Moment Adaptive Fuzzy Control and Residue ... - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 4, AUGUST 2014

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Moment Adaptive Fuzzy Control and Residue Compensation Ted Tao and Shun-Feng Su, Fellow, IEEE

Abstract—In this paper, a novel control scheme adopted from moment control is proposed. In the proposed approach, an adaptive fuzzy system is employed to learn the effective moment. It is easy to see that such an approach can avoid wild guessing for the effective moment, and as shown in our simulation, can have nice control performance. In traditional adaptive fuzzy control approaches, bounds of system functions are required to facilitate supervisory control so as to have the robust control property. It can be expected that when those bounds used in the supervisory controller are not proper, the output may not be able to follow the reference trajectory satisfactorily. With the proposed moment adaptive fuzzy control, the bound needed is only the supremum of the control variance between two consecutive steps. It is much easier to predict. In our study, in order to further relax this requirement, another adaptive system is employed to estimate the residue of the moment adaptive fuzzy control system. It is called residue compensation in this paper. It can be found that with residue compensation, the approach does not need a supervisory controller, but still can quickly track the reference in a satisfactory fashion. Various simulations are conducted to demonstrate the effectiveness of the proposed approaches. Index Terms—Adaptive fuzzy control, cerebellar model articulation controller (CMAC), Lyapunov stability, robust control, supervisory control.

I. INTRODUCTION HE fuzzy set theory was introduced by Zadeh in 1965 [1] and has been applied to a variety of fields in recent years. Fuzzy logic controllers [2]–[6] or, in short, fuzzy controllers were originally proposed to control plants that are poorly understood in mathematic models but are familiar to professional human operators. In recent developments, fuzzy controllers have shown excellent performances in situations where the plant parameters and structures have uncertainties or unknown variations [7]–[9]. Adaptive fuzzy control [10]–[15], [55], [56] is one of those proposed approaches used to deal with system uncertainties. However, it can be found that even though the Lyapunov theorem has guaranteed the stability of adaptive fuzzy control systems [16]–[18], [57], [58], when certain conditions are not

T

Manuscript received September 30, 2012; revised January 27, 2013; accepted June 13, 2013. Date of publication July 30, 2013; date of current version July 31, 2014. T. Tao is with the Department of Computer and Communication Engineering, Chengshih University of Science and Technology, Taipei 112, Taiwan (e-mail: [email protected]). S.-F. Su is with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan (e-mail: sfsu@mail. ntust.edu.tw). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2013.2275168

satisfied, the system may not be able to achieve acceptable control performance [19], [20]. This is because the existence of approximation errors may still make the derivative of the Lyapunov function possibly positive [20], [21]. As shown in [20], when the leaning constant is not properly selected, the system may have stability problems owing to the omission of an error term in the Lyapunov stability proof process. Usually, an extra controller is added in adaptive fuzzy control to resolve the system stability problem in a robust control sense [13], [21]–[25]. The objective of robust control is to maintain certain desired performance of a system despite the existence of uncertainties, such as parameter variation or disturbance. In such a design, robust theories such as the small-gain theorem [26] or the H-infinite tracking design [27]–[29] are employed to redesign the adaptive fuzzy control systems and uncertainties are considered as a disturbance force injection for the input of the system [30]. It can be expected that robust controllers are easy to realize [31], [32] and indeed can have certain effects as expected. Nevertheless, when system parameters and uncertainty bounds are unknown, the system control performance although stable as desired, may not be satisfactory because of improper selections of adaptive constants or control gain constants. Another kind of robust control design for adaptive fuzzy control techniques is supervisory control [33], [34]. Supervisory control approaches are designed to drive the system into a reasonable region when the current control performance is not acceptable. The supervisory controller is first proposed in [35] and many similar supervisory controllers are considered to guarantee the initial control performance [34], [36]–[42] under different original control mechanisms. These supervisory control schemes have a promising advantage of requiring no prior knowledge of exact system dynamics. In those supervisory control systems, the proposed controllers indeed demonstrate good control performance. Nevertheless, a serious drawback of these approaches is that the robust bounds of some system parameters must be anticipatable when implementing supervisory controllers to ensure the stability of the system. In fact, those bounds will be difficult to obtain while the system functions are unknown in practical applications. Thus, the supervisory adaptive fuzzy controllers cannot follow reference trajectories well when those bounds are wrongly predicted. The moment control has been successfully utilized to control an armature in servo motor systems [10], [30], [43]. In such an approach, the so-called effective moment (or generating force) is used to facilitate the controller. With the use of the control action in the previous step [19] or backing step [59], [60], the system indeed can have very nice control performance. However, in this kind of approach, the effective moment must

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be selected heuristically, and in practical applications, it may be difficult to have good guess on this effective moment. Thus, in our study, an adaptive fuzzy system is employed to get the effective moment adaptively. This approach is referred to as the moment fuzzy control scheme. It can be found that such an approach indeed can have nice control performance even though the effective moment is not known. Besides, with the proposed moment adaptive fuzzy control, the bound needed in the supervisory control is only the supremum of the control variance between two consecutive steps. This is much easier to predict than the system function bound used in traditional adaptive fuzzy control. In this study, in order to further relax the requirement of bounds for control or system functions, another adaptive system is employed to estimate the residue of the moment adaptive fuzzy control system. This approach is called residue compensation in our study. Based on the Lyapunov theory and the use of the sliding control structure, the proposed approach is proved to be stable in the Lyapunov sense. It is shown that with the use of residue compensation, the system control performance becomes better. Although some other bounds are required in this approach, their values are usually very small and can be ignored. As a consequence, supervisory control becomes unnecessary in the system. From simulation, it is clearly evident that the system can still have robust stability as required. It is obvious that the proposed approach can easily be realized for any practical systems. The cerebellar model articulation controller (CMAC) [44], [45] has been shown to have several advantages including local generalization [50], [51] and rapid learning convergence [52], [53]. Thus, as shown in [32], the CMAC is a nice mechanism for an on-line learning control. In our implementation, the CMAC [44], [45] is employed to act as the learning mechanism utilized for residue compensation because of its quick learning capability. In the literature, there are also some CMACbased supervisory control approaches [33], [34], [46]. Most of them use CMAC to take the place of the fuzzy estimators in adaptive fuzzy control. In our previous work [19], a CMACbased previous step supervisory control scheme has been proposed to relax the required bound in an adaptive fuzzy control. However, it also used CMAC as the estimators for the controller, and the use of previous control action is only to relax the bound in the supervisory control. In this approach, the moment adaptive fuzzy control is proposed and the CMAC is employed to model the residue in the system. From our simulation, it can be shown that the proposed approaches indeed are effective. This paper is organized as follows. After this introduction section, traditional adaptive fuzzy control systems are introduced and discussed in Section II. The moment adaptive fuzzy control scheme is proposed in Section III. The Lyapunov theorem is also considered to ensure the system stability of the proposed approach. The residue compensation is then introduced in Section IV and the stability is also ensured through the use of the Lyapunov theorem. Simulation examples are provided in Section V to demonstrate the effectiveness of the proposed methods. Finally, Section VI concludes this paper.

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II. TRADITIONAL ADAPTIVE FUZZY CONTROL AND ROBUST BOUNDS In this section, the usual approach considered in traditional adaptive fuzzy control systems is introduced. Consider an nthorder nonlinear system of the form x(n ) = f (x, x , . . . , x(n −1) ) + b(x, x , . . . , x(n −1) )u + d y=x

(1)

where f and b are unknown but bounded continuous functions, d is an external bounded disturbance, and u and y ∈ R are the input and the output, respectively, of the system. Let x ˆ = (x, x , . . . , x(n −1) ) ∈ Rn be the state vector of the system. The control objective is to force the system output to follow a given bounded reference signal r under the constraints that all signals involved must be bounded. Since there are uncertainties and approximate errors, a robust controller is usually required to ensure the stability of the system in a Lyapunov sense. The details are given in the following. First, consider that there is no disturbance in the system (i.e., d = 0). Consider the following controller u∗ =

1 (n ) [r − f (x, x , . . . , x(n −1) ) + kˆT eˆ] b

(2)

where kˆ = [k0 , k1 , k2 , . . . , kn −1 ]T and eˆ = [e, e , e , . . . , e(n −1) ]T , and e(t) is the tracking error defined as e(t) = r(t) − y(t). The controller (2) is usually referred to as the perfect control law. With the perfect control law, we have e(n ) + kˆT eˆ = 0.

(3)

It is easy to verify that if kˆ is selected such that the roots of the characteristic equation as (3) are all in the open left-half plane, then the system will asymptotically track the reference input r. Next, consider the system with an external bounded disturbance (i.e., d = 0). In this situation, the sliding mode control can be employed to ensure the stability of the system [5]. The sliding surface S(t) is defined as the integral of the tracking error characteristic polynomial as  ˙ ˙ where S(t) = e(n ) + kˆT eˆ. (4) S(t) = S(t)dt, Then, a supervisory control force that is denoted as us can be employed to drive the state toward the sliding surface when an external disturbance exists. Such a robust controller is written as u = u∗ + us , us = sign(bS) × Dm ax where Dm ax = d/b∞ = sup |d/b| =

U

d bL

(5)

for

0 < bL ≤ |b| and |d| ≤ dU .

(6)

This kind of approach is usually called the supervisory control. Here, the H∞ norm is utilized to define Dm ax . It can be verified that such a robust control can ensure the system to be stable in a Lyapunov sense [19]. However, in the perfect control law, the functions f and b must be known. When f and b are unknown, the adaptive fuzzy control schemes, as proposed in [13] and [47], can be employed

TAO AND SU: MOMENT ADAPTIVE FUZZY CONTROL AND RESIDUE COMPENSATION

Fig. 1.

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Block diagram of the traditional robust adaptive fuzzy control system.

to resolve the problem. The adaptive fuzzy control scheme is based on the function approximation capability of fuzzy systems and the Lyapunov stability theorem. The fuzzy approximator in these approaches is to approximate either an uncertain control law or an unknown system function [13]. Here, the case of estimating f and b in (2) is introduced. The fuzzy approximator consists of a set of fuzzy IF–THEN rules in which the lth rule is of the form (l) (l) R(l) : IF x1 is F1 , and . . ., and xn is Fn THEN P = θ(l)

(7)

(l) where x1 , . . . , xn are the state variables defined in (1), F1 , . . ., (l) Fn are the corresponding fuzzy labels, P is the output variable (l)

for the fuzzy system, and θ is the corresponding output value for the lth rule. There are two fuzzy systems to approximate f and b, respectively. By using the product operations for the conjunction relations in the premise parts of fuzzy rules, the output of the two fuzzy systems are  N n (l) l=1 i=1 μF i( l ) (xi ) θf  = εT θf (8) fˆ(x|θ) = Pf =   N n l=1 i=1 μF ( l ) (xi ) i

N n

and ˆb(x|θ) = Pb =

 (l) μF ( l ) (xi ) θb i  = εT θ b N n l=1 i=1 μF ( l ) (xi ) l=1

i=1

and εT (x) = [ε(1) , ε(2) , . . . , ε(N ) ]. It is easy to verify [11], [47] with the use of Lyapunov theorem, if the following adaptive update law u∗ θ˙b = −rb Sεˆ

i

where μF ( l ) (xi ) is the membership degree of xi that belongs to

(10)

are employed for the fuzzy approximators, then the system will asymptotically track the reference input r, where rb and rf are adjusting constants, and S(t) is the integral of the tracking error characteristic polynomial, as described in (4). As mentioned in the literature [49], [54], with the use of fuzzy approximators for the perfect control law, it may still have model errors in the system. Those model errors δb and δf can be defined as δb = b − εT θb∗ δf = f − ε

T

θf∗ .

(11) (12)

In order to ensure the stability of the system under uncertainties, the supervisory control us can be selected as us = sign(bS) × Dm ax

(9)

θ˙f = −rf Sε

and

for

∗

Dm ax = (δf + δb u ˆ + d)/b∞ = sup |(δf + δb u ˆ∗ + d)/b| . (13)

i

(l)

the fuzzy label Fi , N is the number of rules n i=1 μF ( l ) (xi ) (l)  , θ = [θ(1) , θ(2) , . . . , θ(N ) ]T ε =   i N n l=1 i=1 μF ( l ) (xi ) i

Finally, for illustration, the block diagram of the aforementioned traditionally robust adaptive fuzzy control system is shown in Fig. 1. By using fuzzy outputs (8) and (9) to replace f and b, respectively, in (2), the fuzzy perfect controller is

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u ˆ∗ =

1 (n ) [r − fˆ(x, x , . . . , x(n −1) ) + kˆT eˆ]. ˆb

(14)

With the use of supervisory control us , the controller is u = u ˆ + us . It is easy to see from (13) that ∗

Dm ax

1 |δb | ≤ sup |(δf + d)| + sup |ˆ u∗ | bL bL

(15)

where bL is the lower bound of b (0 < bL ≤ |b|). Furthermore, it is easy to see that   1 sup |ˆ u∗ | ≤ sup |r(n ) | + f U + |kˆT eˆ| (16) bL where f U is the upper bound of f (|f | ≤ f U ). The aforementioned bounds are the same as those defined in [11], [16], [33], and [34]. In fact, in those traditional supervisory controls, it is always necessary to know the upper bound of the system function f and the lower bound of the system function b so as to estimate the supreme value of the fuzzy perfect control variable u ˆ∗ . However, those bounds sometimes are difficult to obtain, especially in practical systems. In this paper, we propose the moment adaptive fuzzy control system with residue compensation to resolve the aforementioned problems. The proposed method will be discussed in the following sections. III. MOMENT ADAPTIVE FUZZY CONTROL As mentioned, the effective moment (or generating force) has been successfully utilized in the literature [30], [43]. However, in the traditional moment control, the effective moment must be selected heuristically, and usually it may be difficult to make a good guess on this effective moment. Thus, in our study, an adaptive fuzzy system is introduced to get the effective moment adaptively. This approach is referred to as the moment adaptive fuzzy control scheme. The proposed approach is to use a fuzzy approximator to model the effective moment so as to avoid a wild guess for the effective moment. Then, in order to cope with the bound problem as stated in the supervisory control, the previous control input is employed. With the proposed approach, the bound needed is only the supremum of the control variance between two consecutive steps and are much easier to predict. From the simulation shown later, it can be found that the proposed moment adaptive fuzzy control can have a much better control performance when compared with the existing adaptive fuzzy control approach. In our study, in order to further relax this requirement, the residue compensator is also proposed. The details will be introduced in the next section. Now, the moment adaptive fuzzy control mechanism is introduced. First, consider that there is no disturbance. With the use of the kth step perfect control input as uk = 1b (x(n ) − f (x, x , . . . , x(n −1) )), the perfect control law becomes 1 (17) u∗ = uk + (e(n ) + kˆT eˆ). b However, actually, the kth step control variable is not able to get in practice, and thus, the (k–1)th step control variable is used instead. In the moment adaptive fuzzy control, the system

parameter 1/b = M will be estimated by a fuzzy approximator ˆ = θT ε. Then, the moment adaptive fuzzy control is as M M ˆ (e(n ) + kˆT e ) um af = uk −1 + M 

ˆ S. ˙ = uk −1 + M

(18)

With the similar idea to that in adaptive fuzzy control, Theorem 1 is provided to ensure the Lyapunov stability of the moment fuzzy control scheme. Theorem 1: Consider a moment adaptive fuzzy controller as u = um af + us , where um af is defined in (18) and us is defined as us = sign(bS) × Dm ax with ˙ M − d (19) Dm ax = sup Δu + Sδ b ∗ T where δM = M − θM ε is the model error and Δu = uk − uk −1 is the input error. Then, if the adaptive update law for θM in the fuzzy approximator is

˙ θ˙M = αA sign(b)S Sε

(20)

where αA is an adaptive constant, then the system can be ensured to be stable in a Lyapunov sense. Proof: First, define a Lyapunov function as 

1 1 ˜T ˜ 2 V1 = (21) S + θM θM 2 α ∗ − θM with the optimal fuzzy where θ˜M is defined as θ˜M = θM ∗ outputs θM and α is a positive constant. By considering the ∗ T ε, then M can be written as model error δM = M −θM ∗T T T T ˆ + θ˜M M = θM ε + δM = θ M ε + θ˜M ε + δM = M ε + δM (22) ˆ becomes and M

ˆ = M − θ˜T ε − δM . M M

(23)

The moment adaptive fuzzy control scheme (18) can also be rewritten as T ˙ ε − δM )S. um af = uk − Δu + (M − θ˜M

(24)

With u = um af + us , the derivative of the sliding surface equation is S˙ = r(n ) − x(n ) + kˆT eˆ = r(n ) − [f + b(um af + us ) + d] + kˆT eˆ = r(n ) − {f + b[uk − Δu T + (M − θ˜M ε − δM )S˙ + us ] + d} + kˆT eˆ.

(25)

With (2) and (17), we can get r(n ) = f + b[uk + M (e(n ) + kˆT eˆ)] − kˆT eˆ ˙ − kˆT eˆ. = f + b[uk + M S] Substituting it into (25), we have

 d T ˙ M − S˙ θ˜M S˙ = −b us − Δu + − Sδ ε . b

(26)

TAO AND SU: MOMENT ADAPTIVE FUZZY CONTROL AND RESIDUE COMPENSATION

˙ With θ˜M = −θ˙M and (26), the derivative of V1 becomes 1 T ˜˙ θM V˙ 1 = S S˙ + θ˜M α

 d ˙ M − S˙ θ˜T ε − 1 θ˜T θ˙M = −bS us − Δu + − Sδ M b α M



 d 1 ˜T ˙ T ˜ ˙ ˙ = −bS us − Δu + − SδM + bS S θM ε − θM θM . b α ˙ is used, we have When the update law θ˙M = αbS Sε

 d ˙ ˙ V2 = −bS us − Δu + − SδM . b

(27)

If the robust bound Dm ax is selected as (19), then V˙ 1 < 0 is ensured. It should be noted that b is unknown, and thus, the ˙ where adaptive law can be selected as θ˙M = αA sign(b)S Sε, αA = α|b| is a positive constant. In fact, a similar approach can be found in [48]. In that approach, the effects of b are taken out of the adaptive law and become a part of the constant term in the Lyapunov function. The approach used here is a common mechanism in handling the learning constant in adaptive fuzzy control [49]. Hence, Theorem 1 is verified. In Theorem 1, it can be found that the upper bound of the system function f as required in (16) is no longer required in this approach. In fact, because of using previous control input, this approach only needs the bound of Δu. It is easy to verify that the bound of Δu is much smaller and is easy to predict. The effectiveness of the moment adaptive control will be demonstrated in the simulation section. IV. RESIDUE COMPENSATION In the moment adaptive fuzzy control, it can be found that the ˙ M − d is still needed supreme value of the error term Δu + Sδ b to calculate the robust bound Dm ax . This error term that is denoted as Eresidue is called the residue in our study. If the system can find a way to approximate this residue, the system may not need the supervisory control and can still have excellent performance. Thus, we propose a CMAC-based residue compensation in our study. As shown in later simulations, the CMAC can learn this residue very well, and the proposed scheme has excellent tracking performance even without the supervisory control. In CMAC, a set of indexes Cv k = [c1 (vk ), c2 (vk ), . . . , cj (vk ), . . . , cN m (vk )]T is used to address an n-dimension input vector vk = (x1 , x2 , . . . , xn ) to extract the stored weights W from memory cells. Thus, the output of CMAC is Pv k = uCM AC =

cj (vk ) =

CvTk

W =

Nm

if wj is activated

0;

others

The proposed residue compensation control scheme can be written as ˆ (e(n ) + kˆT e ) + uCM AC uin = uk −1 + M T = uk −1 + (θM ε)(e(n ) + kˆT e ) + CvTk W 

cj (vk )wj

(29)

ˆ = θT ε is the output of the moment fuzzy controller where M M ˙ (Theorem 1) whose adaptive fuzzy law is θ˙M = αA sign(b)S Sε and uCM AC is the output of the CMAC-based residue compensator. In our implementation, the input variables for the CMAC ˙ The following learning mechanism considered are Δu and S. theorem provides the update law for the CMAC so as to ensure the Lyapunov stability of the proposed control scheme. Theorem 2: For a system as (1), the controller is u = uin + us , where uin is defined in (29) and us is us = sign(bS) × Dm ax with the robust bound Dm ax = sup |δCM AC |

(30) ∗

where the model error is δCM AC = Eresidue − W . If the update law for the moment adaptive fuzzy control is ˙ (31) θ˙M = αA sign(b)S Sε CvTk

for an adaptive constant αA , and the CMAC update law for residue compensation is ˙ = βA sign(b)SCv (32) W k for an adaptive constant βA , then the system is stable in a Lyapunov sense. Proof: First, define a Lyapunov function as 

1 1 T ˜ 1 ˜T ˜ W (33) V2 = θM + W S 2 + θ˜M 2 α β ˜ are defined as θ˜M = θ∗ − θM and W ˜ = where θ˜M and W M ∗ ∗ ∗ W − W with θM and W are the optimal outputs for θM and W , respectively. Then, Eresidue becomes ˙ M − Eresidue = Δu + Sδ

d b

= CvTk W ∗ + δCM AC ˜ + δCM AC . = CvTk W + CvTk W

(34)

With the proposed controller, the derivative of the sliding surface equation is

 d T ˙ M − S˙ θ˜M S˙ = −b us + uCM AC − Δu + − Sδ ε b T = −b(us + uCM AC − Eresidue − S˙ θ˜M ε)

˜ − S˙ θ˜T ε). = −b(us − δCM AC − CvTk W M

(35)

Similar to Theorem 1, the derivative of V2 is

j =1

1;

807

T ˜ − S˙ θ˜M V˙ 2 = −bS(us − δCM AC − CvTk W ε)

.

(28)

Detailed description of the CMAC can be found in [32], [44], and [45]. Note that since our control cannot define desired outputs, there is no way of using the traditional CMAC learning algorithm. In this approach, the CMAC learning mechanism will be derived from the Lyapunov theorem as shown later.

−

1 T ˙ 1 ˜T ˙ W W − θ˜M θM β α



˜ − 1W ˙ ˜ TW = −bS(us − δ C M A C ) + bSCvTk W β

 1 T ˙ T + bS S˙ θ˜M ε − θ˜M θM . (36) α

808

Fig. 2.

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Block diagram of the moment adaptive fuzzy control system with CMAC-based residue compensation.

˙ and It is easy to prove that if the updating laws θ˙M = αbS Sε ˙ W = βbSCv k are used, (36) becomes V˙ 2 = −bS(us − δCM AC ).

(37)

Again, since b is unknown, thus the adaptive laws can be ˙ and W ˙ = βA sign(b)SCv , selected as θ˙M = αA sign(b)S Sε k where αA = α|b| and βA = β|b| are adaptive constants. If the robust bound Dm ax is selected as (30), then the Lyapunov condition V˙ 2 < 0 is ensured. Thus, Theorem 2 is verified. In fact, as shown in our simulation, the CMAC learning mechanism can learn the residue very well in the moment fuzzy control system, and it can be found that δCM AC will approach to zero. As a result, it is not necessary to add supervisory control us into the proposed control schemes. Finally, in order to illustrate the proposed control algorithm, the block diagram of the CMAC-based residue compensation in the moment adaptive fuzzy control system is shown in Fig. 2. V. SIMULATIONS To illustrate the effectiveness of the proposed schemes, an inverted pendulum system, as used in [11], [38], and [47], is considered in our study first. The inverted pendulum system considered can be described as x˙ 1 = x2 x˙ 2 = f + bu + d ; y = x1 f=

g sin x1 − (mlx22 sin x1 cos x1 )/(mc + m) ; l[4/3 − m cos2 x1 /(mc + m)]

b=

cos x1 /(mc + m) l[4/3 − cos2 x1 /(mc + m)]

(38)

where x1 is the angle of the pole (rad), x2 is the angular velocity of the pole (rad/s2 ), g is the gravity (9.8 m/s2 ), mc is the mass of the cart (1.0 kg), m is the mass of the pole (0.1 kg), u is the force applied to the cart, d is the external disturbance (−5 N ≤ d ≤ 5 N), and the length of the pole l is 0.5 m. Let the reference signal be r(t) and the tracking error e(t) = r(t) − y(t). In order to avoid chattering phenomenon, a saturation function as (39) is introduced in our approach to take the place of sign function. Thus, the supervisory control us is redefined as

sign(bS) × Dm ax us = sat(bS)×Dm ax = sign(bS) × |S|

for |S| ≥ Dm ax

. for |S| < Dm ax (39) First, the traditional robust adaptive fuzzy controller as defined in Section II is adaptively tuned by (10) and 25 rules are used. The membership functions of fuzzy sets used are triangular functions in this implementation. The bound values of fuzzy sets are set to ±1 for the input and ±5 for the output. The fuzzy labels are set to be [−1, − 0.05, 0, 0.05, 1]T for inputs e and e. ˙ The initial values of θf are [−5, −0.625, 0, 0.625, 5]T , and the initial values of θb are [0, 0.875, 1, 1.125, 2]T . The reference π [sin (t) + 0.3 sin (3t)] (rad), and the external signal is r(t) = 10 disturbance is a square wave with its amplitude being 5(N ). Since the robust bound is very difficult to get from (13), it just can be roughly estimated from (6) instead. According to (6), the robust control bound Dm ax is roughly set to be 4. In this example, rb = 0.01 and rf = 0.1 are used for adaptive fuzzy rules in (10). The simulation results of the traditional robust adaptive fuzzy control with supervisory control are shown in Fig. 3 for the initial angle = 0.2(rad) and Fig. 4 for the initial angle = 0.458(rad). In these figures, part (a) shows the reference signal r and the output y, part (b) shows the control variable u and the external

TAO AND SU: MOMENT ADAPTIVE FUZZY CONTROL AND RESIDUE COMPENSATION

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Fig. 3. Performance of using the traditional supervisory adaptive fuzzy control for the initial angle = 0.2(rad). (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

Fig. 4. Performance of using the traditional supervisory adaptive fuzzy control for the initial angle = 0.458(rad). (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

disturbance d, and part (c) shows the system function (f + d) and the estimated system function Pf . Note that the disturbance d also can be seen as the uncertainties of the system function f . From the simulation results, it can be observed that the outputs cannot follow the reference signal very well with traditional adaptive fuzzy control schemes. It is obvious that root mean

square error after 1 s (RMSE1) still has 0.0166 for the initial angle = 0.2(rad) in Fig. 3(a) and 0.0184 for the initial angle = 0.458(rad) in Fig. 4(a). In our study, it is found that if the initial angle is greater than 0.458(rad), then the system becomes unstable. Besides, the control chattering phenomenon still happens in Fig. 4(b) for a large initial angle [0.458(rad)], although a

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Fig. 5. Performance of using the moment adaptive fuzzy control system with supervisory control for D m a x = 4 and the initial angle = 0.2(rad). (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

saturation function as (39) has been used. The estimated system function Pf departs from the actual system function f+d, when an external disturbance occurs as shown in Figs. 3(c) and 4(c), and it even chatters in the beginning of Fig. 4(c) for a larger initial angle. This is because the robust bound is not properly chosen. In fact, the robust bound for the perfect control as (14) is difficult to obtain because the upper bound f U and the lower bound bL are difficult to get from a practical system. In the simulation, the bound Dm ax of the supervisory control is just roughly set to be 4, and the tracking errors will be obvious especially when the external disturbance d rises and falls down or the initial angle is large. Next, the moment adaptive fuzzy control that is proposed in Section III is employed for the same system. In our implementation, αA = 0.01 and the initial values of θM are [0, 0.25, 0.50, 0.75, 1.0]T . Robust bound Dm ax = 4 for the supervisory control is again used. The simulation results are shown in Fig. 5 for the initial angle = 0.2(rad) and in Fig. 6 for the initial angle = 0.458(rad). It can be found that the control performance has been significantly improved. For the initial angle = 0.2(rad), the tracking error is reduced from 0.016618 [see Fig. 3(a)] to 0.0010534 [see Fig. 5(a)], and for the initial angle = 0.458(rad), the tracking error is reduced from 0.018393 [see Fig. 4(a)] to 0.001047 [see Fig. 6(a)]. In these two examples, the errors of the moment adaptive fuzzy control both are far less than the errors of the traditional adaptive fuzzy control. The control chattering phenomenon has also been significantly reduced as shown in Fig. 6(b) for the case of the initial angle = 0.458(rad). Besides, it can be observed that the estimated system functions Pf in Figs. 5(c) and 6(c) become much closer to the system function as compared with those in Figs. 3(c) and 4(c). In fact, other values (0.01 ≤ Dm ax ≤ 100) for the supervisory bound

have been tried in our study. It is concluded that the smaller supervisory bound will make the performance a little worse but their results are all still very good. It should be noted that in the aforementioned simulations, the supervisory control is always used in the system. Finally, the CMC-based residue compensation is considered. All parameters are the same as mentioned earlier i.e., αA = 0.01 and βA = 0.1, which correspond to rb = 0.01 and rf = 0.1 in the traditional robust adaptive fuzzy controller. It should be noted that the supervisory control is not used in this approach. Instead, the CMAC-based residue compensation is employed. The simulation results are shown in Fig. 7 for the initial angle = 0.2(rad) and in Fig. 8 for the initial angle = 0.458(rad). It can be found that the errors are further reduced to 9.75 × 10−4 for the initial angle = 0.2(rad) and to 9.73 × 10−4 for the initial angle = 0.458(rad). Even though, the errors reduced are not so significant, it still has 7% of improvement. Besides, it should be noted that the supervisory control is not used in this approach, so it is not required to guess some bounds. In order to compare the control effects, the errors of using those three approaches are shown in Fig. 9. Fig. 9(a) shows the results for the initial value = 0.2 rad and Fig. 9(b) shows the results for the initial value = 0.458 rad. It is obvious that the errors of using moment fuzzy control (with supervisory control and with CMAC-based residue compensation) are much smaller than the errors using the traditional adaptive fuzzy control systems with supervisory control. Thus, it is clearly evident that the performance and the stability of the proposed method are much better than the traditional robust adaptive method especially when the initial angle is large or external disturbance exists. For illustration, the distribution of weights in the CMAC is shown in Fig. 10(a), and the control variable uCM AC produced

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Fig. 6. Performance of using the moment adaptive fuzzy control system with supervisory control for D m a x = 4 and the initial angle = 0.458(rad). (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

Fig. 7. Performance of using the moment adaptive fuzzy control system with CMAC-based residue compensation for the initial angle = 0.2(rad). (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

from the CMAC learning mechanism is shown in Fig. 10(b). It is evident that the distribution of weights in the CMAC almost depends on S˙ when Δu is significant. From Fig. 10(b), it can be observed that when the initial angle is large or an external disturbance exists, the value of uCM AC produced is large. It indeed not only can take the place of supervisory control variable but also can improve the control performance of the moment adaptive fuzzy control system.

For a better demonstration, another first-order unstable system [20] is considered and is described as x˙ = f + u + d;

f = ex − 1,

y = x.

(40)

It can be found that this system possesses drastically unstable behavior. In our simulation, the reference signal r, the external disturbance d, the initial values for x, and the max bound values

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Fig. 8. Performance of using the moment adaptive fuzzy control system with CMAC-based residue compensation for the initial angle = 0.458(rad). (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) system function (f + d) and the estimated system function Pf .

Fig. 9. Errors of using tradition adaptive fuzzy control systems with supervisory control, the moment fuzzy control systems with supervisory control, and the moment adaptive fuzzy control systems with CMAC-based residue compensation. (a) Initial value = 0.2 rad for an inverted pendulum system. (b) Initial value = 0.458 rad for an inverted pendulum system.

of fuzzy sets, and the membership functions are set to the same as those in the aforementioned example. First, the simulation of using the traditional supervisory adaptive fuzzy control is shown in Fig. 11. The initial central values of the membership functions are set to be [−1, − 0.05, 0, 0.05, 1]T for the input and [−5, −2.5, 0, 2.5, 5]T for the output. It can be observed that the system output cannot follow the reference

signal well when the disturbance exists. This is because the supreme value of the fuzzy perfect control u ˆ∗ is difficult to obtain. Another reason is that the controller will be changed according to the adaptive laws when the errors persistently exist [20]. Next, the moment fuzzy control that is proposed in Section III is employed for the same system. In our implementation, Dm ax = 4 for the supervisory control is again used. The

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Fig. 10.

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(a) Distribution of weights in the CMAC. (b) Control variable u C M A C produced from the CMAC learning mechanism.

Fig. 11. Performance of using the traditional supervisory adaptive fuzzy control for f = ex −1. (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function P f .

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Fig. 12. Performance of using the moment fuzzy control system with supervisory control (D m a x = 4 M = 1) for f = ex −1. (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

Fig. 13. Performance of using the moment adaptive fuzzy control system with CMAC-based residue compensation for f = ex −1. (a) Reference signal r and the output y. (b) Control variable u and the external disturbance d. (c) System function (f + d) and the estimated system function Pf .

performance of using the moment fuzzy control system with supervisory control is shown in Fig. 12. Finally, the performance of using the moment fuzzy control with the CMAC-based residue compensation is shown in Fig. 13. The control performance is, as expected, very good. Similarly, in order to compare the control effects, the errors of using those three approaches are shown in Fig. 14. Thus, it can be concluded that the proposed approach can indeed free bounds required in the moment adaptive fuzzy

control and also can improve the control performance of the system. Thus, the proposed scheme can easily be realized for any practical systems. VI. CONCLUSION Usually, supervisory control is employed to the traditional robust adaptive fuzzy control system such that the control system will be stable in the sense of Lyapunov. In order to avoid

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Fig. 14. Errors for the system as f = ex −1 while using the tradition adaptive fuzzy control systems with supervisory control, the moment fuzzy control systems with supervisory control, and moment adaptive fuzzy control systems with CMAC-based residue compensation.

chattering phenomenon, a saturation function is often used. However, the output may still not be able to follow well the reference command in the traditional robust adaptive fuzzy control systems if the bounds of the system function are improperly selected. The tracking error may also be large especially when the external disturbance changes dramatically. Besides, the chattering phenomenon may be significant when a large initial angle exists, although a saturation function is used. In our study, the moment adaptive fuzzy controller is proposed to overcome the previously mentioned problem. In order to further reduce the need for some bounds in supervisory control, a CMAC-based compensation scheme is proposed. The proposed approach can significantly improve the control performance of adaptive fuzzy control even without supervisory control. Theoretical proofs are also provided to ensure the system to be Lyapunov stable. This approach can free the robust bound limitation so that the proposed approach can easily be realized for practical systems. Finally, simulation results demonstrate that the proposed schemes are effective as claimed. REFERENCES [1] L. A. Zadeh, “Fuzzy set,” Inf. Control, vol. 8, pp. 338–353, 1965. [2] C. C. Lee, “Fuzzy logic in control system: Fuzzy logic controller-part I and part II,” IEEE Trans. Syst., Man, Cybern., vol. 20, no. 2, pp. 404–435, Mar./Apr. 1990. [3] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan./Feb. 1985. [4] W. Pedrycz, Fuzzy Control and Fuzzy Systems, 2nd ed. New York, NY, USA: Wiley, 1993. [5] Y. J. Chen, “Fuzzy sliding mode controller design: Indirect adaptive approach,” Cybern. Syst., vol. 30, no. 1, pp. 9–27, 1999. [6] H. Ying, “The Takagi-Sugeno fuzzy controllers using the simplified linear rules are nonlinear variable gain controllers,” Automatica, vol. 34, no. 2, pp. 157–167, 1998. [7] C. Treesatayapun, “Fuzzy rules emulated network and its application on nonlinear control systems,” Appl. Soft Comput. J., vol. 8, no. 2, pp. 996– 1004, 2008. [8] G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp. 676–697, Oct. 2006.

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Ted Tao was born in Taipei, Taiwan, in 1959. He received the B.S degree from the Industrial Education Department, National Taiwan Normal University, Taipei, Taiwan, in 1984 and the M.S. and Ph.D. degrees from the Electrical Engineering Department, Tatung University, Taipei, in 1990 and 2004, respectively. He was a Lecturer with the Kuang Wu Institute of Technology and Commerce College, Taipei, from 1980 to 2004 and an Associate Professor with the Electrical Engineering Department, Technology and Science Institute of Northern Taiwan, Taipei, from 2004 to 2010. He has been an Associate Professor with the Computer and Communication Engineering Department, Taipei Chengshih University of Science and Technology, since 2010. His research interests include neural network, the cerebellar model articulation controller, learning techniques, fuzzy control systems, discrete-time signal processing, and image processing.

Shun-Feng Su (F’10) received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1983 and the M.S. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, USA, in 1989 and 1991, respectively. He is currently a Chair Professor with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei. He has published more than 160 refereed journal and conference papers in the areas of robotics, intelligent control, fuzzy systems, neural networks, and nonderivative optimization. His current research interests include computational intelligence, machine learning, virtual reality simulation, intelligent transportation systems, smart home, robotics, and intelligent control. Dr. Su is a Chinese Automatic Control Society (CACS) Fellow. He is very active in various international/domestic professional societies. He is the President of the Taiwan Fuzzy System Association and a Vice President of the International Fuzzy Systems Association. He currrently serves on the Board of Governors of the CACS, the Taiwan Society of Robotics, and the Taiwan Association of System Science and Engineering. He has also served as a Program Chair, Program Co-chair, or PC member of various international and domestic conferences. He is currently an Associate Editor for the IEEE TRANSACTIONS ON CYBERNETICS and the IEEE TRANSACTIONS ON SYSTEMS, as well as an Area Editor of the International Journal of Fuzzy Systems.