Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions

Abstract

This paper presents a stability analysis method for discrete-time Takagi-Sugeno fuzzy dynamic systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the fuzzy dynamic system can be established if a piecewise Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. It is also demonstrated via numerical examples that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions.

Full text

Page 1

Stability Analysis of Discrete Time Fuzzy Systems
Based on Piecewise Lyapunov Functions

LOUIS WANG and GANG FENG*
School of Electrical Engineering and Telecommunications
University of New South Wales
Sydney, N.S.W. 2052
AUSTRALIA
*Dept. of Manufacturing Engineering & Engineering Management
City University of Hong Kong
Kowloon
HONG KONG


Abstract: - This paper presents a stability analysis method for the discrete time fuzzy system based on a
piecewise Lyapunov function. It is shown that the stability of the system can be established if a piecewise
Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of Linear
Matrix Inequalities (LMI) that is numerically feasible with commercially available software.

Key-Words: - Discrete time, Fuzzy systems, Linear matrix inequality, Piecewise Lyapunov function, Stability.

1 Introduction
Fuzzy logical control (FLC) has recently proved to
be a successful control approach for certain complex
nonlinear systems, see [1–5] for example. However,
conventional fuzzy control system [1] has proved
extremely difficult to develop a general analysis and
design theory. The reason for this is believed to be
due to the fact that no mathematical model is
available from the conventional fuzzy system.
Recently, there have appeared a number of
stability analysis and controller design results in
fuzzy control literature, where the Takagi-Sugeno’s
fuzzy models are used, see references in [7-11]. The
stability of the overall fuzzy system is determined
by checking a Lyapunov function or a set of Linear
Matrix Inequalities (LMI). It is required that a
common positive definite matrix P can be found to
satisfy the Lyapunov function or the set of LMIs.
However, this is a difficult problem to solve since
such a matrix might not exist in many cases,
especially for highly nonlinear complex systems.
Most recently, a stability result of fuzzy systems
using a piecewise quadratic Lyapunov function has
been reported [14]. It is also demonstrated in the
paper that the piecewise Lyapunov function is a
much richer class of Lyapunov function candidates
than the common Lyapunov function candidates and
thus it is able to deal with a larger class of fuzzy
systems. Further references to piecewise Lyapunov
functions can be found in [15-18].
During the last few years, we have proposed a
number of new methods for the systematic analysis
and design of fuzzy logic controllers based on a so-
called fuzzy dynamic model which is similar to the
Takagi-Sugeno’s model [7-11]. These methods
include designs based on a nominal model, a
common Lyapunov function and a piecewise
Lyapunov function. However, for the methods based
on the piecewise Lyapunov function, certain
restrictive boundary conditions have to be imposed.
Motivated from the results of continuous time
piecewise Lyapunov functions in [14], we develop a
new stability theorem for discrete time fuzzy
systems based on a piecewise Lyapunov function in
this paper. This function is guaranteed to be
decreasing when the state of the system stays within
the subspace or jumps from one subspace to another.
It should be noted that with this kind of piecewise
Lyapunov function, the restrictive boundary
condition existing in our previous analysis can be
removed and global stability of the system can be
easily established. Moreover, the stability checking
procedure is to solve a set of LMIs that is
numerically feasible with commercially available
software.
The rest of the paper is organized as follows.
Section 2 introduces the discrete time fuzzy system.
Section 3 defines the piecewise Lyapunov function
candidate and the S-procedure technique. Then a
new stability theorem for discrete time fuzzy
systems is presented and an example is given to

Page 2

illustrate the theorem. Finally, conclusions are given
in Section 4.


2 Problem Formulation
The fuzzy dynamic model proposed in [7-11] can be
used to represent a complex discrete-time system
with both fuzzy inference rules and local analytic
linear models,
:
R
IF
F1 AND ...
THEN
Atx
=+
) 1(

ml
,...,2 , 1
=
where
R denotes the l-th fuzzy inference rule, m
the number of inference rules,
l
1x is
l
n x is
a
,

l
n F

(1)
ll
tx

+
)(
l
l
j F (j=1,2,...,n) are
fuzzy sets, x(t)
),(
l
A
is the l-th local model of the fuzzy system
(1), and
Let
))(( tx
be the normalized membership
function of the inferred fuzzy set
F =∏
=
j
1
∑
=
l
By using a center-average defuzzifer, product
inference and singleton fuzzifier [7-11], the discrete
time fuzzy system (1) can be expressed by the
following global model,
{
l
l
=
1
n
ℜ∈
the system state variables,
la
la is the offset.
l m
l
F where
l
n
l
j F
, and is defined as
1))((0
≤≤
tx
l m
and
1))((
1
=
tx
m
l m
. (2)
}
l
m
l
atxAtxtx
+⋅=+
∑
)( ))(() 1(
m
,
n
tx
ℜ∈
)( (3)
Remark 2.1: It is noted that the system models
defined in (1) or (3) are in fact affine systems instead
of linear systems. They include an additional offset
term. These models have much improved function
approximation capabilities [7,12].
Define L as the set of subspace indexes,
as the set of indexes for subspaces that contain the
origin and
LL ⊆
1
the set of indexes for the
subspaces that do not contain the origin. Since the
rules of the fuzzy system (1) induce a polyhedral
partition { }
Li
ℜ⊆
∈
of the state space, the fuzzy
system (3) can be viewed as a set of individual
subspaces, which consist of crisp (operating) and
fuzzy (interpolation) subspaces.
The crisp subspace is defined as the subspace
where
1)(
=
x
for some l , all other membership
functions evaluate to zero. The system dynamics of
crisp subspace is given by l-th local model of the
LL ⊆
0

n
iS
l m
fuzzy system (1). On the other hand, the fuzzy
subspace is defined as the subspace where
1)(0
<<
x
and the system dynamics is given by a
convex combination of several affine systems.
In the extreme case where all the subspaces of a
fuzzy system are crisp, that is,
l and all other membership functions are equal to
zero, then fuzzy system (3) becomes a piecewise
linear system,
Atx
=+
) 1(
terms of fuzzy system (3), the membership function,
))(( tx
for some l , could be between 0 and 1.
Thus we are required to find a mean to evaluate this
fuzzy subspace.
In our previous attempts [7-11], we treated the
fuzzy subspace in terms of uncertainties and
modelled them in terms
approximation to perform stability analysis. As
noted in [14], these same subspaces can be
considered as a region with crisp subspaces blend
and overlap each other. Hence, in each subspace, we
can write the fuzzy system (3) as a convex
combination of affine systems
{
∑
∈
)(
iKk
with
1)(0
≤≤
x
k
m
k
subspace )(iK
the system matrices used in the interpolation within
that subspace. For crisp subspace,
single element.
l m
1 ))((
=
tx
l m
for some
ll
atx
+
)(
. However, in
l m
of upper bound
}
+=+
)( ))(( ) 1(
kkk
atxAtxtx
m
,
iSx∈
(4)
, ∑∈
=
)(
1)(
iK
contains the indexes for
kx
m
. For each
iS , the set
)(iK
contains a
Assumption 2.1: We assume that given any initial
condition
0
) 0 (
xx
=
, the global model (4) has a
unique solution for all
≥
t
Assumption 2.2: We also assume that when the state
of the system transits from the subspace
the time t, the dynamics of the system is governed by
the dynamics of the local model of
0
.
iS to
j S at
iS at that time.
For future use, we also define a set Ω that
represents all possible transitions from one subspace
to another, that is,
) 1(,)(|,{:
txStxji
i
+∈=Ω
},
jiS
j
≠∈
(5)
Remark 2.2: Due to the discrete nature of the system,
it is noted that Ω as in (5) could include transitions
occurred between non-adjacent subspaces in one
step.
For convenient notation, we also introduce


00


=
kk
k
aA
A
,




=
1
x
x
, (6)

Page 3

-3
-2
-1
0
1
2
3
-2
0
2
0
0.2
0.4
0.6
0.8
1

1x
2x

m

1S
2 S
4 S
5 S
3 S
where it is assumed that
i∈
. Then using this notation, the system model
(4) can be expressed as
∑
∈
)(
iKk
with
1)(0
≤≤
x
k
m
k
comparison with (3), the state space in fuzzy system
(7) is partitioned and analyzed in terms of subspaces.
The following example illustrates the idea.
0
=
k a
for all )(iKk∈
with
0 L
⋅=+
)())(() 1(
kk
txAtxtx
m
,
iStx
∈
)(
, (7)
, ∑∈
=
)(
1)(
iK
kx
m
. Note in
Example 1: Consider a discrete time fuzzy system
that switches between 3 rules,
:
R
IF
THEN
1
) 1(
Atx
=+
:
R
IF
THEN
A 1)x(t
=+
:
R
IF
THEN
A 1)x(t
=+
where the membership function for “about
negative”, “about zero” and “about positive” is
defined as in Fig.1. The effect of these membership
functions on state space is shown in Fig. 2.
As shown, rather than consider the fuzzy system
as in the form (3),
1
1x is about negative
1
)(
atx
+

2
1x is about zero
22
a x(t)
+

3
1x is about positive
33
a x(t)
+

)( ))

(()())
(
x
(
(
( ) 1
+
(
2211
txAtxt
x
x
A
A
))
t
txtx
m
,
m
+
+
(
t
=


)
33
m
we considered it in this paper as in the form (7),
)() 1
+
) 1
+
) 1
+
) 1
+
) 1
+
(
(
(
(
(
1
t
x
(
t
x
A
x
A
x
A
m
A
m
A
t
t
t
t
t
x
x
x
x
x
=
=
=
=
=
,
t
,
(
,

1 S
S
x∈
x∈
x∈
x∈
x∈





)()(
)
x
)
t
2211
txA
m
+
,
2
2

3 S
S
)()
332
(
2
txA

t
m
+
,
4
3

5 S
so the stability of the system can be checked by a
piecewise Lyapunov function as in section 3.
Remark 2.3: During simulation, subspace
further separated by origin into 2 subregions. This
operation is carried so the S-procedure as in section
3 can be performed.
3 S is


3 Problem Solution
It is demonstrated in [14] that continuous time fuzzy
systems can be analysed by piecewise quadratic
Lyapunov function. We extend the same idea to
discrete time case. The general idea of our approach
is to consider the state space of the fuzzy system (1)










Fig.1. The membership functions for x1 for the fuzzy
system in Example 1 and 2.







Fig.2. The membership functions for x1 divide the state
space into 5 subspaces.
in terms of subspace form as in (7). Then we can use
the piecewise Lyapunov function to check for the
stability of discrete time fuzzy systems in each
subspace.
Due to the fact that the state in discrete time
system may jump between non-adjacent subspaces,
the structural information, like the matrices F’s in
[14], cannot be used to characterize the state
transition from one subspace to another as dealt with
in the case of continuous time systems. More
specifically, it may not be helpful to construct a
piecewise Lyapunov function that is continuous
across boundaries for the discrete time systems to
analyse stability of the system as in [14] for the
continuous time systems. Nevertheless, it may also
be unnecessary to require the piecewise Lyapunov
function to be continuous across boundaries for the
discrete time piecewise linear systems since the state
of such systems may never pass through the
boundaries.
As Lyapunov function candidate we consider
function of the form







-3 -2-10123
0
0.2
0.4
0.6
0.8
1
m
1x
( ) •
negative
about
m
( ) •
zero
about
m
( ) •
positive
about
m

Page 4






∈∈








∈∈
=
1
0
,,
11
,,
)(
LiSx
x
P
x
LiSxxPx
tV
ii
t
ii
t
, (8)
This function combines the power of quadratic
Lyapunov functions near an equilibrium point with
the flexibility of piecewise linear functions in the
large.
Since the matrix
the Lyapunov function in subspace
to use the S-procedure to allow the Lyapunov
function search in a less conservative way. To this
end, construct matrices,
0
=
for
i∈
such that
x
E
ii
∈∈≥

1
It should be noted that the above vector inequality
means that each entry of the vector is nonnegative.
Now, for every symmetric matrix
nonnegative entries, condition (9) implies that
iP or
iP is only used to describe
iS , it is natural
LieEE
iii
∈=
], [ with
ie
0 L
LiSx



,, 0. (9)
i
U with
0
11
>








x
EUE
x
ii
t
i
t
,
iSx∈∀
,
Li∈
. (10)
Remark 3.1: A systematic procedure for constructing
these matrices
LiEi
∈
,
for a given fuzzy system
can be found in [14]. The procedure is directly based
on the information in the fuzzy rule base. The
interested readers please refer to [14] for details.
Then we come to our theorem.
Theorem 3.1: Consider the discrete time fuzzy
system (7). If there exist symmetric matrices,
,,,,
jiPP
ji
∈

,,,
jiPP
ji
U , and
k j i
Q
such that
nonnegative entries, and the following LMIs are
satisfied,
T
ii
EUEP −<
0,
+−
iikik
WEPAPA
for
i∈
, )(iKk∈
and
EUEP −<
0,
+−
iikik
WEPAPA
for
i∈
, )(iKk∈
and
+−
iikjk
QEPAPA
for
Ω∈
ji,,
i∈
,
Lj∈
,
~
+−
iikjk
QEPAPA
0 L
1
L
∈
, symmetric matrices
U , and
k iiW
k iiW
k j i
Q
have
ii
(11)
0
<
ik i
TT
E
, (12)
0 L
ii
T
ii

<
(13)
0
i k i
TT
E
, (14)
1 L
0
and
<
(i
<
ik j i
k∈
TT
E
K
, (15)
1 L
)
0
i k j i
TT
E
, (16)
for
where we define
[
×
=
nj
IP
IP
[
×
=
then the discrete time fuzzy system is globally
exponentially stable, that is, x(t) tends to the origin
exponentially for every trajectory in the state space.
Proof: See Appendix.
Ω∈
ji,,
0 L i∈
,
Lj∈
, )(iKk∈
,
]0[]0
11
×
]
××
nnnj
T
nn
IP
for
0 Lj∈
in (15),
T
nnnjnnnj
IP
0[]0
~
11
×××
for
1 L j∈
in (16),
The above conditions are linear matrix inequalities
in the variables ,,
j
P
,
P
solution to those inequalities ensures V(t) defined in
(8) to be a piecewise Lyapunov function for the
system. The LMI in (11) or (13) for each subspace
guarantees that the function is positive and the LMI
in (12) or (14) guarantees that the function decreases
along all system trajectories in each subspace. The
LMIs (15)-(16) guarantee that the function is
decreasing when the state transits from one subspace
to another. The terms involving
are related to the S-procedure to reduce the
conservatism of those inequalities.
iP
,
jiP

k iiWU ,, and
k j i
Q
. A
k iiii
WUEE
,,,
and
k j i
Q
Remark 3.2: For each fuzzy subspace, we are
seeking for a “common” piecewise quadratic
Lyapunov function which can satisfy all the partial
influencing state matrices, and which decrease in
time within or between the subspace.
Remark 3.3: Matrices
information for each subspace. We exploit this
information even for the case when the state
trajectory
)(tx
transits between different subspaces
as in (15)-(16), based on Assumption 2.2.
i E ,
i E are the structural
Remark 3.4: Due to the discrete nature of the system,
it is noted that transitions could occur between non-
adjacent subspaces in one step. Thus, every subspace
pair, (
j
S , ) as defined by (5), has to be computed
in (15)-(16).
iS
Remark 3.5: It is noted that when the state of the
system does transit across the boundaries, that is,
SStx
∩∈
)( for some t, the result in Theorem 3.1
still holds since the case can be covered by
considering the transition from the subspace
1
+
t
.
Remark 3.6: Theorem 3.1 is only a sufficient
condition for system stability. Thus, the discrete
time fuzzy system may still be stable even if the
piecewise Lyapunov function (8) can not be
identified from the above inequalities. Shall
ji
iS to
j S at the time t or

Page 5

-4 -3-2-101234
-4
-3
-2
-1
0
1
2
3
4
1 x
2 x
Theorem 3.1 fail to generate solutions, one may
refine the partition in order to increase the flexibility
of the Lyapunov function candidate and try anew
[14].
Remark 3.7: The stability test of the discrete time
fuzzy system in (11)-(16) can be easily facilitated by
a commercially available software package Matlab
LMI toolbox [19-20].

Example 2: Consider the discrete time fuzzy system
given in Example 1. The system matrices are given
by


11 . 0


9 . 002. 0


11 . 0
The trajectory of simulations result with initial
conditions
x
] 33 [) 0 (
=
,3 [
3[
−−
indicates that the fuzzy system is stable
though there does not exist a common positive
definite matrix P for the system, see Fig. 3. The
matrices characterizing the subspaces are given by

02

9 . 0
−
=
1 . 0
1 A
,





−
0
=
02. 0

0
0
0
1 a
,


−
=
. 01
2
A
,


. 0
=
2 a
,


9 . 0
−
=
1 . 0
3
A
,



=
02
3 a
.
TT] 3
−
,
T] 33[−
, and
T] 3




−−
1
−
=
30
201
1
E
,




−
2
−
1

1
=
0
0
101
2
E
,



−
=
0
0
0
1
01
3a
E
,



−
1
=
00
3b
E
,




−
=
101
2
4
E
,




−
=
20
301
5
E
.
Using the Theorem 3.1, we can find the following
solutions to those 54 LMIs,











−−
. 8
−
−
188
57
−
=
2465
7338
.3397
3698
. 13962
6622
. 102
45
3397. 14397 . 895689 . 8
3962 . 1025689 . 85426. 41
1P
,









−−−
−
−
=
6759. 201021
−
81
. 17338.57
1021 . 14603.89 3698 . 8
..
2 P
,




−
==
1973
57
.6340 . 2
. 8
6340. 25758. 74
33
ba
PP
,










−
=
6759. 201021 . 1 7338.57
1021 . 14603.893698 . 8
7338. 36986622.45
4 P
,















Fig.3. Simulation and level curves of the computed
piecewise quadratic Lyapunov function in Example 2.


=
3397 . 13962 .102
and thus one can verify that the fuzzy system is
exponentially stable. Note LMI solutions
and
conservatism which requires them to be positive
definite was reduced by the S-procedure. The
piecewise Lyapunov function obtained is shown in
dashed line in Fig. 3.










−
2465.188
3397 . 14397.895689 . 8
3962.102 5689. 85426. 41
5 P
,
1P ,
2 P ,
4 P
5 P are indefinite. This is because the

4 Conclusion
In this paper, a new method is developed to test
stability of discrete time fuzzy system based on a
piecewise Lyapunov function. It is shown that the
stability can be determined by solving a set of LMIs.
The approach can be extended to performance
analysis of such systems as in [14] for their
continuous counterparts.


Acknowledgement
This work was financially supported by the
Australia Research Council, and City University of
Hong Kong, which is gratefully acknowledged.


Appendix
Proof of Theorem 3.1: Consider the Lyapunov
function candidate (8), or in a more compact form
xPxtV
i
=
)(
,
x∈
,
Li∈
t
iS
. (17)
It is obvious from (8) that in an open neighborhood
of the origin there exists a constant 0
>
b
such that