DETERMINING ELLIPSOIDAL BASINS OF ATTRACTION OF

FUZZY SYSTEMS

Carlos Ari

no and Antonio Sala

Systems Engineering and Control Dept., Univ. Polit

ecnica de Valencia

Cno. Vera s/n, 46022 Valencia, Spain

Keywords:

Fuzzy systems, stability, LMI.

Abstract:

This paper discusses how to obtain local stability results from a fuzzy system for which global ones cannot

be obtained, basically due to infeasibility of some associated LMI problems. Two different approaches are

compared: modifying the consequent models vs. setting up some relaxed LMI conditions if bounds on the

memberships are known. Some examples are used to illustrate the approaches.

1 INTRODUCTION

In many literature contributions, LMI stability condi-

tions (Boyd et al., 1994) are devised in order to prove

stability and performance of Takagi-Sugeno (Takagi

and Sugeno, 1985) fuzzy systems; however, such laws

are usually independent of the values of membership

functions, and fulﬁll for any arbitrary shapes of them

(Tanaka and Wang, 2001; Wang et al., 1996). Knowl-

edge of the shape of the membership functions may

allow to lift some conservativeness.

For instance, if the usual Jacobian linearisation in

x = 0 is stable, Lyapunov 1st theorem states that

there exists a region in which the system is locally

stable. The approach in this paper allows to explicitly

deﬁne a minimum spherical zone around the equilib-

rium point where Lyapunov stability conditions are

fulﬁlled, even in the case global quadratic-stability re-

lated LMIs are infeasible. Indeed, (Tanaka and Wang,

2001) shows that the basin of attraction for fuzzy sys-

tems may be membership dependent.

The structure of the paper is as follows: Next sec-

tion discusses notation and widely-known stability

theorems. Section 3 discusses a transformation of

a fuzzy model when the membership functions are

themselves a convex combination of some vertices.

Section 4 applies the results to ﬁnd the largest lo-

cal quadratically stable region. Some examples are

provided in Section 5, and a conclusion section sum-

marises the main results.

2 PRELIMINARIES

Let us consider a Takagi-Sugeno (Takagi and Sugeno,

1985) (TS) fuzzy model:

˙x =

i=1

(x)(A

· x) (1)

where µ

represents membership functions such that:

i=1

(x) = 1, µ

(x) > 0 ∀x i : 1 . . . n

Stability of fuzzy systems

Lyapunov stability theory proves that such a system is

stable if exist a function V (x) such that:

V (x) > 0,

< 0, V (0) = 0, ∀ x 6= 0

The analysis of the Lyapunov stability of TS fuzzy

systems may be approached as a linear matrix in-

equality (LMI) optimization problem (Boyd et al.,

1994). The most popular Lyapunov Functions pro-

posed in literature are quadratic forms: V (x) =

P x. This type of Lyapunov functions fulﬁll the

srability conditions if P is deﬁnite positive and if

V =

i=1

P + P A

)x < 0 (2)

That holds if

P + P A

< 0, i : 1..n (3)

Ariño C. and Sala A. (2006).

DETERMINING ELLIPSOIDAL BASINS OF ATTRACTION OF FUZZY SYSTEMS.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 86-91

DOI: 10.5220/0001203200860091

 SciTePress

The above equation is an LMI, hence widely avali-

able LMI optimization software either ﬁnds a P or

determines that the LMI is infeasible. The reader

is referred to (Tanaka and Wang, 2001) for ample

discussion.

Remark: Note that the membership functions µ

do not appear in the LMI conditions. Hence, the same

P deﬁnes a quadratic Lyapunov function for multi-

ple nonlinear systems with the same “vertex models”

as the original one. Such generality is a too restrictive

condition that in some cases results in infeasibility be-

ing the underlying system actually stable.

When the above LMI problems are unfeasible,

other alternative conditions must be sought. Fuzzy or

piecewise Lyapunov functions are discussed in (Jo-

hansson, 1999), Fuzzy Lyapunov functions are dis-

cussed in (Oliveira et al., 1999).

A different alternative, in the authors’ opinion,

is trying to achieve local stability results in a zone

around the equilibrium as large as possible. Such a

result is motivated on the ﬁrst Lyapunov theorem for

local stability: if the linearised system in x = 0 is ex-

ponentially stable, then so it is the nonlinear one, for

initial conditions in a sufﬁciently small neighborhood

of x = 0.

3 LOCAL FUZZY MODELS

In order to analyze the local stability of a TS fuzzy

model (1) within a region, the original model is

modiﬁed using the information of the membership

functions.

Lemma 1 if the membership functions µ(x) of a

fuzzy system described in (1) in a region of Ω can

be themselves expressed as a convex sum of some

vectors v

µ(x) =

p=1

(x)v

, ∀ x ∈ Ω (4)

where:

µ(x) = [µ

(x), µ

(x), . . . , µ

(x)]

p=1

(x) = 1 β

(x) > 0 ∀x ∈ Ω p : 1 . . . n

Then the system can be transformed to:

˙x =

p=1

(x)A

∗

· x (5)

where

∗

(6)

Proof: The expression (4) can be substituted in the

system equation (1):

µ(x) =

p=1

(x)v

(7)

= [v

, v

, . . . , v

] (8)

(x) =

p=1

(x)v

(9)

˙x =

i=1

p=1

(x)v

· x (10)

˙x =

p=1

(x)

· x (11)

so the local representation of the system in Ω

˙x =

p=1

(x)A

∗

· x ∀x ∈ Ω

where:

p=1

(x) = 1 β

(x) > 0 ∀x ∈ Ω p : 1 . . . n



The convex-combination conditions for the mem-

bership functions required in the above lemmas are

easy to meet. Indeed µ

are assumed known in fuzzy

systems. Then, the result below may be applied to

obtain a (possibly conservative) vertex set.

Note 1 Let us consider a region Ω. If bounds µ

and µ

on the extremum values of the membership

functions in Ω can be computed, in such a way that:

≥ max

x∈Ω

(x) µ

≤ min

x∈Ω

(x) (12)

then there exist a set of β

(x), p = 1, . . . , n

so that

the vector of membership functions

µ(x) = [µ

(x), µ

(x), . . . , µ

(x)]

may be expressed in Ω as:

µ(x) =

p=1

(x)v

, x ∈ Ω (13)

where:

p=1

(x) = 1 β

(x) > 0 ∀x ∈ Ω p : 1 . . . n

Indeed, the linear restrictions µ

≥ µ

= 1 describe a bounded polytope with a ﬁnite

number of vertices (Luenberger, 2003).

DETERMINING ELLIPSOIDAL BASINS OF ATTRACTION OF FUZZY SYSTEMS

Well-known linear-programming-related methods

to obtain the membership vector vertices may be used

(related to the obtention of the basic feasible solutions

in an LP problem (Luenberger, 2003)). A related al-

ternative is described below.

Lemma 2 Consider the set Σ

of at most 2

n−1

vec-

tors deﬁned by:

= {[˜µ

, ..., ˜µ

i−1

, X, ˜µ

i+1

, ..., ˜µ

X = 1 −

1 ≤ j ≤ n

j 6= i

˜µ

such that ˜µ

∈ {µ

, µ

} j 6= i, µ

≤ X ≤ µ

}

(14)

Then, the vectors belonging to the set

Σ =

[

i=1

(15)

satisfy (13) for some β

Indeed, as there is only one equality restriction in

memberships, all except one of them are “free” to

attain an extremum value; the remaining one must

fulﬁll the add-1 restriction and be inside its required

bounds. The above lema produces the union of all the

“all minus one” combinations, and the sought vertices

will belong to such set.

Example. For instance, if three memberships

have minimum and maximum values given by

{0.15,0.3,0.35} and {0.6,0.5,0.4}, the set Σ

is origi-

nated by the four combinations:

{(X

, 0.3, 0.35), (X

, 0.5, 0.35),

, 0.3, 0.4), (X

, 0.5, 0.4)}

with X

= 1 − 0.65 = 0.35, X

= 0.15, X

= 0.3,

= 0.1. As X

is out of the required range, the

candidate vertices kept are:

= {(0.35, 0.3, 0.35), (0.15, 0.5, 0.35),

(0.3, 0.3, 0.4)}

The set Σ

is generated by:

{(0.15, X

, 0.35), (0.6, X

, 0.35),

(0.15, X

, 0.4), (0.6, X

, 0.4)}

with X

= 0.5, X

= 0.05, X

= 0.45 and X

= 0.

Hence,

= {(0.15, 0.5, 0.35), (0.15, 0.45, 0.4)}

Regarding the third membership,

{(0.15, 0.3, X

), (0.6, 0.3, X

(0.15, 0.5, X

), (0.6, 0.5, X

)}

results in

= {(0.15, 0.5, 0.35)}

hence the resulting set of vertices to compute the local

models is:

Σ = {(0.35, 0.3, 0.35), (0.15, 0.5, 0.35),

(0.3, 0.3, 0.4), (0.15, 0.45, 0.4)}

4 STABILITY ANALYSIS IN A

ZONE

The knowledge of the membership functions will al-

low to obtain some local stability analysis results for

a fuzzy systems. Two alternatives may be applied:

the ﬁrst one will use the above deﬁned local models;

the second one will use some relaxations on LMI con-

ditions via additional variables and knowledge of the

minimum and maximum bounds on membership.

4.1 Local Fuzzy Models

By using the transformed models discussed in the pre-

vious section, local stability results may be obtained

by the lemmas in Section 2.

Lemma 3 The ellipsoidal region Ω

∗

⊂ Ω

Ω

∗

= {x \x

P x ≤ V

, P > 0} (16)

is a basin of attraction of the equilibrium point x = 0

of the system (1) if

(17)

≤ min{x

P x \x ∈ ∂Ω} (18)

where ∂Ω denotes the boundary of Ω and P veriﬁes:

∗

P + P A

∗

< 0 p : 1, . . . , n

(19)

i.e., all trajectories with initial state in Ω

∗

converge

assymptotically to x = 0.

Proof: As, by Lemma 1, the system can be ex-

pressed in Ω as:

˙x =

p=1

(x)A

∗

· x

if the LMI (19) is feasible for a positive deﬁnite matrix

P , V (x) = x

P x is a decreasing function with time,

so a Lyapunov function has been obtained ensuring

that Ω

∗

is an invariant set. La Salle’s theorem (Khalil,

1996) ensures that every solution starting in Ω

∗

will

approach x = 0.

As the expression of the local system (5) is not valid

outside Ω, then the local stability can only be proved

in the largest ellipsoid Ω

∗

contained in Ω, which will

be deﬁned by a value of V

equal to the minimum

value of V (x) in the boundary of Ω (∂Ω). 

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

The following lemma is useful in order to set up

an LMI characterisation of the largest ellipsoid in Ω

which is a Lyapunov equipotential

Suppose Ω deﬁned as a symmetric polytope that

contains x = 0:

Ω = {x\|a

x| ≤ 1i : 1, ..., n

} (20)

Lemma 4 Θ = {x\x

−1

x ≤ 1}, Q = Q

> 0

is an ellipsoid contained in Ω which itself contains

the maximum volume sphere centered at x = 0 if the

LMI problem

minimize λ

subject to λI > Q

−1

> 0

Q > 0, a

≤ 1, i = 1, ...n

is feasible. Then, no other ellipsoid in Ω contains a

larger centered sphere. 

The proof appears in (Boyd et al., 1994) chapter 3.

Theorem 1 Consider the system (1). The largest

spherical basin of attraction of x = 0 provable by

a quadratic Lyapunov function in a symmetric poly-

topic region Ω has a radius λ

−

given by the solution

of the following LMI problem:

minimize λ subject to

λI > P > 0 (21)

P > 0 (22)



P a



> 0, j : 1...n

(23)

∗

P + P A

∗

< 0, p : 1...n

(24)

and Ω is deﬁned as (20). The ellipsoid Θ =

{x\x

P x ≤ 1} is, of course, also contained in the

basin of attraction of x = 0.

Proof: Conditions 24 imply that trajectories inside

any equipotential region deﬁned by P converge to the

point x = 0, as shown in Lemma 3.

Applying the Schur complement, the conditions (23)

are equivalent to

−1

< 1, i : 1..n

Then, conditions (23) keep Θ inside Ω and the condi-

tion (21) along with the LMI objective, maximize the

radius of the quadratically invariant sphere contained

in Θ, from Lemma 4. 

Largest is here understood as containing the largest

spherical ball around x = 0, i..e, guaranteeing stability for

the largest initial distance to the origin.

4.2 Relaxed LMI Conditions

Another way to approach the problem is relaxing

the LMI conditions using que information about the

membership functions µ

in the zone Ω in which lo-

cal stability is studied. This will allow to express

some results (possibly more conservative than the pre-

vious one) using the minimum and maximum val-

ues of memberships in the zone (or some bounds on

them), without the need of calculating transformed lo-

cal models.

Assume that, in the zone Ω, the limits of µ

are

≤ µ

(25)

Then, for any positive τ ∈ R:

τ ≤ µ

τ = µ

i=1

τ (26)

where

i=1

= 1 has been used in the equality.

Then, for any positive deﬁnite N

x ≤ µ

i=1

x (27)

Hence, the term

i=1

x − µ

x > 0

may be added to the stability condition (3), so that if

i=1

P + P A

)x +

i=1

−µ

x < 0 (28)

then, the equation (2) holds. Reordering the terms,

the LMI conditions below are obtained:

P + P A

) + µ

< 0, ∀i 6= j (29)

P + P A

) − (1 − µ

< 0, N

> 0 (30)

and adding the condition µ

≤ µ

, for any positive

symmetric matrix N

, the expression

x −

i=1

x > 0 (31)

can be proved analogously to the maximum N

case.

Then (2) is positive if

P + P A

) + µ

−µ

< 0 ∀i 6= j (32)

P + P A

) − (1 − µ

+(1 − µ

< 0, N

, N

> 0 (33)

Note that, in the above expressions, j is a ﬁxed

number. If a bound of µ

is known for all j, the theo-

rem below can be proved.

DETERMINING ELLIPSOIDAL BASINS OF ATTRACTION OF FUZZY SYSTEMS

Theorem 2 Consider the system (1). The largest

spherical basin of attraction of x = 0 provable by

a quadratic Lyapunov function in a symmetric poly-

topic region Ω has a radius λ

−

given by the solution

of the following LMI problem in the variables P , N

minimize λ subject to

λI > P > 0 (34)

P > 0 (35)



P a



> 0, j : 1...n

(36)

P + P A

− (1 − µ

+ (1 − µ

j6=i

(µ

− µ

) < 0, i : 1...n (37)

and Ω is deﬁned as (20). The ellipsoid Θ =

{x\x

P x ≤ 1} is, of course, also contained in the

basin of attraction of x = 0. 

4.3 Algorithm

The results in previous sections may be combined in

order to obtain an algorithm to compute the largest

ball around x = 0 for which attraction is ensured.

Basically, the procedure will ﬁrst check the extreme

cases: (1) checking for feasibility of LMI problems

as stated in Section 2 (2) checking for stability of the

linearised model around x = 0.

If the ﬁrst one is unfeasible but the second one

is feasible, selecting a polytopic region on the state

space and a scaling factor ρ allows to set up a bisec-

tion procedure in order to determine the largest feasi-

ble ρ.

5 EXAMPLES

Example 1. Let us have a fuzzy system given by:

˙x =

i=1

(x)A

x (38)



−0.5 −1

−1 −0.5



(39)



−0.5 1

1 −0.5



(40)

Figure 1 shows the membership functions µ

and µ

which, for simplicity, depend only on x

. The value

of a = 1 will be assumed.

−a

Figure 1: Membership functions µ

), µ

Deﬁne Ω

as a rectangle bounded in x

, unbounded

in x

Ω

= {x\ |(0 1/ρ

)x| ≤ 1}

where k is the iteration number.

Note that the maximum and minimum values of µ

in Ω are easily obtained, and the Lemma 4 can be

applied.

In the proposed procedure, the LMIs for ρ = 1 are

unfeasible. However, the linearised model is:

˙x = (0.5A

+ 0.5A

)x =



−0.5 0

0 −0.5



(41)

which is stable. Hence, there exists a zone around

x = 0 (possibly small) where local stability holds.

The procedures in this paper allow to determine the

largest sphere around x = 0 for which local quadratic

stability holds.

Let us consider for the ﬁst iteration ρ

= 0.1. The

maximum and minimum values of µ are, in that case:

= 0.55, µ

= 0.45, µ

= 0.55, µ

= 0.45

Then the vertices obtained in the region Ω

are:

= [

0.45 0.55

]

= [

0.55 0.45

]

The local fuzzy model from Lemma 1 is described by:

∗



−0.5 0.1

0.1 −0.5



∗



−0.5 −0.1

−0.1 −0.5



And, solving the LMIs:

∗

P + P A

∗

< 0

∗

P + P A

∗

< 0

X > 0

local stability in a certain ellipsoidal region inside Ω

is proved.

When the same procedure is applied to ρ = 0.5 the

LMIs are unfeasible. The LMIs are, however, feasible

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

Ω

Figure 2: The Basin of attraction of x = 0 provable by

quadratic stability (Example 1).

for any ρ < 0.5. for instance, ρ

= 0.499 results in

the following LMI conditions:

∗

P + P A

∗

< 0

∗

P + P A

∗

< 0

1/ρ

0 1/ρ

> 0

P > 0

λI > P

which are feasible for the above value of ρ

and, the

matrix P obtained for the minimum λ deﬁnes an ellip-

soidal basin of atraction Θ = {x\x

P x ≤ 1} with

P =

3.8274 0

0 4.016

Figure 2 depicts the obtained basin of attraction,

and some examples of trajectories from a set of start-

ing points. The maximum circle found by the LMI

toolbox has a radius of 0.499.

Trajectories with starting points inside the ellipsoid

Θ are guaranteed to converge to the origin. Note that

the ellipsoid obtained by the LMIs is not a circle.

Points outside the ellipsoid may lead to either con-

vergent or non-convergent trajectories.

Example 2. Let us considerer the same fuzzy

model as in Example 1. Then applying the theorem

minimize λ subject to

P + P A

) − (1 − µ

+ (1 − µ

+µ

− µ

< 0 ∀i 6= j

P + P A

) − (1 − µ

+ (1 − µ

+µ

− µ

< 0

1/ρ

0 1/ρ

> 0

P , N

, N

> 0

λI > P

Conveniently, we take the same region shape Ω that

in Example 1, the limits µ

and µ

are the mini-

mum and maximum value of µ

in the region Ω. The

maximum ρ obtained is 0.26, i.e. 0.24 units less than

the obtained in the previous example. From this ex-

ample, the conditions discussed in Section 4.2 seem

more conservative than those in Section 4.1.

6 CONCLUSIONS

This paper shows how local stability results (the

largest sphere around x = 0 for which a quadratic

Lyapunov function can be proven via LMI) may

be obtained in fuzzy systems via the knowledge of

the membership functions, even when no feasible

quadratic Lyapunov function can be found to prove

global stability. The found sphere is part of a larger

ellipsoidal guaranteed basin of attraction.

In this way, if the linearised system around the

equilibrium is stable, a precise characterisation of the

local stability region stated in Lyapunov 1st theorem

is achieved.

The approach based on relaxed LMI conditions

from membership bounds yields more conservative

results but it is simpler, without the need of changing

the Takagi-Sugeno consequents.

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DETERMINING ELLIPSOIDAL BASINS OF ATTRACTION OF FUZZY SYSTEMS