Adaptive Fuzzy Control With Guaranteed Convergence ... - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

807

Adaptive Fuzzy Control With Guaranteed Convergence of Optimal Approximation Error Yongping Pan, Meng Joo Er, Senior Member, IEEE, Daoping Huang, and Qinruo Wang

Abstract—With no a priori knowledge of plant boundary functions, a novel direct adaptive fuzzy controller (AFC) for a class of single-input single-output (SISO) uncertain affine nonlinear systems is developed in this paper. Based on the theory of fuzzy logic systems (FLSs) with variable universes of discourse (UDs), sufficient conditions that guarantee that the optimal fuzzy approximation error (FAE) is locally convergent are given. By the use of the output tracking error and its derivatives as input variables and by the selection of suitable adjusting parameters, a variable UD FLS with an optimal FAE local convergence is constructed, and its parameter adaptive law is derived by virtue of the Lyapunov stability theorem. Under the assumption that the optimal FAE is bounded, it is proved that the closed-loop system is asymptotically stable in the sense that all variables are uniformly ultimately bounded and that the tracking errors converge to zero. The proposed approach eliminates the influence of the FAE on the tracking errors by means of the inherent mechanism of the variable UD FLS. Thus, it has the potential to achieve high control performance without additional compensation under only a few fuzzy rules. Simulation studies demonstrate the superiority of the proposed AFC in terms of the settling time, tracking accuracy, smoothness of the control input, and robustness against external disturbances and parameter variations. Index Terms—Approximation error convergence, fuzzy control, uncertain nonlinear system, variable universe of discourse (UD).

I. INTRODUCTION OR uncertain affine nonlinear systems, approximationbased adaptive fuzzy controllers (AFCs) are typically classified into two categories, namely a direct AFC and an indirect AFC [1]. Compared with the indirect scheme that uses two fuzzy logic systems (FLSs) to approximate the plant dynamics, the direct scheme only makes use of one FLS as the controller to approximate an ideal control law. Both AFC schemes must deal with a fuzzy approximation error (FAE) that is an important feature in the system performance analysis. In order to obtain the stability and the tracking error convergence of the closedloop system, the optimal FAE is often assumed to be square

F

Manuscript received August 29, 2010; revised January 22, 2011; accepted March 9, 2011. Date of publication April 21, 2011; date of current version October 10, 2011. Y. Pan and D. Huang are with the School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]; [email protected]). M. J. Er is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). Q. Wang is with the Department of Automation, Guangdong University of Technology, Guangzhou 510006, China (e-mail: [email protected]). 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.2011.2144600

integrable [1]–[3] or sufficiently small [4]–[7]. The square integrable property of the optimal FAE is difficult to verify, especially in cases where an exact mathematical model of the plant is not available [8]. Moreover, according to the fuzzy approximation property [1], the assumption that the optimal FAE is sufficiently small is not always reasonable. A common approach to ensure the stability of the closed-loop system is to introduce an additional supervisory controller in the AFC [9]–[14]. However, it will inevitably result in high gain and chattering of the control input using this type of approach. To circumvent the problem involved in the FAE, one approach that is termed adaptive bounding technique revolves around estimating the supremum of the optimal FAE [15]–[21] or its function [22]–[24]. By the combination of the adaptive bounding technique with sliding mode control, additional sliding control terms with supremum estimators were introduced in the traditional direct AFC [17], traditional indirect AFC [21], general fuzzy-neural-network (FNN) direct controller [23], Takagi–Sugeno–Kang (TSK)-type FNN direct controller [18], recurrent FNN direct controller [19], cerebellar-model-based direct AFC [20], fuzzy-wavelet-network-based direct AFC [22], and dynamically structured FNN indirect controller [24], respectively. Those variations of the AFC in [17]–[24] guarantee the stability of the closed-loop system and effectively eliminate the influence of the FAE on the tracking errors but may cause chattering by the switch function in the sliding control terms. Moreover, the lower bound of the control gain function needs to be known in [19] and [24], whereas nonnegative adaptive laws that make the estimated bounds unbounded are employed to update the supremum estimators in all those approaches. Next, a bounded supremum estimator was presented in [8] by the use of the σ-modification in the updated law, and a novel supremum estimator was developed in [15] and [16] by the use of a saturation function to replace the switch function. However, certain plant boundary functions still require to be known in [8] and [15], and the chattering of the control input is eliminated at the expense of loss of tracking accuracy in [15] and [16]. Another approach to solve the FAE problem is to estimate the optimal FAE directly [25]–[29]. Under the direct AFC scheme, bounded lumped uncertainty (include the optimal FAE) estimators were introduced in [25]–[27]. Subsequently, bounded optimal FAE estimators under the direct and indirect AFC schemes were also developed in [28] and [29], respectively. The estimators in [25]–[29] were obtained without any knowledge of plant boundary functions. Both simulated and experimental studies showed the high efficiency to promote the tracking performance of all those approaches. However, those solutions assume that

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the derivative of the optimal FAE is zero. This assumption is difficult to theoretically explain, especially in uncertain nonlinear control systems. In summary, to the best of our knowledge, all previous solutions of the FAE problem involve some rigorous restrictions. Li [30] introduced an FLS with variable universe of discourse (UD) that has the characteristic of high fuzzy output accuracy. A variable UD FLS essentially is a type of interpolator with dynamic point-by-point convergence. Under the same fuzzy rules, the UDs can be contracted by the adjustment of contraction– expansion (C-E) factors while input variables become smaller, which means increasing fuzzy rules so that the fuzzy output accuracy can be improved. It was pointed out in [31] that the variable UD FLS has the property of the optimal FAE being local convergent. Recently, the set membership methodology was investigated in the approximation of model predictive control laws for linear systems [32]. It was showed in [32] that the approximation error converges to zero if the number of exact solutions of a certain optimization problem tends to infinity. This condition is too strict to achieve the approximation error convergence. The variable UD FLS-based AFC has been successfully applied to many control problems such as balance of quadruple inverted pendulum [33], [34], liquid level regulation [35], stabilization and synchronization of chaos systems [36]–[42], and suppression of aircraft wing rock [43]. However, all these approaches did not make use of the property of the optimal FAE local convergence. Accordingly, it is highly desired to relax the restricted conditions of resolving the FAE problem by using such property in the AFC design. This paper focuses on the design of a direct AFC for a class of single-input single-output (SISO) uncertain affine nonlinear systems without any a priori knowledge of plant boundary functions. First, under the key restriction that the input C-E factors are infinitesimals of higher order (IHOs) of corresponding input variables, it is proved that the optimal FAE of the variable UD FLS has the locally convergent property, i.e., the optimal FAE converges to zero with faster speed than input variables while input variables tend to zero. Next, by the use of the output tracking error and its derivatives as input variables and by the selection of suitable parameters, a variable UD FLS with a locally convergent optimal FAE is constructed, and its parameter adaptive law is derived by the Lyapunov stability theorem. With the assumption that the optimal FAE is bounded, it is prove that the closed-loop system is asymptotically stable in the sense that all variables are uniformly ultimately bounded (UUB) and that the tracking errors converge to zero. Different from the aforementioned solutions to deal with the FAE, we make use of the local convergence property of the optimal FAE to eliminate the influence of the FAE on the tracking errors. Thus it has the potential to achieve favorable control performance without additional compensation under only a few fuzzy rules. The structure of this paper is as follows. The problem under consideration is formulated in Section II. Details of the variable UD FLS are described, and its sufficient conditions to guarantee the optimal FAE being locally convergent are given in Section III. In Section IV, the design procedure of the variable UD FLS-based direct AFC is proposed. Illustrative examples are


demonstrated in Section V. Concluding remarks are summarized in Section VI. II. PROBLEM FORMULATION Consider the following nth-order SISO nonlinear system: (n ) x = f (x) + g(x)u (1) y=x ˙ . . . , x(n −1) ]T ∈ Rn is where x = [x1 , x2 , . . . , xn ]T = [x, x, the measurable state vector; u ∈ R and y ∈ R are the input and output variables, respectively; and f (x) and g(x) are the unknown continuous functions. Assumption 1: For all x ∈ D ⊂ Rn , there exist unknown bound functions f¯(x), g(x), and g¯(x) such that |f (x)| ≤ f¯(x) , and 0 < g(x) ≤ g(x) ≤ g¯(x), where D is a controllable region. (n −1)

Let y=[y, y, ˙ . . . , y (n −1) ]T , and ym =[ym , y˙ m , . . . , ym ]T , where ym denotes a bounded reference input which has the nth-order derivative. Define the output tracking error e = ym − y and the error vector e = ym − y = [e1 , e2 , . . . , en ]T = [e, e, ˙ . . . ,e(n −1) ]T . Choose k = [kn , . . . , k1 ]T ∈ Rn so that n s + k1 sn −1 + · · · + kn is a Hurwitz polynomial, where s is a complex variable. Then, design the following ideal controller: u∗ =

(n )

(−f (x) + ym + kT e) . g(x)

(2)

Substitution of (2) into (1) leads to e(n ) + k1 e(n −1) + · · · + kn e = 0.

(3)

By virtue of the selection of k, all roots of (3) have negative real parts. Thus, limt→∞ |e(t)| = 0. However, since f (x) and g(x) are unknown, the controller in (2) cannot be applied. The objective of this paper is to design a direct AFC without any a priori knowledge of plant boundary functions so that the closed-loop system is stable in the sense that all variables are UUB and that the tracking errors converge to zero. III. FUZZY LOGIC SYSTEMS WITH VARIABLE UNIVERSES OF DISCOURSE A. Description of Fuzzy Logic Systems with Variable Universes of Discourse Consider the FLS with n inputs and a single output. Let Xi = [−Ei , Ei ] and Y = [−U, U ] be UDs of the input variables ei and the output variable u, respectively, where Ei , U ∈ R+ and i = 1, . . . , n. Let Ai = {Alii }(l i =1,...,m ) and B = {B j }(j =1,...,M ) denote fuzzy partitions on Xi and Y , respectively. The terms Ai and B are regarded as linguistic variables so that a fuzzy rule base with M = mn rules is constructed as follows: Rl : If e1 is Al11 and . . . and en is Alnn , then u is B l 1 ···l n

(4)

where l = 1, . . . , M , li = 1, . . . , m, i = 1, . . . , n, and B l 1 ···l n ∈ B. Let μA l i denote the membership function of Alii , and e¯lii and i

li l 1 ···l n u ¯l 1 ···l n denote the peak points , respectively. m of Ai and B Assume that Ai satisfies l i =1 μA l i (ei ) = 1. Thus, by the use i of the singleton fuzzifier, product t-norm, and center-average

PAN et al.: ADAPTIVE FUZZY CONTROL WITH GUARANTEED CONVERGENCE OF OPTIMAL APPROXIMATION ERROR

defuzzifier [1], the FLS can be expressed as follows: m m n ¯l 1 ···l n l 1 =1 · · · l n =1 u i=1 μA l i (ei ) i u = Fˆ (e) = (5) m m n l 1 =1 · · · l n =1 i=1 μA l i (ei ) i m m n where l 1 =1 l n =1 i=1 μA l i (ei ) = 1. i Variable UD means that the UDs Xi and Y can be adjusted by changing the variables ei and u, which are denoted as Xi (ei ) = [−αi (ei )Ei , αi (ei )Ei ] (6) Y (u) = [−β(u)U, β(u)U ] where αi (ei )(i = 1, . . . , n) and β(u) are the C-E factors. Thus, the variable UD form of (5) can be expressed as follows [36]: m m u = Fˆ (e) = β(u) ··· u ¯l 1 ···l n l 1 =1 l n =1 n × μA l i (ei /αi (ei )) . (7) i=1

i

By the definition of the (l1 · · · ln )th fuzzy basic function n ξl 1 ···l n (e) = μA l i (ei /αi (ei )) i=1

(8)

i

(7) can be expressed as u = Fˆ (e|θ) = θ T ξ(e)

(9)

¯M ], and ξ(e) = [ξ1 (e), . . . , ξM (e)]. where θ = [β u ¯1 , . . . , β u Therefore, ξ(x) can be modified by αi (ei ), and β(y) can be adjusted through the adaptive law of θ as in [44].

. . . , m − 1, and i = 1, . . . , n. According to [−Ei , Ei ] = −1 m em , e¯i ], one obtains [¯ e1i , e¯2i ] ∪ · · · ∪ [¯ i −1 m −1 k 1 ···k n . Ωe = ∪m l 1 =1 · · · ∪l n =1 Ωe

Fˆ (e) = β(u)

u ¯l 1 ···l n

n i=1

l n =k n

μA l i (ei /αi (ei )) i

(L1 L2 · · · Ln F ) (e)

w = F (e) − Fˆ (e|θ ∗ )

θ∈Ω θ

k n +1

By the use of (12) and (15), one deduces

=

where θ ∗ is an optimal paremeter vector defined as Δ θ ∗ = arg min supe∈Ω e F (e) − Fˆ (e|θ) .

···

(14) +1 k n +1 n with lk11=k · · · μ = 1. l i (ei /αi (ei )) l n =k n i=1 A 1 i Define linear operators L1 , L2 , . . . , Ln as follows [1]: ⎧ k 1 +1 ⎪ ⎪ (L1 F ) (e) = l 1 =k 1 μA l11 (e1 /α1 (e1 )) ⎪ ⎪ ⎪ ⎪ ×F α1 (e1 )¯ el11 , e2 , . . . , en ⎪ ⎪ ⎪ k 2 +1 ⎪ ⎪ ⎨ (L2 F ) (e) = l 2 =k 2 μA l22 (e2 /α2 (e2 )) l2 (15) ×F e , α (e )¯ e , . . . , e 1 2 2 n 2 ⎪ ⎪ . ⎪ ⎪ .. ⎪ ⎪ ⎪ k n +1 ⎪ ⎪ (Ln F ) (e) = l n =k n μA lnn (en /αn (en )) ⎪ ⎪ ⎩ ×F e1 , e2 , . . . , αn (en )¯ elnn .

=

(10)

k 1 +1 l 1 =k 1

Define the compact sets, i.e., Ωθ = {θ:||θ|| ≤ Mθ } ⊂ R and Ωe = {e : ||e|| ≤ Me } ⊂ Rn , where Me , Mθ ∈ R+ , and Ωe is regarded as the fuzzy approximation region. Let Fˆ in (9) approximate a function F (e) whose analytic expression is unknown. The optimal FAE is given by Δ

(13)

That is, ∀e ∈ Ωe , ∃Ωke 1 ···k n such that e ∈ Ωke 1 ···k n . Now, assume ek1 1 , e¯k1 1 +1 ], . . . ,en ∈ [¯ eknn , e¯knn +1 ]. that e ∈ Ωke 1 ···k n , i.e., e1 ∈ [¯ Since each Ai is normal, consistent, and complete, one can simplify (7) into the following expression:

B. Optimal Fuzzy Approximation Error Convergence Property M

809

k 1 +1 l 1 =k 1

···

k n +1

n

i=1

l n =k n

μA l i (ei /αi (ei )) i

el11 , α2 (e2 )¯ el22 , . . . , αn (en )¯ elnn × F α1 (e1 )¯ k 1 +1 l 1 =k 1

···

k n +1

n

l n =k n

i=1

μA l i (ei /αi (ei )) β(u)¯ ul 1 ···l n . i

(16) (11)

Let Ωe = [−E1 , E1 ] × · · · × [−En , En ]. Now, we state the main result of this paper. Theorem 1: Suppose that F (e) is a twice differentiable unknown but bounded continuous function in Ωe , Ai are normal, consistent, and complete, and peak points u ¯l 1 ···l n satisfy el11 , α2 (e2 )¯ el22 , . . . , αn (en )¯ elnn β(u)¯ ul 1 ···l n = F α1 (e1 )¯ (12) where li = 1, . . . , m, and i = 1, . . . , n. If C-E factors αi (ei ) in (6) are IHOs of ei with αi (0) = 0, where i = 1, . . . , n, then the optimal FAE w → 0 with faster speed than ||e||, while ||e|| → 0, i.e., w has the local convergence property within a small neighborhood of ||e|| = 0. Proof: First, we derive the analytic expression of w. Let ek1 1 , e¯k1 1 +1 ] × · · · × [¯ eknn , e¯knn +1 ], where ki = 1, Ωke 1 ···k n = [¯

It follows from (14) and (16) that (L1 L2 · · · Ln F ) (e) = Fˆ (e).

(17)

From (6), one has (e1 /α1 (e1 )) ∈ X1 . Thus, according to (14) and (15), one obtains k 1 +1 μA l 1 (e1 /α1 (e1 )) F ∞ = F ∞ . L1 F ∞ ≤ l 1 =k 1

1

(18) From (17) and (18), one has F − Fˆ ∞ = F − L1 L2 · · · Ln F ∞ ≤ F − L1 F ∞ + L1 (F − L2 · · · Ln F )∞ ≤ F − L1 F ∞ + F − L2 · · · Ln F ∞ .

810


Consequently, one gets

IV. DESIGN OF DIRECT ADAPTIVE FUZZY CONTROLLER

F − Fˆ ∞ ≤ F − L1 F ∞ + F − L2 F ∞ + · · · + F − Ln F ∞ .

(19)

By the use of the results of single-value linear interpolation in [45], one obtains the following inequality: F − Li F ∞

1 ≤ αi2 (ei )(¯ eki i +1 − e¯ki i )2 ∂ 2 F (ei )/∂e2i ∞ 8

where i = 1, . . . , n. Thus, one concludes that

w = u∗ − u(e|θ ∗ )

(24)

θ ∗ = arg min supe∈Ω e |u∗ − u(e|θ)| .

(25)

where θ ∗ is given by

θ∈Ω θ

After some manipulations of (1)–(3), one obtains the following tracking error dynamic equation:

w ≤ F (e) − Fˆ (e|θ)∞ 1 ∂ 2 F (ei )/∂e2i αi2 (ei )h2i ≤ ∞ 8 i=1

Select the variable UD FLS u(e|θ) in the form of (9) to approximate the ideal controller u∗ given by (2). The optimal FAE is rewritten into

e(n ) = −kT e + g(u∗ − u(e|θ)).

n

(20)

where hi = maxl i =1,...,m −1 |¯ elii +1 − e¯lii |, and i = 1, . . . , n. Second, we derive the local convergence property of w. If ||e|| → 0, then ei → 0, where i = 1, . . . , n. Let wup denote the upper bound of w. From (20), one gets wup =

1 8

n

2 ∂ F (ei )/∂e2i αi2 (ei )h2i . ∞

(21)

i=1

Next, make the following infinitesimal comparison: lim

||e||→0

wup ||e||

n 1 ∂ 2 F (ei )/∂e2i αi2 (ei )h2i ||e|| = lim ∞ 8 i=1 ||e||→0 n 1 ∂ 2 F (ei )/∂e2i ∞ αi2 (ei )h2i = lim e i →0,i=1,...,n 8 e21 + e22 + · · · + e2n i=1 n 1 ∂ 2 F (ei )/∂e2i ∞ h2i αi2 (ei )/ei = lim . 2 2 2 2 e i →0,i=1,...,n 8 i=1 sgn(ei ) (e1 + e2 + · · · + en ) /ei

From the hypothesis in Theorem 1, one has lime i →0 αi (ei )/ei = 0, i.e., lime i →0 αi2 (ei )/ei = 0. By the combination of lim

e i →0, i=1,...,n

w ≤ wup , and αi (0) = 0, one obtains lim (w/||e||) = 0.

(22)

Therefore, w is an IHO of ||e||, i.e., w → 0 with faster speed than ||e|| while ||e|| → 0. Remark 1: Generally, αi (ei ) can be designed as follows [30]: αi (ei ) = 1 − λce exp (−κi e2i )

j =1

where c0 , cj ∈ R+ and j = 1, . . . , M . By the substitution of (27) into (26), one obtains M e(n ) = −kT e + cj θ˜j ξj (e) + c0 w

(28)

where θ˜j = θj∗ − θj , or equivalently M e˙ = Ae + B cj θ˜j ξj (e) + c0 w

(29)

j =1

⎡ ⎤ ⎤ 0 0 1 ··· 0 ⎢ .. ⎥ ⎢ .. .. .. ⎥ . . ⎢ ⎥ ⎢ . . ⎥ . where A = ⎢ . ⎥, B = ⎢ . ⎥. ⎣0⎦ ⎣ 0 0 ··· 1 ⎦ 1 −kn −kn −1 · · · −k1 Since A is a stable matrix, there exists a unique positive definite symmetric matrix P for a given positive definite symmetric matrix Q such that ⎡

AT P + P A = −Q.

(30) (n )

((e21 + e22 + · · · + e2n )/e2i )1/2 ≥ 1

||e||→0

Let θj∗ and θj denote the jth elements of θ ∗ and θ, respectively. It follows from [46] that the following lemma holds. Lemma 1: For the FLS u(e|θ) in (9), there exists θ ∗ given by (25), and finite constants c0 and cj such that M cj θj∗ − θj ξj (e) + c0 w (27) g(u∗ − u(e|θ)) =

j =1

(23)

where λce ∈ (0, 1], κi > R+ , and i = 1, . . . , n. Therefore, αi (ei ) in (23) is an IHO of ei . If λce = 1, then αi (0) = 0. Thus, (23) satisfies the conditions of the optimal FAE being locally convergent in Theorem 1. However, in order to prevent the denominators from becoming zero in (7), λce must be set to a value close to but less than 1 in practice.

(26)

From Assumption 1, (3), and the fact that ym ∈ L∞ , there exists a bounded u∗ in (2) for ∀e(0) ∈ Ωe such that the closedloop system achieves limt→∞ |e(t)| = 0. According to the fuzzy approximation theorem and the boundedness of u∗ , we makes the following assumption as in [47]. Assumption 2: For all e ∈ Ωe , the optimal FAE w in (24) is bounded with a upper bound wup that is given by (21). Redefine Ωθ = {θ:|θj | ≤ mθ + δ, j = 1, . . . , M }, where mθ , δ ∈ R+ . Now, we state the second main result of the paper. Theorem 2: For the nonlinear system in (1), select u(e|θ) in (9) that is equipped with C-E factors in (23) as the controller, and design the parameter adaptive law as follows [48]: ⎧ (1 + (mθ − θj )/δ) γeT P Bξj (e) ⎪ ⎪ ⎪ ⎨ if θj > mθ and eT P B > 0 ˙θj = (1 + (mθ + θj )/δ) γeT P Bξj (e) (31) ⎪ T ⎪ < −m and e P B < 0 if θ ⎪ j θ ⎩ T γe P Bξj (e) otherwise


Fig. 1.

811

Block diagram of a direct AFC.

where γ ∈ R+ is a learning rate. The overall control scheme is shown in Fig. 1. Then, there exist proper κi (i = 1, . . . , n) in (23) to ensure that the closed-loop system is asymptotically stable in the sense that all variables are UUB, and limt→∞ ||e(t)|| = 0. Proof: Construct the following Lyapunov function: V = eT P e/2 +

M j =1

cj θ˜j2 /2γ.

(32)

Differentiating V along (29) and using (30) yield M V˙ = −eT Qe/2 − cj θ˜j θ˙j /γ j =1 M T ˜ + e PB cj θj ξj (e) + c0 w j =1

= −eT Qe/2 + eT P Bc0 w M cj θ˜j (γeT P Bξj (e) − θ˙j )/γ. + j =1

Consider the case of “otherwise” in (31). Substitution of (31) into the aforementioned expression leads to

Fig. 2. Comparisons of two sides of (35). (a) Case that (35) does not always hold. (b) Case that (35) always holds.

V˙ = −eT Qe/2 + eT P Bc0 w ≤ −λm in (Q)||e||2 /2 + pm ax c0 ||e|| · |w|

(33)

where λm in (Q) denotes the minimal eigenvalue of Q, pm ax is the maximal element of pn , and pn is the nth column of P. From the result in [48], one can also obtain (33) by other cases in (31). From Theorem 1, one knows that w is an IHO of ||e||. Hence, according to (20), there exist proper κi (i = 1, . . . , n) in (23) for any given e(0) ∈ Me and ||e|| = 0 such that |w| < λm in (Q)||e||/2pm ax c0

(34)

always holds. By the use of (20), (34) can also be expressed as (35) αi (ei ) < 2 λm in (Q)||e||/ (pm ax c0 nMf h2i ) where Mf = max(||∂ 2 f (ei )/∂e2i ||∞ )(i = 1, . . . , n), and i gets the value that makes αi (ei )hi (i = 1, . . . , n) achieve the maximum value. By substitution of (34) into (33), one has V˙ < 0, ∀||e|| = 0, i.e., the closed-loop system is asymptotically stable. Thus, e, x, θ ∈ L∞ , and limt→∞ ||e(t)|| = 0. By combining (8) and (9), one also has u ∈ L∞ . Hence, all variables are UUB. Remark 2: Because some parameters are unknown in (35), κi (i = 1, . . . , n) in (23) cannot be exactly determined. How-

ever, one has that decreasing κi improves the system stability, but κi that is too small would cause the FLS to be hypersensitive to the input variables. For example, assume that n = 2, hi = 0.4, Mf = 30, c0 = 1, e1 (0) = e2 (0) = 1, Q = diag(10, 10), and A = [0, 1; −1, −2]. Then, it is not difficult to get λm in (Q) = 10, pm ax = 5, and e(0) ≈ 1.414. Comparisons of both sides of (35) with κi = 15, and κi = 5 are shown in Fig. 2. One observes that while ||e|| → 0, (35) does not always hold with κi =15 [see Fig. 2(a)] and always holds with κi = 5 [see Fig. 2(b)]. Remark 3: To improve the control performance under the condition of nonzero initial state variables, we define a modified tracking error as follows: E(t) = e(t) − (a0 + a1 t + · · · + an −1 tn −1 ) exp(−λe t) (36) where ai ∈ R(i = 0, 1, . . . , n − 1), and λe ∈ R+ . The selection methods of ai and λe are given in [49]. Remark 4: For the previous AFCs using variable UD, one main drawback is that they only employ one adjusting parameter in the FLS [35]–[41]. From (12), we know that this drawback would limit the approximation ability of the variable UD FLS. Another main drawback is that they use state variables as the input variables of the FLS [37]–[43], which result in violation

812

Fig. 3.


Inverted pendulum with cart.

of the sufficient conditions for the optimal FAE to be convergent. These two drawbacks are the main reasons the previous approaches cannot make use of the optimal FAE local convergence property. In our approach, we use the center values of consequent fuzzy sets as the adaptive parameters to satisfy (12) and use the tracking error e and its derivatives as the input variables of the FLS. Accordingly, it circumvents the two main drawbacks mentioned earlier. However, in the indirect AFC scheme, since those FLSs cannot use e and its derivatives as input variables, the local convergence of the optimal FAE cannot be achieved. V. ILLUSTRATIVE EXAMPLES A. Inverted Pendulum With Cart To demonstrate the effectiveness of the proposed approach, we consider an inverted pendulum system (see Fig. 3) with the following dynamic model [8], [50]: ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ 4l p l p m p cos 2 x 1 ⎨x˙ 2 = gv sin x1 − m p l p x 22 cos x 1 sin x 1 m c +m p 3 − m c +m p 4l p l p m p cos 2 x 1 cos x 1 ⎪ ⎪ + u + d(t) − ⎪ m c +m p 3 m c +m p ⎩ y = x1 (37) where x1 is the angular position of the pendulum, x2 is the angular velocity of the pendulum, gv = 9.8 m/s2 is the gravitational acceleration, mc is the mass of the cart, mp is the mass of the load, lp is the half-length of the pendulum, and d(t) represents the external disturbance. Choose the initial parameters of the pendulum as mc = 1 kg, mp = 0.1 kg, and lp = 0.5 m. Let x(0) = [π/12, 0]. The control objective is to ensure that the output y tracks ym (t) = (π/6)sin(t) [8]. Construct the AFC for the pendulum as follows. Step 1: Construct the FLS in (9). Select E1 = E2 = π/3, U = 1, and θ(0) = [0,. . .,0]T . Let m = 2 so that there are M = m2 = 4 fuzzy rules for the FLS. Design the membership functions of input variables as μA 1i = 1/(1 + exp(7ei )) μA 2i = 1/(1 + exp(−7ei )), i = 1, 2.

Step 2: Design the controller parameters in (31). Let k1 = 2, k2 = 1, and Q = diag(10, 10). Then, from (30) we get P = [15, 5; 5, 5]. Set mθ = 200, δ = 5, and λe = 5. For the selection of γ, we have that increasing γ improves the tracking performance, but γ that is too large may also cause problems in practical applications [51]. Here, we let γ = 300. Step 3: Select the parameters of the C-E factors in (23). From Remark 1, we have that the tracking accuracy is higher, while λce is closer to 1, but it would significantly increase the computational cost if λce is selected to be too close to 1. The selection method of κ1 and κ2 is shown in Remark 2. Here, we choose λce = 0.99, κ1 = 10, and κ2 = 10 by the trial-and-error solution. Select the latest direct AFCs in [46] (the direct AFC without variable UD) and [28] (the direct AFC with the optimal FAE estimator) that have the same assumptions as the proposed approach to make the fair comparison. Note that the learning rates of the controllers in [46] and [28] are selected to be the same as that of the proposed controller. t Define the tracking indexes J(ITAE) = 0 τ ||e(τ )||dτ and t J(IAE)= 0 ||e(τ )||dτ , as well as the control energy, i.e., Ec = t 2 0 u (τ )dτ . Let ts denote the settling time (5% error limit) and es 1 % and es 2 % denote the percentages of steady-state errors of e1 and e2 , respectively. Simulation studies are carried out in the following three cases. Case 1: Initial condition. Use the initial parameters of the pendulum with d(t) = 0 to carry out the compared simulation. Simulation trajectories are shown in Fig. 4. We see that without having any a priori knowledge of plant boundary functions, the proposed AFC with variable UD successfully controls the pendulum with favorable tracking performance [see Fig. 4(a) and (b)], and superior tracking accuracy compared with those compared controllers is demonstrated [see Fig. 4(c)–(e)]. Moreover, the control input u is smooth without chattering [see Fig. 4(f)]. Case 1 of Table I indicates that the proposed controller achieves the best control performance in terms of the settling time, tracking accuracy, and energy consumption. Case 2: Rejection of external disturbance. Let d(t) = 3 cos(2 t) + 2sin((0.09 t+1)t) as in [50], and keep other conditions the same as those in Case 1. Simulation trajectories are shown in Fig. 5. We observe that the proposed controller still achieves the best tracking accuracy [see Fig. 5(a)] with a smooth control input [see Fig. 5(b)]. Case 2 of Table I shows that the proposed controller achieves the best tracking performance with less control efforts. Comparing Case 2 with Case 1 of Table I, we observe that the influence of the external disturbance on the tracking performance is very small by the use of the proposed AFC. It indicates that the proposed approach has the superior robustness against the external disturbance. Case 3: Adaptation of parameter variation. Change the payload mass into mp = 1 kg, and keep other conditions the same as those in Case 2. Simulation results shown in Fig. 6 are very similar with those in Fig. 5. Case 3 of Table I shows that the proposed controller, obviously, outperforms those compared controllers under the external disturbance and parameter variation. Strong robustness of the proposed AFC against the parameter variation is demonstrated by the comparisons between Case 3 and Case 2 of Table I.


813

Fig. 4. Simulation results of pendulum control in Case 1. (a) Trajectories of x1 tracking y m . (b) Trajectories of x2 tracking y˙ m . (c) Comparison of tracking error e1 . (d) Comparison of tracking error e2 . (e) Comparison of ||e||. (f) Control input u.

B. Robotic Manipulator With Friction Consider a single-link robotic manipulator (see Fig. 7) with the following dynamic model [52]–[54]: ⎧ ⎨ x˙ 1 = x2 x˙ 2 = − (dr x2 + mr gv lr cos(x1 )) /J + (1/J) u + d(t) ⎩ y = x1 (38) where x1 is the angular position of the manipulator, x2 is the angular velocity of the manipulator, mr is the mass of the pay-

load, lr is the length of the manipulator, J = 1.33mr lr2 is the inertia coefficient, and dr is the damping factor. The external disturbance d is a frictional model combined with the Coulomb friction and the viscous friction, which is expressed as d(t) = − (sgn (x2 (t)) cr + vr x2 (t)) /J

(39)

where cr ∈ R+ is the Coulomb friction torque, and vr ∈ R+ is the dynamic friction coefficient. Choose lr = 0.25 m, dr = 2 kg·m2 /s, cr = 1.5, vr = 0.3, and x(0) = [π/6, 0]. Let the payload mass mr = 5 + 4sin(t) kg so that the manipulator

814


Fig. 5.

Simulation results of pendulum control in Case 2. (a) Comparison of ||e||. (b) Control input u.

Fig. 6.

Simulation results of pendulum control in Case 3. (a) Comparison of ||e||. (b) Control input u.

TABLE I PERFORMANCE COMPARISONS OF PENDULUM CONTROL

Fig. 7.

Single-link robotic manipulator.

is a time-varying nonlinear system. The control objective is to ensure that the output y tracks ym (t) = sin(t) [54].


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Fig. 8. Simulation results of manipulator control. (a) Trajectories of x1 tracking y m (b) Trajectories of x2 tracking y˙ m . (c) Comparison of tracking error e1 . (d) Comparison of tracking error e2 . (e) Comparison of ||e||. (f) Control input u.

Construct the AFC for the manipulator as follows. Step 1: Construct the FLS in (9). Select E1 = E2 = π, U = 1, and θ(0) = [0,. . .,0]T . Let m = 2, and design the membership functions of input variables as μA 1i = 1/(1 + exp(1.5ei )) μA 2i = 1/(1 + exp(−1.5ei )), i = 1, 2. Step 2: This step is the same as in Section V-A. Step 3: Let λce = 0.995, κ1 = 10, and κ2 = 10.

Compared controllers are selected to be the same as those in Section V-A. Simulation trajectories are shown in Fig. 8. The proposed controller also demonstrates favorable tracking performance [see Fig. 8(a) and (b)] under smooth control input [see Fig. 8(f)]. Both the proposed AFC and the AFC in [28] obviously outperform the AFC in [46] in terms of the tracking accuracy, and the proposed AFC achieves the best tracking performance [see Fig. 8(c)–(e)]. Table II shows that the proposed approach obviously outperforms those compared approaches

816


TABLE II PERFORMANCE COMPARISONS OF MANIPULATOR CONTROL

with respect to the settling time, steady-state errors, as well as ITAE and IAE tracking indexes, which demonstrates that it has the strong robustness against external disturbances and parameter variations. VI. CONCLUSION In this paper, we have successfully developed a direct AFC for a class of SISO uncertain affine nonlinear systems without any a priori knowledge of plant boundary functions. By virtue of the theory of variable UD FLSs, we gave sufficient conditions that guarantee that the optimal FAE is locally convergent. Next, we proposed the design procedure of the stable direct AFC with guaranteed optimal FAE convergence. Under the assumption that the optimal FAE is bounded, we proved that the closed-loop system is asymptotically stable in the sense that all variables are UUB and that the tracking errors converge to zero. The proposed controller eliminates the influence of the FAE on the tracking errors by means of inherent mechanism of the variable UD FLS. Thus, it has the potential to achieve high control performance without additional compensation under only a few fuzzy rules. Simulated applications in the inverted pendulum and the robotic manipulator demonstrated that the proposed approach obviously outperforms other compared approaches in terms of the comprehensive tracking performance and robustness against external disturbances and parameter variations. Further studies would focus on real-world applications to verify the effectiveness of our approach. ACKNOWLEDGMENT The authors would like to acknowledge the reviewers for their insightful suggestions that have significantly improved the quality of this paper. REFERENCES [1] L. X. Wang, A Course in Fuzzy Systems and Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [2] E. Kim, “Output feedback tracking control of robot manipulators with model uncertainty via adaptive fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 12, no. 3, pp. 368–378, Jun. 2004. [3] C.-H. Wang, T. C. Lin, T. T. Lee, and H. L. Liu, “Adaptive hybrid intelligent control for uncertain nonlinear dynamical systems,” IEEE Trans. Syst. Man Cybern. B, vol. 32, no. 5, pp. 583–597, Oct. 2002. [4] T. J. Koo, “Stable model reference adaptive fuzzy control of a class of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 624–636, Aug. 2001. [5] Y. T. Kim and Z. Z. Bien, “Robust adaptive fuzzy control in the presence of external disturbance and approximation error,” Fuzzy Sets Syst., vol. 148, no. 3, pp. 377–393, Dec. 2004.

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Bei-

Yongping Pan received the B.Eng. degree in automation and M.Eng. (Hons.) degree in control theory and control engineering from the Guangdong University of Technology (GDUT), Guangzhou, China, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree in control theory and control engineering with the South China University of Technology (SCUT), Guangzhou. From 2007 to 2008, he was an R&D Engineer with Santak Electronic (Shenzhen) Co., Ltd., Eaton Corporation. As the first author, he has published more than 20 journal and conference papers. His research interests include approximation-based adaptive control, fuzzy control, neural network control, and embedded control systems. Mr. Pan was the winner of the Rockwell Automation Master Scholarship in 2006, the GDUT Graduate Students’ Academic Award in 2006, and the GDUT Excellent Master Thesis in 2007. He also received the SCUT Innovation Fund of Excellent Doctoral Dissertation and the SCUT Outstanding Ph.D. Student Award in 2010.

Meng Joo Er (S’82–M’87–SM’07) received the B.Eng. and M.Eng. degrees in electrical engineering from the National University of Singapore, Singapore, in 1985 and 1988, respectively, and the Ph.D. degree in systems engineering from the Australian National University, Canberra, Australia, in 1992. From 1987 to 1989, he was an R&D Engineer with Chartered Electronics Industries Pte Ltd. and a Software Engineer with Telerate R&D Pte Ltd., respectively. He served as the Director of the Intelligent Systems Centre—a University Research Centre co-funded by Nanyang Technological University and Singapore Engineering Technologies from 2003 to 2006. He is currently a Full Professor with the School of Electrical and Electronic Engineering (EEE) and the Director of the Renaissance Engineering Program, College of Engineering. He has authored five books, 16 book chapters, and more than 400 journal and conference papers. His research interests include fuzzy logic and neural networks, computational intelligence, robotics and automation, sensor networks, and biomedical engineering. Dr. Er is the Vice Chairman of IEEE Computational Intelligence Society Standards Committee, the Chairman of the IEEE Computational Intelligence Society Singapore Chapter, and the Chairman of the Electrical and Electronic Engineering Technical Committee, Institution of Engineers, Singapore. He serves as the Editor-in-Chief for the IES Journal B on Intelligent Devices and Systems, an Area Editor of the International Journal of Intelligent Systems Science, and an Associate Editor of 12 refereed international journals. He was the winner of the IES Prestigious Publication (Application) Award in 1996 and the IES Prestigious Publication (Theory) Award in 2001. He received the Commonwealth Fellowship tenable from the University of Strathclyde, Glasgow, U.K., in 2000. He received the Teacher of the Year Award from the School of EEE in 1999, the School of EEE Year 2 Teaching Excellence Award in 2008, and the Most Zealous Professor of the Year Award in 2009. He also received the Best Session Presentation Award at the World Congress on Computational Intelligence in 2006. Furthermore, together with his students, he has won more than 30 awards at international and local competitions.

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Daoping Huang received the B.Eng. degree in chemical automation and instruments and the M.Eng. and Ph.D. degrees in automatic control theory and applications, all from the South China University of Technology (SCUT), Guangzhou, China, in 1982, 1986, and 1998, respectively. From 1995 to 1996, he was with the University of Gent, Belgium, as a government sponsored visiting scholar. In 1982, he started his teaching and research career with the Department of Automation, SCUT. Now, as a Full Professor and the Vice Dean of the School of Automation Science and Engineering, he has authored three academic books and more than 150 conference and journal papers. His research interests cover intelligent detection and control, soft-sensing technology, as well as fault diagnosis and accident prediction of industrial process. Dr. Huang serves as the Vice Director of the Education Committee and a member of the Process Control and Application Committees, Chinese Association of Automation. He has directed over 10 research projects. He was granted Third Prize from the National Education Committee in 1992 and Second Prize from the Guangdong Provincial Government in 2005 for his contributions to science and technology.


Qinruo Wang was born in Hainan province, China in 1958. He received the B.Eng. degree in electrical engineering from the Guangdong University of Technology (GDUT), Guangzhou, China, in 1982 and the M.Eng. degree in electrical machines from the Zhejiang University, Hangzhou, China, in 1987. He joined GDUT as a Teacher in 1982 and became an Associate Professor in 1993. He served as the Director of the Rockwell Automation Laboratory and Vice Dean of the Department of Automation from 2000 to 2003. He is currently a Full Professor and the Dean of the Department of Automation. He has authored more than 90 journal and conference papers. His research interests include automatic equipment, mechatronics, network control technology, and new energy technology. Prof. Wang is a Senior Member of the British Association of Engineers and a member of the Expert Consultation Committee, Chinese Association of Automation. He also served as the Vice President of the Guangdong Association of Automation of China. He has taken charge of more than 40 research projects. He received Second Prize from the National Education Committee Scientific and Technological Progress in 1998 and First prize from the National Invention Patents in 2004. He also received the Scientific and Technological Progress Award of Guangdong Province eight times.