disturbance attenuation of nonlinear control systems using an observer-based fuzzy feedback...

Chaos, Solitons and Fractals 33 (2007) 885–900

www.elsevier.com/locate/chaos

Disturbance attenuation of nonlinear control systems usingan observer-based fuzzy feedback linearization control

Chung-Cheng Chen a,*, Chao-Hsing Hsu b, Ying-Jen Chen c, Yen-Feng Lin a

a Department of Electrical Engineering, National Formosa University, 64, Wun-Hwa Road, Huwei, Yunlin 632, Taiwan, ROCb Department of Electronic Engineering, Chienkuo Technology University No. 1, Chieh-Shou North Road, Changhua 500, Taiwan, ROC

c Hung-Jen Girl School, 667, Chung-Hsiao Road, Chiayi 600, Taiwan, ROC

Accepted 4 January 2006

Abstract

The almost disturbance decoupling and trajectory tracking of nonlinear control systems using an observer-basedfuzzy feedback linearization control (FLC) is developed. Because not all of the state variables of the nonlinear dynamicequations are available, a nonlinear state observer is employed to estimate the state variables. The feedback lineariza-tion control guarantees the almost disturbance decoupling performance and the uniform ultimate bounded stability ofthe tracking error system. Once the tracking errors are driven to touch the global final attractor with the desired radius,the fuzzy logic control is immediately applied via human expert’s knowledge to improve the convergence rate. Oneexample, which cannot be solved by the first paper on the almost disturbance decoupling problem, is proposed in thispaper to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved byour proposed approach. In order to demonstrate the practical applicability, the study has investigated a pendulum con-trol system.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Two well-known tasks of stabilization and tracking problem are important topics in the field of control. Trackingproblem is generally more complicated than stabilization problem for nonlinear control systems. Many approaches fornonlinear systems are introduced including feedback linearization, variable structure control (sliding mode control),backstepping, regulation control, nonlinear H1 control, internal model principle and H1 adaptive fuzzy control.Recently, variable structure control has been introduced to deal with nonlinear system [38,44]. However, chatteringbehavior that may create unmodeled high-frequency due to the discontinuous switching and imperfect implementationand even drive system to instability is inevitable for variable structure control scheme. Backstepping has been a pow-erful tool for synthesizing controller for a class of nonlinear systems [4,47]. However, a disadvantage with the backstep-ping approach is the explosion of complexity which is caused by the complicated repeated differentiations of somenonlinear functions [45,34]. An output tracking approach is to utilize the scheme of the output regulation control

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.01.110

* Corresponding author.E-mail address: [email protected] (C.-C. Chen).

mailto:[email protected]

886 C.-C. Chen et al. / Chaos, Solitons and Fractals 33 (2007) 885–900

[13] in which the outputs are assumed to be excited by an exosystem. However, the nonlinear regulation problemrequires to solve the difficult solution of partial-differential algebraic equation. Another problem of the output regula-tion control is that the exosystem states need to be switched to describe changes in the output and this will create tran-sient tracking errors [29]. In general, the nonlinear H1 control has to solve the Hamilton–Jacobi equation, which is adifficult nonlinear partial-differential equation [31,2,14,43]. Only for some particular nonlinear systems we can derive aclosed-form solution [12]. The control approach based on internal model principle converts the tracking problem tononlinear output regulation problem [22]. This approach depends on solving a first-order partial-differential equationof the center manifold [13]. For some special nonlinear systems and desired trajectories, the asymptotic solutions of thisequation via ordinary differential equations have been developed [10,7]. Recently, H1 adaptive fuzzy control has beenproposed to deal with nonlinear systems systematically [5]. The drawback with H1 adaptive fuzzy control is that thecomplex parameter update law makes this approach impractical. During the past decade, significant progress has beenmade in the research of control approaches for nonlinear systems based on the feedback linearization theory[11,27,33,17]. Moreover, feedback linearization approach has been applied successfully to address many real controls.These include the control of electromagnetic suspension system [15], pendulum system [6], spacecraft [32], electrohy-draulic servosystem [1], car-pole system [3] and bank-to-turn missile system [21].

Recently, fuzzy logic control appears to be one that has attracted a great deal attention in the past two decades.Despite the success, many fundamental issues remain unanswered. Almost disturbance decoupling analysis and system-atic design are among the most issues to be further addressed. The almost disturbance decoupling problem, that is thedesign of a controller which attenuates the effect of the disturbance on the output terminal to an arbitrary degree ofaccuracy, was originally developed for linear and nonlinear control systems by Willems [41] and Marino et al. [24],respectively. Henceforth, the problem has attracted considerable attention and many significant results have been devel-oped for both linear and nonlinear control systems [40,25,30]. Marino et al. [24] shows that for nonlinear SISO systemthe almost disturbance decoupling problem may not be solvable as the following example show:

_x1ðtÞ ¼ x2 þH1ðtÞ; _x2ðtÞ ¼ x32H2ðtÞ þ u

y ¼ x1

where u, y denote the input and output, respectively, and H1, H2 are the disturbances.Fuzzy logic control has been applied not only to cement kiln, subway train but also to industrial processes. Its

designing procedure is as follows. First representing the nonlinear system as the famous Takagi–Sugeno fuzzy modeloffers an alternative to conventional model [19,46]. The control design is carried out based on an aggregation of linearcontrollers constructed for each local linear element of the fuzzy model via the parallel distributed compensationscheme [39]. For the stability analysis of fuzzy system, a lot of studies are reported (see, e.g., [36,37,20,35] and the ref-erences therein). The stability and controller design of fuzzy system can be mainly discussed by Tanaka–Sugeno’s the-orem [36]. However, it is difficult to find the common positive definite matrix P for linear matrix inequality (LMI)problem [28,9] even if P is a second-order matrix [16]. To overcome the difficulty of finding the common positive definitematrix P for fuzzy-model approach, we will propose a new method to guarantee that the closed-loop systems is stableand the almost disturbance decoupling performance is achieved. The designing structure is as follows. First, based onthe feedback linearization approach a tracking control is proposed in order to guarantee the almost disturbance decou-pling property and the uniform ultimate bounded stability of the tracking error system. Once the tracking errors aredriven to touch the global final attractor with the desired radius, the conventional fuzzy logic is control immediatelyapplied via human expert’s knowledge to improve the convergence rate. Simulations of the pendulum control systemare carried out to verify the usefulness of the proposed control. Throughout the paper, the notation k Æ k denotes theusual Euclidean norm or the corresponding induced matrix norm.

2. Controller design

In this paper, we consider the following nonlinear control system with disturbances:

_x1

_x2

..

.

_xn

266664377775 ¼

f1ðx1; x2; . . . ; xnÞf2ðx1; x2; . . . ; xnÞ

..

.

fnðx1; x2; . . . ; xnÞ

266664377775þ

g1ðx1; x2; . . . ; xnÞg2ðx1; x2; . . . ; xnÞ

..

.

gnðx1; x2; . . . ; xnÞ

266664377775uþ

Xp

j¼1

q�j hj ð1Þ

yðtÞ ¼ hðx1; x2; . . . ; xnÞ ð2Þ

C.-C. Chen et al. / Chaos, Solitons and Fractals 33 (2007) 885–900 887

i.e.,

_X ðtÞ ¼ f ðX ðtÞÞ þ gðX ðtÞÞuþ Qh ð3ÞyðtÞ ¼ H TX ðtÞ � hðX ðtÞÞ ð4ÞQ � b q�1 q�2 � � � q�p cn�p ð5Þ

where X ðtÞ ¼ ½x1ðtÞx2ðtÞ � � � xnðtÞ�T 2 Rn is the state vector, u 2 R1 is the input, y 2 R1 is the output,h � [h1(t)h2(t) � � �hp(t)]T is a bounded time-varying disturbance vector, f, g, q�1; . . . ; q�p are smooth vector fields on Rn

and hðX ðtÞÞ 2 R1 is a smooth function. The nominal system is then defined as follows:

_X ðtÞ ¼ f ðX ðtÞÞ þ gðX ðtÞÞu ð6ÞyðtÞ ¼ H TX ðtÞ ð7Þ

The nominal system (6) consists of relative degree r [8], i.e., there exists a positive integer 1 6 r <1 such that

LgLkf hðX ðtÞÞ ¼ 0; k < r � 1; ð8Þ

LgLr�1f hðX ðtÞÞ 6¼ 0 ð9Þ

for all X 2 Rn and t 2 [0,1), where the operator L is the Lie derivative [11]. The desired output trajectory yd(t) and itsfirst r derivatives are all uniformly bounded and

k½ydðtÞ; yð1Þd ðtÞ; . . . ; yðrÞd ðtÞ�k 6 Bd ð10Þ

where Bd is some positive constant.The objective of the paper is to propose a control that includes a feedback linearization controller and a nonlinear

state observer. The complete control block diagram is shown in Fig. 1.

Fig. 1. The complete control block diagram.


2.1. Nonlinear state observer

The original nonlinear control system (3) can be rewritten as follows:

_X ðtÞ ¼ f ðX Þ þ gðX Þuþ Qh ¼ F lX ðtÞ þ F nðX Þ þ gðX Þuþ Qh ð11ÞyðtÞ ¼ H TX ðtÞ � hðX ðtÞÞ ð12Þ

where F‘X and Fn(X) denote the linear part and nonlinear part of vector field f(X), respectively.Similarly, the desired nonlinear state observer can be described in the following form:

_bX ðtÞ ¼ f ðbX Þ þ gðbX Þuþ LðyðtÞ � yðtÞÞ ¼ F ‘bX ðtÞ þ F nðbX Þ þ gðbX Þuþ LðyðtÞ � yðtÞÞ ð13Þ

yðtÞ ¼ H T bX ðtÞ � hðbX ðtÞÞ ð14Þ

where L denotes the observer gain. Subtracting (13) from (11) and taking the time derivative yield

_eX ðtÞ ¼ ðF ‘ � LHTÞeX ðtÞ þ ðF nðX Þ � F nðbX ÞÞ þ Qh ð15Þ

where eX � X ðtÞ � bX ðtÞ denotes the state estimation error.It is known that if (F‘,H) are observable, the matrix F‘ � LHT will be Hurwitz for a suitable choice of L. There exists

a positive definite matrix P2 which satisfies

ðF ‘ � LHTÞTP 2 þ P 2ðF ‘ � LH TÞ ¼ �I ð16Þ

Assume that there is a constant M2 P 0 such that the Jacobian satisfies

oF nðbX ÞobX

�� 6 M2 ð17Þ

then F nðbX Þ is Lipschitz [17], i.e.,

kF nðX Þ � F nðbX Þk 6 M2kX � bX k ð18Þ

2.2. Feedback linearization controller

Under the assumption of well-defined relative degree, it has been shown [11] that the mapping

/ : Rn ! Rn ð19Þ

defined as

/iðbX ðtÞÞ � niðtÞ ¼ Li�1f hðbX ðtÞÞ; i ¼ 1; 2; . . . ; r ð20Þ

/kðbX ðtÞÞ � gkðtÞ; k ¼ r þ 1; r þ 2; . . . ; n ð21Þ

and satisfying

Lg/kðbX ðtÞÞ ¼ 0; k ¼ r þ 1; r þ 2; . . . ; n ð22Þ

is a diffeomorphism onto image. For the sake of convenience, define the trajectory error to be

eiðtÞ � niðtÞ � yði�1Þd ðtÞ; i ¼ 1; 2; . . . ; r ð23Þ

eðtÞ � ½e1ðtÞe2ðtÞ . . . erðtÞ�T 2 Rr ð24Þ

The trajectory error is multiplied with some adjustable positive constant e

�eiðtÞ � ei�1eiðtÞ; i ¼ 1; 2; . . . ; r ð25Þ�eðtÞ � �e1ðtÞ�e2ðtÞ � � � �erðtÞ

� �T 2 Rr ð26Þ

and

nðtÞ � ½n1ðtÞn2ðtÞ . . . nrðtÞ�T 2 Rr ð27ÞgðtÞ � ½grþ1ðtÞgrþ2ðtÞ . . . gnðtÞ�T 2 Rn�r ð28ÞqðnðtÞ; gðtÞÞ � ½Lf /rþ1ðtÞLf /rþ2ðtÞ . . . Lf /nðtÞ�

T � qrþ1 qrþ2 . . . qn½ �T ð29Þ


Define a phase-variable canonical matrix Ac to be

Ac �

0 1 0 � � � 0

0 0 1 � � � 0

..

. ...

0 0 0 � � � 1

�a1 �a2 �a3 � � � �ar

26666664

37777775r�r

ð30Þ

where a1,a2, . . . ,ar are any chosen parameters such that Ac is Hurwitz and the vector B to be

B � 0 0 � � � 0 1½ �Tr�1 ð31Þ

Let P1 be the positive definite solution of the following Lyapunov equation:

ATc P 1 þ P 1Ac ¼ �I ð32Þ

kmaxðP 1Þ � the maximum eigenvalue of P 1 ð33ÞkminðP 1Þ � the minimum eigenvalue of P 1 ð34Þ

Assumption 1. For all t P 0, g 2 Rn�r and n 2 Rr, there exists a positive constant M1 such that the following inequalityholds:

kq22ðt; g; �eÞ � q22ðt; g; 0Þk 6 M1ðk�ekÞ ð35Þ

where q22ðt; g; �eÞ � qðn; gÞ.

For the sake of stating precisely the investigated problem, define

d � LgLr�1f hðbX ðtÞÞ; c � Lr

f hðbX ðtÞÞ ð36Þ

Definition 1. [17] A continuous function a : [0,a)! [0,1) is said to belong to class K if it is strictly increasing anda(0) = 0.

Definition 2. [17] A continuous function b : [0,a) · [0,1)! [0,1) is said to belong to class KL if, for each fixed s, themapping b(r, s) belongs to class K with respect to r and, for each fixed r, the mapping b(r, s) is decreasing with respect tos and b(r, s)! 0 as s!1.

Definition 3. [17] Consider the system _x ¼ f ðt; x; hÞ, where f : ½0;1Þ �Rn �Rn ! Rn is a piecewise continuous in t

and locally Lipschitz in x and h. This system is said to be input-to-state stable if there exists a class KL function b,a class K function c and positive constants k1 and k2 such that for any initial state x(t0) with kx(t0)k < k1 and anybounded input h(t) with suptPt0

kh(t)k < k2, the state exists and satisfies

kxðtÞk 6 b kxðt0Þk; t � t0ð Þ þ c supt06s6t

khðsÞk� �

ð37Þ

for all t P t0 P 0. Now we formulate the almost disturbance decoupling problem as follows:

Definition 4. [25] The tracking problem with almost disturbance decoupling is said to be globally solvable by the statefeedback controller u for the transformed-error system by a global diffeomorphism (19), if the controller u enjoys thefollowing properties:

(i) It is the input-to-state stable with respect to disturbance inputs.(ii) For any initial value �xe0 :¼ ½�eðt0Þgðt0Þ�T, any t P t0 and any t0 P 0

jyðtÞ � ydðtÞj 6 b11ðkxðt0Þk; t � t0Þ þ1ffiffiffiffiffiffiffib22

p b33 supt06s6t

khðsÞk� �

ð38Þ


and
Z t
t0

½yðsÞ � ydðsÞ�2ds 6

1

b44

b55ðk�xe0kÞ þZ t

t0

b33ðkhðsÞk2Þds

� ð39Þ

where b22, b44 are some positive constants, b33, b55 are class K functions and b11 is a class KL function.

Theorem 1. Suppose that there exists a continuously differentiable function V : Rn�r ! Rþ such that the following three

inequalities hold for all g 2 Rn�r:

ðaÞ x1kgk26 V ðgÞ 6 x2kgk2

; x1;x2 > 0 ð40ÞðbÞ rtV þ ðrgV ÞTq22ðt; g; 0Þ 6 �2aX V ; aX > 0 ð41ÞðcÞ krgV k 6 -3kgk; -3 > 0 ð42Þ

then the tracking problem with almost disturbance decoupling is globally solvable by the controller defined by

u ¼ ½LgLr�1f hðbX ðtÞÞ��1 �Lr

f hðbX Þ þ yðrÞd � e�ra1½L0f hðbX Þ � yd� � e1�ra2½L1

f hðbX Þ � yð1Þd �n

� � � � � e�1ar½Lr�1f hðbX Þ � yðr�1Þ

d � þ KTxbXo ð43Þ

where Kx ¼ ½kx1 kx2 � � � kxn �Tn�1 and kxi, i = 1,2, . . . , n, are adjustable constants so as to exist a function

b2ð�Þ : Rn ! Rþ such that

kKTxbX k 6 b2k�ek ð44Þ

and the influence of disturbances on the L2 norm of the tracking error can be arbitrarily attenuated by increasing the fol-

lowing adjustable parameter NN2 > 1:

HðeÞ � H 11 H 12

H 12 H 22

�

�

2ax � w3M1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w1k1kminðP 1Þ

p 0

� w3M1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w1k1kminðP 1Þ

p 1

ekmaxðP 1Þ� 2er�1kBTP 1kb2

ekminðP 1Þ0

0 01

ekmaxðP 2Þ� 2k2kP 2k2kQk2

kminðP 2Þ� 1

4

2666666664

3777777775ð45Þ

N � kminðHÞ ð46Þ

N 2 � min x1;k1ðeÞ

2kminðP 1Þ;

k2ðeÞ2

kminðP 2Þ �

ð47Þ

where H is a positive definite matrix and kðeÞ : Rþ ! Rþ is any continuous function satisfying lime!0k(e) = 0 and

lime!0e

kðeÞ ¼ 0. Moreover, the output tracking error of system (1) is exponentially attracted into a global final attractor

Br, r ¼ffiffiffiffiffiffiN�

NN2

q, with an exponential rate of convergence 1

2�NN2

Dmaxþ N�

Dmaxr2

� ;Dmax � max x2;

k1

2kmaxðP 1Þ; k2

2kmaxðP 2Þ

� �, where

N � � 2k2M22kP 2k2

kminðP 2Þþ 1

12sup

t06s6tkhðsÞk

� �ð48Þ

Proof. Applying the co-ordinate transformation (19) to the nonlinear observer system (13) yields

_n1ðtÞ ¼o/1

obX dbXdt¼ ohðbX ðtÞÞ

obX ½f þ g � u� ¼ L1f hðbX ðtÞÞ þ LgL0

f hðbX ðtÞÞu¼ L1

f hðbX ðtÞÞ ¼ n2ðtÞ ð49Þ

..

.

_nr�1ðtÞ ¼o/r�1

obX dbXdt¼

oLr�2f hðbX ðtÞÞ

obX ½f þ g � u� ¼ Lr�1f hðbX ðtÞÞ þ LgLr�2

f hðbX ðtÞÞu¼ Lr�1

f hðbX ðtÞÞ ¼ nrðtÞ ð50Þ


_nrðtÞ ¼o/r

obX dbXdt¼

oLr�1f hðbX ðtÞÞ

obX ½f þ g � u� ¼ Lrf hðbX ðtÞÞ þ LgLr�1

f hðbX ðtÞÞu¼ Lr

f hðbX Þ þ LgLr�1f hðbX Þu ¼ cþ d � u ð51Þ

_gkðtÞ ¼o/ðbX Þ

obX dbXdt¼ o/kðbX Þ

obX ½f þ g � u�

¼ o/kðbX ÞobX f þ o/kðbX Þ

obX g � u ¼ Lf /k ; k ¼ r þ 1; r þ 2; . . . ; n ð52Þ

Since

cðnðtÞ; gðtÞÞ � Lrf hðbX ðtÞÞ ð53Þ

dðnðtÞ; gðtÞÞ � LgLr�1f hðbX ðtÞÞ ð54Þ

qkðnðtÞ; gðtÞÞ ¼ Lf /kðbX Þ; k ¼ r þ 1; r þ 2; . . . ; n ð55Þ

the dynamic equations of the nonlinear observer system (13) in the new co-ordinates are as follows:

_niðtÞ ¼ niþ1ðtÞ; i ¼ 1; 2; . . . ; r � 1 ð56Þ_nrðtÞ ¼ cðnðtÞ; gðtÞÞ þ dðnðtÞ; gðtÞÞu ð57Þ_gkðtÞ ¼ qkðnðtÞ; gðtÞÞ; k ¼ r þ 1; . . . ; n ð58ÞyðtÞ ¼ n1ðtÞ ð59Þ

Define

m � yðrÞd � e�ra1bL0f hðbX Þ � ydc � e1�ra2bL1

f hðbX Þ � yð1Þd c � � � � � e�1ar Lr�1f hðbX Þ � yðr�1Þ

d

h iþ KT

xbX ð60Þ

According to Eqs. (53), (54) and (60), the tracking controller (43) can be rewritten as

u ¼ d�1½�cþ m� ð61Þ

Substituting Eq. (61) into (57), the dynamic equations of the nonlinear observer system (13) can be shown as follows:

_n1ðtÞ_n2ðtÞ

..

.

_nr�1ðtÞ_nrðtÞ

266666666664

377777777775¼

0 1 0 � � � 0

0 0 1 0 � � � 0

..

. ...

0 0 0 � � � 1

0 0 0 � � � 0

266666664

377777775

n1ðtÞn2ðtÞ

..

.

nr�1ðtÞnrðtÞ

2666666664

3777777775þ

0

0

..

.

0

1

266666664

377777775m ð62Þ

_grþ1ðtÞ_grþ2ðtÞ

..

.

_gn�1ðtÞ_gnðtÞ

2666666664

3777777775¼

qrþ1ðtÞqrþ2ðtÞ

..

.

qn�1ðtÞqnðtÞ

2666666664

3777777775ð63Þ

y ¼ ½ 1 0 � � � 0 0 �r�1

n1ðtÞn2ðtÞ

..

.

nr�1ðtÞnrðtÞ

2666666664

3777777775r�1

¼ n1ðtÞ ð64Þ

Combining Eqs. (23), (25), (30) and (60), it can be easily verified that Eqs. (62)–(64) can be transformed into the fol-lowing forms:


_gðtÞ ¼ qðnðtÞ; gðtÞÞ � q22ðt; gðtÞ; _eÞ ð65Þ

e _�eðtÞ ¼ Ac�eþ BerKT

xbX ð66Þ

yðtÞ ¼ n1ðtÞ ð67Þ

We consider Lðg; �e; eX Þ defined by a weighted sum of V ðgÞ;W 1ð�eÞ and W 2ðeX Þ,
Lðg; �e; eX Þ � V ðgÞ þ k1ðeÞW 1ð�eÞ þ k2ðeÞW 2ðeX Þ ð68Þ
as a composite Lyapunov function of the subsystems (15), (65) and (66) [18,23], where W 1ð�eÞ and W 2ðeX Þ satisfy

W 1ð�eÞ �1

2�eTP 1

�e ð69Þ

W 2ðeX Þ � 1

2eX TP 2

eX ð70Þ

In view of (32), (35), (40), (41), (42), (43) and (44), the derivative of L along the trajectories of (15), (65) and (66) is givenby

_L ¼ rtV þ ðrgV ÞT _gh i

þ k1

2_�e� T

P 1�eþ �eTP 1ð _�eÞ

� þ k2

2ð _eX ÞTP 2

eX þ eX TP 2ð _eX Þh i¼ ½rtV þ ðrgV ÞT _g� þ k1

2

1

eAc

�eþ Ber�1ðKTxbX Þ� T

P 1�eþ �eTP 1

1

eAc

�eþ Ber�1ðKTxbX Þ� ( )

þ k2

2ðF l � LHTÞeX þ F nðX Þ � F nðbX Þ�

þ Qhh iT

P 2eX þ eX TP 2 ðF l � LHTÞeX þ F nðX Þ � F nðbX Þ�

þ Qhh i �

¼ rtV þ ðrgV ÞT _gh i

þ k1

2e�eTðAT

c P 1 þ P 1AcÞ�e� �

þ k1

2er�1 2BTP 1

�eKTxbXh i

þ k2

2eX T ðF l � LHTÞTP 2 þ P 2ðF l � LHTÞh ieXn o

þ k2

2ðF nðX Þ � F nðbX ÞÞTP 2 þ P 2ðF nðX Þ � F nðbX ÞÞ þ hTQTP 2

eX þ eX TP 2Qhn o

¼ rtV þ ðrgV ÞTq22ðt; g; �eÞh i

� k1

ekmaxðP 1ÞW 1 þ k1e

r�1ðBTP 1�eKT

xbX Þ � k2

ekmaxðP 2ÞW 2

þ k2

2ðF nðX Þ � F nðbX ÞÞTP 2 þ P 2ðF nðX Þ � F nðbX ÞÞ þ 2hTQTP 2

eXn o6 rtV þ ðrgV ÞT q22ðt; g; �eÞ � q22ðt; g; 0Þ þ q22ðt; g; 0Þ

� �h i� k1

ekmaxðP 1ÞW 1 �

k2

ekmaxðP 2ÞW 2

þ k1er�1kBTP 1kk�ekkKT

xbX k þ k2

22kF nðX Þ � F nðbX ÞkkP 2k þ 2khkkQkkP 2kkeX kn o

6 �2aX V þ x3kgkM1k�ek �k1

ekmaxðP 1ÞW 1 �

k2

ekmaxðP 2ÞW 2 þ k1e

r�1kBTP 1kk�ekb2k�ek

þ k2fM2keX kkP 2k þ khkkQkkP 2kkeX kg6 �2aX V þ x3kgkM1k�ek �

k1

ekmaxðP 1ÞW 1 �

k2

ekmaxðP 2ÞW 2

þ k1er�1kBTP 1kb2

W 1

12kminðP 1Þ

þ k2M2kP 2kkeX k þ k2kP 2kkhkkQkkeX k6 �ð2aX Þð

ffiffiffiffiVpÞ2 þ 2

x3M1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kminðP 1Þx1k1

p" # ffiffiffiffiVp ffiffiffiffiffiffiffiffiffiffiffi

k1W 1

p� 1

ekmaxðP 1Þ� 2er�1kBTP 1kb2

kminðP 1Þ

� ðffiffiffiffiffiffiffiffiffiffiffik1W 1

pÞ2

� 1

ekmaxðP 2Þ

� ffiffiffiffiffiffiffiffiffiffiffik2W 2

p� 2

þ 2k2M22kP 2k2

kminðP 2Þþ 1

4

ffiffiffiffiffiffiffiffiffiffiffik2W 2

p� 2

þ 2k2kP 2k2kQk2

kminðP 2Þffiffiffiffiffiffiffiffiffiffiffik2W 2

p� 2

þ 1

4khk2

¼ �ffiffiffiffiVp ffiffiffiffiffiffiffiffiffiffiffi

k1W 1

p ffiffiffiffiffiffiffiffiffiffiffik2W 2

p� �H

ffiffiffiffiVpffiffiffiffiffiffiffiffiffiffiffik1W 1

pffiffiffiffiffiffiffiffiffiffiffik2W 2

p

264375þ N 1 þ

1

4khk2 ð71Þ


where

N 1 �2k2M2

2kP 2k2

kminðP 2Þð72Þ

i.e.

_L 6 �kminðHÞLþ N 1 þ1

4khk2 ð73Þ

where kmin(H) denotes the minimum eigenvalue of the matrix H. Utilizing (40), (69) and (70) yields

_L 6 �kminðHÞ x1kgk2 þ k1

2kminðP 1Þk�ek2 þ k2

2kminðP 2ÞkeX k2

� þ N 1 þ

1

4khk2

6 �NN 2 kgk2 þ k�ek2 þ keX k2h i

þ N 1 þ1

4khk2 ð74Þ

Define

�e1r �

�e2

..

.

�er

26643775 ð75Þ

Hence

_L 6 �NN 2 kgk2 þ k�e1k2 þ k�e1rk2�

þ N 1 þ1

4khk2 ð76Þ

From (76), we obtain
Z t
t0

ðyðsÞ � ydðsÞÞ2ds 6

Lðt0ÞNN 2

þ 1

4NN 2

Z t

t0

ðN 1 þ khðsÞk2Þds ð77Þ

so that statement (39) is satisfied. From (74), we get

_L 6 �NN 2ðkytotalk2Þ þ N 1 þ1

4khk2 ð78Þ

where

kytotalk2 � k�ek2 þ kgk2 þ keX k2. ð79Þ

By virtue of Khalil [17, Theorem 5.2], (78) implies the input-to-state stability for the closed-loop system. Furthermore, itis easy to see that

Dmin k�ek2 þ kgk2 þ keX k2�

6 L 6 Dmax k�ek2 þ kgk2 þ keX k2�

ð80Þ

i.e.

Dmin � kytotalk2�

6 L 6 Dmax kytotalk2�

ð81Þ

where

Dmin � min x1;k1

2kminðP 1Þ;

k2

2kminðP 2Þ

�
and
Dmax � max x2;k1

2kmaxðP 1Þ;

k2

2kmaxðP 2Þ

�
From (74) and (81), we get
_L 6 � NN 2

Dmax

Lþ N 1 þ1

4sup

t06s6tkhðsÞk

� �2" #

ð82Þ


Hence,

LðtÞ 6 Lðt0ÞeNN2Dmax

ðt�t0Þ þ Dmax

NN 2

N 1 þ1

4sup

t06s6tkhðsÞk

� �2" #

; t P t0 ð83Þ

which implies

jy � ydj ¼ je1ðtÞj 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Lðt0Þ

k1kminðP 1Þ

se

NN22Dmax

ðt�t0Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Dmax

k1kminðP 1ÞNN 2

N 1 þ1

4sup

t06s6tkhðsÞk

� �2" #vuut ð84Þ

so that statement (38) is proved and then the tracking problem with almost disturbance decoupling is globally solved.Finally, we will prove that the sphere Br is a global attractor for the output tracking error of system (1). From (48) and(78), we get

_L 6 �NN 2ðkytotalk2Þ þ N � ð85Þ
For kytotalk > r, we have _L < 0. Hence any sphere defined by
Br ��e

g

" #: k�ek2 þ kgk2

6 r

( )ð86Þ

is a global final attractor for the tracking error system of the nonlinear control systems (1). Furthermore, it is easy rou-tine to see that, for y 62 B�r ,we have

_LL6�NN 2kytotalk2 þ N �

L6�NN 2kytotalk2 þ N �

Dmaxkytotalk26�NN 2

Dmax

þ N �

Dmaxkytotalk26�NN 2

Dmax

þ N �

Dmaxr2� �a� ð87Þ

i.e.,

_L 6 �a�L ð88Þ
According to the comparison theorem [26], we get
LðytotalðtÞÞ 6 Lðytotalðt0ÞÞ exp½�a�ðt � t0Þ� ð89Þ
Therefore,
Dminkytotalk26 LðytotalðtÞÞ 6 Lðytotalðt0ÞÞ exp½�a�ðt � t0Þ� 6 Dmaxkytotalðt0Þk2 exp½�a�ðt � t0Þ� ð90Þ

Consequently, we get

kytotalk 6ffiffiffiffiffiffiffiffiffiffiDmax

Dmin

rkytotalðt0Þk exp � 1

2a�ðt � t0Þ

� ð91Þ

i.e., the convergence rate toward the sphere Br is equal to a*/2. This completes our proof. h

2.3. Fuzzy logic controller

After the feedback linearization control is utilized as a guarantee of uniform ultimate bounded stability, the multipleinput/single output fuzzy control design can be technically applied via human expert’s knowledge to improve the con-

Fig. 2. Fuzzy logic controller.

Fig. 3. Membership functions for (a) e(t), (b) _eðtÞ and (c) ufuzzy.

Table 1Fuzzy control rule base

_eðtÞ e(t)

NB NM NS ZE PS PM PB

NB PB PB PB PB PM PS ZENM PB PB PB PM PS ZE NSNS PB PB PM PS ZE NS NMZE PB PM PS ZE NS NM NBPS PM PS ZE NS NM NB NBPM PS ZE NS NM NB NB NBPB ZE NS NM NB NB NB NB


vergence rate of tracking error. The block diagram of the fuzzy control is shown in Fig. 2. In general, the tracking errore(t) and its time derivative h are utilized as the input fuzzy variables of the IF-THEN control rules and the output is thecontrol variable ufuzzy. For the sake of easy computation, the membership functions of the linguistic terms for e(t), _eðtÞand ufuzzy are all chosen to be the triangular shape function. We define seven linguistic terms: PB (positive big), PM(positive medium), PS (positive small), ZE (zero), NS (negative small), NM (negative medium) and NB (negativebig), for each fuzzy variable as shown in Fig. 3.

Fuzzy controllable table for ufuzzy is shown in Table 1. The rule base is heuristically built by the standard Macvicar–Whelan rule base [42] for usual servo control systems. The Mamdani method is used for fuzzy inference. The defuzz-ification of the output set membership value is obtained by the centroid method. Therefore, we can combine the designsof feedback linearization control and fuzzy control to construct the overall controller as follows:

ufeþfu � ufeedbackusðtÞ þ ufuzzyusðt � t1Þ

¼ ½LgLr�1f hðbX ðtÞÞ��1 �Lr

f hðbX Þ þ yðrÞd � e�ra1½L0f hðbX Þ � yd�

n�e1�ra2½L1

f hðbX Þ � yð1Þd � � � � � � e�1ar½Lr�1f hðbX Þ � yðr�1Þ

d � þ KTxbXousðtÞ þ ufuzzyusðt � t1Þ ð92Þ

where us(t) denotes the unit step function and t1 is the time that the tracking error of system touches the global finalattractor Br.

3. Illustrative example

Consider the simple pendulum system with friction subjected to a control moment u (per unit mass) and a distur-bance shown in Fig. 4. From Khalil [17], the equations of motion are

_x1ðtÞ_x2ðtÞ

� ¼

x2

� g‘

sin x1ðtÞ � km x2

" #þ

01

m‘2

" #T þ

0

�0:1 sinðtÞ cos x1ðtÞl þ sinðt � 12Þ

" #ð93Þ

yðtÞ ¼ x1ðtÞ þ x2ðtÞ :¼ hðX ðtÞÞ ð94Þ

where m denotes the mass of the bob, h1 denotes the angle subtended by the rod, g denotes the gravity acceleration, k

denotes the coefficient of friction, T denotes the supplied torque, x1 � h1; x2 � _h1 and u = T. The following physical

Fig. 4. The simple pendulum system with friction.

Fig. 5. The output trajectory of uncontrolled pendulum system with friction.


parameters are chosen in our simulation: g‘¼ 1; 1

m‘2 ¼ 0:01 and km ¼ 0:5. Hence the mathematical model can be rewritten

as

_x1ðtÞ_x2ðtÞ

� ¼

x2

� sin x1 � 0:5x2

� þ

0

0:01

� uþ

0

�ð0:1 sin tÞ cos x1ðtÞ þ sinðt � 12Þ

� ð95Þ

yðtÞ ¼ x1 þ x2ðtÞ � hðX ðtÞÞ ð96Þ

The output trajectory of uncontrolled pendulum system with disturbance is shown in Fig. 5. Now we will show how toexplicitly construct a controller that tracks the desired signal yd = 0 and attenuates the disturbance’s effect on the out-put terminal to an arbitrary degree of accuracy. Let us arbitrarily choose a1 = 0.007 such that Ac = �0.007 andP1 = 71.43. The original system (95) is a system of relative degree one. Its co-ordinate transformation can be chosen as

/ðbX ðtÞÞ ¼ ½n1ðtÞn2ðtÞ�T ¼ ½x1 þ x20:1x1�T ð97Þ

From (92), we obtain the desired tracking controller

ufeþfu ¼ 100f�2:7x1 þ sin x1 � 3:2x2gusðtÞ þ ufuzzyusðt � t1Þ ð98Þ

It can be verified that the relative conditions of Theorem 1 are satisfied with e = 0.0025, Bd = 0, b2 = 0.1, HT = [1 1],L = [1 1], x3 = 2, V ¼ g2

2, Kx ¼ ½0:1 0:1�T, M1 = 0.1, aX ¼ 1, x1 = x2 = 1, k2 ¼ 100ffiffiep

, k1 ¼ffiffiep

, N2 = 0.7, N = 2 and

P 2 ¼0:633 �0:133

�0:133 0:333

�

Fig. 6. The tracking error driven by ufe+fu for system (95).

Fig. 7. The tracking error driven by ufeedback for system (95).


Hence the tracking controller (98) will steer the output tracking error of the closed-loop system, starting from any initialvalue, to be attenuated to zero by virtue of Theorem 1. The tracking errors driven by ufe+fu and ufeedback or (95) aredepicted in Figs. 6 and 7, respectively. It is easy to see that the convergence rate of pendulum system driven by bothufeedback and ufuzzy, i.e., ufe+fu, is better than only by ufeedback.

4. Comparative example

Marino et al. [24] exploits the fact that for nonlinear single-input single-output systems with the disturbancesH1(t) = H2(t) = 0.5sin t the almost disturbance decoupling problem can not be solved, as the following example shows:

_x1ðtÞ_x2ðtÞ

� ¼

x2

0

� þ

0

1

� uþ

H1ðtÞx3

2H2ðtÞ

� ; ð99Þ

yðtÞ ¼ x1ðtÞ ð100Þ

Fig. 8. The tracking error driven by ufe+fu for system (99).

Fig. 9. The tracking error driven by ufeedback for system (99).


where u, y denote the input and output, respectively. The feedback control algorithm proposed in this paper will solve itperfectly. Applying the same design procedures of Theorem 1 yields the desired tracking and almost disturbance decou-pling controller as follows:

ufeþfu ¼ f� sin t � ð0:03Þ�2ðx1 � sin tÞ � ð0:03Þ�1ðx2 � cos tÞgusðtÞ þ ufuzzyusðt � t1Þ ð101Þ

The tracking controller will drive the tracking error to zero. The tracking errors driven by ufe+fu, and ufeedback for (99)are depicted in Figs. 8 and 9, respectively. It is obvious to see that the almost disturbance decoupling performance isachieved and the convergence rate of system driven by ufe+fu is better than only by ufeedback.

5. Conclusion

In this paper, we have constructed a fuzzy feedback linearization control algorithm which globally solves the track-ing problem with almost disturbance decoupling for some class of nonlinear control systems. A feedback linearizationcontroller is designed by a coordinate transformation with estimated states. One comparative example is proposed toshow the significant contribution of this paper with respect to some existing approaches. Moreover, a practical exampleof pendulum system with friction demonstrates the applicability of the proposed feedback linearization approach and


the fuzzy logic approach. Based on the simulation results, the proposed control scheme offers an effective trajectorytracking and almost disturbance decoupling performances of the controlled systems.

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