FUZZY CONTROL - CiteSeerX

International Journal of Innovative Computing, Information and Control Volume 3, Number 2, April 2007

c ICIC International °2007 ISSN 1349-4198 pp. 319—333

RETURNING TO THE STARTING POINT OF THE “FUZZY CONTROL” Hugang Han and Akira Ikuta Faculty of Management and Information Systems Prefectural University of Hiroshima 1-1-71, Ujina, Minami-ku, Hiroshima-city, Hiroshima 734-8558, Japan {hhan, ikuta}@pu-hiroshima.ac.jp

Received June 2006; revised October 2006 Abstract. The primary idea of fuzzy control is to employ the knowledge of experts to control a plant instead of the algorithm derived from the mathematical model of the plant. If the parameters in the fuzzy rules, which are based on experts’ views, are changed at all, then fuzzy control loses its initiative. In this paper, we consider a few cases to show how to develop a real fuzzy control system with stability in which it persistently maintains the fuzzy rules in accordance with experts’ views. Keywords: Fuzzy control, System stability, Experts’ view, Lyapunov function

1. Introduction. The applications of fuzzy set theory to control systems have had innumerable successes in the industrial world. This shows that fuzzy control is a very useful approach to develop a control system. However, some researchers, especially those who have got used to using the traditional control theory such as adaptive control to design a system under an entirely theoretical proof of its stabilities, are always concerned if the fuzzy control system designed will continue to work stably all the time till the proof is given, even if the designed fuzzy system has worked well as expected so far. This is an important reason why active research on adaptive fuzzy control system, in which the guarantee of the stability of the control system is the first task to be solved, has been conducted. In the last decade or more, a large quantity of research on the adaptive fuzzy control system has achieved success in a sense. In order to develop a stable fuzzy control system, there are, in general, two ways: (1) After using the so-called Takagi-Sugeno (T-S), also known as the Takagi-SugenoKang (TSK) fuzzy models [1, 2], to represent certain complex nonlinear systems, the control design is carried out based on the fuzzy models by the so-called parallel distributed compensation (PDC) scheme [3]. For each local linear model, a linear feedback control is designed. The resulting overall controller is a fuzzy blending of the individual linear controllers. A sufficient condition for the system stability of the T-S fuzzy systems are obtained by solving a Linear Matrix Inequality (LMI) [4, 5]. Actually, in this case this is nothing to do with experts’ knowledge except for the T-S fuzzy models, which also is difficult to be described by experts’ views. (2) Another way is to adopt the Lyapunov synthesis approach, and considering the fuzzy rules involved in the approach as some approximators to deal with some unknown parts (functions) in the plant to be controlled. Furthermore, the traditional adaptive control theory has been merged to compose the 319

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update law of parameters involved in the system including the fuzzy rules in terms of system stability [6]—[12]. Herein let us consider a question: “what is the initial purpose for using fuzzy control?” One, of the key reasons is that, instead of the computational controllers like a PID obtained from the mathematical models of the plant to be controlled, the control strategy is described in the human-being-oriented form (i.e., if-then rule) obtained by experienced experts. In other words, experts can take reasonable control actions, even in the timevarying conditions of the process, nonlinearities, and existing disturbances. This means that knowledge about the plant and the control process is well represented in the if-then rules. Apparently, in an adaptive fuzzy control system, if the fuzzy rules are only considered as the (fuzzy) approximators and finally their parameters are tuned completely differently from their original faces, then there are two questions that arise: (1) why fuzzy rules? If we just want to approximate some unknown functions in the plant, some other soft computing methods like neural networks seem more attractive; and (2) why fuzzy control? If the experienced fuzzy rules are almost all changed in the control process, then where is the knowledge of experts going? Moreover, we should note that in the case of (1), we tune the parameters in the fuzzy approximators only from the standpoint of system stability, and as a result, the fuzzy approximators, in most cases, are not real approximators of their corresponding functions at all, because the errors between the two are sometimes very huge. Therefore, once we say fuzzy control, it seems that we should keep the original fuzzy rules. In this way, however, how to maintain the stability of the whole control system is a question we have to answer. In this paper, we consider a few cases to show how to develop a real fuzzy control system with stability in which it persistently maintains the fuzzy rules in accordance with the experts’ views. Also, computer simulations will support the approach proposed in this paper. 2. Problem Statement and Fuzzy Approximator. 2.1. Problem statement. This paper focuses on the design of adaptive fuzzy control algorithms for a class of nonlinear systems whose equation of motion can be expressed in the canonical form: (1) x(n) (t) = f (X(t)) + b(X(t))u(t) £ ¤ where X T (t) = x(t), x(t), ˙ . . . , x(n−1) (t) is the state vector, u(t) is the control input, f is an unknown linear or nonlinear function of X, and b is the control gain. It should be noted that more general classes of nonlinear systems could be transformed into this structure [13]. For the sake of simplicity throughout the remainder of this paper, we will omit time t unless a confusion arises. The control objective is to force the output x to follow a specified desired trajectory. Let xd be the desired trajectory and define the tracking error, x˜ = x − xd

(2)

The problem is thus to design a controller u for (1) which ensures that x˜ → 0 as t → ∞ while maintaining the stability of all signals involved in the system. Here in this paper, we take b = 1 for the sake of stressing a clear design procedure.

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Now, we adopt the variable structure theory to construct our adaptive fuzzy control system. The sliding mode hyperplane is firstly defined as µ ¶n−1 d x˜ with λ > 0 (3) s= +λ dt where λ defines the bandwidth of the error dynamics of the system. The equation defines a time-varying hyperplane in Rn on which the tracking error x˜ decays exponentially to zero, so that perfect tracking can be asymptotically obtained by maintaining this condition. In this case, the control objective becomes the design of controller to force s → 0 as t → ∞. The time derivative of the error metric can be written as (n) ˜ (4) s˙ = f + u − xd + ΛT X £ ¤ ˜ T = x˜, x˜˙ , . . . , x˜(n−1) . where ΛT = [0, n−1 Cn−1 λn−1 , n−2 Cn−1 λn−2 , . . . , 1 Cn−1 λ ], X Referring to system (1), it naturally suggests that when f is known, a controller of form, (n) ˜ u = −ks − f + xd − ΛT X, k>0 (5) 2 leads to ss ˙ = −ks , and hence, x˜ → 0 as t → ∞, where k > 0. However, as mentioned, the function f is unavailable. Therefore, the problem is how u can be determined when a system involves such an unknown function. Here, let us classify what cases can exist in such a system before we determine the controller. Generally, there are two cases: the first case is that we do not have any information available about the functions so that we need to approximate them from scratch; the second case is that there is a so-called expert— fuzzy-controller given by experts about the control process. In this case, the controller, clearly, should be utilized as much as possible in the development of the control system. In what follows, we will give details to show how to develop a stable controller in the two cases, but at first, the fuzzy system is reviewed briefly. 2.2. Fuzzy approximator. Generally, a fuzzy system consists of three main parts: the rule base, the inference engine module, and the difuzzification model. The rule base, also known as the knowledge base, stores the rules, which contain the qualitative and heuristic knowledge, in the form of a collection of fuzzy IF antecedent THEN consequent rules that can be interpreted form a point of view of automatic control as IF condition THEN action. Herein, we consider a subset U ∈ Rn of a fuzzy system with singleton consequent, product inference, and Gaussian membership function. Hence, such a fuzzy system can be written as F (X) = W T · G(X)

(6)

where X T = [x1 , x2 , . . . , xn ], W T = [ω1 , ω2 , . . . , ωN ] withQωj being the connection weight,

GT (X) = [g1 (X), g2 (X), . . . , gN (X)], and gj (X) = Gaussian membership function, defined by

n i=1

μAi (xi ) j Qn j=1 i=1 μAi (xi )

PN

" µ ¶2 # xi − ξji μAij (xi ) = exp − σji

where μAij (xi ) is a

j

(7)

where ξji indicates the position, and σji indicates the variance of the membership function. We now can show an important property on the above fuzzy system [14].

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Theorem 2.1. For any given real continuous function f on the compact set U ∈ Rn and arbitrary ε∗ , there exists an optimal fuzzy system expansion F ∗ (X) = W ∗T · G(X) with, such that sup |f (X) − F ∗ (X)| < ε∗

(8)

X∈U

where, ∗

W = arg

min

W ∈Ω⊂RN

½

sup X∈U ⊂Rn

¾ |f (X) − W · G(X)| T

The theorem above shows that the fuzzy system F can be viewed as an approximator to approximate a real continuous function f . In this paper such an approximator is referred to be as fuzzy approximator. 3. Fuzzy Controllers. 3.1. Controller from scratch. It is an obstacle to have such an unknown function f on the process of system development. In order to deal with it, one way is to approximate the unknown functions. Thinking of the approximator, a lot of literature employs the fuzzy approximator as shown in the previous subsection. Here we adopt this idea for the time being. Let us denote f ∗ (X) = Wf∗T Gf (X) to be the fuzzy approximator of the unknown function f . Then, there is a small positive value ε∗f such that, the error, εf = f − f ∗

(9)

which is referred to as reconstruction error, satisfies the following inequality, |εf | < ε∗f

(10)

u = uf d + uet + uec

(11)

Apparently, the optimal vector Wf∗ is unknown while Gf (X) is an available vector function in the fuzzy approximator f ∗ (X) = Wf∗T Gf (X), therefore, as usual, its estimate, ˆ T Gf (X) is adopted, and will be tuned based on the error dynamics denoted as fˆ(X) = W f s. Now, we are ready to develop our control system. Inspired by the above control structure in (5), our controller is now described as

where uf d is a linear feedback component described as (n)

˜ − ksφ uf d = x d − Λ T X s sφ = s − φ · sat( ) φ ⎧ ⎨ 1, s > φ s s , |s| ≤ φ sat( ) = ⎩ φ φ −1, s < −φ

(12) (13)

(14)

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and constant φ describes the width of a boundary layer, which is used to prevent discontinuous control transitions. uet is an estimate component for the unknown function f and described as ˆ T Gf (X) (15) uet = −fˆ = −W f ˆ f is a tunable parameter vector and tuned by adaptive law, where W ˆ˙ f = Γf Gf (X)sφ W

(16)

ˆ f is the estimate of W ∗ , and Γf is an appropriate symmetric positive definite where W f matrix which determines the adaptation rate. And uec is a complement component, which complements the error between f and fˆ, described as s uec = −ˆ εf sat( ) (17) φ where εˆf is the estimate of ε∗f . Actually, instead of the saturation function sat(s/φ), a sign function sgn(s) could be used in (17). The purpose of adopting such a saturation function is to eliminate the chattering phenomenon by smoothing out the discontinuous sign function in a neighborhood. In (17), εˆf also has a purpose of avoiding a prior knowledge about the reconstruction error. εˆf is estimated by, εˆ˙f = γf |sφ | (18) where γf is the adaptation rate. A block diagram of this controller structure is depicted in Figure 1, where the block (n) ˜ and s from signals xd , labeled “Filter” is an operation system which generates xd , X, and x.

Figure 1. Closed-loop control system without knowledge available The stability of the closed-loop system described by (1)-(3), and (11)-(18) is established in the following theorem. Theorem 3.1. If the plant (1) is controlled by (11)-(15) and (17) with the adaptive law (16), and (18), then tracking error (2) will be shrunk to zero while maintaining all signals involved in the system bounded.

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Proof: Consider the following Lyapunov function candidate: ´ 1³ 2 ˜ T Γ−1 W ˜ f + γ −1 ε˜2 V = sφ + W f f f f 2

(19)

where

ˆf − W∗ ˜f = W W f ε˜f = εˆf −

ε∗f

(20) (21)

From (4), (9) and (11), s˙ can be rewritten as s s˙ = −ksφ + f − fˆ − εˆf sat( ) φ s = −ksφ + εf + f ∗ − fˆ − εˆf sat( ) φ s = −ksφ − f˜ + εf − εˆf sat( ) φ

(22)

While s˙ φ is not defined when |s| = φ, d/dt s2φ is well defined and continuous anywhere and can be rewritten d/dt s2φ = 2sφ s. ˙ Taking the derivative of both sides of (19), and substituting (16), (18), and (22) into it, we have V˙

˜ T Γ−1 W ˆ˙ f + γ −1 ε˜f εˆ˙f = sφ s˙ + W f f f 2 = −ks − f˜sφ + εf sφ − εˆf |sφ | + f˜sφ + ε˜f |sφ |

≤

≤

φ −ks2φ −ks2φ

+ εf sφ − ε∗f |sφ |

(23)

ˆ f , εˆf ) are where the fact that sφ · sat(s/φ) = |sφ | is used. It means all signals (sφ , W bounded, and s → 0 as t → ∞. Furthermore, tracking error x˜ → 0 as t → ∞. Q.E.D. Remark 3.1. It should be noted that the bound of ε∗f is not necessary to be known in the approach above. In this way, in the approximation for f the necessity to assume a prior knowledge of the bound on the reconstruction error can be removed. Of course, if the bound ε∗f is given, then the complement component (17) can be replaced by s uec = −ε∗f sat( ) φ Consequently, the estimate εˆf (18) is no longer needed, and stability is surely guaranteed. Remark 3.2. The parameters in the fuzzy approximator are completely tuned by adaptive law (16) only based on the demand of the system stability. In other words, the fuzzy approximator does not involve any knowledge from the experts, and consequently the approximatedness is sometimes very poor though the control performance is good. Therefore, strictly speaking, in our view the control system, even if it involves fuzzy rules, is no longer a fuzzy control system, but an adaptive control system.

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Figure 2. The general fuzzy control system 3.2. Controller based on experts. Generally, a fuzzy control system can be depicted as Figure 2 where the block Filter generates input signals like tracking error or its difference from xd , and x, and the block F C represents the fuzzy controller, which generates control moments u to the plant from all the input signals. In quite a few cases, experts or operators may have very good knowledge on the control process in terms of fuzzy rules, i.e., F C (≡ uF ), to effectively control a specific plant. Therefore, the F C should be involved in the development of the control system as much as possible if it is available. However, even if the experienced F C has worked as good as expected so far, there is still a case that the F C only works well in a limited region, say Rf , and the system stability could not be guaranteed when control state travels in RfC , which is the complement of Rf in the state space, say R ⊂ X n , where Rf = {X | ||X −X0 ||p,π } ≤ r with a weighted p-norm defined by (29), a specified positive constant r and X0 fixing the absolute location of the sets in R (Figure 3). Therefore, the design of a controller, which is referred as stable controller uS with consideration of stability, is only necessary when X ∈ RfC . Hence, we can introduce a state dependent modulation function m so that it allows the system controller to shift between uF and uS . It means that, in general, the system is controlled by uF , and the controller is shifted from uF to uS when the system stability is threatened. (n) ˜ (≡ uT ), From (5), the theoretical controller is composed of −ksφ − f + xd − ΛT X and here the control input (Figure 2) is determined by F C. This means uF is a kind of description (approximation) of uT . Define the error between uF and uT as follows: εu = uF − uT

(24)

Because uF can effectively control the plant as well, it is reasonable to suppose that there exists a positive value ε∗u such that (25) |εu | ≤ ε∗u Referring to (11), the controller involving the F C is composed of

m=

u = m · uF + (1 − m) · uS

(26)

s uS = uF − εˆu sat( ) φ

(27)

⎧ ⎨ ⎩

1, ||X−X0 ||p,π −r , ψ

0,

X ∈ Rf X ∈ Rψ − Rf X ∈ RψC

(28)

where Rψ = {X | ||X − X0 ||p,π } ≤ r + ψ with ψ ≥ 0, region (Rψ − Rf ) is a transition region in which it is easy to check that 0 < m < 1, ψ is a positive constant representing

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the width of the transition region, and the weighted p-norm is defined as follows. ( n µ )1/p X |xi | ¶p ||X||p,π = πi i=1

(29)

for a set of strictly positive weights {πi }ni=1 . In (27), the coefficient εˆu is estimated by εˆ˙u = (1 − m)γu |sφ | (30)

where γu is the adaptation rate. The purpose of setting such a transition region is to avoid the discontinuous controllers shifting between uF and uS (Figure 3). A block diagram of this controller structure is depicted in Figure 4.

Figure 3. The disposition of regions Rf , Rψ − Rf , and RψC

Figure 4. Closed-loop control system with the well-tuned-fuzzy-controller The stability of the closed-loop system described by (1)-(3), and (26)-(30) is established in the following theorem. Theorem 3.2. If the plant (1) is controlled by (26) with (27)-(30), then the system will behave stably under the experienced expert fuzzy controller. Proof: Define the following Lyapunov function candidate: ¢ 1¡ 2 sφ + γu−1 ε˜2u V = 2 where ε˜u = εˆu − ε∗u

(31) (32)

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Taking the derivative of both sides of (31), and substituting (30) into it, we have εu − ε∗u ) |sφ | (33) V˙ = sφ s˙ + (1 − m) (ˆ

As shown in Figure 3, we can divide the state space into two regions: RψC , and R − RψC . Clearly, while the region R − RψC is compact, the region RψC maybe is so big that the system become unstable when the control state traveled in. Therefore, we consider two cases (RψC , R − RψC ) in this proof, and show the system stability in the two cases. (I) In the case of RψC . In this case, from (28) we have m = 0, and u in (26) becomes s u = uF − εˆu sat( ) φ s = εu + uT − εˆu sat( ) φ (n) ˜ − εˆu sat( s ) = εu − ksφ − f + xd − ΛT X (34) φ Substituting (34) into (4), one has s s˙ = −ksφ + εu − εˆu sat( ) (35) φ From (33) and (35), we get straightforwardly (36) V˙ ≤ k · s2φ where m = 0 in (33). It means all signals involved are bounded, and sφ will shrink into the compact region R − RψC . (II) In the case of R − RψC . When the control state travels into this region, clearly sφ is bounded, i.e., |sφ | ≤ s∗φ (37) where s∗φ is a positive constant, which is not necessary to be known. Also, u in (26) can be rewritten as s εu sat( ) u = uF − (1 − m)ˆ φ (n) ˜ = εu − ksφ − f + xd − ΛT X s −(1 − m)ˆ εu sat( ) (38) φ Substituting (4), (38) into (33) and taking (37) into account, we have V˙ ≤ k · s2 + mε∗ |sφ | ≤ k·

φ s2φ

u

+c

(39)

where m ≤ 1, and c = ε∗u s∗φ that is a positive constant. Obviously, in this case all signals Q.E.D. are bounded and sφ → 0 as t → ∞. Remark 3.3. In the control structure above, in order to avoid the chattering phenomenon, two kinds of smoothing boundary regions were adopted. One is the saturation function sat(s/φ) instead of sign (switching) function sgn(s). The other one is m in order to smooth the shift between uF in Rf and uS in RfC by inserting the boundary region (Rψ − Rf ).

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4. Simulation Study. To illustrate and clarify the proposed design procedures, we apply them developed in the previous section to control a nonlinear system: x(t) ˙ = where d is disturbance described as d=

1 − e−x(t) + d + u(t) 1 + e−x(t)

(40)

½

(41)

3, 1 ≤ t ≤ 2 0, otherwise

Without the control, i.e., u(t) = 0, it can be easily seen that the system is unstable, −x(t) −x(t) > 0 for x(t) > 0, and f = 1−e < 0 for x(t) < 0. The control because of f = 1−e 1+e−x(t) 1+e−x(t) objective is to force the system state x(t) to the origin, i.e., xd (t) = 0. −x(t) First, let’s consider that we do not have any knowledge about the function f = 1−e , 1+e−x(t) and the control process. Therefore, we have to approximate the unknown function f from scratch. In this simulation, the fuzzy approximator is conducted with seven fuzzy rules: Rj :

IF x is Aj

T HEN f is wj

where j(= 1, 2, . . . , 7) is rule’s number, h Aj is2 ia fuzzy set that is characterized by fuzzy (x−ξ ) membership function μAj (x) = exp − 2 j with ξ· = −3, −2, −1, 0, 1, 2, 3 for j = 1, 2, . . . , 7, respectively (Figure 5), and wj is a singleton value. Because we do not have any knowledge about f , we are supposed not to give any appropriate values about wj . Therefore, the initial values for wj (j = 1, 2, . . . , 7) are just assigned 0.5, uniformly. Then for the demand of system stability, wj are tuned by the adaptive law (16) where Γf = 0.1I with I being an appropriate identity symmetric matrix, and φ = 0.1. The feedback component uf d is calculated by (12) where k = 0.1. To determine the complement component uec , (18) is used where γf = 0.2.

Figure 5. Membership functions in the antecedents of fuzzy rules The control results are shown in Figures 6 and 7 where Figure 6 shows the state’s evolution and Figure 7 shows the amount of the control input u. Clearly, a great control

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performance is observed, even though we do not have any knowledge about the unknown function f and estimate it from scratch. However, because we tune the parameters wj (j = 1, 2, . . . , 7) only based on the standpoint of the “stability”, and not from the viewpoint of “approximatedness”, the performance of estimator, as shown in Figure 8, is quite poor. Obviously, the error between f and its fuzzy estimate fˆ is covered by the linear feedback component uf d and the complement component uec .

Figure 6. State evolution in the case that has no knowledge available

Figure 7. Control moments in the case that has no knowledge available Next, suppose that we have following knowledge F C on the control process before the design of control system. Rij :

IF x˜ is Ai , ∆˜ x is Bj

T HEN u is wij

where, x˜ = x − xd ; ∆˜ x = x˜ − x˜0 with x˜0 being the tracking error in the previous sampling time. Ai ’s membership functions are the same as shown in Figure 5. For

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Figure 8. Function f and its estimated value h i (x−ξ )2 Bj ’s, their fuzzy membership functions are characterized by μBj (x) = exp − 2 j with ξ· = −1, −0.6, −0.3, 0, 0.3, 0.6, 1 for j = 1, 2, . . . , 7, respectively. wij ’s are given by Table 1. The control performance driven by this F C are shown in Figures 9 and 10. We should note that it is different from the performance in Figure 6, which could not be obtained without the mathematical model of (40), the performance in Figure 9 is obtained only based on the experience as shown in Table 1, which is the true meaning of the “fuzzy control.” Table 1. The experts’ fuzzy controller

Now, involving this F C, the plant (40) is controlled by (26)-(30) where r = 0.5, ψ = 0.1, p = π = 1 and γu = 0.2. Figures 11 and 12 depict the performance. In comparison with the performance only driven by the F C in Figure 9, the performance in Figure 11 is much better. In this case, r is set 0.5. It means uF only worked when |x| < 0.5 while

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Figure 9. State evolution only driven by F C

Figure 10. Controller of F C uS = 0. The detail is shown in Figure 13, where we could confirm that uS was activated when the state x(t) entered in the region RψC . In other words, uS only works when the system stability is threatened, otherwise the controller is only determined by F C. 5. Conclusion. Directly employing the experienced experts’ knowledge about the control process is the most important reason why fuzzy control has been greatly welcomed practically and theoretically for more than two decades. However, in most cases the adaptive fuzzy control systems proposed so far laid stress on system stabilities via the relevant parameters’ tuning, rather than the involvement of experts’ knowledge. In this paper, we gave details showing how to develop an adaptive fuzzy control system in which experts’ knowledge is taken into account wherever possible. Also, computer simulations confirmed the approach proposed in this paper.

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Figure 11. State evolution with the controller based on experts

Figure 12. Controller based on experts Acknowledgment. The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation. REFERENCES [1] Takagi, T. and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern., vol.15, pp.116-132, 1985. [2] Sugeno, M. and G. T. Kang, Structure identification of fuzzy model, Fuzzy Sets Syst., vol.28, no.10, pp.15-33, 1988. [3] Tanaka, K. and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems, vol.45, no.2, pp.135-156, 1992. [4] Tanaka, K., T. Ikeda and H. O. Wang, Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based design, IEEE Trans. on Fuzzy Systems, vol.6, pp.250-265, 1998. [5] Han, H., Theoretical Analysis for a design of delay-dependent fuzzy control system with input saturation, Proc. of the IEEE International Conference on Fuzzy Systems, Vancouver, 2006.

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Figure 13. Controller moments of uF and uS [6] Su, C.-Y. and Y. Stepanenko, Adaptive control of a class of nonlinear systems with fuzzy logic, IEEE Trans. on Fuzzy Systems, vol.2, pp.258-294, 1994. [7] Polycarpou, M. M. and M. J. Mears, Stable adaptive tracking of uncertain systems using nonlinear parametrized on-line approximators, Int. J. Control, vol.73, no.3, pp.363-384, 1998. [8] Han, H., C.-Y. Su and Y. Stepanenko, Adaptive control of a class of nonlinear systems with nonlinearly parameterized fuzzy approximators, IEEE Trans. on Fuzzy Systems, vol.9, no.2, pp.315-323, 2001. [9] Tong, S. and H.-X. Li, Fuzzy adaptive sliding-mode control for MIMO nonlinear systems, IEEE Trans. on Fuzzy Systems, vol.11, no.3, pp.354-360, 2003. [10] Han, H. Observer-based fuzzy control scheme for a class of nonlinear systems, IEEJ Trans. on EIS, vol.125, no.3, pp.435-441, 2005. [11] Han, H., Adaptive fuzzy controller for a class of nonlinear systems, International Journal of Innovative Computing, Information & Control, vol.1, no.4, pp.727-742, 2005. [12] Han, H., Adaptive fuzzy control for a class of nonlinear systems with state observer, Journal of Advanced Computational Intelligence and Intelligent Informatics, vol.10, no.2, pp.225-233, 2006. [13] Sanner, R. M. and J.-J. Slotine, Gaussian network for direct adaptive control, IEEE Trans. on Neural Networks, vol.3, no.6, pp.837-863, 1992. [14] Wang, L.-X. and J. M. Mendal, Fuzzy basis function, universal approximation, and orthogonal least square learning, IEEE Trans. on Neural networks, vol.3, pp.807-814, 1992. [15] Han, H., H. Kawabata and S. Murakami, A stable controller design with fuzzy rules in accordance with experts’ knowledge, Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, vol.16, no.2, pp.83-92, 2004.