴F , and ∂˚Gl /∂␴G are defined as ∂˚Fj ∂mF



T =

0· · ·0

 (j−1)×p

∂˚Fj ∂mF1j

···

∂˚Fj ∂mFpj

 0· · ·0

 (m−j)×p

∂˚Fj

,

(46)

∂˚Gl ∂␴G



T

0· · ·0



=

∂˚Gl ∂˚Gl ··· ∂mG1l ∂mGpl

(l−1)×p



T

0· · ·0



=

∂␴F

or





∂˚Gl ∂mG

=

0· · ·0

 (l−1)×p



,

(47)

(n−l)×p

∂˚Fj ∂F1j

···

∂˚Fj ∂Fpj

(j−1)×p

T

 0· · ·0

 0· · ·0



,

(48)

(m−j)×p

∂˚Gl ∂˚Gl ··· ∂G1l ∂Gnl

 0· · ·0



.

(49)

(n−l)×p

Substituting (43) and (45) into (40) and (41), it yields T ˆF+m ˜ F + εF , ˜ TF  ˆF+␴ ˆF+w ˜ TF AF w ˆ F OF + w ˜ TF  ˜ TF BF w F˜ = w

(50)

T T ˜ ˆG+m ˜ =w ˜ TG AG w ˆ G OG + w ˆ G ˆG+␴ ˆG+w ˜ TG ˚ ˜ TG BG w G G + εG .

(51)

The structure of ASCFNN identifier is shown in Fig. 1. 4.2. Design of the computation controller and robust controller In this paper, the proposed ASCFNN controller u is comprised of a computation controller ucc and a robust controller ur . Thus, the total control law can be rearranged as u = ucc + ur ,

(52)

H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

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Differentiating (55) with respect to time and using (35), (52), (53) and (54), it yields ˜ ˜ G, m ˜ F, m ˜ G , ˜ F , ˜ G , ı) ˜ F, w V˙ (s, w ˆ ˜ − ısgn(s)] = sgN (x)[F˜ + Gh + +

1 1 ˜ Tw ˜ Tw ˜˙ F + ˜˙ G w w wF F wG G

.

(56)

1 1 1 T˙ 1 T˙ 1 ˜ ˜˙ ˜ Tm ˜ Tm ˜˙ F + ˜˙ G + m m ˜ ˜ F + ˜ ˜ G + ıı mF F mG G F F G G ı

From (50) and (51), Eq. (56) can be rewritten as ˜ ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, w ˜ F, ␴ ˜ G , ı) V˙ (s, w T ˆF +m ˜ F + εF ) ˆF +␴ ˆF +w ˜ TF  ˜ TF AF w ˆ F OF + w ˜ TF  ˜ TF BF w = sgN (x)[(w T

T

T

T

ˆG +m ˜ G + εG )h − ıˆ sgn(s)] ˜ G ˜ G AG w ˆ G OG + w ˜ G ˆG +␴ ˆG +w ˜ TG BG w +(w

Fig. 2. Structure of the computation controller.

+

where ucc and ur are given as ˆ ucc = −Fˆ − Gh,

(53)

ˆ ur = −ısgn(s),

(54)

where ıˆ is an estimation of this uncertainty bound. ˆ the outputs of ASCFNN identifier, are used to estimate Fˆ and G, the unknown dynamic functions F and G, respectively. The structure of computation controller and robust controller are also shown in Figs. 2 and 3, respectively.

In order to design the adaptive laws and estimation law for the ASCFNN controller and robust controller, these laws are derived based on the sense of Lyapunov. Then, the stability of the system can be guaranteed as the following. Stability Theorem: Consider a non-linear dynamic system represented by Eq. (34). If the adaptive ASCFNN controller is designed as (52), robust stability of the closed-loop system in the presence of model uncertainties and disturbances is guaranteed. Proof: In order to design parameter learning algorithms and prove the robustness and stability of the system, the following Lyapunov function is considered. ˜ ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) V (s, w 1 2 1 1 1 1 ˜ Tw ˜F+ ˜ Tw ˜G+ ˜ Tm ˜F+ ˜ Tm ˜G w w m m s + 2 2wF F 2wG G 2mF F 2mG G

+

1 1 1 ˜2 ˜ T␴ ˜F+ ˜ T␴ ˜G+ ı , ␴ ␴ 2F F 2G G 2ı

1 1 1 ˜ ˜˙ ˜ T␴ ˜ T␴ ˜˙ F + ˜˙ G + ␴ ␴ ıı, F F G G ı

reformulate (57) and it can be rewritten as ˜ ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) V˙ (s, w T

˜F =w T

˜F +m ˜ TF +␴







ˆF+ sgN (x)

1 ˙ ˜F w wF



T



˜G +w

ˆF+ sgN (x)AF w

1 ˙ ˜F m mF

ˆF + sgN (x)BF w

1 ˙ ˜F ␴ F





ˆ Gh + sgN (x)

T



˜G +m

ˆ Gh + sgN (x)AG w



˜ TG +␴

1 ˙ ˜G w wG

ˆ Gh + sgN (x)BG w



1 ˙ ˜G m mG

1 ˙ ˜G ␴ G

 (58)



1 ˜ ˜˙ +sgN (x) − |s|gN (x)ıˆ + ıı, ı

5. Stability analysis

=

(57)

1 1 1 1 ˜ Tw ˜ Tw ˜ Tm ˜ Tm ˜˙ F + ˜˙ G + ˜˙ F + ˜˙ G + w w m m wF F wG G mF F mG G

(55)

where ı˜ ≡ ı − ıˆ is the estimation error of uncertainty bound in which ı is a constant of the uncertainty bound.

T

T

˜ F + εF + (w ˆ F OF + w ˜ TF  ˆ G OG + where the uncertain term  = w T ˜ ˜ G + εG )h, and assume that   ≤ ı, where ı is a positive conw G

stant and represents the upper bound of uncertain term. Then the ıˆ is utilized to estimate this unknown uncertainty bound ı. The adaptive laws are derived based on the sense of Lyapunov so that the stability of the system can be guaranteed. It is known that the parameter learning in ASCFNN identifier is to on-line optimize the link weight and the parameters of the membership functions by using the adaptive laws. Thus, the adaptive laws and estimation law are chosen as ˆ F, ˆ˙ F = −wF sgN (x) ˜˙ F = −w w

(59)

ˆ G h, ˜˙ G = −w ˆ˙ G = −wG sgN (x) w

(60)

˜˙ F = −m ˆ F, ˆ˙ F = −mF sgN (x)AF w m

(61)

ˆ˙ G = −mG sgN (x)AG w ˜˙ G = −m ˆ G , h, m

(62)

ˆ F, ˆ˙ F = −F sgN (x)BF w ˜˙ F = −␴ ␴

(63)

ˆ G h, ˜˙ G = −␴ ˆ˙ G = −G sgN (x)BG w ␴

(64)

˙ ı˜˙ = −ıˆ = −ı |s| gN (x)

(65)

In order to guarantee the stability of the system, the adaptive laws (59)–(65) as above are designed to update the parameters. By substituting (59)–(65) into (58), (66) can be obtained ˜ = −gN (X)|s|(ı − | |) ≤ 0 . ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) V˙ (s, w

(66)

  In this paper, these terms   ≤ ı, |s(t)| > 0 and gN (x) is positive.

Fig. 3. Structure of the robust controller.

˜ ≤ ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) From (66), it can be known that V˙ (s, w ˜ G, m ˜ F, ˜ F, w 0 and is negative semidenfinite. It implies that s(t), w ˜ G, ␴ ˜ F, ␴ ˜ G and ı˜ are bounded. Define a function as ˝(t) ≤ m ˜ and ˝(t) ≡ gN (x)|s|(ı − | |). ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, w ˜ F, ␴ ˜ G , ı) −V˙ (s, w

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Fig. 4. Framework of the ASCFNN controller.

H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

Fig. 5. Flow chart of structure and parameter learning algorithms for ASCFNN.

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Fig. 6. Simulation results of [39] for periodic sinusoidal command: (a) the tracking response of angle, (b) the tracking response of angular velocity, (c) tracking error of angle, (d) tracking error of angular velocity, (e) control effort.

Now, integrating this function with respect to time, then it is shown that



t

˝() d

(67)

0

˜ − V˙ (s(t), w ˜ ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) ˜ F, ␴ ˜ G , ı). ≤ V˙ (s(0), w

˜ is bounded, ˜ F, w ˜ G, m ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) As a result of V˙ (s(0), w ˜ is non-increasing and ˜ F, w ˜ G, w ˜ F, m ˜ G, ␴ ˜ F, ␴ ˜ G , ı) and V˙ → (s(t), w bounded, the following result can be obtained.



˝() d < ∞

lim

t→∞

t

(68)

0

˙ From (68), it is known that ˝(t) is bounded and lim ˝(t) = 0 can t→∞

be concluded by using the Barbalat’s Lemma [37]. Therefore, the stability of the proposed ASCFNN controller design can be guaranteed. If the nominal dynamic term is unavailable in practical applicaˆ −1 in the adaptive laws and tion, the term gN (x) is replaced with −G estimation law (59)–(65). Therefore, the new adaptive laws can be rewritten as follows: ˆ F, ˆ −1  ˜˙ F = −w ˆ˙ F = wF sG w

(69)

ˆ −1 G h, ˆ˙ G = wG sG ˜˙ G = −w w

(70)

ˆ −1 AF w ˆ F, ˆ˙ F = mF sG ˜˙ F = −m m

(71)

ˆ −1 AG w ˆ˙ G = mG sG ˆ G h, ˜˙ G = −m m

(72)

ˆ −1 BF w ˆ F, ˜˙ F = −␴ ˆ˙ F = F sG ␴

(73)

Fig. 7. Simulation results of [8] for periodic sinusoidal command: (a) the tracking response of angle, (b) the tracking response of angular velocity, (c) tracking error of angle, (d) tracking error of angular velocity, (e) control effort, (f) rule number.

ˆ −1 BG w ˆ G h, ˆ˙ G = G sG ˜˙ G = −␴ ␴

(74)

˙ ˆ −1 ı˜˙ = −ıˆ = ı |s|G

(75)

6. Summary In view of Figs. 1–3, the framework of the ASCFNN controller can be obtained as shown in Fig. 4. In this figure, the tracking error vector E is defined as x − xd , where x and xd represent the state vector and reference trajectory vector, respectively. The sliding surface is chosen to guarantee that control system is asymptotically ˆ of the system are estimated by stable. The parameters (Fˆ and G) using ASCFNN identifier. And the structure learning is employed to determine whether the fuzzy rules of the ASCFNN identifier are generated/eliminated or not. The optimum parameters of the ASCFNN identifier are obtained from the adaptive laws (69)–(74). Next, the computation controller is used to sum up the outputs of the ASCFNN identifier and h(t) = xdn − KT E as shown in Fig. 2. The robust controller is adopted to compensate the uncertainties of the system parameters and achieve robust stability as shown in Fig. 3. Furthermore, the estimation law is used to achieve the uncertainty bound of the tracking performance. Then, the final control effort u of the ASCFNN controller is comprised of a computation controller ucc and a robust controller ur . Finally, Fig. 5 shows the flow chart of the structure learning phase and adaptive law for ASCFNN identifier. Based on the above discussions, the design procedures can be summarized as follows.

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Step1) First, the ASCFNN identifier has only layer 1 and layer 4. The layer 2 and layer 3 are generated automatically and dynamically in the learning process according to the online incoming data by performing the structure learning process. If xi is the first data, then initialize the predefined parameters of ASCFNN identifier and generate a new rule, or go to structure learning. In the first generated rule, the means of uF 1 (1), uF 1 (1), uG1 (1) and uG1 (1) are 1

2

1

2

˙ given as m11 = x1 (1) = e(1), m12 = x2 (1) = e(1), m11 = ˙ respectively. Next, x1 (1) = e(1) and m12 = x2 (1) = e(1), ˙ e(1) is the position error of the first data and e(1) is the derivative of the position error of the first data. Further, the standard deviations (Fnew and Gnew ) of uF 1 (1), uF 1 (1), uG1 (1), uG2 (1) and the link weights 1

2

1

1

(wFnew and wGnew ) of uF 1 (1), uF 1 (1), uG1 (1), uG2 (1) are all 1

2

1

1

preset constants, which are selected through the trail experience. Step2) Determine whether the process will go to the structure ˙ where emin learning phase by using emin ≤ |e| or e˙ min ≤ |e|, and e˙ min are the preset positive constants. Step3) Determine whether the process will cancel an existing node by using Dmax,F > ka,F or Dmax,G > ka,G and determine whether the process will add a new node by using Dmin,F > kd,F or Dmin,G > kd,G , where ka,F , ka,G , kd,F and kd,G are the preset positive constants. The ka,F , ka,G , kd,F , and kd,G are important constants which affect the number of rules. If kd is set to be a small value, the rule number will be added easily and if ka is set to be a small value, the rule number will be canceled easily. In the generated new rule, the link weights as mean and the standard deviation of the new membership function are assigned according to the input variables. Therefore, the means of u j (k) and uGl (k) are selected as F

i

i

˙ mj1 = ml1 = x1 (k) = e(k) and mj2 = ml2 = x2 (k) = e(k), respec˙ tively, in which e(k) and e(k) are the position error and the derivative of the position error at the sampling period k. The other procedure of generating new fuzzy rule is the same as Step1. ˆ F,  ˆ G, m ˆ F, w ˆ G,  ˆ F, m ˆ G, ␴ ˆ F , and ␴ ˆ G ) of Step4) The parameters (w ASCFNN identifier are updated by adaptive laws (69)–(74) ˆ of robust controller is estimated and the parameters (ı) by estimation law (74). In the meanwhile, the output of ˆ are given as (16) and (17). ASCFNN identifier Fˆ and G Step5) Determine xi whether it is the last data or not. If xi is the last data, the process will return to Step2 or go to the end. 7. Simulation and experimental results In the following, the simulation and experiment results are used to verify the performance of the proposed controller. 7.1. Simulation results

Fig. 8. Simulation results of ASCFNN controller for periodic sinusoidal command: (a) the tracking response of angle, (b) the tracking response of angular velocity, (c) tracking error of angle, (d) tracking error of angular velocity, (e) control effort, (f) rule number of F, (g) rule number of G, (h) estimation of system parameter F, and (i) estimation of system parameter G.

7.1.1. Example1: inverted pendulum system Consider the non-linear dynamic system as the inverted pendulum system in [38]. It is seen that the control object is to balance the inverted pendulum and, at the same time, to guarantee that the state x1 and x2 are bounded. The considered inverted pendulum system is defined as



x˙ 1 x˙ 2

y=





=

  1 0



0 0 x1

1 0





x1 x2

x2 ,

   +

0 1

(f (x) + g(x)u + d) (76)

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a

Command ASCFNN SAFNC

1

state x

0.5

0

-0.5

-1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time(sec)

b

2

Command ASCFNN SAFNC

1.5

state x-dot

1 0.5 0 -0.5 -1 -1.5 -2 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time(sec)

c

6

ASCFNN SAFNC

rule number

5 4 3 2 1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec) Fig. 10. Simulation results of a second-order chaotic system of the tracking response, rule number and tracking errors with q = 7 from ASCFNN and SAFNC controller.

Fig. 9. Simulation results of ASCFNN controller without uGl and Gl for periodic

and the state x1 = ␪ denotes the angle of the pendulum, the state x2 = ˙ is the angular velocity, gv is the acceleration due to gravity constant, mc is the mass of cart, mp is the mass of pendulum, l is the half-length of pendulum, u is the control effort, and d is external disturbance. Here, the parameters are given as mc = 1 kg, mp = 0.1 kg, l = 0.5 m, gv = 9.8 m/s2 in the following simulations. To investigate the effectiveness of the proposed controller, two simulated conditions including external disturbance or not are considered in the following:

i

sinusoidal command: (a) the tracking response of angle, (b) the tracking response of angular velocity, (c) tracking error of angle, (d) tracking error of angular velocity, (e) control effort, (f) rule number of F, (g) estimation of system parameter F, and (h) estimation of system parameter G.

where mp lx2 sin x1 cos x1 − (mc + mp )gv sin x1 4 mp l cos2 x1 − l(mc + mp ) 3 − cos x1 g(x) = , 4 2 mp l cos x1 − l(mc + mp ) 3

f (x) =

Condition 1: d = 0, without external disturbance before 10 s. Condition 2: d = 10 · sin (t), with external disturbance after 10 s. The parameters for ASCFNN controller are selected as: k1 = 150, k2 = 150, e = 0.005, e = 0.001, kd,F = 0.001, ka,F = 0.008, kd,G = 0.001, ka,G = 0.005, Fnew = 0.2, wFnew = 0, Gnew = 0.3, wGnew = −0.2, wF = mF = 10, F = 1, wG = mG = 10, G = 15, and ı = 5. These parameters are selected through the trail experience. The simulation is carried out by using Matlab software. The sampling time is set to be 1 ms. The reference command is a periodic sinusoidal signal xc = (/30) sin(t) with the initial conditions x1 (0) = 0.1, and x2 (0) = 0.1. The proposed ASCFNN controller is com-

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Table 1 Performance comparison of various existing controllers. Controller

Condition Before 10 s (MSE)

Fuzzy controller [39] SAFNC controller [8] ASCFNN controller

After 10 s (MSE)

e1 (rad) ×10

e2 (rad/s) ×10

Rule number

e1 (rad) ×103

e2 (rad/s) ×103

Rule number

9.84 9.21 6.02

15.86 15.31 13.55

18 45 13

24.38 0.71 0.36

42.79 2.09 0.62

18 49 13

3

3

pared with Fuzzy controller [39] and self-organizing adaptive fuzzy neural control (SAFNC) controller [8]. The simulation results of [39,8] and the proposed ASCFNN controller are shown in Figs. 6–8. In condition 1, Fig. 6 has acceptable performance with fixed fuzzy rules. In Fig. 7, the error of tracking response of angle is small. However, 45 rule numbers are required which show that more memory and computational time are required. In Fig. 8, it also has satisfactory performance of tracking response of angle. Furthermore, the proposed controller uses only 13 rule numbers. From the simulation results, we can see that the proposed ASCFNN controller not only uses less fuzzy rules, but also owns the smaller MSE of tracking response than the other two controllers in steady state because of the system dynamics can be on-line estimated. In condition 2, due to the fixed fuzzy rule number and membership functions, Ref. [39] does not have the capability of learning for the change of external disturbance as shown in Fig. 6. Therefore, we can see that the error of tracking performance is bigger and bigger. Ref. [8] has learning ability for the change of external disturbance. Thus the error of tracking performance is better than Ref. [39]. ˆ the However, due to the unknown system dynamic parameter G, rule numbers are required more to cover the unknown influence of ˆ as shown in Fig. 7. From Fig. 8, we can observe that the parameter G ASCFNN eliminates 2 fuzzy rules for Fˆ and generates 2 fuzzy rules ˆ according to the change of external disturbance. Therefore, for G the rule number and tracking error will have a satisfactory perforˆ mance. Furthermore, the estimation of system dynamic Fˆ and G, the rule numbers and the MSE value of the proposed controller can be proved to be more effectively. Finally these simulation results

are summarized in Table 1. From Table 1, the proposed ASCFNN controller has the best performance than the other two controllers in tracking capability and rule numbers. From these simulation results, we can observe that the robust tracking performance can be achieved without any knowledge of the controlled system dynamics. In Fig. 8, the simulation result shows not only perfect tracking reference command can be achieved but also hat an appropriate network size can be obtained since the proposed ASCFNN controller and the on-line learning algorithms are useful. Moreover, the controlled system dynamics F and G can be estimated by the proposed ASCFNN identifier under the conditions with/without external disturbance. Finally, we tune more in detail exploring different kinds of solutions for the ASCFNN identifier. The ASCFNN identifier is used without uGl and Gl . Then, Eq. (17) can be rewritten as yG = wG T F . i

In Fig. 9, the simulation result shows not only rule numbers are required more than the proposed ASCFNN identifier but also the controlled system dynamics G is not estimated accurately. Although controlled system dynamics G cannot be estimated accurately, the estimated error can be eliminated by the characteristic of the robust controller. Thus, the favorable tracking performance is obtained.

7.1.2. Example2: second-order chaotic system Consider the Duffing’s equation of a second-order chaotic system [8] and it is represented as x¨ = f (x) + u,

Fig. 11. Schematic diagram of the hardware implementation.

(77)

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Fig. 12. Diagram of hardware implementation of an inverted pendulum system.

where f (x) = −c x˙ − c1 x − c2 x3 + q cos(wt) is the system dynamics, u is the control effort, w is the frequency, t is the time variable, and c, c1 , c2 , and q are real constants [8]. Depending on the different choice of these constants, the solutions of (77) may exhibit periodic, almost periodic, and chaotic behavior. The open-loop system behavior was simulated with c = 0.4, c1 = −1.1, c2 = 1.0, and w = 1.8 in order to observe the unpredictable behavior of chaotic system



T



T

under the conditions u = 0 and x x˙ = 0 0 . It is seen that Fig. 10 shows the simulation results of the tracking response and rule number with q = 7 from ASCFNN and SAFNC controller. According to the simulation results, excellent tracking responses and robust control characteristics can be obtained for both controllers. However, the rule number and tracking errors of SAFNC controller are larger than the proposed controller. Furthermore, the simulation results also demonstrate that the proposed ASCFNN controller has better performances in tracking ability and less rule number.

Fig. 13. Experimental results of ASCFNN controller for zero command: (a) the tracking response, (b) the distance of LIM.

7.2. Experimental results Fig. 11 shows the experimental schematic diagram of the hardware implementation. In view of it, the hardware of the experimental apparatus is composed of a linear induction motor (LIM), an inverted pendulum (IP) system, a rotary encoder, PCI capture card, main PC and a digital drive. The system is actuated by an LIM of the distance of 300,000 steps and the linear encoder, which is set on the LIM, is used to get the position (x), velocity (v) and acceleration (a) from LIM. The pendulum consists of a 0.5 kg stainless-steel rod of cross-sectional area and its total length is 0.4 m. At the pivot, the rotary encoder with 5000 pulses per resolution is installed to measure the angular displacement ( ). The PCI counter is a data acquisition card, which uses to receive the pulse of rotary encoder. A personal computer with Intel Pentium 4 CPU 3 GHz processor and 1G RAM is used to execute the control program. Then, the proposed ASCFNN controller is demonstrated by the practical inverted pendulum system. The diagram of hardware implementation of an inverted pendulum system is shown in Fig. 12. The tracing target is a zero command xc = 0 and the initial angle is (0) = −5◦ . Fig. 13 shows a real-time experiment on the inverted pendulum system with ASCFNN controller. The pendulum swings into the upright position in 6 s. During the swinging of the pendulum, the ASCFNN controller can be appropriate to catch the pendulum at the upright position. After reaching the upright position, the controller is stable but the pendulum pole oscillates around the equilibrium point and the LIM periodically moves its

position. Form the experimental results; the proposed ASCFNN controller can be applied to control the practical inverted pendulum system. In view of the simulation and experiment results, it shows that the ASCFNN controller works quite well for the position control of the inverted pendulum (IP) system. Position tracking control of the linear induction motor (LIM) while balancing the pendulum has been successfully performed.

8. Conclusions For the practical implementation of self-constructing FNNbased control, the number of fuzzy rules will be increased while in heavy load condition. In this paper, an ASCFNN controller is proposed, in which an ASCFNN identifier is used to online estiˆ effectively. Therefore, mate the controlled system dynamics Fˆ and G the proposed control strategy performs even better than the controller that was proposed by [8] or [39]. Furthermore, the proposed ASCFNN identifier design, a dynamic rule generating/elimination mechanism is developed to cope with the approximation accuracy and computational loading. Moreover, all the adaptive laws are derived via Lyapunov synthesis method, so that the stability of the closed-loop system can be guaranteed. Finally, the design methodology has been applied to control the integrated architecture of LIM and IP system successfully. Simulation and experimental results have demonstrated the efficiency of the proposed method.

H.-C. Lu et al. / Applied Soft Computing 11 (2011) 3962–3975

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[25] T. Suji, T. Ohnishi, K.A., Control of biped robot which applies inverted pendulum mode with virtual supporting point, in: Proceedings of the 7th International Workshop on Advanced Motion Control, July, 2002, pp. 478– 483. [26] K. Pathak, J. Franch, S.A. Agrawal, Velocity and position control of a wheeled inverted pendulum, IEEE Transactions on Robotics 21 (June (3)) (2005) 505–514. [27] P. Hsu, Dynamics and control design project offers taste of real world, IEEE Control Systems Magazine 12 (June (3)) (1992) 31–39. [28] H.T. Cho, S. Jung, Neural network position tracking control of an inverted pendulum an X–Y table robot, Proceedings of IEEE Conference on Intelligent Robots and Systems 2 (October) (2003) 1210–1215. [29] H.T. Cho, S. Jung, Balancing and position tracking control of an inverted pendulum on a x-y plane using decentralized neural networks, in: Proceedings of IEEE/ASME Conference on Advanced Intelligent Mechatronics, vol. 1, July, 2003, pp. 181–186. [30] P. Wang, C.-Z. Xu, Z. Fan, Evolutionary linear control strategies of triple inverted pendulums and simulation studies, in: Proceedings of WCICA Fifth World Congress on Intelligent Control and Automation, vol. 3, June, 2004, pp. 2365–2368. [34] J.J.E. Slotine, W.P. Li, Applied Non-linear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. [35] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1994. [36] C.M. Lin, C.F. Hsu, Neural-network hybrid control for antilock braking systems, IEEE Transactions on Neural Networks 14 (March (2)) (2003) 351–359. [37] J.M. Mendel, A Prelude to Neural Networks: Adaptive and Learning Systems, Prentice-Hall, Englewood Cliffs, NJ, 1994. [38] S.C. Tong, H.X. Li, W. Wang, Observer-based adaptive fuzzy control for SISO non-linear systems, Fuzzy Sets and System 148 (2004) 355–376. [39] N. Musˇkinja, B. Tovornik, Swinging up and stabilization of a real inverted pendulum, IEEE Transactions on Industrial Electronics 53 (April (2)) (2006) 631–639. Lu Hung-Ching was born in Taipei, Taiwan, R.O.C. in 1959. He received the M.S. and Ph.D. degrees in Electrical Engineering from Tatung Institute of Technology, Taipei, Taiwan in 1986 and 1989, respectively. Now, he is currently professor of Electrical Engineering of Tatung University. His researches focus on fuzzy control, adaptive control, intelligent control and their application for industrial design.

Ming-Hung Chang received the B.S. degree in Department of Communications and Guidance Engineering, National Taiwan Ocean University, Keelung, Taiwan, R.O.C., in 2004, and the M.S. degree in Department of Electrical Engineering, Tatung University, Taipei, Taiwan, R.O.C., in 2006. His research interests are fuzzy system, neural networks, fuzzy neural networks and intelligent control.

Cheng-Hung Tsai received his B.S., M.S., and Ph.D. degrees in electrical engineering from Tatung University, Taipei, Taiwan in 1994, 1996, and 2000, respectively. Since 2002, he has been with the Department of Electrical Engineering, China University of Science and Technology, Taipei, Taiwan, where he is currently an Assistant Professor. His current interests include neural networks and intelligent control on induction and DC motor.