observer-based fuzzy adaptive control for strict-feedback nonlinear systems

Fuzzy Sets and Systems 160 (2009) 1749–1764www.elsevier.com/locate/fss

Observer-based fuzzy adaptive control for strict-feedbacknonlinear systems�

Shaocheng Tong∗, Yongming LiDepartment of Basic Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121001, China

Received 27 January 2008; received in revised form 13 August 2008; accepted 2 September 2008Available online 25 September 2008

Abstract

In this paper, a new fuzzy adaptive control approach is developed for a class of SISO strict-feedback nonlinear systems withunmeasured states. Using fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy adaptive observer isintroduced for state estimation as well as system identification. Under the framework of the backstepping design, fuzzy adaptiveoutput feedback control is constructed recursively. It is proven that the proposed fuzzy adaptive control approach guarantees thesemi-global boundedness property for all the signals and the tracking error to a small neighborhood of the origin. Simulation studiesare included to illustrate the effectiveness of the proposed approach.© 2008 Elsevier B.V. All rights reserved.

Keywords: Fuzzy control; Adaptive control; Nonlinear systems; State observer; Backstepping; Stability

1. Introduction

Fuzzy control methodologies have emerged in recent years as promising ways to approach nonlinear control prob-lems. Fuzzy control, in particular, has had an impact in the control community because of the simple approach itprovides to use heuristic control knowledge for nonlinear control problem. In very complicated situations, where theplant parameters are subject to perturbations or when the dynamics of the systems are too complex for a mathemat-ical model to describe, adaptive schemes have to be used online to gather data and adjust the control parametersautomatically. Based on the universal approximation theorem and by incorporating fuzzy logic systems (FLS) intoadaptive control schemes, a stable fuzzy adaptive controller was first developed to control unknown nonlinear sys-tems [20]. Afterwards, various adaptive fuzzy control approaches have been introduced for controlling nonlinearsystems [4–8,11–17,19,22,23], among them [4,11,14–17,22] for single-input single-output (SISO) nonlinear systemsand [5,6,7,12,13,19,23] for multiple-input multiple-output (MIMO) nonlinear systems. Generally, the adaptive fuzzycontrol approaches can have nice performance. However, these approaches have been applied only to a relatively simpleclass of nonlinear systems. The key requirement is that the unknown nonlinearities appear on the same equation as

� This workwas supported by theNational Natural Science Foundation of China (no. 60674056), theNational KeyBasic Research andDevelopmentProgram of China (no. 2002CB312200) and the Outstanding Youth Funds of Liaoning Province (no. 2005219001) and Educational Department ofLiaoning Province (nos. 2006R29 and 2007T80).

∗Corresponding author. Tel.: +864164199101; fax: +864164199415.E-mail addresses: [email protected] (S. Tong), [email protected] (Y. Li).

0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.09.004

http://www.elsevier.com/locate/fss

mailto:[email protected]

mailto:[email protected]

1750 S. Tong, Y. Li / Fuzzy Sets and Systems 160 (2009) 1749–1764

the control input in a state space representation. Such restrictions on the location of the uncertain nonlinear functionsare usually referred to as matching conditions. If physical systems are subject to some unknown nonlinear functionswhich do not satisfy the matching conditions, the adaptive fuzzy control approaches mentioned above cannot beimplemented.With the development of adaptive and robust backstepping designs in nonlinear systems [10], many fuzzy adaptive

control schemes have been reported that combined the backstepping technique with adaptive FLS for nonlinear systems[2,3,9,18,21,24–26]. Recently, fuzzy adaptive backstepping control schemes can provide a systematic framework forthe design of tracking or regulation strategies, in which the FLS are used to approximate the unknown nonlinearfunctions, and an adaptive fuzzy controller is constructed recursively. These approaches are suitable for some uncertainnonlinear systems which do not satisfy the requirement of matching conditions, or require the linearity, in the parameterassumption of nonlinear systems. It is due to that fuzzy adaptive backstepping control techniques have attracted moreattention in the fuzzy control fields. However, the existing fuzzy adaptive backstepping control methods are all basedon the assumption that the states of the systems are directly measured. As pointed out in [1,6,7,11,12,17,22], inpractice, state variables are often unmeasured for many nonlinear systems. Therefore, the problem of fuzzy adaptivebackstepping control for nonlinear systems with state unmeasured is very important in both theory and real worldapplications. Recently, Refs. [1,8,11,17,22] have developed fuzzy adaptive output feedback control approaches basedon the state observers for the uncertain SISO nonlinear systems. In [1,11,17,22], FLS are utilized to approximate theunknown functions, the error observers are designed. Based on feedback linearizationmethod, the fuzzy adaptive outputfeedback control approaches have been proposed. However, these two approaches both require that the systems are ofuncertain affine nonlinear systems [1], whichmust satisfy thematching condition. In [8], a kind of fuzzy adaptive outputfeedback controller is first developed for a class of uncertain nonlinear systems with no match condition. It is designedon the variable structure and backstepping design technique. It takes the unknown function as the external disturbances,and developed a sliding mode controller. In order to overcome the high gain and discontinuous phenomenon existingin the controller, B-spine-type membership functions are introduced into the controller to form a “soft” fuzzy slidingmode output controller, which is called a fuzzy adaptive output controller.The present paper will address the fuzzy adaptive output feedback control problem in the framework of Refs.

[9,10,15,20,21,24–26]. In such problems, the system does not satisfy the matching condition and the state vector is notdirectly measured. Using FLS to approximate the unknown nonlinear functions, a fuzzy adaptive observer is introducedfor state estimation as well as system identification. Combining backstepping design with FLS, fuzzy adaptive outputfeedback control controller is constructed recursively, and the adjusting parameter vectors are derived on the Lyapunovfunctions. It is proved that the proposed fuzzy adaptive control approach guarantees the semi-global boundednessproperty for all the signals and the tracking error converges to a small neighborhood of the origin.

2. Preliminaries

2.1. Nonlinear control problem

Consider the following SISO strict-feedback nonlinear system:

x1 = x2 + f1(x1) + d1

x2 = x3 + f2(x1, x2) + d2

...

xn−1 = xn + fn−1(x1, . . . , xn−1) + dn−1

xn = u + fn(x1, . . . , xn) + dn

y = x1 (1)

where X = (x1, x2, . . . , xn)T ∈ Rn is the system state vector, u ∈ R is the control input, y ∈ R is the output of system,fi (Xi ) (i=1, 2, . . . , n) are unknown smooth functions and di (i=1, 2, . . . , n) are the bounded disturbance uncertaintiesof the system. Let Xi = (x1, x2, . . . , xi )T and it is assumed that Xi (i�2) are unmeasured.

S. Tong, Y. Li / Fuzzy Sets and Systems 160 (2009) 1749–1764 1751

Rewriting (1) in the following form:

X = AX + Ky +n∑

i=1

Bi [ fi (Xi ) + di ] + Bu (2)

where

A =⎡⎣−k1

... I−kn 0 . . . 0

⎤⎦ , K =

⎡⎣k1

...

kn

⎤⎦ , B =

⎡⎣0

...

1

⎤⎦ , Bi = [0 . . . 1 . . . 0]T

and K is chosen such that A is a strict Hurwitz matrix. Thus, given a Q> 0, there exists a P> 0 satisfying

ATP + PA = −2Q (3)

Let yr be a reference signal and contain finite derivatives up to the nth order. Define the output tracking error as�1 = y − yr.Control objectives: Utilizing FLS, reference signal yr, the output of the system y and the states estimates Xi to

determine a fuzzy controller and parameters adaptive laws such that all the signals involved in the closed-loop systemare boundedness and the tracking error �1 is as small as possible.

2.2. Fuzzy logic systems

An FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules,and the defuzzifier. The knowledge base for FLS comprises a collection of fuzzy if–then rules of the following form:

Rl : if x1 is Fl1 and x2 is F

l2 and . . . and xn is F

ln, then y isGl , l = 1, 2, . . . , N (4)

where x = (x1, . . . , xn)T and y are the FLS input and output, respectively. Fuzzy sets Fli and Gl , associated with the

fuzzy functions �Fli(xi ) and �Gl (y), respectively. N is the rules number.

Through singleton function, center average defuzzification and product inference [19], the FLS can be expressed as

y(x) =∑N

l=1 yl∏n

i=1�Fli(xi )∑N

l=1[∏n

i=1�Fli(xi )]

(5)

where yl = maxy∈R �Gl (y).Define the fuzzy basis functions as

�l =∏n

i=1�Fli(xi )∑N

l=1(∏n

i=1�Fli(xi ))

(6)

Denoting �T = [y1, y2, . . . , yN ] = [�1, �2, . . . , �N ] and �(x) = [�1(x), . . . , �N (x)]T, then FLS (5) can be rewritten as

y(x) = �T�(x) (7)

Lemma 1 (Boulkroune et al. [1]). Let f(x) be a continuous function defined on a compact set �. Then for any constant� > 0, there exists an FLS (7) such as

supx∈�

| f (x) − �T�(x)|�� (8)

3. Fuzzy adaptive control design

In this section, it is assumed that the states of system (1) are not available for feedback, in this situation, a state observershould be established to estimate the states, and then fuzzy adaptive output feedback control scheme is investigated.


By Lemma 1, FLS are universal approximators, i.e., they can approximate any smooth function on a compact space.Due to this approximation capability, we can assume that the nonlinear terms in (1) can be approximated as

fi (Xi |�i ) = �Ti �i (Xi ), fi (Xi |�i ) = �Ti �i (Xi ), 1� i�n (9)

where Xi = (x1, x2, . . . , xi )T.According to Refs. [10,16], define the optimal parameter vectors �∗

i as

�∗i = argmin

�i∈�i

[sup

Xi∈Ui1,Xi∈Ui2

| fi (Xi |�i ) − fi (Xi )|]

, 1� i�n (10)

where�i,Ui1 andUi2 are compact regions for �i, Xi and Xi , respectively. Also the FLSminimum approximation errors�i and approximation errors �i are defined as

�i = fi (Xi ) − fi (Xi |�∗i ), �i = fi (Xi ) − fi (Xi |�i ) (11)

Denote wi = �i + di and �′i = �i + di , i = 1, 2, . . . , n.

Assumption 1. There exist known constants �i0 and �i0, i = 1, 2, . . . , n, such that |wi |��i0 and |�′i |��′

i0.

Since the state variables are not available, state observer should be designed to estimate the states.Design fuzzy state observer as

˙X = AX + Ky +n∑

i=1

Bi fi (Xi |�i ) + Bu

y = C X (12)

where C = [1 . . . 0 . . . 0].Let e = X − X be observer error, then from (2) and (12), we have the observer errors equation

e = Ae +n∑

i=1

Bi [ fi (Xi ) − fi (Xi |�i ) + di ]

= Ae +n∑

i=1

Bi [�i + di ]

= Ae + � (13)

where � = [�′1, . . . , �n]

T.The ensuring systematic design procedure involves, at each step, augmenting an integrator and the design of a virtual

control law to stabilize the augmented system, until the actual control u appears at the nth step. The detailed designprocedures of fuzzy adaptive output feedback controller are described in the following steps:Step 1. Define the tracking error for the system as

�1 = y − yr (14)

Expressing x2 in terms of its estimate as x2 = x2 + e2, we obtain

�1 = x1 − yr= x2 + f1(x1) + d1 − yr= x2 + f1(x1) + d1 − yr + e2

= x2 + �T1�1(x1) − yr + e2 + �T1�1(x1) + w1 (15)

where �1 = �∗1 − �1.


Taking x2 as a virtual control, and define

�2 = x2 − 1(x1, �1, y, yr) − yr (16)

Then we have

�1 = �2 + 1(x1, �1, y, yr ) + �T1�1(x1) + e2 + �T1�1(x1) + w1 (17)

Consider the following Lyapunov function:

V1 = 1

2eTPe + 1

2�21 + 1

21�T1 �1 (18)

where 1 > 0 is a design constant.From (13) and (17), the time derivative of V1 is

V1 = 1

2eTPe + 1

2eTPe + �1�1 + 1

1�T1˙�1

= 1

2eT[PAT + AP]e + eTP� + �1�1 + 1

1�T1˙�1

= − eTQe + eTP� + �1[�2 + 1(x1, �1, y, yr) + �T1�1(x1) + w1] + e2�1 + �T1�1(x1)�1 + 1

1�T1˙�1 (19)

By using the inequality 2ab�a2 + b2, we have

eTP� + e2�1�12‖e‖2 + 1

2‖P�‖2 + 12 |e2|2 + 1

2�21

� ‖e‖2 + 12�

21 + 1

2‖P�‖2 (20)

Substituting (20) into (19) yields

V1� − eTQe + �1[�2 + �1/2 + 1 + �T1�1(x1) + w1] + ‖e‖2 + 1

2‖P�‖2 + �

T1�1(x1)�1 + 1

1�T1˙�1

� − (�min(Q) − 1)‖e‖2 + �1[�2 + �1/2 + 1(x1, �1, y, yr)

+ �T1�1(x1) + w1] + 1

2‖P�‖2 + 1

1�T1 (1�1(x1)�1 − �1) (21)

Since variables �1 and x1 are available, the intermediate control function 1(x1, �1, y, yr) and the adaptation function�1 are chosen as

1 = −c1�1 − �1/2 − �T1�1(x1) − �10 tanh(�10�1/�) (22)

�1 = 1�1(x1)�1 − (�1 − �10) (23)

where c1 > 0, � > 0, > 0, and �10 are design parameters.In view of (22), use the following nice property with regard to function tanh(·), i.e.,

�1w1 − �1�10 tanh(�10�1/�)�0.2785� = �′

Substituting (22) and (23) into (21) results in

V1� − (�min(Q) − 1)‖e‖2 − c1�21 + �1�2 + 1

2‖P�‖2 +

1�T1 (�1 − �10) + �′ (24)


Step 2. Differentiating (16) yields

�2 = ˙x2 − 1(x1, �1, y, yr) − yr

= x3 + k2e1 + �T2�2(X2) + �T2�2(X2) + w2 − �′

2 − �1�x1

˙x1 − �1��1

�1 − �1�yr

yr − yr − �1�y

y

= x3 + k2e1 + �T2�2(X2) + �T2�2(X2) + w2 − �′

2 − �1�x1

[x2 + �T1�1(x1) + k1e1] − �1��1

�1

− �1�yr

yr − yr − �1�y

[x2 + �T1�1(x1) + e2 + �′1]

= x3 + H2(x1, x2, �1, �2, y, yr, yr) − yr − �1�y

e2 − �1�y

�′1 + �

T2�2(X2) + w2 − �′

2 (25)

where

H2 = �T2�2(X2) − �1�x1

[x2 + �T1�1(x1) + k1e1] − �1��1

�1 − �1�yr

yr + k2e1 − �1�y

[x2 + �T1�1(x1)]

Consider the following Lyapunov function:

V2 = V1 + 1

2�22 + 1

22�T2 �2 (26)

where 2 > 0 is a constant.By (24) and (25), we have the time derivative of V2

V2 = V1 + �2�2 + 1

2�T2˙�2

� − (�min(Q) − 1)‖e‖2 − c1�21 + �1�2 + 1

2‖P�‖2 + �′

+

1�T1 (�1 − �10) + �2

[x3 + H2 − yr + w2 − �′

2 − �1�y

e2 − �1�y

�′1

]+ 1

2�T2 (2�2�2(X2) − �2) (27)

By using the inequality 2ab�a2 + b2, we have

−�2�′2 − �2

�1�y

e2 − �2�1�y

�′1�

1

2�22 + ‖e‖2 +

(�1�y

)2

�22 + 1

2�′21 + 1

2�′22 (28)

Substituting (28) into (27) yields

V2� − (�min(Q) − 2)‖e‖2 − c1�21 + �2

[x3 + �1 + 1

2�2 +

(�1�y

)2

�2 + H2 − yr + w2

]

+ 1

2�T2 (2�2�2(X2) − �2) + 1

2�′21 + 1

2�′22 + 1

2‖P�‖2 +

1�T1 (�1 − �10) + �′ (29)

Taking x3 as a virtual control and introducing the variable

�3 = x3 − 2(x1, x2, �1, �2, y, yr, yr) − yr

It is noted that variables �2 and X2 are available, choosing intermediate control function 2 and the adaptation function�2 as

2 = −�1 − c2�2 − 1

2�2 −

(�1�y

)2

�2 − H2 − �20 tanh(�20�2/�) (30)

�2 = 2�2�2(X2) − (�2 − �20) (31)

where �20 is a design parameter vector.


It is noted that �2w2 − �2�20 tanh(�20�2/�)�0.2785� = �′, and from (30) to (31), we have

V2� − (�min(Q) − 2)‖e‖2 −2∑

k=1

ck�2k + �2�3 + 1

2�′21 + 1

2�′22 + 1

2‖P�‖2 + 2�′ +

2∑k=1

i�Tk (�k − �k0) (32)

Step i. (3� i�n − 1): a similar procedure is employed recursively at each step. By defining

�i = xi − i−1(x1, x2, . . . , xi−1, �1, �2, . . . , �i−1, y, yr, . . . , y(i−2)r ) − y(i−1)

r

we have

�i = ˙xi − i−1

= xi+1 + ki e1 + �Ti �i (Xi ) + �Ti �i (Xi ) + wi − �′

i

− �i−1

�yy −

i−1∑k=1

�i−1

�xk˙xk −

i−1∑k=1

�i−1

��k�k −

i−1∑k=1

�i−1

�y(k−1)r

y(k)r − y(i)r

= xi+1 + ki e1 + �Ti �i (Xi ) + �Ti �i (Xi ) + wi − �′

i −i−1∑j=1

�i−1

�x j[x j+1 + �Tj � j (X j ) + k j e1]

−i−1∑k=1

�i−1

��k�k −

i−1∑k=1

�i−1

�y(k−1)r

y(k)r − y(i)r − �i−1

�y[x2 + �T1 �1(x1) + e2 + �′

1]

= xi+1 + Hi (x1, x2, . . . , xi−1, �1, �2, . . . , �i−1, y, yr, . . . , y(i−1)r )

− y(i)r + �Ti �i (Xi ) + wi − �′

i − �i−1

�ye2 − �i−1

�y�′1 (33)

where

Hi = ki e1 + �Ti �i (Xi ) −i−1∑k=1

�i−1

�xk[xk+1 + �Tk �k(Xk)] −

i−1∑j=1

k j�i−1

�x je1 −

i−1∑k=1

�i−1

��k�k

−i−1∑k=1

�i−1

�y(k−1)r

y(k)r − �1�y

[x2 + �T1�1(x1)]

Consider the following Lyapunov function

Vi = Vi−1 + 1

2�2i + 1

2i�Ti �i (34)

The time derivative of Vi is

Vi = Vi−1 + �i �i + 1

i�Ti˙�i � Vi−1 + �i

[xi+1 + Hi − y(i)r + wi − �′

i − �i−1

�ye2 − �i−1

�y�′1

]

+ 1

i�Ti (i�i�i (Xi ) − �i ) (35)

Again by using the inequality 2ab�a2 + b2, we have

−�i�′i − �i

�i−1

�ye2 − �i

�i−1

�y�′1�

1

2�2i + ‖e‖2 +

(�i−1

�y

)2

�2i + 1

2�′21 + 1

2�′2i (36)


Substituting (36) into (35) results in

Vi � − (�min(Q) − i)‖e‖2 −i−1∑k=1

ck�2k + �i−1�i + �i

[xi+1 + 1

2�i +

(�i−1

�y

)2

�i + Hi − y(i)r + wi − �′i

]

+ 1

i�Ti (i�i�i (y) − �i ) +

i−1∑k=1

k�Ti (�k − �k0) + (i − 1)

2�′21 + 1

2

i∑k=2

�′2k + 1

2‖P�‖2 + (i − 1)�′ (37)

Introducing the variable

�i+1 = xi+1 − i (x1, x2, . . . , xi , �1, �2, . . . , �i−1, y, yr, . . . , y(i−1)r ) − y(i)r (38)

Since variables �i and Xi are available, choose intermediate control function i adaptation function �i as

i = −�i−1 − ci�i − Hi − 1

2�i −

(�i−1

�y

)2

�i − �i0 tanh(�i0�i/�)

�i = i�i (Xi )�i − (�i − �i0) (39)

where i > 0 and �i0 are design constants. By (37)–(39), (37) becomes

Vi � − (�min(Q) − i)‖e‖2 −i∑

k=1

ck�2k + �i�i+1 + 1

2‖P�‖2 + (i − 1)

2�′21

+ 1

2

i∑k=2

�′2k +

i∑k=1

k�Tk (�k − �k0) + i�′ (40)

Step n. In the final design step, the actual control input u appears. We consider the overall Lyapunov function as

Vn = Vn−1 + 1

2�2n + 1

2n�Tn �n (41)

By setting i= n, the control u and adaptation functions �n described by

u = −�n−1 − cn�n − Hn − 1

2�n −

(�n−1

�y

)2

�n − �n0 tanh(�n0�n/�) + y(n)r (42)

�n = n�n(Xn)�n − (�n − �n0) (43)

We can obtain that the time derivative of Vn is

Vn � − (�min(Q) − n)‖e‖2 −n∑

k=1

ck�2k +

n∑k=1

k�Tk (�k − �k0) + (n − 1)

2�′21 + 1

2

n∑k=2

�′2k + 1

2‖P�‖2 + n�′ (44)

By Assumption 1, (44) can be rewritten as

Vn � − (�min(Q) − n)‖e‖2 −n∑

k=1

ck�2k −

n∑k=1

k�Tk �k +

2

n∑k=1

1

k‖�∗

k − �k0‖2

+ (n − 1)

2�′210 + 1

2(1 + ‖P‖2)

n∑k=2

�′2k0 + n�′ (45)


Denote

c = min{2(�min(Q) − n)/�min(P), 2ci , 2 ; i = 1, . . . , n} (46)

� =

2

n∑k=1

1

k‖�∗

k − �k0‖2 + (n − 1)

2�′201 + 1

2(2 + ‖P‖2)

n∑k=1

�′2k0 + n�′ (47)

Then (45) becomes

Vn � − cVn + � (48)

From (48), we can have

Vn(t)�Vn(t0)e−c(t−t0) + �

c(49)

If chosen a positivematrixQ such that �min(Q)−n > 0, then from (49), it can be shown for each i=1, 2, . . . , n, the signalsxi (t), xi (t), �1(t), �i and u(t) are globally uniformly ultimately bounded, and that |y(t)− yr (t)|�

√2Vn(t0)e−(c/2)(t−t0)+√

2�/c. In order to achieve the tracking error convergences to a small neighborhood around zero, the parameters ci, and Q should be chosen appropriately, then it is possible to make (2�/c)1/2 as small as desired. Denotes �> (2�/c)1/2.Since as t → ∞, e−(c/2)(t−t0) → 0, therefore, it follows that there exists T, when t�T , |y(t) − yr(t)|��.

The above design and analysis procedure is summarized in the following theorem:

Theorem 1. Suppose the bounding Assumptions 1 holds. Then the fuzzy adaptive output tracking design describedby the state observer (12), control law (42) and parameter adaptive laws (23), (31), (39) and (43) guarantee that theclosed-loop system is semi-global stability and the output tracking error converges to a small neighborhood of theorigin.

4. Simulation example

In this section, the feasibility of the proposed method and the control performances are illustrated by two examples.

Example 1. Consider the duffing forced oscillation system as

x1 = x2x2 = −0.1x2 − x31 + 12 cos t + u

y = x1 (50)

where f (x1, x2) = −0.1x2 − x31 + 12 cos t . The reference signal is defined as yr = sin(t).

Since (50) is the nonlinear system which satisfies the matching condition, we use the proposed approach and the onein Ref. [11] to control it.

Choosing fuzzy membership functions as

�F1(x1, x2) = 1

1 + exp[5(x1 + 0.6)]∗ 1

1 + exp[5(x2 + 0.6)]

�F2(x1, x2) = exp[−(x1 + 0.4)2] ∗ exp[−(x2 + 0.4)2]

�F3(x1, x2) = exp[−(x1 + 0.2)2] ∗ exp[−(x2 + 0.2)2]

�F4(x1, x2) = exp[−(x1)2] ∗ exp[−(x2)

2]

�F5(x1, x2) = exp[−(x1 − 0.2)2] ∗ exp[−(x2 − 0.2)2]

�F6(x1, x2) = exp[−(x1 − 0.4)2] ∗ exp[−(x2 − 0.4)2]


�F7(x1, x2) = 1

1 + exp[−5(x1 − 0.6)]∗ 1

1 + exp[−5(x2 − 0.6)]

Defining fuzzy basis functions as

� j (x1, x2) =�F j

(x1, x2)∑7l=1�Fl (x1, x2)

, j = 1, . . . , 7

The FLS can be expressed in the form

�T�(x1, x2) =7∑j=1

�Tj � j (x1, x2)

where

�T = [�1, �2, �3, �4, �5, �6, �7]

�(x1, x2) = [�1(x1, x2), �2(x1, x2), �3(x1, x2), �4(x1, x2), �5(x1, x2), �6(x1, x2), �7(x1, x2)]T

Case 1. Using adaptive fuzzy output control method in Ref. [11] to control (50).Choose

KTc = [144, 24], KT

0 = [1, 1], L−1(s) = 1/(s + 2), Q1 = Q2 = diag[10, 10]

Design parameters in controller and in adaptive laws are chosen as

r1 = 0.5, r2 = 0.5, = 0.005, � = 20

The initial conditions are chosen as

x1(0) = 0.2, x2(0) = 0.2, x1(0) = 1.5, x2(0) = 1.5 and �(0) = [0, 0, 0, 0, 0, 0, 0]

The simulation results are expressed by Figs. 1–4.Case 2. Using the proposed adaptive fuzzy output control method to control (50).Choose design parameters in controller and in adaptive laws are chosen as

k1 = 144, k2 = 24, = 0.2, �20 = 0.1, � = 0.01, c1 = 0.5, c2 = 0.5

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

seconds

Fig. 1. The trajectories of the tracking errors x1−yr. “Dash–dotted” is using the controller of Ref. [11], “solid line” is our approach.


0 5 10 15 20 25 30−1.5

−1

−0.5

0

0.5

1

1.5

2

seconds

Fig. 2. The trajectories of the estimation errors x1 − x1. “Dash–dotted” is using the controller of Ref. [11], “solid line” is our approach.

0 5 10 15 20 25 30−2

−1

0

1

2

3

4

5

seconds

Fig. 3. The trajectories of the estimation errors x2 − x2. “Dash–dotted” is using the controller of Ref. [11], “solid line” is our approach.

we obtain

K = [k1, k2]T = [144, 24]T, and A =

[−k1 1−k2 0

]=

[−144 1−24 0

]

For given the symmetric positive matrix Q=diag[10, 10], by solving Lyapunov equation (3), we have the followingsymmetric positive matrix P:

P =[1.7361 −10−10 60.0723

]

The initial conditions are chosen as the same as case 1. Then we obtain the simulation results, which are shown byFigs. 1–4.


0 5 10 15 20 25 30−50

0

50

100

150

200

seconds

Fig. 4. The trajectory of control input u. “Dash–dotted” is using the controller of Ref. [11], “solid line” is our approach.

By comparing the simulation results shown in Figs. 1–4, it is concluded that the proposed fuzzy control approachcannot only guarantee the boundedness of the signals x1, x1, x2, x2 and u, the tracking error �1 = x1 − yr converges toa small neighborhood of the origin, but also achieve the control performance as good as Ref. [11].

Example 2. Consider a system governed by the following form:

x1(t) = x2(t) + x1e−0.5x1

x2(t) = u(t) + x1 sin(x22 )

y(t) = x1(t) (51)

where f1(x1)=x1e−0.5x1 and f2(x1, x2)=x1 sin(x22 ) are unknown functions.Assume that state x1 and x2 are unmeasured,the given tracking reference signal is yr = 1

2 sin(t).

It is noted that system (51) is the nonlinear system which does not satisfy the matching condition, the adaptive fuzzyoutput feedback control method in Ref. [11] cannot control (51), but we can use our proposed method to deal with it.

Choosing fuzzy membership functions as

�Fl1(x1) = exp

[− (x1 − 3 + l)2

16

], l = 1, . . . , 5

�Fl2(x1, x2) = exp

[− (x1 − 3 + l)2

4

]∗ exp

[− (x2 − 3 + l)2

16

], l = 1, . . . , 5

Defining fuzzy basis functions as

�1 j (x1) =exp

[− (x1 − 3 + j)2

16

]∑5

n=1 exp

[− (x1 − 3 + n)2

16

] , j = 1, . . . , 5

�2 j (x1, x2) =exp

[− (x1 − 3 + j)2

4

]∑5

n=1 exp

[− (x1 − 3 + l)2

4

]∗ exp

[− (x2 − 3 + l)2

16

] , j = 1, . . . , 5


0 5 10 15 20 25 30 35 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

seconds

Fig. 5. The trajectories of x1 “solid line” and yr “dash–dotted”.

0 5 10 15 20 25 30 35 40−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

seconds

Fig. 6. The trajectories of x1 “solid line” and x1 “dash–dotted”.

The FLS can be expressed in the form

�T1�1(x1) =5∑j=1

�T1 j�1 j (x1) (52)

�T2�2(x1, x2) =5∑j=1

�T2 j�2 j (x1, x2) (53)

where

�T1 = [�11, �12, �13, �14, �15], �T2 = [�21, �22, �23, �24, �25]

�1(x1) = [�11(x1), �12(x1), �13(x1), �14(x1), �15(x1)]T

�2(x1, x2) = [�21(x1, x2), �22(x1, x2), �23(x1, x2), �24(x1, x2), �25(x1, x2)]T


0 5 10 15 20 25 30 35 40−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

seconds

Fig. 7. The trajectories of x2 “solid line” and x2 “dash–dotted”.

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

seconds

Fig. 8. The trajectory of u.

Design parameters in controller and in adaptive laws are chosen as

k1 = 2, k2 = 1, 1 = 2 = 0.1, = 0.2, �10 = �20 = 0.1, c1 = 5, c2 = 5

We obtain

K = [k1, k2]T = [2, 1]T and A =

[−k1 1−k2 0

]=

[−2 1−1 0

]

For the given symmetric positive matrix Q = diag[4, 4], by solving Lyapunov equation (3), we have the followingsymmetric positive matrix P:

P =[

4 −4−4 12

]


If the initial conditions are chosen as

x1(0) = 0, x2(0) = −0.2, x1(0) = 0, x2(0) = 0.3

�1(0) = [�11(0), �12(0), �13(0), �14(0), �15(0)] = [−0.6, 0, 0.5, 0, 0.6]

�2(0) = [�21(0), �22(0), �23(0), �24(0), �25(0)] = [0.4, 0, 0.1, −0.1, 0]

Then we obtain the simulation results, which are shown by Figs. 5–8. Fig. 5 shows the trajectories of state x1 and yr.Fig. 6 shows the trajectories of state x1 and its estimate x1. Fig. 7 shows the trajectories of state x2 and its estimate x2and Fig. 8 shows the trajectory of input u. From Figs. 5–8, we can conclude that the proposed fuzzy control approachcan guarantee the boundedness of the signals x1, x1, x2, x2 and u. Especially, the tracking error �1 = x1 − yr convergesto a small neighborhood of the origin.From Examples 1 and 2, we can conclude that the proposed fuzzy controller not only control SISO nonlinear systems

with unmeasured states and matching conditions, but also can deal with a class of nonlinear systems with unmeasuredstates and no matching conditions.

5. Conclusion

In this paper, an adaptive robust fuzzy control approach based on backstepping design has been proposed for SISOstrict-feedback nonlinear systems. The main contribution of this paper is that by designing a fuzzy adaptive stateobserver, the application of adaptive fuzzy backstepping control is extended to a new class of nonlinear systems withstates unmeasured. In addition, the stability of the closed-loop has been proved by using Lyapunov method, i.e., theproposed adaptive fuzzy control scheme can guarantee the closed-loop system is semi-global stability and the trackingerror of the system converges to a small neighborhood of the origin.

References

[1] A. Boulkroune, M. Tadjine, M. M Saad, M. Farza, How to design a fuzzy adaptive controller based on observers for uncertain affine nonlinearsystems, Fuzzy Sets and Systems 159 (2008) 926–948.

[2] B. Chen, X.P. Liu, Fuzzy approximate disturbance decoupling of MIMO nonlinear systems by backstepping and application to chemicalprocesses, IEEE Trans. Fuzzy Systems 13 (2005) 832–847.

[3] B. Chen, X.P. Liu, S.C. Tong, Fuzzy approximate disturbance decoupling of MIMO nonlinear systems by backstepping approach, Fuzzy Setsand Systems 158 (2007) 1097–1125.

[4] B.S. Chen, C.H. Lee, Y.C. Chang, H∞ tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach, IEEE Trans. FuzzySystems 4 (1996) 32–43.

[5] C.S. Chiu, Mixed feedforward/feedback based adaptive fuzzy control for a class of MIMO nonlinear systems, IEEE Trans. Fuzzy Systems 14(2006) 716–727.

[6] N. Golea, A. Golea, K. Barra, Observer-based adaptive control of robot manipulators: fuzzy systems approach, Appl. Comput. 8 (2008)778–787.

[7] N. Golea, A. Golea, K. Benmahammed, Stable indirect fuzzy adaptive control, Fuzzy Sets and Systems 127 (2003) 353–366.[8] F.Y. Hsu, L.C. Fu, A novel adaptive fuzzy variable structure control for a class of nonlinear uncertain systems via backstepping, Fuzzy Sets

and Systems 22 (2001) 83–106.[9] S. Jagannnathan, F.L. Lewis, Robust backstepping control of a class of nonlinear systems using fuzzy logic, Inform. Sci. 123 (2000) 223–240.[10] M. Kristic, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995.[11] Y.G. Leu, W.Y. Wang, Observer-based adaptive fuzzy–neural control for unknown nonlinear dynamical systems, IEEE Trans. Systems Man

Cybernet. 29 (1999) 583–591.[12] H.X. Li, S.C. Tong, A hybrid adaptive fuzzy control for a class of nonlinear MIMO systems, IEEE Trans. Fuzzy Systems 11 (2003) 24–34.[13] R. Ordones, K.M. Passino, Stable multi-input multi-output adaptive fuzzy/neural control, IEEE Trans. Fuzzy Systems 7 (1999) 345–353.[14] J.T. Spooner, K.M. Passino, Stable adaptive control of a class of nonlinear systems and neural network, IEEE Trans. Fuzzy Systems 4 (1996)

339–359.[15] C.Y. Sue, Y. Stepanenko, Adaptive control of a class of nonlinear systems with fuzzy logic, IEEE Trans. Fuzzy Systems 2 (1994) 285–294.[16] S.C. Tong, T.Y. Chai, Fuzzy adaptive control for a class of nonlinear systems, Fuzzy Sets and Systems 101 (1999) 31–39.[17] S.C. Tong, H.X. Li, W. Wang, Observer-based adaptive fuzzy control for SISO nonlinear systems, Fuzzy Sets and Systems 148 (2004)

355–376.[18] S.C. Tong, Y.M. Li, Direct adaptive fuzzy backstepping control for a class nonlinear systems, Internat. J. Innovative Comput. Informat. Control

3 (2007) 887–896.[19] S.C. Tong, J.T. Tang, T. Wang, Fuzzy adaptive control of multivariable nonlinear systems, Fuzzy Sets and Systems 111 (2000) 153–167.


[20] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1994.[21] M. Wang, B. Chen, Direct adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear systems, Fuzzy Sets and Systems

158 (2007) 2655–2670.[22] W.Y. Wang, Y.G. Leu, T.T. Lee, Output-feedback control of nonlinear systems using direct adaptive fuzzy–neural controller, Fuzzy Sets and

Systems 140 (2003) 341–358.[23] Y.C. Yang, Robust tracking control of nonlinear MIMO systems via fuzzy approaches, Automatica 36 (2000) 1535–1545.[24] Y.S. Yang, G. Feng, J.S. Ren, A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear

systems, IEEE Trans. Systems Man Cybernet. Part A: Systems Humans 34 (2004) 406–420.[25] Y.S. Yang, C.J. Zhou, Adaptive fuzzy H∞ stabilization for strict-feedback canonical nonlinear systems via backstepping and small-gain

approach, IEEE Trans. Fuzzy Systems 13 (2005) 104–114.[26] S.S. Zhou, G. Feng, C.B. Feng, Robust control for a class of uncertain nonlinear systems: adaptive fuzzy approach based on backstepping,

Fuzzy Sets and Systems 151 (2005) 1–20.

observer-based fuzzy adaptive control for strict-feedback nonlinear systems

Documents