Adaptive Fuzzy Robust Tracking Controller Design via Small Gain Approach and Its Application

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 783
Adaptive Fuzzy Robust Tracking Controller Design
via Small Gain Approach and Its Application
Yansheng Yang and Junsheng Ren
Abstract—A novel adaptive fuzzy robust tracking control
(AFRTC) algorithm is proposed for a class of nonlinear systems
with the uncertain system function and uncertain gain function,
which are all the unstructured (or nonrepeatable) state-dependent
unknown nonlinear functions arising from modeling errors
and external disturbances. The Takagi–Sugeno type fuzzy logic
systems are used to approximate unknown uncertain functions
and the AFRTC algorithm is designed by use of the input-to-state
stability approach and small gain theorem. The algorithm is
highlighted by three advantages: 1) the uniform ultimate bound-
edness of the closed-loop adaptive systems in the presence of
nonrepeatable uncertainties can be guaranteed; 2) the possible
controller singularity problem in some of the existing adaptive
control schemes met with feedback linearization techniques
can be removed; and 3) the adaptive mechanism with minimal
learning parameterizations can be obtained. The performance
and limitations of the proposed method are discussed. The uses
of the AFRTC for the tracking control design of a pole-balancing
robot system and a ship autopilot system to maintain the ship on a
predetermined heading are demonstrated through two numerical
examples. Simulation results show the effectiveness of the control
scheme.
Index Terms—Adaptive robust tracking, fuzzy control, input-to-
state stability (ISS), nonlinear systems, small gain theorem.
I. INTRODUCTION
I
N RECENT years, interest in designing robust tracking

control for uncertain nonlinear systems has been ever
increasing, and many significant research attentions have been
attracted. Most results addressing this problem are available
in the control literature, e.g., Kokotovic and Arcak [1] and
references therein. And many powerful methodologies for
designing tracking controllers are proposed for uncertain
nonlinear systems. The uncertain nonlinear systems may
be subjected to the following two types of uncertainties:
structured uncertainties (repeatable unknown nonlinearities),
which are linearly parameterized and referred to as parametric
uncertainties, and unstructured uncertainties (nonrepeatable
unknown nonlinearities), which are arising from modeling
errors and external disturbances. To handle the parametric
uncertainties, adaptive control method, which has undergone
rapid developments in the past decade, e.g., [2]–[7] can be used.
Manuscript received June 28, 2001; revised July 9, 2002 and January 15,
2003. This work was supported in part by the Research Fund for the Doctoral
Program of Higher Education under Grant 20020151005, the Science Founda-
tion under Grant 95-06-02-22, and the Young Investigator Foundation under
Grant 95-05-05-31 of the National Ministry of Communications of China.
The authors are with the Navigation College, Dalian Maritime University
(DMU), Dalian 116026, China (e-mail: ).
Digital Object Identifier 10.1109/TFUZZ.2003.819837
As for unstructured uncertainties, if there is a prior knowledge
of the bounded functions, deterministic robust control method,
e.g., [8]–[12] can be used. Unfortunately, in industrial control
environment, there are some controlled systems with the
unstructured uncertainties where none of prior knowledge of
the bounded functions is available, then the adaptive control
method and the deterministic robust control method can not

be used to design controller for those systems. A solution to
that problem is presented that the neural networks (NNs) are
used to approximate the continuous unstructured uncertain
functions in the systems and Lyapunov’s stability theory is
applied in designing adaptive NN controller. Several stable
adaptive NN control approaches are developed by [13]–[19]
which guarantee uniform ultimate boundedness in the presence
of both unstructured uncertainties and unknown nonlinearities.
As an alternative to NN control approaches, the intensive
research has been carried out on fuzzy control for uncertain
nonlinear systems. The fuzzy systems are used to uniformly
approximatetheunstructureduncertainfunctionsinthedesigned
system by use of the universal approximation properties of
the certain classes of fuzzy systems, which are proposed by
[20] and [21], and a Lyapunov based learning law is used, and
several stable adaptive fuzzy controllers that ensure the stability
of the overall system are developed by [22]–[26]. Recently, an
adaptive fuzzy-based controller combined with VSS and
control technique has been studied in [27] and [28]. However,
there is a substantial restriction in the aforementioned works:
A lot of parameters are needed to be tuned in the learning
laws when there are many state variables in the designed
system and many rule bases have to be used in the fuzzy
system for approximating the nonlinear uncertain functions,
so that the learning times tend to become unacceptably large
for the systems of higher order and time-consuming process is
unavoidable when the fuzzy logic controllers are implemented.
This problem has been pointed out in [26].
In this paper, we will present a novel approach for that
problem. A new systematic procedure is developed for the

synthesis of stable adaptive fuzzy robust controller for a class
of continuous uncertain systems, and Takagi–Sugeno (T–S)
type fuzzy logic systems [29] are used to approximate the un-
known unstructured uncertain functions in the systems and the
adaptive mechanism with minimal learning parameterizations
can be achieved by use of input-to-state stability (ISS) theory
first proposed by Sontag [31] and small gain approach given
in [32]. The outstanding features of the algorithm proposed
in the paper are: i) that only one function is needed to be
approximated by T–S fuzzy systems and no matter how many
states in the designed system are investigated and how many
1063-6706/03$17.00 © 2003 IEEE
784 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003
rules in the fuzzy system are used, only two parameters needed
to be adapted on-line, such that the burdensome computation of
the algorithm can be lightened increasingly and it is convenient
to realize this algorithm in engineering; and 2) the possible
controller singularity problem in some of the existing adaptive
control schemes met with feedback linearization techniques
can be avoided.
This paper is organized as follows. In Section II, we will
give the problem formulation, the description of a class of
nonlinear systems and tracking control problem of nonlinear
systems. Section III contains some needed definitions of ISS,
small gain theorem and preliminary results. In Section IV, a
systematic procedure for the synthesis of adaptive fuzzy robust
tracking controller (AFRTC) is developed. In Section V, two
application examples for designing the tracking control for
the pole-balancing robot system and the ship autopilot system
by use of the AFRTC are included and numerical simulation

results are presented. The final section contains conclusions.
II. P
ROBLEM FORMULATION
A. System Description
Consider the
th-order uncertainnonlinear systemsof thefol-
lowing form:
(1)
where
and represent the control input and the
output of thesystem, respectively.
is comprised of the states which are assumed to be available, the
integer
denotes the dimension of the system. and
are unknown smooth uncertain functions and may contain non-
repeatable nonlinearities.
is the disturbance, unknown
but bounded, e.g.,
, where is an unknown
constant.
Throughout this paper, the following assumption is made on
(1).
Assumption 1: The sign of
is known, and there exists a
constant
such that , .
This assumption implies that smooth function
is strictly
either positive or negative. From now onwards, without loss of
generality, we shall assume

, . As-
sumption 1 is reasonable because
being away from zero
is the controllable conditions of system (1). It should be em-
phasized that the low bound
is only required for analytical
purposes, its true value is not necessarily known. Some stability
is needed to proceed.
Definition 1: It is said that the solution of (1) is uniformly
ultimately bounded (UUB) if for any
, a compact subset of
, and all , there exists an and a
number
such that for all .
We represent
as any suitable vector norm. In this paper,
vector
norm is Euclidean, i.e., and given a
matrix
, matrix norm is defined by
where denotes the operation of taking the max-
imum (minimum) eigenvalue. The norm
denoted by
throughout this paper unless specified explicitly, is nothing but
the vector two-norm over the space defined by stacking the ma-
trix columns into a vector, so that it is compatible withthe vector
two-norm, i.e.,
.
The primary goal of this paper is to track a given reference
signal

while keeping the states and control bounded. That
is, the output tracking error
should be small.
The given reference signal
is assumed to be bounded and
has bounded derivatives up to the
th order for all , and
is piecewise continuous.
Let
such that is bounded. Sup-
pose
. The (1) can be transformed into
(2)
In thispaper, we present a methodfor the adaptive robust con-
trol design for system (2) in the present of unstructured uncer-
tainties. Our design objective is to find an AFRTC
of the
form
(3)
(4)
where
is theknown fuzzy base functions.In sucha way that
all the solutions of the closed-loop system (2)–(4) are uniformly
ultimately bounded. Furthermore, the output tracking error of
the system can be steered to a small neighborhood of origin.
B. T–S Fuzzy Systems
In this section, we briefly describe the structure of fuzzy sys-
tems. Let
denote the real numbers, the real -vectors,
the real matrices. Let be a compact simply

connected set in
. With map , define to
be the function space such that
is continuous. A fuzzy system
can be employed to approximate the function
in order to
design the adaptive fuzzy robust control law, thus the configu-
ration of T–S type fuzzy logic system called T–S fuzzy system
for short [29] and approximation theorem are discussed first as
follows.
Consider a T–S fuzzy system to uniformly approximate a
continuous multidimensional function
that has a com-
plicated formulation,where
is inputvector with independent
. The domain of is . It fol-
lows that the domain of
is
In order to construct a fuzzy system, the interval [ ]is
divided into
subintervals
On each interval , continuous
input fuzzy sets, denoted by
, are defined
to fuzzify
. The membership function of is denoted by
, which can be represented by triangular, trapezoid, gen-
eralized bell or Gaussian type and so on.
YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 785
Generally, T–S fuzzy system can be constructed by the fol-

lowing
fuzzy rules:
where , , are the unknown
constants. The product fuzzy inference is employed to evaluate
the
ANDs in the fuzzy rules. After being defuzzified by a typical
center average defuzzifier, the output of the fuzzy system is
(5)
where
and
, which is called a
fuzzy base function. When the membership function
in
is denoted by some type of membership function, is
a known continuous function. So, restructuring (5) as follows:
Let ,
,
.
.
.
.
.
.
.
.
.
.
.
.
, then

the (5) can be easily rewritten as
(6)
where
, and
.
.
.
.
.
.
.
.
.
.
.
.
.
When the fuzzy systemis used to approximate thecontinuous
function, two questions of interest may be asked: whether there
exists a fuzzy system to approximate any nonlinear function to
an arbitrary accuracy? how to determine the parameters in the
fuzzy system if such a fuzzy system does exist. The following
lemma [30] gives a positive answer to the first question.
Lemma 1: Suppose that the input universal of discourse
is
a compactset in
.Then, foranygivenreal continuousfunction
on and , there exists a fuzzy system in the
form of expression (6) such that
(7)

III. M
ATHEMATICAL PRELIMINARIES
The concept of ISS and ISS-Lyapunov function due to
Stontag [31], [33] and Sontag and Wang [34] have recently
been used in various control problems such as nonlinear
stabilization, robust control and observer designs (see, e.g.,
[35]–[40]). In order to ease the discussion of the design of
AFRTC scheme, the variants of those notions are reviewed
in the following. First, we begin with the definitions of class
, and functions which are standard in the stability
literature; see [41].
Definition 2:
• A function
is said of class if it is
continuous, strictly increasing and
. It is of class
if it is of class and is unbounded.
• A function
is said of
class
if, for each , is of class , and,
for each
, is strictly decreasing and satisfies
, and is a class function if and
only if there exist two class
functions and such
that
We consider the following system:
(8)
where

is the state and is the input. For this system, we give
the definition of input-to-state stable in the following.
Definition 3: For (8), it is said to be input-to state practically
stable (ISpS) if there exist a function
of class , called the
nonlinear
gain, and a function of class such that, for
any initial condition
, each measurable essentially bounded
control
defined for all and a nonnegative constant ,
the associated solutions
are defined on [0, ) and satisfy
(9)
where
is the truncated function of at and stands
for the
supremum norm.
When
in (9), the ISpS property collapses to the ISS
property introduced in [33].
Definition 4: A
function is said to be an ISpS-Lya-
punov function for (8) if
• there exist functions
, of class such that
(10)
• there exist functions
, of class and a constant
such that

(11)
When (11) holds with
, is referred to as an ISS-
Lyapunov function.
Then it holds that one may pick a nonlinear
gain in (9)
of the form, which is given in [35]
(12)
For the purpose of application studied in this paper, we intro-
duce the sequel notion of exp-ISpS Lyapunov function.
Definition 5: A
function is said to be an exp-ISpS Lya-
punov function for system (8) if
• there exist functions
, of class such that
(13)
786 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003
Fig. 1. Feedback connection of composite systems.
• there exist two constants , and a class
function such that
(14)
When (14) holds with
, the function is referred to as
an exp-ISS Lyapunov function.
The three previous definitions are equivalent from [34] and
[39]. Namely, the following.
Proposition 1: For any control system (8), the following
properties are equivalent:
i) it is ISpS;
ii) it has an ISpS-Lyapunov function;

iii) it has an exp-ISpS Lyapunov function.
Consider the stability of the closed-loop interconnection of
two systems shown in Fig. 1.
A trivial refinement of the proof of the generalized small
gain theorem given in [32] and [40] yields the following variant
which is suited for our applications here.
Theorem 1: Consider a system in composite feedback form
(cf. Fig. 1)
(15)
(16)
of two ISpS systems. In particular, there exist two constants
, , and let and of class , and and
of class be such that, for each in the supremum
norm, each
in the supremum norm, each and
each
, all the solutions and are de-
fined on [0,
) and satisfy, for almost all
(17)
(18)
Under these conditions
(19)
the solution of the composite systems (15) and (16) is ISpS.
IV. D
ESIGN OF ADAPTIVE FUZZY ROBUST TRACKING CONTROL
Using the pole-placement approach, we consider a term
where , the ’s are chosen such that all
roots of polynomial
lie in the

left-half complex plane, leads to the exponentially stable dy-
namics. Then, the (2) can be transformed into
(20)
where
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Because
is stable, a positive–definite solution of
the Lyapunov equation
(21)
always exists and
is specified by the designer.
For this control problem, if both functions
and in
(20) are available for feedback, the technique of the feedback
linearization can be used to design a well-defined controller,

which is usually given in the form of
for some auxiliary control input with
being nonzero for all time, such that the resulting closed-loop
system can be shown to achieve a satisfactory tracking per-
formance. However, in many practical control systems, plant
uncertainties that contain structured (or parametric) uncertain-
ties and unstructured uncertainties (or nonrepeatable uncer-
tainties) are inevitable. Hence, both
and may not
be available directly in the robust control design. Obtaining a
simple control algorithm as before is impossible. Moreover,
if any adaptation scheme is implemented to estimate
and as and respectively, the simple control
algorithm aforementioned can be also used for substituting
and for and , so the extra precaution
is required to guarantee that
for all time. At the
present stage, no effective method is available in the litera-
ture. In this paper, we develop a semi-globally stable adaptive
fuzzy robust controller which does not require to estimate
the unknown function
, and therefore avoids the possible
controller singularity problem.
In this paper, the effects due to plant uncertainties and
external disturbances will be considered simultaneously. The
philosophy of our tracking controller design is expected that
T–S fuzzy approximators equipped with adaptive algorithms
are introduced first to learn the behaviors of uncertain dy-
namics. Here, only uncertain function
is needed to be

considered.
For
is an unknown continuous function, by Lemma 1,
T–S fuzzy system
with input vector for some
compact set
is proposed here to approximate the un-
certain term
where is a matrix containing the approxi-
mating parameters. Then,
can be expressed as
(22)
where
is a parameter with respect to approximating accuracy.
YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 787
Substituting (22) into (20), we get
(23)
Let
, such that and
. It follows that (23) reduces to
(24)
In order to design the adaptive fuzzy robust controller easily
by use of the small gain theorem, the following output equation
can be obtained by comparing (24) with (15):
Then, the feedback equation is given as follows:
So, (24) can be rewritten as (15) and (16)
(25)
(26)
Then, the feedback connection using the (25) and (26) can be
implemented using the block diagram shown in Fig. 2.

From Fig. 2, we observe that the system
should be made
to satisfy ISpS condition of the system through designing the
controller
. In (25), is an unknown, and there exist
some parameters with boundedness. According to these prop-
erties, an adaptive fuzzy robust tracking control algorithm will
be proposed, which not only gives the controller
to
make the system
meet ISpS condition but also the online
adaptive law for
and the other parameters in the (25). For this
purpose, we will discuss it in the following.
Construct an adaptive fuzzy robust tracking controller as
follows:
(27)
where
denotes a certainty equivalent controller and de-
notes a supervisory controller for the disturbance, approxima-
tion error and other bounded items. Those will be given in the
following.
Substituting (27) into (25) yields
(28)
Based on the aforementioned condition, we can get
(29)
where
, ,
, ,
, and . denotes

the largest term with unknown constant in all boundedness. In
order to design the controller, we can get
.
Fig. 2. Feedback connection of fuzzy system.
Let and be the parameter estimate of and
, respectively. We propose an adaptive fuzzy robust tracking
controller (AFRTC) as follows:
(30)
where
will be specified by designer, and is the gain
of
to be chosen later on.
The adaptive laws for
and are now chosen as
(31)
where
, , 2 are the updating rates chosen by
designer, and
, ,2, and are design con-
stants. Adaptive laws (31) incorporate leakage term based on
a variant of the
-modification proposed by Polycarpou and
Ioannou [42], which can prevent parameter drift of the system.
Theorem 2: Consider the system (20), suppose that As-
sumption 1 is satisfied and the
can be approximated by
T–S fuzzy system. If we pick
and in
(21), then the control scheme (30) with adaptive laws (31) is
an AFRTC which can make all the solutions (

)of
the derived closed loop system uniformly ultimately bounded.
Furthermore, given any
and bounds on and ,we
can tune our controller parameters such that the output error
satisfies .
Before proving Theorem 2,the following lemma given in [42]
is reviewed first.
Lemma 2: The following inequality holds for any
and
any
:
(32)
where
is a constant that satisfies , i.e.,
.
The proof of Theorem 2 can be divided into twofold. First, let
the constant
and set as the input of the system
, to prove the satisfaction of ISpS for the system by
788 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003
use of the adaptive fuzzy robust tracking controller, and then to
prove uniform ultimate boundedness of the composite of two
systems with the feedback system
by use of small
gain theorem.
Choose the Lyapunov function as
(33)
where
, , .

The time derivative of
along the error trajectory (28) is
(34)
We deal with relative items in (34), substitute (30) into the
relative items shown before, and obtain
(35)
(36)
and substituting (30) into the relative items of (34), we get
Substituting (29) into the aforementioned equation yields
Let , by use of Lemma 2, the previous equa-
tion can be rewritten as
(37)
Substituting (35)–(37) into (34), such that
(38)
Substituting (31) into (38), we get
(39)
where
. If we pick
, we get
By Definition 4, we propose the adaptive fuzzy robust
tracking controller such that the requirement of ISpS for
system
can be satisfied with the functions and
of class . By Definition 3 and the (12), we can
get a gain function
of system as follows:
where .
For system
, it is a static system such that we have
(40)

Then, the gain function
for system is .
According to the requirement
of small gain
Theorem 1, we can get
YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 789
Owing to , the condition of the small gain
theorem 1 can be satisfied by choosing
, so that it can be
proven that the composite closed-loop system is ISpS. There-
fore, a direct application of Definition 3 yields that the com-
posite closed-loop system has bounded solutions over [0,
).
More precisely, there exist a class
-function and a posi-
tive constant
such that
This, in turn, implies that the tracking error is bounded
over [0,
). By Proposition 1, there exists an ISpS-Lyapunov
function for the composite closed-loop system. By substituting
(40) into (39), the ISpS-Lyapunov function is satisfied as
follows:
(41)
where
. From
(41), we get
It results that the solutions of composite closed-loop system
are uniformly ultimately bounded, and implies that, for any
, there exists a constant such that

for all . The last statement of Theorem
2 follows readily since
can be made arbitrarily
small if the design parameters
, , , , are chosen
appropriately.
Remark 1: It is interesting to note that most of the available
adaptive fuzzy controllers inthe literature arebased on feedback
linearization techniques, whose structures are usually taken the
form
with and be the estimates
of
and , respectively, and be a new control variable.
To avoid singularity problem when
, several modified
adaptive methods were provided by [44], [25], and [28]. In this
paper, the adaptive fuzzy robust tracking controller developed
before has the following properties:
where and .According to those properties, it is easy
to show that we does not require to estimate the unknown gain
function
. In such a way we can not only reduce the number
of parameters needed to be adapted on-line for
and but
also avoid the possible controller singularity problem usually
met with feedback linearization design when the adaptive fuzzy
control is executed.
Remark 2: Since the function approximation property
of fuzzy systems is only guaranteed within a compact set,
the stability result proposed in this paper is semiglobal in

the sense that, for any compact set, there exists a controller
with sufficiently large number of fuzzy rules such that all
the closed-loop signals are bounded when the initial states
are within this compact set. In practical applications, the
number of fuzzy rules usually can not be chosen too large due
to the possible computation problem. This implies that the
fuzzy system approximation capability is limited, that is, the
approximating accuracy
in (22) for the estimated function
will be greater when chosen small number of fuzzy rules.
However, we can choose appropriately the design parameters
, , , , to improve both stability and performance of
the closed-loop systems.
V. A
PPLICATION EXAMPLES
Now, we will reveal the control performance of the proposed
AFRTC via application examples. Two examples on designing
tracking controller for pole-balancing robot system and ship au-
topilot system are given in this section. The former has an un-
known input gain function
and the latter unknown input
gain constant
. We shall find the adaptive fuzzy robust tracking
controllers by following the design procedures given in the pre-
vious section. Simulation results will be presented.
A. Pole-Balancing Robot System
To demonstrate the effectiveness of the proposed algorithms,
a pole-balancing robot is used for simulation. The Fig. 3 shows
the plant composed of a pole and a cart. The cart moves on the
rail tracksin horizontal direction. Thecontrol objective isto bal-

ance the pole starting froman arbitrary condition by supplying a
suitable force to the cart. The same case studied has been given
in [43]. The dynamic equations are described by
(42)
where
is the angular position from the equilibrium position
and
. Suppose that the trajectory planning problem
for a weight-lifting operation is considered and this pole-bal-
ancing robot system suffers from uncertainties and exogenous
disturbances. The desired angle trajectory is assumed here
to be
. Here, denotes the mass of the
pendulum,
is the mass of the vehicle, is the length of the
pendulum and
is the applied force. Here, we use the parame-
ters for simulations
, , .
790 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003
Fig. 3. Pole-balancing robot system.
Define five fuzzy sets for each , with labels (NL),
(NM), (ZE), (PM), (PL) which are character-
ized by the following membership functions:
(43)
where
.
Twenty-five fuzzy rules for the fuzzy system are included in
the fuzzy rule bases. Hence, the function
is approximated

by T–S fuzzy system as follows:
(44)
where
.
.
.
,
.
.
.
.
.
.
, can be defined
as (6).
We select
and , then the solution
of Lyapunov expression (21) is obtained by
If picking in (30), we can obtain the adaptive fuzzy
robust tracking controller for pole-balancing robot system as
follows:
(45)
(b)
Fig. 4. Simulation results for proposed AFRTC algorithm in this paper.
(a) Position of pole-balancing robot system (Solid line: actual position, Dashed
line: reference position). (b) Control force.
Fig. 5. Simulation results for the adaptive parameters when employing
AFRTC algorithm. (a) Adaptive parameter
. (b) Adaptive parameter .
where

For the convenience of simulation, choose the initial condi-
tion
, , , . The simula-
tion results are shown in Figs. 4 and 5.
YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 791
Before presenting the outstanding advantages of AFRTC
developed in this paper, we will briefly review the control
law proposed in [44] as follows:
(46)
where
, , ,
and . Then, we solve the Lya-
punov equation and obtain
where if (which is a constant specified by the
designer),
if , and .
Define five fuzzy sets the same as those in (43) for each
,
, twenty-five fuzzy rules for the fuzzy systems and
, respectively, and singleton fuzzifier, the product infer-
ence and the center-average defuzzification are used. Hence, the
functions
and can be approximated by the fuzzy sys-
tems
and where
with components
and
and the construction of is similar to .
(b)
Fig. 6. Simulation results for Control algorithm in (46). (a) Position of

pole-balancing robot system (Solid line: actual position, Dashed line: reference
position). (b) Control force.
In Wang [44], use the following adaptive law to adjust pa-
rameter vector
; see (47) at the bottom of the page, where the
projection operation
is defined as
In [44], use the following adaptive law to adjust parameter
vector
:
if
if
(48)
Here, forthe parameters
, , and , pleaserefer to Wang
[44]. The simulation results are shown in Fig. 6.
Fig. 7 shows the simulation results of tracking errors by use
of the proposed AFRTC and the controller given in (46), respec-
tively. Fromthe results,we cansee thatthe controlperformances
are almost the same. Hence, we can state that the AFRTC satis-
fies the following advantages that have been described in Sec-
tion IV: only one function
is needed to be approximated
by T–S fuzzy systems and no matter how many states in the
system are investigated and how many rules in the fuzzy system
are used, only two parameters are needed to be adapted on-line
in AFRTC. However, for the traditional methodology (e.g., the
control law proposed in [44]), even based on five fuzzy sets for
each state variable and singleton fuzzy model aforementioned,
there are 50 parameters needed to be adapted online for the

fuzzy system
and when the fuzzy logic con-
troller is implemented. And the traditional methodology can
cause the increase of the number of parameters needed to be
if( )or( and )
if( and )
(47)
792 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003
(b)
Fig. 7. Simulation results of the tracking errors. (a) Proposed AFRTC
algorithm. (b) Control algorithm in (46).
adapted exponentially with that of number of state variables
or fuzzy sets. The computational complexity can be lessened
dramatically and the learning time can be reduced vastly when
using AFRTC developed in this paper. Then AFRTC has the
potential to provide uncomplicated, reliable and easy-to-under-
stand solutions fora large variety of nonlinearcontrol tasks even
for higher order systems.
B. Ship Autopilot System
Many of thepresent generation of autopilots installedin ships
are designed for the course keeping. They aim at maintaining
the ship on a predetermined course and thus require directional
information. Developments in the last 20 y include variants of
the analogue proportional-integral-derivative (PID) controller.
In the recent years, some sophisticated autopilots are proposed
based on advanced control engineering concepts whereby the
gain settings for the proportional, derivative and integral terms
of heading are adjusted automatically to suit the dynamics of
the ship and environmental conditions such as model reference
adaptive control [46], self-tuning [47], optimal [48],

theo-
ries [49] and adaptive robust fuzzy control [50].
In this paper, the adaptive fuzzy robust tracking controller
proposed above will be used for designing ship autopilot. Be-
fore considering the designs of the autopilots, it is of interest to
describe the dynamics of the ship. The mathematical model re-
lating the rudder angle
to the heading of the ship is found to
be of the form
(49)
where
( ) and (s) are parame-
ters which are function of ship’s constant forward velocity and
its length.
is a nonlinear function of .The function
can be found from the relationship between and in steady
TABLE I
F
UZZY IF–THEN RULES
state such that . An experiment known as the
“spiral test” has shown that
can be approximated by
(50)
where
and are real valued constants.
In normal steering, a ship often makes only small deviations
from its desired direction. The coefficient
in the (50) could be
equal to0 suchthat alinear modelis used asthe designmodel for
designing the autopilot, but in this paper, let both

and be not
equal to 0, a nonlinear model (50) is used as the design model
for designing the adaptive fuzzy robust controller as following.
Let the statevariables be
, and controlvariable
be
, then the (49) can be rewritten in the state–space form
(51)
Without lossof generality,we assumethat thefunction
in the (50) can be defined in the function which is un-
known with a continuous complicatedformulation system func-
tion, T–S fuzzy system can be constructed to approximate the
function
by the following nine fuzzy IF–THEN rules
in Table I.
In TableI, weselect
, .
denotes the fuzzy set “Positive”, denotes the fuzzy set “Zero”
and
denotes the fuzzy set “Negative”. They can be character-
ized by the membership functions as follows
For the previous example, we may use fuzzy sets on the normal-
ized universes of discourse as shown in Fig. 8.
Using the center average defuzzifier and the product infer-
ence engine, the fuzzy system is obtained as follows:
(52)
where
.
.
.

and
.
.
.
.
.
.
. can be defined
as the (6).
To demonstrate the availability of the proposed scheme, we
take ageneral cargo shipwith the length 126m and the displace-
ment 11 200 tons as an example for simulation.
YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 793
Fig. 8. Fuzzy member ship functions.
The reference model is chosen so as to represent somewhat
realistic performance requirement as
(53)
where
specifies the desired system performance for the
ship heading
.
When let
, and , then the
solution of Lyapunov expression (21) is obtained by
If the gain is , the adaptive fuzzy robust tracking
control scheme can be obtained for ship autopilot as follows:
(54)
where
,
Generally, the first step in the controller design procedure is

construction of a “truth model” of the dynamics of the process
to be controlled. The truth model is a simulation model that in-
cludes all the relevant characteristics of the process. The truth
model is too complicated for use in the controller design. Thus,
we need develop a simplified modelcalled the designmodel that
can be used to design the controller. In this paper, we take the
(29) as the design model. In order to verify the performance of
the ATRFC proposed above by use of simulation, a truth model
(b)
Fig. 9. Simulation results for AFRTC algorithm when employed for a cargo
ship. (a) Ship heading [ship heading
(solid line) and reference course
(dashed line)]. (b) Rudder angle .
Fig. 10. Simulation results for AFRTC algorithm when employed for a cargo
ship: Tracking error [deg].
which accurately represents the characteristics of the ship is
used as follows:
(55)
where
and are the velocity components of the ship and
is the angular rate of yaw angle with respect to time, and ,
, , , and are mass, added mass, inertia moment
and added inertia moment of ship.
, are the components of
hydrodynamic force acting on ship in the bodyfixed axis system
and
is amoment bythe aforementionedforces. Thesubscripts
in the left of the (55) mean that
denotes the bare hull, is
screw,

is rudder, is wind and wave which produce the ex-
ternal forces and moments acting on ship. The methods of cal-
culating external forcesand moments above have beenproposed
in [51].
Simulation results based on the Matlab Simulink package are
shown in Figs. 9–11.
794 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003
(b)
Fig. 11. Simulation results for AFRTC algorithm when employed for a cargo
ship. (a) Adaptive parameter
. (b) Adaptive parameter .
The controller is trained by applying a signal , which
changes its value in the interval (0, 30
) every 300 s. In Fig. 9,
the initial state of the reference signal begins with 20
and the
initial state of the ship heading
is 0 . After 100 s, the plant
output and the referencesignal are practically indistinguishable,
and it can be seen that the plant output converges rapidly to
the reference signal in Fig. 10. Fig. 11 shows the adaption of
parameters in AFRTC algorithm.
VI. C
ONCLUSION
In this paper, the tracking control problem has been con-
sidered for a class of nonlinear uncertain systems with the
unknown system function and unknown gain function, and
T–S type fuzzy logic systems have been used to approximate
unknown system function and an AFRTC algorithm, that
can guarantee the closed-loop in the presence of nonrepeat-

able uncertainties is uniformly ultimately bounded, and the
output tracking error of the system can be steered to a small
neighborhood around
, has been achieved by use
of the ISS and general small-gain approach. The outstanding
features of the algorithm proposed in this paper are that it can
avoid the possible controller singularity problem in some of
existing adaptive control schemes with feedback linearization
techniques and the adaptive mechanism with minimal learning
parameterizations, e.g., no matter how many states in the
system are investigated and how many rules in the fuzzy system
are used, only two parameters, which are a parameter of fuzzy
system and a bounded value including approximation error
and disturbance, are needed to be adapted on-line, so that the
computation load of the algorithm can be reduced and it is
convenient to realize this algorithm in engineering. In order
to make study of efficiency of the algorithm proposed in this
paper, it has been applied to a pole-balancing robot system and
ship autopilot system. In accordance with unknown parameters
of the ship model and the structure of uncertain function of
the system, the T–S fuzzy system is made to approximate the
uncertain function, then an AFRTC scheme for ship autopilot
is proposed. Simulation results have shown the effectiveness of
the control scheme.
A
CKNOWLEDGMENT
The authors would like to thank the anonymous referees for
their valuable comments and criticism which have helped to im-
prove this paper.
R

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Yansheng Yang was born in Jiangsu Province, P. R.
China, in 1957.Hereceived the B.S., M.S.,and Ph.D.
degrees from the Department of Navigation, Dalian
Maritime University, Dalian, China, in 1982, 1985,
and 2000, respectively.
From 1995 to 1996, he was a Visiting Scholar at
Hiroshima University, Hiroshima, Japan. In 1998, he
became a Professor with the Navigation College at
Dalian Maritime University. His research interest in-
cludes robust control and fuzzy control for nonlinear
system and their applications in marine control.
Junsheng Ren was born in Henan province, P.R.

China, in 1976. He received the B.S. degree from the
Navigation College at Dalian Maritime University,
Dalian, in 1999. He is currently working toward the
Ph.D. degree at the same university.
His research interests includesship motion simula-
tion, robust control theory, and its application to ship
motion.