Adaptive Output-Feedback Fuzzy Tracking Control for a Class of Nonlinear Systems

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

Adaptive Output-Feedback Fuzzy Tracking Controlfor a Class of Nonlinear SystemsQi Zhou, Peng Shi, Senior Member, IEEE, Jinjun Lu, and Shengyuan Xu

Abstract—This paper is concerned with the problem of adaptivefuzzy tracking control via output feedback for a class of uncertainsingle-input single-output (SISO) strict-feedback nonlinear sys-tems. The dynamic feedback strategy begins with an input-drivenfilter. By utilizing fuzzy logic systems to approximate unknownand desired control input signals directly instead of the unknownnonlinear functions, an output-feedback fuzzy tracking controlleris designed via a backstepping approach. It is shown that the pro-posed fuzzy adaptive output controller can guarantee that all thesignals remain bounded and that the tracking error converges to asmall neighborhood of the origin. Simulations results are presentedto demonstrate the effectiveness of the proposed methods.

Index Terms—Adaptive control, backstepping, fuzzy control,nonlinear systems, strict-feedback systems.

I. INTRODUCTION

ADAPTIVE control for nonlinear systems with paramet-ric uncertainties has received a lot of attention over the

past few decades (see, for example, [1]–[5] and the referencestherein). However, the early stages of the research were basedon the assumption that the uncertain nonlinearities are either tohave a prior knowledge of the bound or to be linearly parame-terized. These assumptions cannot always be satisfied becausein some practical systems, it is inevitable that they will con-tain some uncertain elements that cannot be modeled; then, thetechniques that are developed in [1]–[3] are not applicable.

It is well known that neural networks and fuzzy logic sys-tems have been found to be particularly powerful tools to con-trol nonlinear systems because of their universal approximationproperties. Over the past few decades, the issues of utilizingneural networks and fuzzy logic systems to approximate un-

Manuscript received January 11, 2011; accepted April 25, 2011. Date of pub-lication June 7, 2011; date of current version October 10, 2011. This work wassupported by the National Natural Science Foundation of China under Grant61074043 and Grant 61074008, the Qing Lan Project, the Natural Science Foun-dation of Jiangsu Province under Grant BK2008047 and Grant BK2008188,the Engineering and Physical Sciences Research Council, U.K., under GrantEP/F029195, and the China Scholarship Council.

Q. Zhou is with the School of Automation, Nanjing University of Scienceand Technology, Nanjing 210094, China, and also with the School of Engineer-ing and Science, Victoria University, Melbourne, Vic. 8001, Australia (e-mail:[email protected]).

P. Shi is with the Department of Computing and Mathematical Sciences, Uni-versity of Glamorgan, Pontypridd, CF37 IDL, U.K., and also with the School ofEngineering and Science, Victoria University, Melbourne, Vic. 8001, Australia(e-mail: [email protected]).

J. Lu is with the School of Electrical Engineering, Nantong University, Nan-tong 226019, China (e-mail: [email protected]).

S. Xu is with the School of Automation, Nanjing University of Science andTechnology, Nanjing 210094, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2011.2158652

known nonlinearities have been well investigated (see, for ex-ample, [6], [7], and the references therein). However, the sys-tems that are considered in these works are restricted to satisfymatching conditions, that is, the unknown functions must appearon the same equation as the control input channel. Therefore,these schemes are not feasible for more general systems such asthe strict-feedback or pure-feedback systems.

Recently, a backstepping technique has been successfully ap-plied to analyze the problem of stability and to design adaptivecontrollers for nonlinear systems. Many researchers have fo-cused on the triangular nonlinear systems and developed somesignificant results such as [8]–[13] via a backstepping tech-nique and neural networks methods. The controllers that aredesigned in [8]–[13] cannot only achieve good performance butalso guarantee that all the signals in the closed-loop system to beuniformly ultimate bounded. In the meantime, many researchersresorted to a backstepping technique and fuzzy logic systems tosolve the adaptive controller design problems for these triangu-lar nonlinear systems and to achieve the performance [14]–[21].At the same time, the backstepping technique has been usedto solve a class of nonlinear stochastic systems (see, for exam-ple, [22] and [23]). In [14]–[21], it was assumed that the statesof the systems are directly measurable. When the state of thenonlinear systems is unmeasured, the adaptive output-feedbackcontrol method may be an effective way to control the systems.In [24], the problem of robust adaptive output-feedback con-trol for a class of nonlinear systems with time-varying actuatorfaults is investigated, and in [25], the problem of adaptive fuzzyoutput-feedback control for nonlinear systems with unknownsign of high-frequency gain is researched. By exploiting us-ing fuzzy logic systems to approximate the unknown nonlinearfunctions and combining backstepping approach, a fuzzy adap-tive output-feedback controller is designed in [26] for single-input single-output (SISO) strict-feedback nonlinear systemsand in [27] and [28] for multiple-input multiple-output nonlin-ear systems, respectively. In addition, by combining neural net-works with backstepping approach, an adaptive output-feedbackcontroller is designed for a class of large-scale nonlinear sys-tems in [29]. The weakness of the aforementioned adaptivecontrol methods in [8]–[12], [14]– [16], [24], and [26]–[29]is that many adaptive parameters need to be tuned by onlinelearning laws. When the number of neural network nodes orthe number of the fuzzy logic rule bases increase, the numberof adaptation laws increase accordingly. Therefore, the learn-ing time tends to be large inevitably. More recently, a newadaptive fuzzy control method has been presented in [30] toreduce the adaptive parameters. The work in [31] developed anew direct adaptive fuzzy control for nonlinear strict-feedback

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ZHOU et al.: ADAPTIVE OUTPUT-FEEDBACK FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS 973

systems, in which Mamdani-type fuzzy systems are used to di-rectly approximate the desired control input signals instead ofthe unknown nonlinearities. And the computation burden hasbeen greatly reduced because only one parameter needs to betune. However, the results in [31] hold only when the state ofthe nonlinear system is measurable.

Motivated by the previous discussion, in this paper, we willdevelop a direct adaptive output-feedback control approachfor strict-feedback nonlinear systems. The dynamic feedbackstrategy begins with an input-driven filter, which is differentfrom [26]–[29], and by combining fuzzy logic systems with abackstepping technique, fuzzy adaptive output-feedback con-troller is constructed recursively. It is proved that the controllerthat is designed in this paper guarantees that all the signals in theclosed-loop remain bounded and achieves good tracking perfor-mance. In addition, the method of direct adaptive control that isused in this paper can reduce the computation burden; therefore,the algorithm is convenient to realize in engineering.

The remaining of this paper is organized as follows. Theproblem formulation and preliminaries are presented in Sec-tion II, and a direct adaptive output-feedback fuzzy trackingcontroller is designed by a recursive procedure in Section III.Two simulation examples are given in Section IV to demon-strate the effectiveness of the proposed scheme, and the paperis concluded in Section V.

II. PROBLEM FORMULATION AND PRELIMINARIES

In this section, the nonlinear control problem is first formu-lated, and then in order to develop an adaptive dynamic feedbackcontrol design procedure, an input-driven filter is introduced. Fi-nally, to approximate the desired control signals, the fuzzy logicsystems are given.

A. Nonlinear Control Problem

Consider an SISO nonlinear dynamic system in the followingform:

xi = xi+1 + fi (xi) + di (t) , 1 ≤ i ≤ n − 1

xn = u + fn (x) + dn (t)

y = x1 (1)

where x = [x1 , x2 , . . . , xn ]T ∈ Rn denotes state vector of thesystem; u ∈ R and y ∈ R are input and output of the system,respectively. xi = [x1 , x2 , . . . , xi ]T , i = 1, 2 . . . , n − 1. fi(.),i = 1, 2 . . . , n stand for the unknown smooth system functionswith fi(0) = 0. di(t), i = 1, 2 . . . , n are the external distur-bance uncertainties of the system, which satisfy |di(t)| ≤ di ,with di as a constant.

B. Dynamic Feedback Design

The dynamic feedback strategy begins with an input-drivenfilter [32] as follows:

.

x1 = x2 − l1 x1.

x2 = x3 − l2 x1

....

xn = u − ln x1 (2)

where xi is the estimate of xi , (i = 1, . . . , n), and li is the designparameter such that the matrix

Ac =

⎡⎢⎢⎣

−l1

... In−1

−ln 0 . . . 0

⎤⎥⎥⎦

is a strict Hurwitz matrix, which means for a given matrix Q > 0,there exists a matrix P > 0 that satisfies the following equation:

ATc P + PAc = −Q. (3)

Define ei = xi − xi , (i = 1, . . . , n). Then, we have

e1 = e2 − l1e1 + f1 (x1) + d1 (t) + l1y

e2 = e3 − l2e1 + f2 (x2) + d2 (t) + l2y

...

en = −ln e1 + fn (xn ) + dn (t) + lny

which can be rewritten as

e = Ace + F (x) + D (t) (4)

with

e = (e1 , e2 , . . . , en )T

F (x) = (f1 (x1) + l1y, . . . , fn (xn ) + lny)T

�= (F1 (x) , . . . , Fn (x))T

D (t) = (d1 (t) , . . . , dn (t))T .

After combining (1), (2), and (4), the system can be taken as

e = Ace + F (x) + D (t)

y = x2 + e2 + f1 (x1) + d1 (t).

x2 = x3 − l2 x1

....

xn = u − ln x1 (5)

where y and xi , (i = 1, . . . , n) are available for control design.The control objective of this paper is to design a direct adap-

tive fuzzy controller such that the system output y can trackthe reference signal yd , while all the signals in the derivedclosed-loop system remain bounded. The reference signal yd isassumed to be available together with its ith time derivative y

(i)d

for i = 1, 2, . . . , n, y(i)d =

(yd, y

(1)d , . . . , y

(i)d

)T

.

C. Fuzzy Logic Systems

Construct fuzzy logic systems with the following IF–THEN

rules:

Ri : If x1 is F i1 and . . . and xn is F i

n .

Then y is Bi, i = 1, 2, . . . , N.


The fuzzy logic systems with the singleton fuzzifier, productinference, and center average defuzzifier can be formulated as

y(x) =

∑Ni=1 Φi

∏nj=1 μF i

j(xj )

∑Ni=1

[∏nj=1 μF i

j(xj )

]

where x = [x1 , x2 , . . . , xn ]T ∈ Rn , μF ij(xj ) is the membership

of F ij , Φi = max

y∈RμB i (y). Let

pi(x) =

∏nj=1 μF i

j(xj )

∑Ni=1

[∏nj=1 μF i

j(xj )

]

ξ(x) = [p1(x), p2(x), . . . , pN (x)]T , and Φ = [Φ1 , Φ2 , . . . ,ΦN ]T . Then, the fuzzy logic systems can be rewritten as follows:

y(x) = ΦT ξ(x). (6)

Lemma 1: [33] Let f(x) be a continuous function that isdefined on a compact set Ω; then, for any given constant ε > 0,there exists a fuzzy logic system y(x) in the form of (6) suchthat

supx∈Ω

|f(x) − y(x)| ≤ ε.

III. FUZZY ADAPTIVE CONTROL DESIGN AND

STABILITY ANALYSIS

The backstepping design procedure contains n steps. In eachstep, a virtual control function αi should be developed using anappropriate Lyapunov function Vi , and the real tracking controllaw u will finally be designed. To begin with the backsteppingdesign procedure, let us define a constant

θ = max{‖Φi‖2 : i = 0, 1, 2, . . . , n

}.

From the definition of θ, we know that θ is an unknown constant,and we define θ as the estimate of θ. The feasible virtual controlsignal is designed as

αi (Xi) = − 12a2

i

χi θξTi (Xi) ξi (Xi) , i = 1, . . . , n (7)

where X1 =(x1 , θ, y

(1)d

)T

, Xi =(

x1 ,−xi , θ, y

(i)d

)T

, with

−xi = (x1 , . . . , xi)

T for i = 2, . . . , n, and χi will be defined atthe proof of Theorem 1.

Theorem 1: Consider the nonlinear dynamic system in (1), ifwe choose the input driven filter (2), and a control law as

u = − 12a2

n

χn θξTn (Xn ) ξn (Xn ) (8)

with the intermediate virtual control signals αi that are describedas (7), and the adaptive law that is defined as

.

θ =n∑

i=1

r

2a2i

χ2i ξ

Ti (Xi) ξi (Xi) − k0 θ (9)

where positive constants ai (i = 1, . . . , n), r, and k0 are designparameters so that the closed-loop system can be guaranteed

to be semiglobal stable, and for any given scalar ε > 0, thetracking error satisfies lim supt→∞ χ2

1 ≤ ε2 by appropriatelychoosing the design parameters.

The proof of Theorem 1 is presented in the following with abackstepping approach.

Proof:Step 1: For the reference signal yd , introduce the tracking

error variable χ1 = y − yd , and choose the following Lyapunovfunction candidate:

V1 = eT Pe +12χ2

1 +12r

θ2 (10)

where θ = θ − θ. The time derivative of V1 is given by

V1 = −eT Qe + 2eT P (F + D)

+ χ1 (x2 + e2 + f1 + d1 − yd) −1rθ

.

θ. (11)

As F�= (F1 (x) , . . . , Fn (x))T , and Fi (x) , i = 1, 2, . . . , n is

an unknown function, by Lemma 1, for any given εi0 > 0, thereexists a fuzzy logic system ΦT

i0ξ0(X0) such that

Fi(X0) = ΦTi0ξ0(X0) + δi0 (X0)

|δi0 (X0)| ≤ εi0

where X0 = x. Therefore

F (X0) = ΦT0 ξ0(X0) + δ0 (X0)

‖δ0 (X0)‖ ≤ ε0

where

Φ0 = (Φ10 , . . . ,Φn0)

δ0 (X0) = (δ10 (X0) , . . . , δn0 (X0))T .

As ξT0 ξ0 ≤ 1, and according to the definition of θ, we know

‖Φ0‖2 ≤ θ. Therefore, the following inequality holds:

2eT PF = 2eT P(ΦT

0 ξ0(X0) + δ0 (X0))

≤ ‖e‖2 + ‖P‖2 θ + ‖P‖2 ε20 . (12)

By the definition of D, we have

2eT PD ≤ ‖e‖2 + ‖P‖2 ∥∥D∥∥2

(13)

where D =(d1 , d2 , . . . , dn

)T.

Similarly

χ1d1 ≤ 12ρ−2χ2

1 +12ρ2 d 2

1 (14)

χ1e2 ≤ e22 +

14χ2

1 (15)

where ρ is a positive constant.Substituting (3) and the inequalities (12)–(15) into (11), the

time derivative of V1 is rewritten as

V1 ≤ − [λmin (Q) − 3] ‖e‖2

+ χ1

(x2 + f1 +

12ρ−2χ1 +

14χ1 − yd

)

+ ‖P‖2 θ + ‖P‖2 ε20


+ ‖P‖2 ∥∥D∥∥2 +

12ρ2 d2

1 −1rθ

.

θ

= − [λmin (Q) − 3] ‖e‖2 + χ1(x2 + f1

)

+ ‖P‖2 θ + ‖P‖2 ε20 + ‖P‖2 ∥∥D

∥∥2

+12ρ2 d2

1 −12χ2

1 −1rθ

.

θ (16)

where

f1 (X1) = f1 +12ρ−2χ1 +

14χ1 − yd +

12χ1 .

Now, take the intermediate control signal α1(X1) as

α1(X1) = −(k1χ1 + f1

)

where k1 > 0. Then, we have

V1 ≤ − [λmin (Q) − 3] ‖e‖2 + χ1 (x2 − α1)

− k1χ21 + ‖P‖2 θ + ‖P‖2 ε2

0 + ‖P‖2 ∥∥D∥∥2

+12ρ2 d 2

1 − 12χ2

1 −1rθ

.

θ. (17)

However, α1(X1) is an unknown nonlinear function as it con-tains f1(x1), which cannot be implemented in practice. There-fore, according to Lemma 1, for any given constant ε1 > 0, thereexists a fuzzy logic system ΦT

1 ξ1(X1) such that

α1(X1) = ΦT1 ξ1(X1) + δ1(X1)

|δ1 (X1)| ≤ ε1 . (18)

From the definitions of θ and α1 , we have

−χ1 α1 = −χ1ΦT

1

‖Φ1‖ξ1 ‖Φ1‖ − χ1δ1

≤ 12a2

1χ2

1θξT1 ξ1 +

12a2

1 +12χ2

1 +12ε2

1 (19)

χ1α1 = − 12a2

1χ2

1 θξT1 ξ1 . (20)

Then, the substitution of (19) and (20) into (17) yields

V1 ≤ − [λmin (Q) − 3] ‖e‖2 + χ1 (x2 − α1) − k1χ21

+1rθ

(r

2a21χ2

1ξT1 ξ1 −

.

θ

)+ Δ1 (21)

where

Δ1 = ‖P‖2 θ + ‖P‖2 ε20 + ‖P‖2 ∥∥D

∥∥2

+12ρ2 d 2

1 +12a2

1 +12ε2

1 .

Step 2: Define the variable χ2 = x2 − α1 , and consider thefollowing Lyapunov function:

V2 = V1 +12χ2

2 . (22)

The time derivative of V2 is

V2 = V1 + χ2 χ2

= V1 + χ2

[x3 − l2 x1 −

∂α1

∂x1(x2 + e2 + f1 + d1)

−∂α1

∂θ

.

θ − ∂α1

∂ydyd − ∂α1

∂ydy

(2)d

]. (23)

Using a similar way as (14) and (15) in Step 1, the followinginequalities can be obtained:

−χ2∂α1

∂x1e2 ≤ e2

2 +14

(∂α1

∂x1

)2

χ22 (24)

−χ2∂α1

∂x1d1 ≤ 1

2ρ−2

(∂α1

∂x1

)2

χ22 +

12ρ2 d 2

1 . (25)

Then, the substitution of (24) and (25) into (23) gives

V2 ≤ − [λmin (Q) − 4] ‖e‖2 − k1χ21 + χ1χ2

+1rθ

(r

2a21χ2

1ξT1 ξ1 −

.

θ

)+ Δ1

+12ρ2 d2

1 + χ2

[x3 − l2 x1

− ∂α1

∂x1(x2 + f1) +

14

(∂α1

∂x1

)2

χ2

+12ρ−2

(∂α1

∂x1

)2

χ2 −∂α1

∂θ

.

θ

−∂α1

∂ydyd − ∂α1

∂ydy

(2)d

]

= − [λmin (Q) − 4] ‖e‖2 − k1χ21 + χ1χ2

+1rθ

(r

2a21χ2

1ξT1 ξ1 −

.

θ

)+ Δ1

+12ρ2 d2

1 + χ2(x3 + f2

)

+ χ2

(ϕ2 (X2) −

∂α1

∂θ

.

θ

)− 1

2χ2

2 (26)

where

f2 (X2) = −l2 x1 −∂α1

∂x1(x2 + f1)

+14

(∂α1

∂x1

)2

χ2 +12ρ−2

(∂α1

∂x1

)2

χ2

− ∂α1

∂ydyd − ∂α1

∂ydy

(2)d +

12χ2 − ϕ2 (X2)

with

ϕ2 (X2) = −k0 θ∂α1

∂θ− χ2

r

2a22

∣∣∣∣χ2∂α1

∂θ

∣∣∣∣

+∂α1

∂θ

r

2a21χ2

1ξT1 ξ1 .

Now, take the intermediate control signal α2(X2) as

α2(X2) = −(k2χ2 + χ1 + f2

)


where k2 > 0; then, addition and subtraction of α2(X2) in (26)yield

V2 ≤ − [λmin (Q) − 4] ‖e‖2 −2∑

i=1

kiχ2i

+ χ2 (x3 − α2) +1rθ

(r

2a21χ2

1ξT1 ξ1 −

.

θ

)

+ Δ1 +12ρ2 d2

1

+ χ2

(ϕ2 (X2) −

∂α1

∂θ

.

θ

)− 1

2χ2

2 . (27)

Since α2(X2) is also an unknown nonlinear function, thefuzzy logic system ΦT

2 ξ2(X2) is now employed to approximateit. For any given constant ε2 > 0, there exists a fuzzy logicsystem ΦT

2 ξ2(X2) such that

α2(X2) = ΦT2 ξ2(X2) + δ2(X2)

|δ2 (X2)| ≤ ε2 . (28)

Using a similar procedure as in (19) and (20), we can get

−χ2 α2 ≤ 12a2

2χ2

2θξT2 ξ2 +

12a2

2 +12χ2

2 +12ε2

2 (29)

χ2α2 = − 12a2

2χ2

2 θξT2 ξ2 . (30)

Then, by substituting (29) and (30) into (27), we have

V2 ≤ − [λmin (Q) − 4] ‖e‖2 −2∑

i=1

kiχ2i + χ2 (x3 − α2)

+1rθ

(2∑

i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+ χ2

(ϕ2 (X2) −

∂α1

∂θ

.

θ

)+ Δ2 (31)

where

Δ2 = ‖P‖2 θ + ‖P‖2 ε20 + ‖P‖2 ∥∥D

∥∥2

+2∑

i=1

i

2ρ2 d2

1 +12

2∑i=1

a2i +

12

2∑i=1

ε2i .

Step m: Similarly, for each step m (3 ≤ m ≤ n − 1), defineχm = xm − αm−1, and choose the Lyapunov function candi-date

Vm = Vm−1 +12χ2

m . (32)

The time derivative of Vm is

Vm = Vm−1 + χm [xm+1 − lm x1

− ∂αm−1

∂x1(x2 + e2 + f1 + d1)

−m−1∑i=1

∂αm−1

∂xi(xi+1 − li x1)

−∂αm−1

∂θ

.

θ −m∑

i=1

∂αm−1

∂y(i−1)d

y(i)d

]. (33)

Using a similar way as in Step 1 and Step 2, the followinginequalities can be obtained:

−χm∂αm−1

∂x1e2 ≤ e2

2 +14

(∂αm−1

∂x1

)2

χ2m (34)

−χm∂αm−1

∂x1d1 ≤ 1

2ρ−2

(∂αm−1

∂x1

)2

χ2m +

12ρ2 d2

1 . (35)

Substituting (34) and (35) into (33), we get

Vm ≤ − [λmin (Q) − (2 + m)] ‖e‖2 −m−1∑i=1

kiχ2i

+ χm−1χm + Δm−1 +12ρ2 d2

1

+1rθ

(m−1∑i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+m−1∑i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ χm

[xm+1 − lm x1 −

∂αm−1

∂x1(x2 + f1)

−m−1∑i=1

∂αm−1


− ∂αm−1

∂θ

.

θ −m∑

i=1

∂αm−1

∂y(i−1)d

y(i)d

+14

(∂αm−1

∂x1

)2

χm +12ρ−2

(∂αm−1

∂x1

)2

χm

]

= − [λmin (Q) − (2 + m)] ‖e‖2 −m−1∑i=1

kiχ2i

+ χm−1χm + Δm−1 +12ρ2 d2

1

+1rθ

(m−1∑i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+m∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ χm

(xm+1 + fm

)− 1

2χ2

m (36)

where

fm (Xm ) = −lm x1 −∂αm−1

∂x1(x2 + f1)

−m−1∑i=1

∂αm−1



−m∑

i=1

∂αm−1

∂y(i−1)d

y(i)d +

14

(∂αm−1

∂x1

)2

χm

+12ρ−2

(∂αm−1

∂x1

)2

χm

+12χm − ϕm (Xm )

with

ϕm (Xm ) = −k0 θ∂αm−1

∂θ−

m∑j=2

χmr

2a2m

∣∣∣∣χj∂αj−1

∂θ

∣∣∣∣

+m−1∑j=1

∂αm−1

∂θ

r

2a2j

χ2j ξ

Tj ξj .

Take the intermediate control signal αm (Xm ) as

αm = −(km χm + χm−1 + fm

)

where km > 0; then, addition and subtraction of αm (Xm ) in(36) yield

Vm ≤ − [λmin (Q) − (2 + m)] ‖e‖2

−m∑

i=1

kiχ2i + χm (xm+1 − αm )

+1rθ

(m−1∑i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+m∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ Δm−1 +12ρ2 d2

1 −12χ2

m . (37)

Similarly, αm (Xm ) can be approximated by the fuzzy logicsystem ΦT

m ξm (Xm ) as

αm (Xm ) = ΦTm ξm (Xm ) + δm (Xm )

|δm (Xm )| ≤ εm . (38)

We can also get

−χm αm ≤ 12a2

m

χ2m θξT

m ξm +12a2

m +12χ2

m +12ε2m (39)

χm αm = − 12a2

m

χ2m θξT

m ξm . (40)


Vm ≤ − [λmin (Q) − (2 + m)] ‖e‖2

−m∑

i=1

kiχ2i + χm (xm+1 − αm )

+1rθ

(m∑

i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+m∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ Δm (41)

where

ϕi (Xi) = −k0 θ∂αi−1

∂θ−

i∑j=2

χir

2a2i

∣∣∣∣χj∂αj−1

∂θ

∣∣∣∣

+i−1∑j=1

∂αi−1

∂θ

r

2a2j

χ2j ξ

Tj ξj

Δm = ‖P‖2 θ + ‖P‖2 ε20 + ‖P‖2 ∥∥D

∥∥2

+m∑

i=1

i

2ρ2 d2

1 +12

m∑i=1

a2i +

12

m∑i=1

ε2i .

Step n: Define the variable as χn = xn − αn−1 , and considerthe following Lyapunov function:

Vn = Vn−1 +12χ2

n . (42)

The time derivative of Vn is given by

Vn = Vn−1 + χn

[u − ln x1 −

∂αn−1

∂x1(x2 + e2 + f1 + d1)

−n−1∑i=1

∂αn−1

∂xi(xi+1 − li x1) −

∂αn−1

∂θ

.

θ

−n∑

i=1

∂αn−1

∂y(i−1)d

y(i)d

]. (43)

Similar to the aforementioned steps, we can obtain the followinginequalities:

−χn∂αn−1

∂x1e2 ≤ e2

2 +14

(∂αn−1

∂x1

)2

χ2n (44)

−χn∂αn−1

∂x1d1 ≤ 1

2ρ−2

(∂αn−1

∂x1

)2

χ2n +

12ρ2 d2

1 . (45)

Substituting (44) and (45) into (43), we get

Vn ≤ − [λmin (Q) − (2 + n)] ‖e‖2 −n−1∑i=1

kiχ2i

+ χn−1χn + Δn−1 +12ρ2 d2

1

+1rθ

(n−1∑i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+n−1∑i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ χn

[u − ln x1 −

∂αn−1

∂x1(x2 + f1)

−n−1∑i=1

∂αn−1


− ∂αn−1

∂θ

.

θ −n∑

i=1

∂αn−1

∂y(i−1)d

y(i)d


+14

(∂αn−1

∂x1

)2

χn +12ρ−2

(∂αn−1

∂x1

)2

χn

]

= − [λmin (Q) − (2 + n)] ‖e‖2 −n−1∑i=1

kiχ2i

+ χn−1χn + Δn−1 +12ρ2 d2

1

+1rθ

(n−1∑i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+n∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ χn

(u + fn

)− 1

2χ2

n (46)

where

fn (Xn ) = −ln x1 −∂αn−1

∂x1(x2 + f1)

−n−1∑i=1

∂αn−1


−n∑

i=1

∂αn−1

∂y(i−1)d

y(i)d +

14

(∂αn−1

∂x1

)2

χn

+12ρ−2

(∂αn−1

∂x1

)2

χn +12χn − ϕn (Xn ) .

Take the intermediate control signal αn (Xn ) as

αn = −(knχn + χn−1 + fn

)

where kn > 0; then, addition and subtraction of αn (Xn ) in (46)yield

Vn ≤ − [λmin (Q) − (2 + n)] ‖e‖2

−n∑

i=1

kiχ2i + χn (u − αn )

+1rθ

(n−1∑i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+n∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

+ Δn−1 +12ρ2 d2

1 −12χ2

n . (47)

Similar to the aforementioned steps, αn (Xn ) can be approxi-mated by the fuzzy logic system ΦT

n ξn (Xn ) as

αn (Xn ) = ΦTn ξn (Xn ) + δn (Xn )

|δn (Xn )| ≤ εn . (48)

Following the similar procedure and by the definition of u, wehave

−χnαn ≤ 12a2

n

χ2nθξT

n ξn +12a2

n +12χ2

n +12ε2n (49)

χnu = − 12a2

n

χ2n θξT

n ξn . (50)


Vn ≤ − [λmin (Q) − (2 + n)] ‖e‖2 −n∑

i=1

kiχ2i

+1rθ

(n∑

i=1

r

2a2i

χ2i ξ

Ti ξi −

.

θ

)

+n∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)+ Δn

where

Δn = ‖P‖2 θ + ‖P‖2 ε20 + ‖P‖2 ∥∥D

∥∥2

+n∑

i=1

i

2ρ2 d2

1 +12

n∑i=1

a2i +

12

n∑i=1

ε2i .

By the definition of.

θ, we can get

Vn ≤ − [λmin (Q) − (2 + n)] ‖e‖2 −n∑

i=1

kiχ2i +

k0

rθθ

+ Δn +n∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

≤ − [λmin (Q) − (2 + n)] ‖e‖2 −n∑

i=1

kiχ2i −

k0

2rθ2

+ Δn +n∑

i=2

χi

(ϕi (Xi) −

∂αi−1

∂θ

.

θ

)

where

Δn = Δn +k0

2rθ2 .

From the work in [31], it can be shown thatn∑

i=2χi

(ϕi (Xi) − ∂αi−1

∂ θ

.

θ

)≤ 0; therefore

Vn ≤ − [λmin (Q) − (2 + n)] ‖e‖2 −n∑

i=1

kiχ2i −

k0

2rθ2 + Δn .

Denote

c = min{

λmin (Q) − (2 + n)λmax (P )

, 2ki, k0 ; i = 1, . . . , n

}.

Then, we get

Vn ≤ −cVn + Δn .

Therefore, the following inequality holds:

Vn (t) ≤(

Vn (0) − Δn

c

)e−ct +

Δn

c, t ≥ 0. (51)


Choose λmin (Q) − (3 + n) > 0; then, (51) implies that the sig-nals xi, xi , χi, θ, αi , and u are bounded. Therefore, for anygiven ε > 0, by appropriately choosing the design parameters ki

and k0 and choosing parameters ai, εi , and ρ to be sufficientlysmall, as well as r to be sufficiently large, it is possible to make(Δn/c) ≤ (ε2/2). Therefore, from (51), the following holds:

χ21 ≤ 2

(Vn (0) − Δn

c

)e−ct + 2

Δn

c

which means

lim supt→∞

χ21 ≤ 2

Δn

c≤ ε2 .

Remark 1: It should be pointed out that the fuzzy logic systemis directly used to approximate the intermediate control signalαi (i = 1, 2, . . . , n) rather than the unknown nonlinear functionfi (i = 1, 2, . . . , n). In addition, if the design parameters ai ,k0 , ki , and r are chosen appropriately, the tracking error canconverge to a small neighborhood of the origin.

Remark 2: The main contribution of this paper is utilizing aninput-driven filter to design a fuzzy adaptive output-feedbackcontroller for SISO strict-feedback nonlinear systems. In addi-tion, a direct adaptive fuzzy control method is used, which canreduce the computation burden, because only one parameterneeds to be estimated online.

IV. SIMULATION

In this section, two examples are presented to demonstratethe effectiveness of our main results.

Example 1 [26]: Consider the following system:

x1 = x2

x2 = −0.1x2 − x31 + 12 cos (t) + u

y = x1 . (52)

The reference signal is given as yd = sin (t).Choose the following fuzzy membership functions:

μF 1i(x) = exp

[(x + 1.5)2

2

], μF 2

i(x) = exp

[−(x + 1)2

2

]

μF 3i(x) = exp

[−(x + 0.5)2

2

], μF 4

i(x) = exp

[−x2

2

]

μF 5i(x) = exp

[−(x − 0.5)2

2

], μF 6

i(x) = exp

[−(x − 1)2

2

]

μF 7i(x) = exp

[−(x − 1.5)2

2

]. (53)

According to Theorem 1, the virtual control function α1 andthe true control law u are chosen, respectively, as

α1 = − 12a2

1χ1 θξ

T1 ξ1 , u = − 1

2a22χ2 θξ

T2 ξ2 (54)

where χ1 = y − yd , and χ2 = x2 − α1 .

Fig. 1. System output y (“-”) and reference signal yd (“-.-”) of Example 1.

Fig. 2. Trajectory of the tracking error y − yd of Example 1.

The adaptive law is given as

.

θ =2∑

i=1

r

2a2i

χ2i ξ

Ti (Xi) ξi (Xi) − k0 θ. (55)

In the simulation, we choose design parameters l1 = 144,l2 = 24, a1 = 0.305, a2 = 0.215, r = 8.5, k0 = 0.005, andthe same initial conditions [x1(0), x2(0), x1(0), x2(0)]T =[0.2, 0.2, 1.5, 1.5]T as in [26, Ex. 1] and θ (0) = 0. The sim-ulation results are shown by Figs. 1– 4, respectively. Fig. 1shows the system output y and reference signal yd . Fig. 2 de-picts the trajectory of the tracking error y − yd . Fig. 3 plots thetrajectory of input u. Fig. 4 illustrates the trajectory of adaptiveparameter θ. By comparing the simulation results with the onesin [26], it can be seen that the method in this paper cannot onlyachieve the tracking performance but requires a smaller controlgain as well. In addition, the number of adaptive parameter isonly one in our paper, while the number is 7 in [26].

Example 2 [26]: To further show the effectiveness of ourresult, we consider the following nonlinear dynamic system:

x1 = x2 + x1e−0.5x1

x2 = u + x1 sin(x2

2)

y = x1 . (56)


Fig. 3. Control law u of Example 1.

Fig. 4. Adaptive parameter θ of Example 1.

Fig. 5. System output y (“-”) and reference signal yd (“-.-”) of Example 2.

The tracking reference signal is yd = (1/2) sin(t). For thissystem, we still consider the fuzzy membership functions thatare defined in (53). Similarly, the direct adaptive fuzzy con-troller for system (56) is designed by Theorem 1. The virtualcontrol function α1 , the true control law u, and the adaptive lawθ are chosen as (54) and (55). In the simulation, the design pa-rameters are chosen as l1 = 5, l2 = 1, a1 = 0.298, a2 = 0.215,

Fig. 6. Trajectories of the tracking errors y − yd of Example 2.

Fig. 7. Control law u of Example 2.

Fig. 8. Adaptive parameter θ of Example 2.

r = 8.6, and k0 = 0.005. The initial conditions are chosen as[x1(0), x2(0), x1(0), x2(0)]T = [0,−0.2, 0, 0.3]T , which is thesame as [26, Ex. 2]. The simulation results are shown by Figs.5– 8, from which we can see that the tracking performance isachieved well as in [26]. However, the computation burden isreduced, because in our paper, the adaptive parameter is onlyone, while the number is 10 in [26].


From the two examples, we can see that the direct adaptivefuzzy control method that is proposed in this paper can achievethe tracking performance well. It is worth mentioning that theadaption law that is proposed by this method is only one. There-fore, the computation burden can be significantly reduced.

V. CONCLUSION

By using Fuzzy logic systems and backstepping approach,a direct adaptive fuzzy tracking control scheme has been pro-posed in this paper. Additionally, by introducing the input-drivenfilter, a fuzzy adaptive output-feedback controller has been con-structed. The method that has been proposed in this paper canbe applied to a class of systems with unmeasurable states. Inaddition, as the number of the online adaptive parameter is onlyone, the computation burden can be reduced accordingly; there-fore, it is convenient to implement this algorithm in practicalsystems. Finally, two simulation examples have been presentedto illustrate the effectiveness of the method that is proposed inthis paper.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor and thereviewers for their very constructive comments and suggestionsthat have helped improve the quality and presentation of thispaper.

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Qi Zhou received the B.S. and M.S. degrees in math-ematics from Bohai University, Jinzhou, China, in2006 and 2009, respectively. She is currently workingtoward the Ph.D. degree with the School of Automa-tion, Nanjing University of Science and Technology,Nanjing, China.

She is a Visiting Student with Victoria University,Melbourne, Australia. Her research interest includesfuzzy control, stochastic control, and robust control.


Peng Shi (SM’97) received the B.Sc. degree in math-ematics from the Harbin Institute of Technology,Harbin, China, in 1982, the M.E. degree in systemsengineering from Harbin Engineering University in1985, the Ph.D. degree in electrical engineering fromthe University of Newcastle, Newcastle, Australia, in1994, the Ph.D. degree in mathematics from the Uni-versity of South Australia, Mount Gambier, Australia,in 1998, and the D.Sc. degree from the University ofGlamorgan, Pontypridd, U.K., in 2006.

He was a Lecturer with Heilongjiang University,Harbin, during 1985–1989 and with the University of South Australia during1997–1999, and he was a Senior Scientist with Defence Science and Technol-ogy Organization, Department of Defence, Australia, during 1999–2005. Hejoined the University of Glamorgan as a Professor in 2004. Since 2008, he hasalso been a Professor with Victoria University, Melbourne, Australia. In addi-tion, he is a coauthor of three research monographs: Analysis and Synthesis ofSystems with Time-Delays (Berlin, Germany: Springer, 2009), Fuzzy Controland Filtering Design for Uncertain Fuzzy Systems (Berlin, Germany: Springer,2006), and Methodologies for Control of Jump Time-Delay Systems (Boston,MA: Kluwer, 2003). His research interests include control system design, faultdetection techniques, Markov decision processes, and operational research. Hehas published a number of papers in these areas.

Dr. Shi is the Editor-in-Chief of the International Journal of InnovativeComputing, Information and Control. He is also an Advisory Board Member,an Associate Editor, and an Editorial Board Member for a number of interna-tional journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL,the IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS–PART B, theIEEE TRANSACTIONS ON FUZZY SYSTEMS, Information Sciences, and the In-ternational Journal of Systems Science. He is the recipient of the Most CitedPaper Award of Signal Processing in 2009. He is a Fellow of the Institute ofEngineering and Technology (U.K.) and the Institute of Mathematics and itsApplications (U.K.).

Jinjun Lu was born in 1964. He received the B.S.and M.S. degrees in mathematics from Nanjing Uni-versity of Technology, Nanjing, China, and HohaiUniversity, respectively.

He is currently a Professor with Nantong Voca-tional College and College of Electrical Engineer-ing, Nantong University, Nantong, China. His currentresearch interests include congestion control, robustcontrol, and fault-tolerant control.

Shengyuan Xu received the B.Sc. degree from theHangzhou Normal University, Hangzhou, China, in1990, the M.Sc. degree from the Qufu Normal Uni-versity, Qufu, China, in 1996, and the Ph.D. degreefrom the Nanjing University of Science and Technol-ogy, Nanjing, China, in 1999.

From 1999 to 2000, he was a Research Associatewith the Department of Mechanical Engineering, TheUniversity of Hong Kong, Hong Kong. From Decem-ber 2000 to November 2001 and December 2001 toSeptember 2002, he was a Postdoctoral Researcher

with the Center for Systems Engineering and Applied Mechanics, CatholicUniversity of Louvain, Louvain-la-Neuve, Belgium, and the Department ofElectrical and Computer Engineering, University of Alberta, Edmonton, AB,Canada, respectively. From September 2002 to September 2003 and September2003 to September 2004, he was a William Mong Young Researcher and anHonorary Associate Professor, respectively, with the Department of Mechani-cal Engineering, The University of Hong Kong, Hong Kong. Since November2002, he has been a Professor with the School of Automation, Nanjing Univer-sity of Science and Technology. His research interests include robust filteringand control, singular systems, time-delay systems, neural networks, and multi-dimensional and nonlinear systems.

Dr. Xu was a recipient of the National Excellent Doctoral Dissertation Awardin 2002 from the Ministry of Education of China. He obtained a grant from theNational Science Foundation for Distinguished Young Scholars of China in2006. He received a Cheung Kong Professorship in 2008 from the Ministry ofEducation of China. He is a member of the Editorial Boards of MultidimensionalSystems and Signal Processing and Circuits, Systems, and Signal Processing.

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Adaptive Output-Feedback Fuzzy Tracking Control for a Class of Nonlinear Systems

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