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406 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000
Robust
H
Disturbance Attenuation for a Class of Uncertain Discrete-Time
Fuzzy Systems
Yong-Yan Cao and P. M. Frank
Abstract—This paper is concerned with robust stabilization and
control of a class of uncertain discrete-time fuzzy systems.
The class of uncertain systems is described by a state space
Takagi–Sugeno (TS) fuzzy models with linear nominal parts and
norm-bounded parameter uncertainties in the state and output
equations. First, a sufficient condition on robust stability of the
fuzzy models is proposed. Then, -disturbance attenuation
performance of the fuzzy models is analyzed. Some sufficient
conditions are derived on robust -disturbance attenuation in
which both robust stability and a prescribed performance
are required to be achieved, irrespective of the uncertainties.
Numerical example shows the use of the results on the stabilization
and -disturbance attenuation of a class of discrete-time
nonlinear systems via fuzzy switch.
Index Terms—Discrete-time nonlinear systems, fuzzy control,
fuzzy switch, -disturbance attenuation, linear matrix in-
equality (LMI), robust control.
I. INTRODUCTION
MODELING and control of nonlinear dynamical systems
is one of the most challenging areas of system and con-
trol theory. Fuzzy logic control [21] is an effective approach to
obtain nonlinear control systems, especially in the presence of
incomplete knowledge of the plant or even of the precise control
action appropriate to a given situation. It has found successful
applications not only in consumer products but also in industrial
processes (see, e.g., [5], [8] and the references cited therein). Re-
cently, a nonlocal approach, which is conceptually simple and
straightforward is proposed for nonlinear systems design via
fuzzy control [12], [13], [18]. The procedure is as follows. First,
the nonlinear plant is represented by a so-called Takagi–Sugeno
(TS) type fuzzy model. In this type of fuzzy model, local dy-
namics in different state space regions are represented by linear
models. The overall model of the system is achieved by fuzzy
“blending” of these fuzzy models. The control design is car-
ried out based on the fuzzy models via the so-called parallel
distributed compensation (PDC) scheme [13], [18]. The idea
is that for each local linear model, a linear feedback control is
designed. The resulting overall controller, which is, in general,
nonlinear is again a fuzzy blending of each individual linear con-
troller. Their works introducethe model-based analysis methods
into fuzzy logic control.
Manuscript received November 9, 1999; revised March 16, 2000. This work
was supported by the Alexander von Humboldt Foundation.
Y. Y. Cao is with the Department of Electrical Engineering, University of Vir-
ginia, Charlottesville, VA 22903 USA. He is on leave from the National Labo-
ratory of Industrial Control Technology, Institute of Industrial Process Control,
Zhejiang University, Hangzhou, 310027, P.R. China.
P. M. Frank is with the Department of Measurement and Control, Faculty of
Electrical Engineering, Duisburg University, Duisburg,47048 Germany (e-mail:
yycao@iipc.zju.edu.cn).
Publisher Item Identifier S 1063-6706(00)06589-9.
One of the most important requirements for a control system
is the so-called robustness. In the past 20 years, considerable at-
tention has been paid to the problems of robust stabilization and
robust performance of uncertain dynamical systems (see, e.g.,
[22], [24]). Since the pioneering work on the so-called -op-
timal control theory [22], there has been a dramatic progress
in -control theory [24]. Recently, the problem of nonlinear
control has been intensively studied (see, e.g., [17], [11],
[10]). On the other hand, robust control for both linear
and nonlinear systems has recently been studied (see, e.g., [6],
[4], [19], [20] and the references therein). For example, the
problem of robust stabilization and robust disturbance attenu-
ation is investigated by [19], [2] for continuous-time systems
with linear nominal parts and norm-bounded nonlinear uncer-
tainties on both state and control inputs using the Riccati equa-
tion approach and LMI-based approach.
Since TS fuzzy models are usually used to describe com-
plex nonlinear systems, there may be uncertainties resulting not
only from the modeling procedure, but also from inherent un-
certainties in the real system. In this paper, we will consider
robust control for a class of TS fuzzy models including nom-
inal model which has been constructed or identified and per-
turbed models differing from the nominal one by some type of
perturbations. The class of uncertain models are described by a
state–space model with linear nominal parts and time-varying
norm-bounded parameter uncertainty in the system matrices.
In [19], it is shown that some nonlinear uncertainties can also
be described by this form (see also [6], [4], [20]). In [7] and
[23], the modeling error associated with fuzzy models has been
characterized as a norm-bounded uncertainty. In fact, existence
of modeling error may be a potential source of instability for
control designs that have been based on the assumption that
the fuzzy model exactly matches the plant [9]. In this paper,
we will address the issue of robust stability in the presence
of norm-bounded uncertainty. Some sufficient conditions under
which the fuzzy model-based system is robust against the norm-
bounded uncertainty will be derived. Another important propose
of this paper is the robust -performance analysis and de-
sign of the fuzzy model-based discrete-time nonlinear systems.
We address the problem of robust -disturbance attenuation
for a class of discrete-time uncertain TS fuzzy models in which
both robust stability and a prescribed performance are re-
quired to be achieved, irrespective of the uncertainties. In [14],
the authors considered the robust stabilization problem using
-control approach for continuous-time TS fuzzy models,
i.e., the robust stabilization problem has been transformed to an
associated -control problem. In this paper, the robust sta-
bilization problem and robust -control problem will be di-
rectly addressed using linear matrix inequality approach.
1063–6706/00$10.00 © 2000 IEEE
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000 407
Thepaper isorganizedas follows.Definitions and preliminary
results are given in Section II for uncertain discrete-time fuzzy
nonlinear systems. Then robust stability and robust -distur-
banceattenuationperformanceareanalyzedin Sections III. Some
sufficient conditions on stabilizability and disturbance at-
tenuation of the fuzzy nonlinear systems are derived in Sections
IV. In Section V, results are extended to robust stabilization and
-disturbance attenuation for the uncertain fuzzy nonlinear
systems. The development is based on the feasibility of certain
LMI’s. In Section VI, a numerical example is proposed to show
the effectiveness of results on stabilization and -disturbance
attenuationfor thenonlinear systemsvia fuzzyswitch. Thepaper
is concluded in Section VII.
Notations: The following notations will be used throughout
the paper. denotes the set of real numbers, is the set
of nonnegative real numbers, denotes the dimensional
Euclidean space, and denotes the set of all real
matrices. In the sequel, if not explicitly stated, matrices are as-
sumed to have compatible dimensions. The notation
is used to denote a symmetric positive definite (pos-
itive semidefinite, negative definite, negative semidefinite, re-
spectively) matrix. , denote the minimum and
the maximum eigenvalue of the corresponding matrix, respec-
tively. ( may be finite or infinite) denotes the space
of square summable vector sequence over , i.e., space
formed by the sequence with
and such that
We shall use without distinction to denote the norm in
, whenever the context makes it clear to which one we
are referring.
II. PROBLEM STATEMENT
A general multivariable uncertain nonlinear system can be
represented as a fuzzy plant model with uncertainties. As in
[12], a TS fuzzy model with uncertainty is composed of plant
rules that can be represented as [7], [9], and [23].
Plant Rule :IF is and and is THEN
(1)
(2)
(3)
where fuzzy set, ;
the state;
control input;
the output;
exogenous disturbance input with
.
, , , , , and are appropriately dimen-
sioned real-valued matrices. is the number of IF-THEN rules,
are the premise variables. It is assumed in
this paper that the premise variables do not explicitly depend
on the input variables .
We assume that the time-varying uncertainties enter the
system matrices in (4) and (5) as shown at the bottom of the
page, where , , , , and are some constant
matrices of compatible dimensions and , ,
, , , and are real-valued
matrix functions of compatible dimensions representing
time-varying parameter uncertainties. The uncertainties are
assumed to be norm-bounded and are given by
(6)
where , , , and are known constant matrices
with compatible dimensions and are un-
known nonlinear time-varying matrix functions satisfying
(7)
It is assumed that the elements of are Lebesgue measur-
able. This type of uncertainty is an effective representation of
some nonlinear uncertainties (see [4], [6], [19] and the refer-
ences therein).
Given a pair of , the final outputs of the fuzzy system
are inferred as follows:
(8)
(9)
where and
is the grade of membership of in . It is easy
to find that
(4)
(5)
408 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000
and
for all . Therefore, for and
for all .
Based on the parallel distributed compensation (PDC) [13],
[18], we consider the following fuzzy control law for the fuzzy
model (8) and (9).
Regulator Rule :IF is and and is
THEN
(10)
The overall state feedback fuzzy control law is represented by
(11)
The design of state feedback fuzzy controller is to determine
the local feedback gains such that the following closed-loop
system is stable in large:
(12)
(13)
Remark 1: It can be seen that the closed-loop fuzzy system
described by (12) and (13) can be further simplified if the sub-
system in each rule of the fuzzy plant model possesses common
input matrices and , namely , for all
. At this time, the closed-loop system can be simplified as
For simplicity, when system (8) and (9) is such that ,it
is referred to as an unforced fuzzy system, while when ,
it is referred to as a disturbance-free fuzzy system. In the case
when the uncertainties , it is referred to as the nom-
inal fuzzy system. In [12], [13], and [18], the authors considered
the stability problem of the nominal disturbance-free fuzzy sys-
tems. In this paper, we will first consider the robust stability and
the performance analysis of the unforced fuzzy system (8)
and (9) for (the open-loop case). The stability of
equilibrium point will be studied.
Definition 1: The unforced nominal fuzzy system (8) and
(9) is said to be stable with -disturbance attenuation if for all
, , the equilibrium of the disturbance-free
fuzzy system is asymptotically stable and the response of
the system under the zero initial condition satisfies
(14)
where is the prescribed level of disturbance attenuation.
The unforced uncertain fuzzy system (8) and (9) is said to be
robust stable with -disturbance attenuation if it is stable with
-disturbance attenuation for all admissible uncertainty .
Definition 2: For a given control law (11) and a prescribed
level of disturbance attenuation to be achieved, the nom-
inal fuzzy system (8) and (9) is said to be stabilizable with
-disturbance attenuation if for all , , the
closed-loop disturbance-free system (8), (9), and (11) is asymp-
totically stable and the response under the zero initial con-
dition satisfies (14). The uncertain closed-loop fuzzy system (8),
(9), and (11) is said to be robust stabilizable with -disturbance
attenuation if it is stabilizable with -disturbance attenuation for
all admissible uncertainty .
First we will derive the robust stability condition of unforced
disturbance-free fuzzy systems (8), i.e.,
(15)
Some sufficient conditions for ensuring robust stability of fuzzy
system (15) will be derived using Lyapunov approach as in [13].
Amongst other things, we are concerned with the design of a
fuzzy controller (11) such that the closed-loop fuzzy system (8),
(9), and (11) is robust stable with -disturbance attenuation. In
the paper, we assume that perfect observation of are available
at time .
To facilitate development, we state the following lemma
which can be established easily [3].
Lemma 1: Let , , and be real matrices of appropriate
dimensions with . Then for and scalar
satisfying ,wehave
III. ROBUST STABILITY AND -PERFORMANCE ANALYSIS
In this section, we will first analyze the stability and per-
formance of nominal unforced fuzzy system (8) and (9). Then a
sufficient condition on robust stabilization and robust -dis-
turbance attenuation performance will be derived for the uncer-
tain unforced fuzzy systems. The following lemma was proved
in [13].
Lemma 2: The equilibrium of the nominal unforced distur-
bance-free fuzzy system (15) is globally asymptotically stable
if there exists a common matrix such that
(16)
The following result is an extension of Lemma 4.1 of [13].
Lemma 3: Given two matrices , and
two positive semi-definite matrices , ,
such that
and
then
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000 409
Theorem 1: The nominal unforced fuzzy system is stable
with -disturbance attenuation, that is for all
nonzero under the zero initial condition, if there
exists a common such that
(17)
Proof: Obviously, inequality (17) implies the following
inequality:
This means that the inequality (16) holds and, hence, the
stability of the disturbance-free fuzzy system is proven from
Lemma 2.
The nominal unforced fuzzy system can be rewritten as
(18)
(19)
Select the Lyapunov function as
(20)
and define
(21)
In the following, without loss of generality we assume zero ini-
tial condition . Similar to the derivation of [4], we can
find that for any nonzero
(22)
where , , ,
, . Applying Lemma
3, we have
From the Schur complement, if the LMI’s (17) hold.
Therefore, the dissipativity inequality (14) holds for all .
In other words, we have that , for any nonzero
, and .
Corollary 1: The unforced disturbance-free uncertain
fuzzy system is robust stable if there exist scalars ,
, and a common matrix satisfying the
following LMIs
(23)
Corollary 2: The unforced uncertain fuzzy system is robust
stable with -disturbance attenuation, that is
for all , , if there exist a common matrix
and constants , , satisfying
the following coupled LMIs:
(24)
where
The above results can be easily proven based on the technique
of [20] and, hence, the proofs are omitted.
IV. FUZZY CONTROL DESIGN
In this section, we consider the control design for the
nominal fuzzy system. Under control law (10), the closed-loop
fuzzy system becomes
where , . In the fol-
lowing, we will define , .
Theorem 2: For the nominal fuzzy system, there exists a
state feedback fuzzy control law (11) such that the closed-loop
system is stable with -disturbance attenuation; that is,
for all , , if there exists a common
matrix satisfying the following coupled matrix inequali-
ties:
(25)
(26)
for all and except the pairs such that ,
.
410 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000
Proof: Select the Lyapunov function as in (20) and
define in (21). As in the last section, under zero initial condi-
tion we can find that for any nonzero
(27)
Applying Lemma 2, we have
(28)
Obviously, if , then closed-loop system is stable with
-disturbance attenuation, that is for all
, .
It can be easily found that
So, if the matrix inequalities (25) and (26) hold, and,
hence, the dissipativity inequality (14) holds for all .
Theorem 3: For the nominal fuzzy system, there exists a
state feedback fuzzy control law (11) such that the closed-loop
system is stable with -disturbance attenuation; that is,
for all , , if there exist matrices
and , , satisfying the following LMIs:
(29)
(30)
where represents blocks that are readily inferred by symmetry,
for all and except the pairs such that
, . A robust stabilizing controller to provide -disturbance
attenuation can be constructed as .
Proof: This result can be proved based on the Schur com-
plement from Theorem 4.
The above theorems are derived from the stability and per-
formance analysis of the closed-loop fuzzy system. The state
feedback fuzzy control law design is reduced to a problem of
finding a common Lyapunov matrix satisfying
some (maximum: ) matrix inequalities. If is large, it
might be difficult to find the matrix satisfying the condition of
the above theorems. In [15], a relaxed stability conditions is pro-
posed for fuzzy control design. In the following, we extend the
results to fuzzy control design. The following two lemmas
were proved in [15].
Lemma 4:
where ,for all .
Lemma 5: If the number of rules that fire for all is less than
or equal to where , then
where , for all .
The following result can be easily derived from the above
theorems and lemmas.
Corollary 3: For the nominal fuzzy system, assume that the
number of rules that fire for all is less than or equal to where
. There exists a state feedback fuzzy control law (11)
such that the closed-loop system is stable with -disturbance
attenuation; that is, for all , ,
if there exist matrices and , , and two
common matrices , satisfying the following
LMIs:
(31)
(32)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000 411
for all and except the pairs such that
, .
V. ROBUST FUZZY CONTROL DESIGN FOR UNCERTAIN
FUZZY SYSTEMS
In this section, we consider the robust control design
for the uncertain fuzzy system. Under the control law (10), the
closed-loop uncertain fuzzy system becomes
where
Similar to the derivation of last section, the uncertain
closed-loop system is robust stable with -disturbance attenua-
tion if the following inequalities hold
where
Hence, the closed-loop fuzzy system is robust stable with
-disturbance attenuation if
(33)
(34)
Obviously
where
diag
From (7) we have
From Lemma 1, we can find matrix inequalities (33) and (34)
hold if
respectively, where
which are equivalent to
(35)
(36)
respectively, where
Let , and define diag .
Pre- and postmultiplying (35) by and , respectively, we
find that (35) are equivalent to the matrix inequalities as
seen in (37) at the bottom of the next page.for .
Similarly, we can prove that (36) are equivalent to the ma-
trix inequalities as seen in (38), at the bottom of the next
page, where ,
for .
Theorem 4: For the uncertain fuzzy system, there exists a
state feedback fuzzy control law (11) such that the closed-loop
fuzzy system is robust stable with -disturbance attenuation;
that is, for all , , if there
exist matrices and , , and constants ,
412 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000
, , , satisfying the LMIs (37) and (38)
for all and except the pairs such that
, . A robust stabilizing controller to provide -disturbance
attenuation can be constructed as .
Corollary 4: For the uncertain fuzzy system, assume that the
number of rules that fire for all is less than or equal to , where
. There exists a state feedback fuzzy control law (11)
such that the closed-loop uncertain fuzzy system is robust stable
with -disturbance attenuation, that is for all
, , if there exist matrices and ,
, and two common matrices , , and
constants , , , , satisfying the LMIs,
as shown in (39) and (40) at the bottom of the page. for all and
except the pairs such that , .
(37)
(38)
(39)
(40)
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000 413
VI. NUMERICAL EXAMPLE
To illustrate the proposed results, consider the backing-up
control of a computer-simulated truck-trailer, which is borrowed
from [16], [1]. The truck-trailer model is formulated as
(41)
(42)
(43)
Equations (41) and (42) are linear, but (43) is nonlinear. The
model parameters are given as , , ,
. Weuse the following fuzzy models to design the fuzzy
controller:
Rule 1: IF is about
THEN
Rule 2: IF is about or
THEN
where
(44)
(45)
and denotes the offset in the second model. As in
[16], we set and the membership functions as
When we do not consider the uncertainties and , the
nonlinear fuzzy model is given by
(46)
(47)
(48)
Fig. 1. Model error
e
(
)
,
2
[
0
;
]
.
Obviously, fuzzy model (48) is an universal approximation of
(43). The modeling error between (43) and (48) is
(49)
which is shown in Fig. 1.
In fact, we can capture the dynamics of a nonlinear system
within any specified accuracy, i.e., , is a
predetermined tolerance. The higher the precision of the model,
however, the larger the number of the local models in the ag-
gregation. Attempts to maintain a relatively small number of
local models inevitably introduce modeling error into the fuzzy
models (8) and (9). In most of the references, such as [9], [12],
[13], only the stability is addressed for the fuzzy models (8), but
not for the original nonlinear systems. In [1], [16] fuzzy control
law is designed for the fuzzy models (46)–(48), but not directly
for the original nonlinear system . The closed-loop stability
of the fuzzy system (46)–(48) and fuzzy control law (11) can
be guaranteed by their design methods, however, the stability of
the closed-loop nonlinear system (41)–(43) and fuzzy control
law (11) cannot be guaranteed in general. In above references,
it is only checked by simulation. In the following, we will give
a designed approach to guarantee that the closed-loop nonlinear
system (41)–(43) and fuzzy control law (11) is stable.
In the above uncertain fuzzy models, the uncertainty to de-
scribe the modeling error is assumed to be in the form
where
The parameters and are to be determined. For simplicity,
we assume that . Hence, the uncertain nonlinear
fuzzy system is
(50)
414 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000
Fig. 2.
e
(
)
=
,
2
[
0
;
]
i.e.,
(51)
If we let
then (51) will match the nonlinear equation (43). This will make
is very complex. Fortunately, if we let
then the original nonlinear system will be a subsystem of the
set of the uncertain fuzzy system (50), i.e.,
System(50)
So the robust stability of uncertain system (50) will guarantee
the stability of original nonlinear system . Fig. 2 shows that
. This means that .
Applying Theorem 9, we get a feasible solution
Fig. 3 shows the state response of the nonlinear system
(41)–(43) under the above fuzzy switch control when the initial
condition of the truck-trailer is . The
figure shows that the position response is much better than
that of paper [1].
We assume that the nonlinear system with disturbance is
given by
(52)
Fig. 3. State response under initial condition
(0
:
5
;
0
:
75
;
0
10)
.
Fig. 4. State response under stochastic disturbance
(53)
(54)
(55)
is the measured output and is the exogenous disturbance
input with . This means that
Applying Theorem 10, we get a feasible solution
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 4, AUGUST 2000 415
Fig. 5. The stochastic disturbance
w
and the output response
z
when . Fig. 4 shows the state responses of the un-
certain system when the initial condition of the truck-trailer
is and is a stochastic disturbance
which is generated by MATLAB function
, which belongs to . Fig. 5 shows the disturbance
and the output response
VII. CONCLUSION
In this paper, we have studied robust stabilization and
-disturbance attenuation for uncertain discrete-time fuzzy
systems via linear matrix inequality approach. Sufficient con-
ditions on robust stabilizability and -disturbance attenuation
are presented based on coupled LMIs. The results can be used
to the stabilization and -disturbance attenuation for a class
of discrete-time nonlinear systems. Numerical example is
presented to show the effectiveness. The results can also be
easily extended to the systems with time-delay using so-called
Lyapunov–Krasovskii functional approach [2], [3].
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