Fuzzy Robust Tracking Control for Uncertain System

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Fuzzy robust tracking control for
uncertain nonlinear systems
Shaocheng Tong
a,
*
, Tao Wang
a
, Han-Xiong Li
b
a
Department of Basic Mathematics, Liaoning Institute of Technology, Jinzhou, 121001, PR China
b
Department of Manufacturing Engineering and Engineering Management, City University of
Hong Kong, Hong Kong, PR China
Received 1 September 2001; accepted 1 November 2001
Abstract
A robust output tracking control technique for nonlinear systems is developed. First,
the Takagi and Sugeno (T–S) Fuzzy model with parametric uncertainties is employed to
represent a nonlinear system. Based on (T–S) fuzzy model, fuzzy robust state feedback
output tracking controller and fuzzy robust observer-based output tracking controller
are proposed. Sufficient conditions are derived for robust asymptotic output tracking
controllers in the format of linear matrix inequalities (LMIs), which can be very effi-
ciently solved by using LMI optimization techniques. The effectiveness of the proposed
fuzzy tracking controllers is finally demonstrated through numerical simulations on an
inverted pendulum. Ó 2002 Elsevier Science Inc. All rights reserved.
Keywords: Fuzzy control; Fuzzy plant model; Fuzzy observer; Parametric uncertainties;
Stability; Robustness
1. Introduction
Fuzzy control is one of the useful control techniques for uncertain and ill-
defined nonlinear systems. Control actions of the fuzzy controller are designed
by some linguistic rules. This property makes the control algorithm to be
understood easily. The early design of fuzzy controllers is heuristics. It
International Journal of Approximate Reasoning 30 (2002) 73–90
www.elsevier.com/locate/ijar
*
Corresponding author.
E-mail address: [email protected] (S. Tong).
0888-613X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved.
PII: S0888- 613X( 02) 00061- 0
incorporates the experience or knowledge of the designed into the rules of the
fuzzy controller, which is fine-tuned based on trial and error. A fuzzy controller
implemented by neural network was proposed in [2,3]. Through the use of
tuning methods, fuzzy rules can be generated automatically, which makes the
design very simple. In spite of the usefulness of the fuzzy control, its main
drawback comes from the lack of a systematic control design methodology.
Particularly, stability analysis, robustness and good performance of a fuzzy
system are not easy.
To solve these problems, the idea that a linear system is adopted as the
consequent part of a fuzzy rule has evolved into the innovative Takagi and
Sugeno (T–S) model [1], which becomes quite popular today. For a few years,
the trend of fuzzy control has been to develop some systematic design algo-
rithms so as to guarantee the control performance and system stability for the
T–S fuzzy model-based control technique is simple and effective in the control
of complex systems with nonlinearity, such as the inverted pendulum [5] and
chaotic systems [6].
Besides stabilization problem, tracking control designs are also important
issues for practical applications: for example, in robotic tracking control,
missile tracking control and attitude tracking control of aircraft. However,
there are very few studies concerning with tracking control design based on the
T–S fuzzy model, especially for continuous-time systems [10]. In general,
tracking control design is more general and more difficult than the stabilization
control design. In [7], feedback linearization technique is proposed to sys-
tematically design a fuzzy tracking controller for discrete-time systems. As
pointed out in [8], the controller derived by feedback linearization may not be
bounded, i.e., the fuzzy controller is not guaranteed to be stable for non-
minimum phase system. Lam and co-workers [9] studied a model following
control and addressed the robustness of the fuzzy controller on the assumption
that the state variables are available. In practice, this assumption does not
hold. Tseng et al. [10] proposed fuzzy tracking control design for nonlinear
dynamic systems via T–S fuzzy model. However, their works did only concern
on the tracking control of nominal T–S fuzzy systems without considering T–S
fuzzy systems in the presence of norm-bounded time-varying uncertainty. So
the robustness of the whole control tracking control system cannot be guar-
anteed.
Motivated by the aforementioned concerns, this paper discusses the robust
output tracking control design for nonlinear systems in the presence of the
norm-bounded time-varying uncertainty. The T–S fuzzy model with para-
metric uncertainties is first employed to represent a nonlinear system. Based on
(T–S) fuzzy robust observer-based output tracking controller are proposed.
Sufficient conditions are derived for robust asymptotic output tracking con-
trollers in the format of linear matrix inequalities (LMIs), which can be very
efficiently solved by using LMI optimization techniques. The overall proposed
74 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
design methodology presents a systematic and effective framework for an in-
verted pendulum.
2. Problem formulation
A fuzzy dynamic model proposed by Takagi and Sugeno [1] is often used to
represent a nonlinear system. The T–S fuzzy model is a piecewise interpolation
of several linear models through membership functions. The fuzzy model is
described by fuzzy If–Then rules and will be employed here to deal with the
control design problem for the nonlinear system. The ith rule of the fuzzy
model for the nonlinear system is of the following form [6]:
T–S fuzzy model:
Plant Rule i:
IF z
1
ðtÞ is M
i
1
and z
2
ðtÞ is M
i
2
and . . . z
n
ðtÞ is M
i
n
;
THEN
_ xxðtÞ ¼ ðA
i
þDA
i
ÞxðtÞ þðB
i
þDB
i
ÞuðtÞ þw;
yðtÞ ¼ C
i
xðtÞ;
i ¼ 1; 2; . . . ; q; ð1Þ
where M
i
j
is a fuzzy set (j ¼ 1; 2; . . . ; n), zðtÞ ¼ ½z
1
ðtÞ; . . . ; z
n
ðtފ
T
is the premise
variable vector. xðtÞ ¼ ½x
1
ðtÞ; x
2
ðtÞ; . . . ; x
n
ðtފ
T
2 R
n
is the state vector, uðtÞ 2 R
m
is the control input vector, uðtÞ ¼ ½u
1
ðtÞ; u
2
ðtÞ; . . . ; u
m
ðtފ
T
2 R
m
is the control
input vector, and yðtÞ 2 R
l
is the system output vector, wðtÞ denotes unknown
but bounded disturbance. A
i
2 R
nÂn
, B
i
2 R
nÂm
and C
i
2 R
lÂn
are system matrix,
input matrix and output matrix, respectively, DA
i
and DB
i
are time-varying
matrices with appropriate dimensions, which represent parametric uncertainties
in the plant model, and q is the number of rules of this T–S fuzzy model.
The fuzzy system is inferred as follows:
_ xxðtÞ ¼

q
i¼1
l
i
ðzðtÞÞ½A
i
xðtÞ þB
i
uðtފ þ

q
i¼1
l
i
ðzðtÞÞ½DA
i
xðtÞ þDB
i
uðtÞ þwŠ;
yðtÞ ¼

q
i¼1
l
i
ðzðtÞÞC
i
xðtÞ;
ð2Þ
where
w
i
ðzðtÞÞ ¼

n
j¼1
M
i
j
ðz
j
ðtÞÞ; l
i
ðzðtÞÞ ¼
w
i
ðzðtÞÞ

q
i¼1
w
i
ðzðtÞÞ
; i ¼ 1; 2; . . . ; q;
in which M
i
j
ðz
j
ðtÞÞ is the grade of membership of z
j
ðtÞ in M
i
j
. Some basic
properties are:
w
i
ðtÞ P0;

q
i¼1
w
i
ðtÞ > 0; i ¼ 1; 2; . . . ; q:
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 75
It is clear that
l
i
ðzðtÞÞ P0;

q
i¼1
l
i
ðzðtÞÞ ¼ 1; i ¼ 1; 2; . . . ; q:
The T–S uncertain fuzzy model in (2) is a general nonlinear time-varying
equation with parametric uncertainties and has been used to model the be-
haviors of complex nonlinear dynamic systems [6].
Consider a reference mode as follows [10]:
_ xx
r
ðtÞ ¼ A
r
x
r
ðtÞ þrðtÞ; ð3Þ
where x
r
ðtÞ is reference state, A
r
specific asymptotically stable matrix, rðtÞ
bounded reference input.
Define the tracking error as
e
r
ðtÞ ¼ xðtÞ Àx
r
ðtÞ:
Then the objective is to design a T–S fuzzy model-based controller, which
stabilizes the fuzzy system (2) and achieves the H
1
tracking performance re-
lated to tracking error e
r
ðtÞ as follows [10]:
_
t
f
0
e
T
r
ðtÞQe
r
ðtÞ dt 6q
2
_
t
f
0
ww
T
wwdt; ð4Þ
where ww ¼ ½ wðtÞ rðtÞ Š
T
for both reference input rðtÞ and external disturbance
wðtÞ, t
f
terminal time of control, Q positive definite weighting matrix, q a
prescribed attenuation level.
The physical meaning of (4) is that the effect of any wwðtÞ on tracking error
e
r
ðtÞ must be attenuated below a desired level q forms the viewpoint of energy,
no matter what wwðtÞ is, i.e., the L
2
gain from wwðtÞ to e
r
ðtÞ must be equal to or
less than a prescribed value q
2
.
Since the dynamic system (2) has time-varying uncertain matrices, it is not
easy to design the controller gain matrices. In order to find these gain matrices,
K
i
, the uncertain matrices should be removed under some reasonable as-
sumptions. Therefore, we assume, as usual, that the uncertain matrices DA
i
and
DB
i
are admissibly norm-bounded and structured.
Assumption 1. The parameter uncertainties considered here are norm-bounded,
in the form [6]:
½DA
i
; DB
i
Š ¼ D
i
F
i
ðtÞ½E
1i
; E
2i
Š;
F
T
i
ðtÞF
i
ðtÞ 6I;
where D
i
, E
1i
, and E
2i
are known real constant matrices of appropriate di-
mension, and F
i
ðtÞ is an unknown matrix function with Lebesgue-measurable
elements, I is the identity matrix of appropriate dimension.
76 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
3. Fuzzy state feedback tracking control design
In this section, we assume that the all state variables are available for
feedback control. Later in Section 4, we will relax this assumption. In this case,
the fuzzy controller is designed as:
IF z
1
ðtÞ is M
i
1
and z
2
ðtÞ is M
i
2
and . . . andz
n
ðtÞ is M
i
n
;
THEN uðtÞ ¼ ÀK
i
e
r
ðtÞ; i ¼ 1; 2; . . . ; q; ð5Þ
where K
i
2 R
mÂn
is a constant feedback gain to be determined.
Hence, the overall fuzzy controller is given by
uðtÞ ¼

q
i¼1
l
i
ðzðtÞÞK
i
e
r
ðtÞ: ð6Þ
Substituting (6) into (2) yields
_ xxðtÞ ¼

q
i¼1

q
j¼1
l
i
ðzðtÞÞl
j
ðzðtÞÞððA
i
þDA
i
ÞxðtÞ þðB
i
þDB
i
ÞK
j
e
r
ðtÞÞ
þwðtÞ: ð7Þ
Define the augmented state vector and external disturbance vector as follows
xxðtÞ ¼ ½ xðtÞ x
r
ðtÞ Š
T
; wwðtÞ ¼ ½ wðtÞ rðtÞ Š
T
:
Then (2) and (7) become, on using (6),
_
xx xxðtÞ ¼

q
i¼1

q
j¼1
l
i
ðzðtÞÞl
j
ðzðtÞÞð

AA
ij
þD

AA
ij
ÞxxðtÞ þ wwðtÞ; ð8Þ
where

AA
ij
¼
A
i
þB
i
K
j
ÀB
i
K
j
0 A
r
_ _
; D

AA
ij
¼
DA
i
þDB
i
K
j
ÀDB
i
K
j
0 0
_ _
:
The main results on the fuzzy tracking control design of T–S fuzzy model,
with parametric uncertainties, are summarized in the following theorem:
Theorem 1. If there exist a symmetric and positive definite P, some matrices K
i
,
(i ¼ 1; 2; . . . ; q) such that the following matrix inequalities are satisfied, then for a
prescribed q
2
, H
1
tracking control performance in (4) is guaranteed via the T–S
fuzzy model-based state-feedback controller (6):

AA
T
ij
P þP

AA
ij
þ
1
q
2
PP þ

QQ þ

EE
T
ij

EE
ij
þP

DD
i

DD
T
i
P < 0; ð9Þ
where

QQ ¼
Q ÀQ
ÀQ Q
_ _
; E
ij
¼
E
i1
ÀE
i2
K
j
ÀE
i2
K
j
0 0
_ _
;
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 77

DD
i
¼
D
i
0
0 D
i
_ _
;

FF
i
ðtÞ ¼
F
i
ðtÞ 0
0 F
i
ðtÞ
_ _
:
Proof. Consider the Lyapunov function candidate
V ðtÞ ¼ xxðtÞ
T
PxxðtÞ; ð10Þ
where the weighting matrix P ¼ P
T
> 0. The time derivative of V ðtÞ is
V ðtÞ ¼
_
xx xxðtÞ
T
PxxðtÞ þxxðtÞ
T
P
_
xx xxðtÞ: ð11Þ
By substituting (8) into (11), we get
_
VV ðtÞ ¼

q
i¼1

q
j¼1
l
i
l
j
xx
T
ðtÞ

AA
T
ij
P
_
þP

AA
ij
þD

AA
T
ij
P þPD

AA
ij
_
xxðtÞ
þ ww
T
PxxðtÞ þxx
T
ðtÞP –w–wðtÞ
¼

q
i¼1

q
j¼1
l
i
l
j
xx
T
ðtÞ

AA
T
ij
P
_
þP

AA
ij
þD

AA
T
ij
P þPD

AA
ij
_
xxðtÞ þ ww
T
PxxðtÞ
þxx
T
ðtÞP –w–wðtÞ Àq
2
ww
T
wwÀ
1
q
2
ww
T
wwþ
1
q
2
ww
T
wwþq
2
ww
T
ww
¼

q
i¼1

q
j¼1
l
i
l
j
xx
T
ðtÞ

AA
T
ij
P
_
þP

AA
ij
þD

AA
T
ij
P þPD

AA
ij
_
xxðtÞ
À
1
q
PxxðtÞ Àq wwðtÞ
_ _
T
1
q
PxxðtÞ Àq wwðtÞ
_ _
þ
1
q
2
xx
T
ðtÞPPxxðtÞ þq
2
ww
T
ww
6

q
i¼1

q
j¼1
l
i
l
j
xx
T
ðtÞ

AA
T
ij
P
_
þP

AA
ij
þD

AA
T
ij
P þPD

AA
ij
_
xxðtÞ
þ
1
q
2
xx
T
ðtÞPPxxðtÞ þq
2
ww
T
ww
¼

q
i¼1

q
j¼1
l
i
l
j
xx
T
ðtÞ

AA
T
ij
P þP

AA
ij
þD

AA
T
ij
P þPD

AA
ij
þ
1
q
2
PP
_ _
ÂxxðtÞ þq
2
ww
T
ww: Ã ð12Þ
Lemma 1 [14]. Given constant matrices X and Y of appropriate dimensions, the
following inequality holds:
X
T
Y þY
T
X 6X
T
X þY
T
Y :
78 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
Applying Lemma 1 to xx
T
ðtÞðD

AA
T
ij
P þPD

AA
ij
Þx
T
ðtÞ, we have
xx
T
ðtÞ D

AA
T
ij
P
_
þ PD

AA
ij
_
xxðtÞ 6xx
T
ðtÞ

EE
T
ij

FF
T
i

DD
T
i
PxxðtÞ þxx
T
ðtÞP

DD
i

FF
i

EE
ij
PxxðtÞ
6xx
T
ðtÞ

EE
T
ij

FF
T
i

FF
i

EE
ij
xxðtÞ þxx
T
ðtÞP

DD
i

DD
T
i
PxxðtÞ
6xx
T
ðtÞ

EE
T
ij

EE
ij
xxðtÞ þxx
T
ðtÞP

DD
i

DD
T
i
PxxðtÞ: ð13Þ
Substituting (13) into (12) yields:
_
VV 6

q
i¼1

q
j¼1
l
i
l
j
xx
T
ðtÞ

AA
T
ij
P
_
þ P

AA
ij
þ

EE
T
ij

EE
ij
þ
1
q
2
PP þ

QQ þ P

DD
i

DD
T
i
P
_
xxðtÞ
þq
2
ww
T
ww: ð14Þ
From (9), we get
_
VV ðtÞ 6 Àxx
T
ðtÞ

QQxxðtÞ þq
2
ww
T
wwðtÞ: ð15Þ
Integrating (15) from t ¼ 0 to t ¼ t
f
yields:
V ðt
f
Þ ÀV ð0Þ 6 À
_
t
f
0
xx
T
ðtÞ

QQxxðtÞ dt þq
2
_
t
f
0
ww
T
wwdt
¼ À
_
t
f
0
e
T
r
ðtÞ

QQe
r
ðtÞ dt þq
2
_
t
f
0
ww
T
ww dt: ð16Þ
Or equivalently
_
t
f
0
e
T
r
ðtÞ

QQe
r
ðtÞ dt 6e
T
r
ð0ÞPe
r
ð0Þ þq
2
_
t
f
0
ww
T
ww dt: ð17Þ
That is (4) and the H
1
control performance is achieved with a prescribed q
2
.
It is noted that Theorem 1 gives the sufficient condition of ensuring the
stability of the fuzzy system (2) and achieving the H
1
tracking performance
(4). However, it does not give the methods of obtaining the solution of a
common positive matrix

PP for (9). In general, it is not easy to analytically
determine such a common positive matrix, fortunately (9) can be transferred
into LMIs, which can be solved in a computationally efficient manner using
convex optimization techniques such as the interior point method [4].
For the convenience of the design, we assume
P ¼
P
11
0
0 P
22
_ _
: ð18Þ
By substituting (18) into (9), we obtain
F
11
F
12
F
21
F
22
_ _
< 0;
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 79
where
F
11
¼ ðA
i
þB
i
K
j
Þ
T
P
11
þP
11
ðA
i
þB
i
K
j
Þ þ
1
q
2
P
11
P
11
þQ
þðE
i1
þE
i2
K
j
Þ
T
ðE
i1
þE
i2
K
j
Þ þP
11
D
i
D
T
i
P
11
;
F
12
¼ F
T
21
¼ ÀðE
i1
þE
i2
K
j
Þ
T
ðB
i
K
j
Þ ÀQ;
F
22
¼ A
T
r
P
22
þP
22
A
r
þQ þ
1
q
2
P
22
P
22
þP
12
D
i
D
T
i
P
22
:
ð19Þ
By the Schur complement, (19) is equivalent to
H
11
H
12
0
H
21
H
22
P
22
0 P
22
Àq
2
I
_ _
< 0;
where
H
11
¼ ðA
i
þB
i
K
j
Þ
T
P
11
þP
11
ðA
i
þB
i
K
j
Þ þ
1
q
2
P
11
P
11
þQ
þðE
i1
þE
i2
K
j
Þ
T
ðE
i1
þE
i2
K
j
Þ þP
11
D
i
D
T
i
P
11
;
H
12
¼ H
T
21
¼ ÀðE
i1
þE
i2
K
j
Þ
T
ðB
i
K
j
Þ ÀQ;
H
22
¼ A
T
r
P
22
þP
22
A
r
þQ þ
1
q
2
P
22
P
22
þP
12
D
i
D
T
i
P
22
:
ð20Þ
In the following, we can solve P
11
, P
22
and K
j
by the following two-step pro-
cedures:
In the first step, note that (20) implies that H
11
< 0, i.e.,
ðA
i
þB
i
K
j
Þ
T
P
11
þP
11
ðA
i
þB
i
K
j
Þ þ
1
q
2
P
11
P
11
þQ
þðE
i1
þE
i2
K
j
Þ
T
ðE
i1
þE
i2
K
j
Þ þP
11
D
i
D
T
i
P
11
< 0: ð21Þ
By introducing new variables W ¼ P
À1
11
and Y
j
¼ K
j
W , then (21) is equivalent
to the following matrix inequalities:
WA
T
i
þðA
i
W þB
i
Y
j
Þ þY
j
B
i
þ
1
q
2
I þD
i
D
T
i
þWQW
þ E
i1
W ð þE
i2
Y
j
_
T
E
i1
W ð þE
i2
Y
j
_
< 0: ð22Þ
By the Schur complement, (22) is equivalent to the following LMIs:
U
ij
W E
i1
W þE
i2
Y
j
W
T
ÀQ
À1
0
ðE
i1
W þE
i2
Y
j
Þ
T
0 ÀI
_
_
_
_
< 0; ð23Þ
where
U
ij
¼ WA
T
i
þA
i
W þB
i
Y
j
þY
j
B
i
þ
1
q
2
I þD
i
D
T
i
:
80 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
The parameters W and Y
i
(thus P
11
¼ W
À1
, K
j
¼ Y
j
W
À1
) can be obtained by
solving the LMI (23).
The second step, by substituting P
11
and K
j
into (21), then (21) becomes a
standard LMI’s. Similarly, we can easily solve P
22
.
4. Fuzzy observer-based tracking control
In the previous section, fuzzy tracking control requires that all the state
variables are available. In practice, this assumption often does not hold. In this
situation, we need to estimate state vector x from output y for feedback control.
Suppose the following fuzzy observer is proposed to deal with the state
estimation of the fuzzy system (2)
IF z
1
ðtÞ is M
i
1
and z
2
ðtÞ is M
i
2
and . . . and z
n
ðtÞ is M
i
n
THEN
_
^ xx^ xxðtÞ ¼ A
i
^ xxðtÞ þB
i
uðtÞ þG
i
½yðtÞ À ^ yyðtފ;
^ yyðtÞ ¼ C
i
^ xxðtÞ;
i ¼ 1; 2; . . . ; q; ð24Þ
where G
i
2 R
nÂl
is constant observer gain to be determined.
The overall fuzzy observer is represented as follows
_
^ xx^ xxðtÞ ¼

q
i¼1
l
i
ðzðtÞÞA
i
^ xxðtÞ þ

q
i¼1
l
i
ðzðtÞÞB
i
uðtÞ þ

q
i¼1
l
i
ðzðtÞÞG
i
½yðtÞ À ^ yyðtފ;
^ yyðtÞ ¼

q
i¼1
l
i
ðzðtÞÞC
i
^ xxðtÞ: ð25Þ
Define observation error as
eðtÞ ¼ xðtÞ À ^ xxðtÞ: ð26Þ
Design the observer-based fuzzy controller in the form
uðtÞ ¼

q
i¼1
l
i
ðzðtÞÞK
i
ð^ xxðtÞ Àx
r
ðtÞÞ: ð27Þ
From systems (7), (25) and (26), we obtain
_ eeðtÞ ¼

q
i¼1

q
j¼1
l
i
ðzðtÞÞl
j
ðzðtÞÞðA
i
ÀG
i
C
j
þDB
i
K
j
ÞeðtÞ
þ

q
i¼1

q
j¼1
l
i
ðzðtÞÞl
j
ðzðtÞÞDA
i
ÞxðtÞ þwðtÞ: ð28Þ
The augmented system composed of (2), (3) and (28), on using (27), can be
expressed in the following form:
_
~xx ~xxðtÞ ¼

q
i¼1

q
j¼1
l
i
ðzðtÞÞl
j
ðzðtÞÞð
~
AA
ij
þD
~
AA
ij
Þ~xxðtÞ þ
~
EE
i
~ wwðtÞ; ð29Þ
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 81
where
~xxðtÞ ¼ ½ eðtÞ xðtÞ x
r
ðtÞ Š
T
; ~ wwðtÞ ¼ ½ 0 wðtÞ rðtÞ Š
T
;
~
AA
ij
¼
A
i
ÀG
i
C
j
0 0
ÀB
i
K
j
A
i
þB
i
K
j
ÀB
i
K
j
0 0 A
r
_
¸
_
_
¸
_;
D
~
AA
ij
¼
DB
i
K
j
DA
i
0
0 DA
i
þDB
i
K
j
ÀDB
i
K
j
0 0 0
_
¸
_
_
¸
_;
~
EE
i
¼
I I 0
0 0 I
_ _
T
:
The main result on fuzzy observer-based tracking control for the T–S fuzzy
system with norm-bonded uncertainties is summarized in the following theo-
rem.
Theorem 2. If there exist symmetric and positive definite matrix
~
PP, some matrices
K
i
and G
i
(i ¼ 1; . . . ; q), such that the following matrix inequality are satisfied,
then for a prescribed q
2
, H
1
tracking control performance in (4) is guaranteed via
fuzzy observer-based controller (27):
~
AA
T
ij
~
PP þ
~
PP
~
AA
ij
þ
1
q
2
~
PPE
i
E
T
i
~
PP þ
~
QQ þ
~
EE
T
ij
~
EE
ij
þ
~
PP
~
DD
i
~
DD
T
i
~
PP < 0; ð30Þ
where
~
DD
i
¼
D
i
0 0
0 D
i
0
0 0 D
i
_
_
_
_
;
~
EE
ij
¼
E
i2
K
j
E
i1
0
0 E
i1
þE
i2
K
j
ÀE
12
K
j
0 0 0
_
_
_
_
;
~
FF ðtÞ ¼
F
i
ðtÞ 0 0
0 F
i
ðtÞ 0
0 0 F
i
ðtÞ
_
_
_
_
;
~
QQ ¼
0 0 0
0 Q ÀQ
0 ÀQ Q
_
_
_
_
:
Proof. Consider the Lyapunov function candidate
V ðtÞ ¼ ~xx
T
ðtÞ
~
PP~xxðtÞ; ð31Þ
where the weighting matrix
~
PP ¼
~
PP
T
> 0. The time derivative of V ðtÞ is
V ðtÞ ¼
_
~xx ~xx
T
ðtÞ
~
PP~xxðtÞ þ~xx
T
ðtÞ
~
PP
_
~xx ~xxðtÞ: ð32Þ
By substituting (29) into (32), and repeating the procedures of proof of The-
orem 1, we obtain
82 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
_
VV ðtÞ 6

q
i¼1

q
j¼1
l
i
l
j
~xx
T
~
AA
T
ij
~
PP þ
~
PP
~
AA
ij
þD
~
AA
T
ij
~
PP þ
~
PPD
~
AA
ij
þ
1
q
2
~
PPE
i
E
T
i
~
PP
_ _
~xxðtÞ
þq
2
~ ww
T
~ ww: ð33Þ
Applying Lemma 1 to ~xx
T
ðtÞðD
~
AA
T
ij
P þPD
~
AA
ij
Þ~xxðtÞ, we have
~xx
T
ðtÞðD
~
AA
T
ij
~
PP þ
~
PPD
~
AA
ij
Þ~xxðtÞ ¼ ~xx
T
ðtÞ
~
EE
T
ij
~
FF
T
i
~
DD
T
i
~
PP~xxðtÞ þ~xx
T
ðtÞ
~
PP
~
DD
i
~
FF
i
~
EE
ij
~xxðtÞ
6~xx
T
ðtÞ
~
EE
T
ij
~
FF
T
i
~
FF
i
~
EE
ij
~xxðtÞ þ~xx
T
ðtÞ
~
PP
~
DD
i
~
DD
T
i
~
PP~xxðtÞ
6~xx
T
ðtÞ
~
EE
T
ij
~
EE
ij
~xxðtÞ þ~xx
T
ðtÞ
~
PP
~
DD
i
~
DD
T
i
~
PP~xxðtÞ: ð34Þ
Substituting (34) into (33) yields:
_
VV ðtÞ 6

q
i¼1

q
j¼1
l
i
l
j
~xx
T
~
AA
T
ij
~
PP þ
~
PP
~
AA
ij
þ
~
EE
T
ij
~
EE
ij
þ
~
PP
~
DD
i
~
DD
T
i
~
PP þ
1
q
2
~
PPE
i
E
T
i
~
PP
_ _
~xxðtÞ
þq
2
~ ww
T
~ ww: ð35Þ
Form (30), we get
_
VV ðtÞ 6 À~xx
T
ðtÞ
~
QQ~xxðtÞ þq
2
~ ww
T
ðtÞ ~ wwðtÞ: ð36Þ
In the same manipulation as in the previous section, we get
_
t
f
0
e
r
ðtÞ
T
Qe
r
ðtÞ dt 6e
r
ð0Þ
T
Pe
r
ð0Þ þq
2
_
t
f
0
~ ww
T
~ ww dt: ð37Þ
That is (4) and the H
1
control performance is achieved with a prescribed q
2
.
In the following, we transfer matrix inequality (30) into LMI’s, and obtain
the symmetric definite matrix
~
PP.
We suppose
~
PP ¼
P
11
P
22
P
33
_
_
_
_
: ð38Þ
By substituting (38) into (30), we obtain
S
11
S
12
0
S
21
S
22
S
23
0 S
32
S
33
_
_
_
_
< 0; ð39Þ
where
S
11
¼ðA
i
ÀG
i
C
j
Þ
T
P
11
þP
11
ðA
i
ÀG
j
C
j
Þ þK
T
j
E
T
i2
E
i2
K
j
þP
11
D
i
D
T
i
P
11
þ
1
q
2
P
11
P
11
;
S
12
¼ S
T
21
¼ ÀðB
i
K
j
Þ
T
P
22
þðE
i2
K
j
Þ
T
E
i1
þP
11
P
22
;
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 83
S
22
¼ ðA
i
þB
i
K
j
Þ
T
P
22
þP
22
ðA
i
þB
j
K
j
Þ þE
T
i1
E
i1
þðE
i1
þE
i2
K
j
Þ
T
ðE
i1
þE
i2
K
j
Þ þP
22
D
i
D
T
i
P
22
þ
1
q
2
P
22
P
22
;
S
23
¼ S
T
32
¼ ÀP
22
B
i
K
j
ÀðE
i1
þE
i2
K
j
ÞE
i2
K
j
;
S
11
¼ A
T
r
P
33
þP
33
A
r
þðE
i2
K
j
Þ
T
ðE
i2
K
j
Þ þP
33
D
i
D
T
i
P
33
þ
1
q
2
P
33
P
33
:
By the Schur complement, (39) is equivalent to
M
11
P
11
P
11
D
i
M
41
0 0
P
11
Àq
2
I 0 0 0 0
ðP
11
D
i
Þ
T
0 ÀI 0 0 0
M
T
41
0 0 M
44
M
45
0
0 0 0 M
T
45
M
55
P
33
0 0 0 0 P
33
À
1
q
2
I þD
i
D
T
i
_ _
À1
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
< 0; ð40Þ
where
M
11
¼ A
T
i
P
11
þP
11
A
i
ÀðG
i
C
j
Þ
T
P
11
þP
11
G
i
C
j
þK
T
j
E
T
i2
E
i2
K
j
M
14
¼ M
T
41
¼ S
12
; M
44
¼ S
22
; M
45
¼ M
T
54
¼ S
23
;
M
55
¼ A
T
r
P
33
þP
33
A
r
þðE
i2
K
j
Þ
T
ðE
i2
K
j
Þ:
Since five parameters P
11
, P
22
, P
33
; K
j
and G
i
should be determined from (39),
there are no effective algorithms for solving them simultaneously. However, we
can solve them by the following two-step procedures.
In the first step, note that (40) implies that M
44
< 0, i.e.,
ðA
i
þB
i
K
j
Þ
T
P
22
þP
22
ðA
i
þB
j
K
j
Þ þE
T
i1
E
i1
þðE
i1
þE
i2
K
j
Þ
T
ðE
i1
þE
i2
K
j
Þ þP
22
D
i
D
T
i
P
22
þ
1
q
2
P
22
P
22
< 0: ð41Þ
With W
22
¼ P
À1
22
and Y
j
¼ K
j
W
22
, (41) is equivalent to
W
22
A
T
i
þA
i
W
22
þB
i
Y
j
þY
T
j
B
T
i
þ
1
q
2
I þD
i
D
T
i
þW
22
ðE
T
i1
E
i1
þQÞW
22
þðE
i1
W
22
þE
i2
Y
j
Þ
T
ðE
i1
W
22
þE
i2
Y
j
Þ < 0: ð42Þ
By the Schur complement, (42) is equivalent to the following LMI’s:
W
ij
W
22
E
i1
W
22
þE
i2
Y
j
W
T
22
ÀðE
T
i1
E
i1
þQÞ
À1
0
ðE
i1
W
22
þE
i2
Y
j
Þ
T
0 ÀI
_
_
_
_
; ð43Þ
84 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
where
W
ij
¼ W
22
A
T
i
þA
i
W
22
þB
i
Y
j
þY
T
j
B
T
i
þ
1
q
2
I þD
i
D
T
i
:
The parameters W
22
and Y
j
, i.e., P
22
¼ W
À1
22
and K
j
¼ Y
j
W
À1
22
are obtained by
solving LMI in (43). In the second step, by substituting P
22
and K
j
into (41),
(41) becomes a standard LMIs. Similarly, we can easily solve P
11
, P
33
and G
i
from (41). Ã
5. Simulation example
To illustrate the proposed fuzzy robust tracking control approach, a control
problem of balancing an inverted pendulum on a cart is considered. Denote
x
1
¼ x, x
2
¼ _ xx, then the equation of the motion is given by [9] _ xx
1
¼ x
2
_ xx
2
¼
g sinðx
1
Þ Àamlx
2
2
sinð2x
1
Þ=2 Àa cosðx
1
Þu
4l=3 Àaml cos
2
ðx
1
Þ
; ð44Þ
where x and _ xx are the angular displacement about the vertical axis (in rad) and
the angular velocity (in rad s
À1
), respectively, g ¼ 9:8 m=s
2
is the acceleration
Fig. 1. The trajectories of the state variable x
1
and the reference state variable x
r1
xð0Þ ¼ ½60°; 0Š and
^ xx
r
ð0Þ ¼ ½0; 0Š.
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 85
due to gravity, a ¼ 1=m þM, m ¼ 2 kg is the mass of the pendulum, M ¼ 8 kg
is the mass of the cart, 2l ¼ 1 m is the length of the pendulum, and u is the
force applied to the cart. It was reported in [9] that the system of (44) can be
approximated by a fuzzy plant model with the following two rules:
Plant rule 1:
IF x
1
is about 0
THEN _ xx ¼ ðA
1
þDA
1
ÞxðtÞ þB
1
uðtÞ þw
y
1
ðtÞ ¼ C
1
xðtÞ:
Plant rule 2:
IF x
1
is about Æp=2
THEN _ xx ¼ ðA
2
þDA
2
ÞxðtÞ þB
2
uðtÞ þw
y
2
ðtÞ ¼ C
2
xðtÞ;
where
A
1
¼
0 1
g
4l=3Àaml
0
_ _
; A
2
¼
0 1
2g
pð4l=3Àamlb
2
Þ
0
_ _
;
Fig. 2. The trajectories of the state variable x
2
and the reference state variable x
r2
xð0Þ ¼ ½60°; 0Š and
^ xx
r
ð0Þ ¼ ½0; 0Š.
86 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
B
1
¼
0
Àa
4l=3Àaml
_ _
; B
2
¼
0
ab
4l=3Àamlb
2
_ _
; C
1
¼ C
2
¼
1
0
_ _
T
;
b ¼ cosð88°Þ; x 2 À
p
2
;
p
2
_ _
;
where DA
1
and DA
2
represent the system parameters uncertainties but bounded,
the elements of DA
1
and DA
2
randomly achieve the values within 30% of their
nominal values corresponding to A
1
and A
2
; DB
1
¼ DB
2
¼ 0. Based on as-
sumption 1, we define
D
1
¼ D
2
¼
0:3 0
0 0:3
_ _
; E
1
¼ E
2
¼
15 0
0 15
_ _
;
E
21
¼ E
22
¼ 0;
w ¼ ½0 Dg sinðx
1
Þ=ð4l=3 Àaml cos
2
ðx
1
Þފ
T
;
where Dg ¼ gð6370 Â10
3
=6370 Â10
3
þHÞ, H 2 ½0; 100Š.
Fig. 3. The trajectories of the state variable x
1
and the reference state variable x
r1
xð0Þ ¼ ½60°; 0Š and
^ xx
r
ð0Þ ¼ ½0; 0Š.
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 87
The membership functions are chosen as:
l
1
ðxðtÞÞ ¼
1 þ
2x
1
p
; À
p
2
6x
1
60
1 À
2x
1
p
; 0 6x
1
6
p
2
_
; l
2
ðxðtÞÞ ¼
À
2x
1
p
; À
p
2
6x
1
60;
2x
1
p
; 0 6x
1
6
p
2
:
_
The reference model given by
_ xx
r1
_ xx
r2
_ _
¼
0 1
À4 À3
_ _
x
r1
x
r2
_ _
þ
0
sinðtÞ
_ _
: ð45Þ
In the following simulation, we only give the results in the case of the
variables unavailable. For Q ¼ I, and q ¼ 0:06, solving LMIs (43) and (40) by
LMI optimization algorithm [11], feedback and observer gain matrices can be
obtained as
K
1
¼ ½ 1900:13 2483:11 Š; K
2
¼ ½ 3:33 7:12 Š;
G
1
¼ ½ 1019:89 483:00 Š
T
; G
2
¼ ½ 405:37 170:41 Š
T
:
The initial values of the states are chosen xð0Þ ¼ ½60°; 0Š, ^ xxð0Þ ¼ ½0; 0Š,
x
r
ð0Þ ¼ ½0; 0Š and xð0Þ ¼ ½À60°; 0Š, ^ xxð0Þ ¼ ½0; 0Š, x
r
ð0Þ ¼ ½0; 0Š. Figs. 1–4
Fig. 4. The trajectories of the state variable x
2
and the reference state variable x
r2
xð0Þ ¼ ½À60°; 0Š
and ^ xx
r
ð0Þ ¼ ½0; 0Š.
88 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90
illustrate the closed-loop system behaviors. The simulation results show that
the T–S fuzzy model-based controller through fuzzy observer is robust against
norm-bounded parametric uncertainties.
6. Conclusions
In this paper, we have developed a robust fuzzy tracking control design
methodology for T–S fuzzy model with parameter uncertainties in both the
conditions of the state variables available or unavailable. The basic approach is
based on the rigorous Lyapunov stability theory, and the basic tool is LMI.
Some sufficient conditions for robust stabilization of the fuzzy system and
achieving the H
1
tracking performance are formulated in the LMIs format. To
demonstrate the effectiveness of the proposed controller design method, the
design scheme is applied to controlling the inverted pendulum system. The
simulation results have verified the effectiveness of the proposed control design
method.
Acknowledgements
Supported by the natural Science foundation of China and Liaoning
province.
References
[1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and
control, IEEE Trans. Syst. Man Cybern. SMC-15 (1) (1985) 116–132.
[2] Y.C. Chen, C.C. Teng, A model reference control structure using fuzzy neural network, Fuzzy
Sets and Systems 73 (1995) 291–312.
[3] J.S.R. Jang, C.T. Sun, Neural-fuzzy modeling and control, Proc. IEEE 38 (3) (1995) 378–405.
[4] H.O. Wang, K. Tanaka, M.F. Griffin, An approach to fuzzy control of nonlinear systems:
stability and design issues, IEEE Trans. Fuzzy Syst. 4 (1) (1996) 14–23.
[5] L.K. Wang, F.H.F. Leung, P.K.S. Tam, fuzzy model-based design of fuzzy logic controllers
and its application on combining controllers, IEEE Trans. Ind. Elect. 45 (3) (1998) 502–509.
[6] H.J. Lee, J.B. Park, G. Chen, Robust fuzzy control of nonlinear systems with parametric
uncertainties, IEEE Trans. Fuzzy Syst. 9 (2) (2001) 369–379.
[7] C.C. Kung, H.H. Li, Tracking control of nonlinear systems by fuzzy model-based controller,
in: Pro. IEEE Int. Conf., vol. 2, 1997, pp. 623–628.
[8] H. Ying, Analytical analysis and feedback linearization tracking control of the general Takagi–
Sugeno fuzzy dynamical systems, IEEE Trans. Syst. Man. Cybern. 29 (1999) 290–298.
[9] F.H.F. Leung, H.K. Lam, P.K.S. Tam, fuzzy control of a class of multivariable nonlinear
systems subject to parameter uncertainties: model reference approach, Int. J. Approximate
Reasoning 26 (2001) 129–144.
S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90 89
[10] C.-S. Tseng, B.-S. Chen, H.-J. Uang, Fuzzy tracking control design for nonlinear dynamic
systems via T–S fuzzy model, IEEE Trans. Fuzzy Syst. 9 (3) (2001) 381–392.
[11] P.P. Khargonekar, I.R. Persen, K. Zhou, Robust stabilization of uncertainty linear systems:
quadratic stability and H
1
control theory, IEEE Trans. Autom. Control 35 (3) (1990) 356–
361.
90 S. Tong et al. / Internat. J. Approx. Reason. 30 (2002) 73–90

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