Automatica 45 (2009) 1530–1535

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Direct adaptive fuzzy control of nonlinear strict-feedback systemsI

Bing Chen a,∗, Xiaoping Liu b, Kefu Liu b, Chong Lin aa Institute of Complexity Science, Qingdao University, Qingdao, 266071, PR Chinab Faculty of Engineering, Lakehead University, Thunder Bay, On, P7B 5E1, Canada

a r t i c l e i n f o

Article history:Received 2 February 2007Received in revised form4 January 2009Accepted 16 February 2009Available online 29 March 2009

Keywords:Nonlinear systemsBacksteppingAdaptive fuzzy controlOutput tracking

a b s t r a c t

This paper focuses on adaptive fuzzy tracking control for a class of uncertain single-input /single-output nonlinear strict-feedback systems. Fuzzy logic systems are directly used to approximate unknownand desired control signals and a novel direct adaptive fuzzy tracking controller is constructed viabackstepping. The proposed adaptive fuzzy controller guarantees that the output of the closed-loopsystem converges to a small neighborhood of the reference signal and all the signals in the closed-loopsystem remain bounded. Amain advantage of the proposed controller is that it contains only one adaptiveparameter that needs to be updated online. Finally, an example is used to show the effectiveness of theproposed approach.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In the past years, backstepping-based nonlinear adaptive con-trol has been paid considerable attention (Freeman & Kokotovic,1996; Jiang & Hill, 1999; Kanellakopoulos, Kokotovic, & Morse,1991; Krstic, Kanellakopoulos, &Kokotovic, 1992; Lin&Qian, 2001;Liu, Gu, & Zhou, 1999; Marino & Tomei, 1993a,b; Schwartz, Isidori,& Tarn, 1999). All the aforementioned works are based on theassumption that the uncertain nonlinearities in systems are ei-ther known functions whose parameters are unknown and lin-ear with respect to those known functions, or bounded by knownnonlinear functions. Thus, if such a prior knowledge of the struc-ture or the upper-bounds of these unknown nonlinearities is notavailable, these approaches become infeasible. Such restrictionshave been removed by using adaptive neural network control (Ge,Hang, & Zhang, 2000; Ge, Lee, & Harris, 1998; Ge & Wang, 2002;Kwan & Lewis, 2000; Lewis, Yesildirek, & Liu, 2000; Zhang, Ge, &Hang, 2000; Zhang, Peng, & Jiang, 2000), or adaptive fuzzy con-trol (Wang & Mendel, 1992) (Chen, Li, & Chang, 1996; Tong &Li, 2003; Wang, 1993; Wang, Chan, & Liu, 2000). In the afore-mentioned literatures, neural networks or fuzzy logic systems are

I The material in this paper was not presented at any conference. This paper wasrecommended for publication in revised form by Associate Editor Shuzhi Sam Geunder the direction of Editor Miroslav Krstic, Edito.∗ Corresponding author. Tel.: +86 0532 85953607; fax: +86 532 85953672.E-mail addresses: chenbing1958@yahoo.com.cn (B. Chen),

xiaoping.liu@lakeheadu.ca (X. Liu), kfiu@lakeheadu.ca (K. Liu),linchong-2004@hotmail.com (C. Lin).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.02.025

employed to approximate the unknown nonlinearities and thebackstepping technique is implemented to construct controllers.The proposed adaptive controllers guarantee the uniform ultimateboundedness of all the signals in the closed-loop system. However,a common weakness of these control methods is that the num-ber of adaptation laws depends on the number of the neuralnetwork nodes or the number of the fuzzy rule bases. With anincrease of neural network nodes or fuzzy rules, the number ofparameters to be estimated will increase significantly. As a result,the on-line learning time becomes prohibitively large. To solve thisproblem, Yang, Feng, and Ren (2004) and Yang, Zhou, and Ren(2003) considered the norm of the ideal weighting vector in fuzzylogic systems as the estimation parameter instead of the elementsof weighting vector. Thus, the number of adaptation laws isreduced considerably.Inspired by thework of Yang et al. (2003, 2004), wewill develop

a new direct adaptive fuzzy control approach in this paper. Unlikeall the existing results on adaptive fuzzy control, Mamdani typefuzzy systems are used to directly approximate the desired controlinput signals instead of the unknown nonlinearities in systems. Inthis way, a direct adaptive fuzzy control method is developed. Themain advantage of the developed method is that for an n-th orderstrict feedback nonlinear system, only one parameter is needed tobe estimated on-line regardless of the number of fuzzy rule basesused. Therefore, the computation burden is significantly reducedand the algorithm is easily realized in practice.

2. Preliminaries and problem formulation

Consider the following SISO nonlinear system.xi = fi(xi)+ gi(xi)xi+1 + di(t), 1 ≤ i ≤ n− 1,

B. Chen et al. / Automatica 45 (2009) 1530–1535 1531

xn = fn(x)+ gn(x)u+ dn(t),y = x1, (1)where x = [x1, . . . , xn]T ∈ Rn is the state vector, u ∈ Rand y ∈ R are the control input and output, respectively. xi =[x1, . . . , xi]T, fi(.) and gi(.), i = 1, 2, . . . , n are unknown nonlinearsmooth functions with fi(0) = 0, and di(.), i = 1, 2, . . . , n, arethe unknown external disturbances and satisfy |di(t)| ≤ di with dibeing a constant.In this paper, a fuzzy logic systemwill be used to approximate a

continuous function f (x) defined on some compact set. Adopt thesingleton fuzzifier, the product inference, and the center-averagedefuzzifier to deduce the following fuzzy rules:Ri: IF x1 is F i1 and . . . and xn is F

in

THEN y is Bi (i = 1, 2, . . . ,N),where x = [x1, . . . , xn]T ∈ Rn and y ∈ R are the input and output ofthe fuzzy system, respectively, F ji andB

i are fuzzy sets inR. Since thestrategy of singleton fuzzification, center-average defuzzificationand product inference is used, the output of the fuzzy system canbe formulated as

y(x) =

N∑j=1Φj

n∏i=1µF ji(xi)

N∑j=1[

n∏i=1µF ji(xi)]

,

where Φj is the point at which fuzzymembership functionµBj(Φj)achieves its maximum value, which is assumed to be 1. Let pj(x) =∏n

i=1 µF ji(xi)∑N

j=1[∏ni=1 µF ji

(xi)], P(x) = [p1(x), p2(x), . . . , pN(x)]T and Φ =

[Φ1, . . . , ΦN ]T. Then the fuzzy logic system can be rewritten as

y(x) = ΦTP(x). (2)If all memberships are chosen as Gaussian functions, the followinglemma holds.

Lemma 1 (Wang &Mendel, 1992). Let f (x) be a continuous functiondefined on a compact set Ω . Then for any given constant ε > 0, thereexists a fuzzy logic system (2) such that

supx∈Ω

∣∣f (x)− ΦTP(x)∣∣ ≤ ε.Assumption 1. The sign of gi(xi) does not change and there existconstants bm and bM such that for i = 1, . . . , n,

0 < bm ≤ |gi(xi)| ≤ bM .

Assumption 1 means that the unknown functions gi(.) areeither strictly positive or negative. Without loss of generality, it isfurther assumed that gi ≥ bm. In addition, because the constantsbm and bM are not used for controller design, they can be unknown.

Assumption 2. The reference signal yd(t) and its time derivativesup to the n-th order are continuous and bounded.

3. Main result

For the system (1), backstepping-based design procedurecontains n steps. At Step i (1 ≤ i ≤ n), the desired but unknowncontrol signal is first considered to stabilize the first i subsystemstheoretically. Then, a fuzzy logic systemΦTi Pi(Xi)will be employedto approximate this unknown control signal, consequently, toconstruct the virtual control signal. The real tracking control law uwill be designed at the last step. To develop a backstepping-baseddesign procedure, we first define a constant as follows:

θ = max1bm‖Φi‖

2: i = 1, 2, . . . , n

. (3)

Obviously, θ is an unknown positive constant because bm and ‖Φi‖are unknown.At Step i, the feasible virtual control signal is constructed as

αi(Xi) =−12a2ieiθPTi (Xi)Pi(Xi) (4)

where ei = xi − αi−1 with α0 = yd, Xi = [xTi , θ , y(i)Td ]

T, y(i)d denotesthe vector of yd and up to its i-th order time derivative. The virtualcontrol signals αi will be specified at Step i.

Theorem 1. For the reference signal yd(t), consider the system (1)satisfying Assumptions 1 and 2. Suppose that for 1 ≤ i ≤n, the packaged unknown functions αi, i = 1, 2, . . . , n, can beapproximated by the fuzzy logic systems in the sense that theapproximating errors are bounded. If a control law is chosen as

u = −12a2nenθPTn (Xn)Pn(Xn), (5)

with the intermediate virtual control signals αi defined by (4) and theadaptive law

˙θ =

n∑i=1

r2a2i

(xi − αi−1)2PTi (Xi)Pi(Xi)− k0θ , (6)

where the constants r > 0, k0 > 0, and ai > 0 (i = 1, 2, . . . , n)are design parameters, then all the signals in the closed-loop systemremain bounded. Furthermore, for any given scalar ε > 0, thecontroller parameters can be tuned such that

limt→∞‖y− yd‖2 ≤ ε2,

The proof of Theorem 1 consists of two steps. First, a systematicdesign scheme is presented based on the backstepping approach.Then, the stability analysis of the closed-loop system is carried out.

3.1. Adaptive fuzzy control design

In the following, for the purpose of simplicity, the time variablet and the state vector xi will be omitted from function expressionsand let Pi = Pi(Xi). In addition, it is easy to prove from (6) that ifθ (0) ≥ 0, then θ (t) ≥ 0 for all t ≥ 0. In fact, it is always reasonableto choose θ (0) ≥ 0 in a practical situation, as θ is an estimation ofθ . This conclusion will be used in each design step.

Step 1. Define the tracking error as e1 = x1 − yd and consider aLyapunov function candidate as

V1 =12e21 +

bm2rθ2, (7)

where θ = θ − θ . Differentiating V1 yields

V1 = e1 (f1 + g1x2 − yd + d1)−bmrθ˙θ

≤ e1(g1x2 + f1

)+12ρ2d21 −

12g−21 e

21 −

bmrθ˙θ, (8)

where f1(X1) = f1 + 12ρ−2e1 + 1

2g−21 e1 − yd and the completion of

squares are used for the term e1d1 with ρ > 0 being a constant.To stabilize this system, take the intermediate control signal asα1(X1) = −g−11

k1e1 + f1

with k1 being a positive constant.

Further, add and subtract g1α1 in the last bracket in (8) to obtainthe following inequality.V1 ≤ −k1e21 + e1g1

(x2 − α1

)+12ρ2d21 −

12g−21 e

21 −

bmrθ˙θ. (9)

However, α1 cannot be implemented in practice as it contains theunknown functions f1 and g1. Thus, according to Lemma 1, for anygiven ε1 > 0, there exists a fuzzy logic systemΦT1P1(X1) such that

1532 B. Chen et al. / Automatica 45 (2009) 1530–1535

α1 = ΦT1P1(X1)+ δ1(X1),

ε1 ≥ |δ1(X1)| , (10)where δ1(X1) denotes the approximation error. Consequently, itfollows from (10), the completion of squares and the definition ofθ that

− e1g1α1 = −e1g1ΦT1

‖Φ1‖P1 ‖Φ1‖ − e1g1δ1

≤ +bm2a21e21θP

T1P1 +

12g21 e

21 +

bM2a21 +

12ε21. (11)

With g1 > 0 and θ ≥ 0, the following inequality can be verifiedeasily.

e1g1α1 ≤ −bm2a21e21θP

T1P1. (12)

Now, substituting (11) and (12) into (9) gives

V1 ≤ −k1e21 +bM2a21 +

12ε21 + e1g1 (x2 − α1)

+12ρ2d21 +

bmrθ

(r2a21e21P

T1P1 −

˙θ

). (13)

Remark 1. Unlike the methods proposed in Yang et al. (2003,2004) and Chen and Liu (2005), the desired control signal, i.e., α1,is first constructed by using the unknown nonlinear functionsand constants. Further, the fuzzy logic system is directly used toapproximate the unknown control signal α1 rather than f (x1) andg1(x1). After that, the feasible intermediate control signal α1 isdesigned. Therefore, such a method is called the direct adaptivefuzzy control method.

Step k. (2 ≤ k ≤ n − 1) In general, assume that for Vk−1 =Vk−2 + 1

2 e2k−1 the inequality below holds.

Vk−1 ≤ Ψ (k− 1)+ ek−1gk−1ek, (14)where

Ψ (k) = −k∑i=1

kie2i +k∑i=1

12

(bMa2i + ε

2i

)+

k∑i=1

ρ2

2(k− i+ 1)d2i

+

k∑i=2

ei

(ϕi(Xi)−

∂αi−1

∂θ

˙θ

)

+bmrθ

(k∑i=1

r2a2ie2i P

Ti Pi −

˙θ

). (15)

Choose a Lyapunov function candidate as

Vk = Vk−1 +12e2k .

Differentiating Vk results in

Vk = Vk−1 + ek (fk + gkxk+1 − αk−1 + dk) , (16)where

− ekαk−1 = −ekk−1∑i=1

∂αk−1

∂xi(fi + gixi+1 + di)

− ekk−1∑i=0

∂αk−1

∂y(i)dy(i+1)d − ek

∂αk−1

∂θ

˙θ. (17)

From the completion of squares, the following inequality holds.

− ek∂αk−1

∂xidi ≤

12ρ2e2k

[∂αk−1

∂xi

]2+12ρ2d2i (18)

with ρ > 0 being a constant.

From (6), ˙θ is evidently a function of all state variables. Hence,the term ∂αk−1

∂θ

˙θ cannot be dealt with in the same way as in Yang

et al. (2003, 2004), where it is treated as part of fk. To compensatethis term, introduce a function ϕi as follows

ϕi(Xi) = −k0θ∂αi−1

∂θ−

i∑j=2

eir2a2i

∣∣∣∣ej ∂αj−1∂θ

∣∣∣∣+

i−1∑j=1

∂αi−1

∂θ

r2a2je2j P

Tj Pj.

Now define a function fk as

fk(Xk) = fk + gk−1ek−1 +12ρ2ekk−1∑i=1

[∂αk−1

∂xi

]2− ek

k−1∑i=1

∂αk−1

∂xi(fi + gixi+1)− ϕk

− ekk−1∑i=0

∂αk−1

∂y(i)dy(i+1)d +

12

(g2k + ρ

−2) ek.Substituting (14), (17) and (18) into (16) produces

Vk ≤ Ψ (k− 1)+k∑i=1

ρ2

2d2i −

12g2k e

2k

+ ek

(ϕk(Xk)−

∂αk−1

∂θ

˙θ

)+bmrθ

(r2a2ke2kP

Tk Pk −

˙θ

)+ ek(fk + gkxk+1). (19)

Apparently, if we choose

αk(Xk) = −g−1k

(kkek + fk +

12g2k ek

),

then (19) can be rewritten in the following form.

Vk ≤ Ψ (k− 1)− kke2k +k∑i=1

ρ2

2d2i

+ ek

(ϕk(Xk)−

∂αk−1

∂θ

˙θ

)−12g2k e

2k + ekgk(xk+1 − αk). (20)

Currently, the fuzzy logic systemΦTk Pk(Xk) is used to approximatethe unknown control input αk such that for any given constantεk > 0,αk = Φ

Tk PTk (Xk)+ δk(Xk), ‖δk‖ ≤ εk.

At the present stage, following a similar line as used in theprocedure from (10) to (12) yields

−ekgkαk ≤bm2a2ke2kθP

Tk Pk +

12g2k e

2k +

12

(bMa2k + ε

2k

), (21)

ekgkαk ≤−12a2kekθPTk Pk. (22)

Thus, it follows immediately from substituting (21) and (22) into(20) that.

Vk = Ψ (k)+ ekgkek+1. (23)Step n. Choose a Lyapunov function candidate as:

Vn = Vn−1 +12e2n.

It follows from differentiating Vn and taking (20) with k = n intoaccount that

B. Chen et al. / Automatica 45 (2009) 1530–1535 1533

Vn ≤ Ψ (n− 1)− kne2k +n∑i=1

ρ2

2d2i + en

(ϕn(Xn)−

∂αn−1

∂θ

˙θ

)−12g2n e

2n + engn(u− αn). (24)

Similarly, for any given positive constant εn, the fuzzy logic systemΦTnPn(Xn) is utilized to approximate the unknown αn(Xn). Again,following the same line as used in the procedure from (10) to (12)gives the following inequalities.

− engnαn ≤bm2a2ne2nθP

TnPn +

12g2n e

2n +

12

(a2nbM + ε

2n

), (25)

and

engnu ≤ −bm2a2ne2nθP

TnPn. (26)

Hence, combining (24) with (25) and (26) gives

Vn = Ψ (n), (27)

where Ψ (n) is defined by (15) with k = n.

3.2. Analysis of stability

For the stability analysis of the closed-loop system, choose aLyapunov candidate function as V = Vn. From (27) and (6), thetime derivative of V is given by

V ≤ −n∑i=1

kie2i +n∑i=1

12

(bMa2i + ε

2i

)+

n∑i=1

ρ2

2(n− i+ 1)d2i +

k0bmrθ θ

+

n∑i=2

ei

(ϕi(Xi)−

∂αi−1

∂θ

˙θ

). (28)

For the term bmr k0θ θ , the following holds.

bmrk0θ θ ≤ −

bmk02r

θ2 +bmk02r

θ2. (29)

In the following, it will be shown that the last term in (28) isnegative. With 0 < PTi Pi ≤ 1 and the definition of

˙θ , we have

−

n∑i=2

ei∂αi−1

∂θ

˙θ =

n∑i=2

k0eiθ∂αi−1

∂θ

−

n∑i=2

ei∂αi−1

∂θ

(n∑j=1

r2a2je2j P

Tj Pj

)

=

n∑i=2

k0eiθ∂αi−1

∂θ

−

n∑i=2

ei∂αi−1

∂θ

(i−1∑j=1

r2a2je2j P

Tj Pj +

n∑j=i

r2a2je2j P

Tj Pj

)

=

n∑i=2

k0eiθ∂αi−1

∂θ−

n∑i=2

ei∂αi−1

∂θ

(i−1∑j=1

r2a2je2j P

Tj Pj

)

−

n∑i=2

ei∂αi−1

∂θ

(n∑j=i

r2a2je2j P

Tj Pj

). (30)

For the last term in (30), by rearranging its sequence and using thefact 0 < PTj Pj ≤ 1 for all j, we have

−

n∑i=2

ei∂αi−1

∂θ

(n∑j=i

r2a2je2j P

Tj Pj

)

≤

n∑i=2

∣∣∣∣ei ∂αi−1∂θ

∣∣∣∣(

n∑j=i

r2a2je2j

)

=

∣∣∣∣e2 ∂α1∂θ

∣∣∣∣ r2a22 e22 +∣∣∣∣e2 ∂α1

∂θ

∣∣∣∣ r2a23 e23+ · · · +

∣∣∣∣e2 ∂α1∂θ

∣∣∣∣ r2a2n e2n+

∣∣∣∣e3 ∂α2∂θ

∣∣∣∣ r2a23 e23 +∣∣∣∣e3 ∂α2

∂θ

∣∣∣∣ r2a24 e24+ · · · +

∣∣∣∣e3 ∂α2∂θ

∣∣∣∣ r2a2n e2n+ · · · +

∣∣∣∣ek ∂αk−1∂θ

∣∣∣∣ r2a2k e2k +∣∣∣∣ek ∂αk−1

∂θ

∣∣∣∣ r2a2k+1

e2k+1

+ · · · +

∣∣∣∣ek ∂αk−1∂θ

∣∣∣∣ r2a2n e2n+ · · · +

∣∣∣∣en−1 ∂αn−2∂θ

∣∣∣∣ r2a2n−1

e2n−1

+

∣∣∣∣en−1 ∂αn−2∂θ

∣∣∣∣ r2a2n e2n +∣∣∣∣en ∂αn−1

∂θ

∣∣∣∣ r2a2n e2n=

n∑i=2

r2a2ie2i

(i∑j=2

∣∣∣∣ej ∂αj−1∂θ

∣∣∣∣). (31)

Combining (30) and (31) shows

−

n∑i=2

ei∂αi−1

∂θ

˙θ

≤

n∑i=2

k0eiθ∂αi−1

∂θ−

n∑i=2

ei∂αi−1

∂θ

(i−1∑j=1

r2a2je2j P

Tj Pj

)

+

n∑i=2

r2a2ie2i

(i∑j=2

∣∣∣∣ej ∂αj−1∂θ

∣∣∣∣)

=

n∑i=2

ei

[k0θ

∂αi−1

∂θ−∂αi−1

∂θ

(i−1∑j=1

r2a2je2j P

Tj Pj

)

+r2a2iei

(i∑j=2

∣∣∣∣ej ∂αj−1∂θ

∣∣∣∣)]

= −

n∑i=2

eiϕi(Xi), (32)

which implies that the last term in (28) is negative. Furthermore,let a0 = min 2ki, k0, i = 1, 2, . . . , n and b0 =

bmk02r θ

2+∑ni=1

12

(bMa2i + ε

2i

)+∑ni=1

ρ2

2 (n− i+ 1)d2i . From (28) we have

V (t) ≤ −a0V (t)+ b0, t ≥ 0. (33)So far, it follows immediately from (33) and Lemma 1.1 in Ge andWang (2004) that V (t) is bounded and

V (t) ≤(V (0)−

b0a0

)e−a0t +

b0a0, t ≥ 0, (34)

which implies that for t ≥ 0, e = [e1, . . . , en, θ ]T belongs to thecompact set

Ω =

e|V (e(t)) ≤ V (0)+

b0a0

.

1534 B. Chen et al. / Automatica 45 (2009) 1530–1535

Therefore, ei (i = 1, 2, . . . , n) and θ are bounded. Since θ is aconstant, thus θ is bounded. Consequently, αi are also boundedbecause PTi (Xi)Pi(Xi) ≤ 1. Hence, we conclude that xi are bounded.This proves that all the signals in the closed-loop system arebounded. On the other hand, note that ki, r, ai and k0 are designparameters, and θ , bm and bM are constants. Theoretically, for anygiven ε > 0, first choosing the design parameters ki and k0, thenchoosing ai, εi and ρ sufficiently small and r sufficiently large, theinequality b0a0 ≤

ε2

2 can be obtained. In addition, from (34) we have

e21 ≤ 2(V (0)− b0

a0

)e−a0t + 2 b0a0 , which implies that

limt→∞

e21 ≤ 2b0a0≤ ε2.

Remark 2. The above analysis shows that tracking error dependson θ, εi, bm, bM , r, k0, ki and ai. Because θ, bm and bM are unknown,an explicit estimation of the tracking error is impossible. However,it is clear that reducing ai and k0, meanwhile increasing ki and r ,will diminish the tracking error. In addition, it is worth noting thatthe parameters kis are not used in the controller design, which areintroduced just for stability analysis.

4. Simulation

In this section, an example will be used to test the effectivenessof the proposed controller.Example. Consider the following nonlinear system.

x1 = 0.5x21 + ln(10+ x21)x2 + 2 sin(t),

x2 =x1x2

1+ x21 + x22+

(1+ e−x

21−x

22

)x3 + 0.5 cos(2t),

x3 = x3x21

(2+

x221+ x23 + x

22

)u− 1.5 sin(1.5t).

Both the adaptive neural control controller in Zhang, Ge et al.(2000) and the proposed adaptive fuzzy controller are imple-mented. The reference signal is given as yd = 0.5(sin(t) +sin(1.5t)). For the fuzzy control, seven fuzzy sets are characterizedby the following membership functions.

µF1i= exp

[−(x+ 1.5)2

2

], µF2i

= exp[−(x+ 1)2

2

],

µF3i= exp

[−(x+ 0.5)2

2

], µF4i

= exp[−(x+ 0)2

2

],

µF4i= exp

[−(x− 0.5)2

2

], µF6i

= exp[−(x− 1)2

2

],

µF7i= exp

[−(x− 1.5)2

2

].

The control law (5), the virtual control signal (4) and the adaptionlaw (6) are used with a1 = 0.218, a2 = 0.2, r = 15, andk0 = 0.00125. For the neural control proposed in Zhang, Ge et al.(2000), take k1 = k2 = 21, k3 = 35,Γi = 1.5, σi = 0.5, i = 1,2, 3, respectively, and the width of Guassian function is 2. 5nodes are used for each input. The simulations are run with theinitial conditions x(0) = [−0.5 0.5 0.5 ]Tand θ (0) = 0. Thesimulation results are shown in Figs. 1–4. From the simulationresults, it is seen that for this system both adaptive fuzzy controland adaptive neural control achieve the tracking performance.The fuzzy control scheme requires a smaller control gain thanthe neural control method. In addition, the neural control schemerequires 15 adaption laws, but the proposed fuzzy control methodneeds only one.

Fig. 1. y (‘‘–’’) and yd (‘‘- -’’) for Fuzzy control.

Fig. 2. y (‘‘–’’) and yd (‘‘- -’’) for NN control.

Fig. 3. uF (‘‘–’’) and 0.1uNN (‘‘- -’’).

5. Conclusion

In this paper, the output tracking control problem has beenaddressed for a class of SISO nonlinear uncertain systems witha strict-feedback structure. The nonlinear functions in systemsare completely unknown, and are not linearly parameterized.A direct adaptive fuzzy control scheme is presented via thebackstepping technique. The proposed controller guarantees thatthe output of the closed-loop system follows the reference signal,and all the signals in the closed-loop system remain bounded.The main feature of the control scheme proposed in this paper

B. Chen et al. / Automatica 45 (2009) 1530–1535 1535

Fig. 4. Adaptive parameter curve.

is simplicity. In particular, no matter how many fuzzy rulebases are used, there is only one parameter that needs to beestimated. It should be pointed out that as a classical functionapproximator, neural networks can be used to replace fuzzy logicsystems to approximate nonlinear functions. Hence, the approachproposed in this paper is also suitable for neural networks control.In addition, because fuzzy logic systems are just used as thefunction approximators, the characteristic ability of fuzzy controlto describe a system or generate control actions based on expertknowledge is not shown.

Acknowledgements

This work is partially supported by the Natural Sciences andEngineering Research Council of Canada, and the Natural ScienceFoundation of China (60674055, 60774047), the Taishan scholarprogram and theNatural Science Foundation of Shandong Province(No. Y2006G04).

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Bing Chen received the B.A. degree in mathematicsfrom Liaoning University, PR China, the M.A. degree inmathematics from Harbin Institute of Technology, PRChina, and the Ph.D. degree in electrical engineering fromNortheastern University, PR China, in 1982, 1991 and1998, respectively. Currently, he is a Professor with theInstitute of Complexity Science, Qingdao University,Qingdao, PR China. His research interest includes non-linear control systems, robust control, and fuzzy controltheory.

Xiaoping Liu obtained his B.Sci., M.Sci., and Ph.D. degreein electrical engineering from Northeastern University, PRChina, in 1984, 1987, and 1989, respectively. He spentmore than 10 years in the School of Information Scienceand Engineering at Northeastern University, PR China. In2001, he joined the Department of Electrical Engineeringat Lakehead University, Canada. His research interests arenonlinear control systems, singular systems, and robustcontrol. He is a member of the Professional Engineers ofOntario

Kefu Liu is a Full Professor in the Department ofMechanical Engineering at Lakehead University, Canada.He receivedhis B.Eng. andM.Sc. inMechanical Engineeringfrom Central South University of Technology, China in1981 and 1984, respectively, and his Ph.D. degree inMechanical Engineering fromTechnical University of NovaScotia, Canada in 1992. He was Assistant Professor atSt. Mary’s University, Canada, from 1993 to 1995, andat Dalhousie University, Canada, from 1995 to 1998. Hejoined Lakehead University in 1998. Dr. Liu is a memberof Professional Engineers Ontario. His research interests

include vibration control, control of nonlinear systems, and mechatronics.

Chong Lin (SM’2006) received the B.Sci. and the M.Sci. inApplied Mathematics from the Northeastern University,PR China, in 1989 and 1992, respectively, and the Ph.D.in Electrical and Electronic Engineering from the NanyangTechnological University, Singapore, in 1999. He was aResearch Associate with the Department of MechanicalEngineering, University of Hong Kong, in 1999. From 2000to 2006, he was a Research Fellowwith the Department ofElectrical and Computer Engineering, National Universityof Singapore. Since 2006, he has been a professor with theInstitute of Complexity Science, QingdaoUniversity, China.

He has published more than 60 research papers and co-authored two monographs.His current research interests are mainly in systems analysis and control, robustcontrol and fuzzy control.