On Controllability and Observability of Fuzzy Dynamical ...downloads.hindawi.com/journals/afs/2008/421525.pdf · lites are well known. In general, fuzzy systems are mainly classiﬁed

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2008, Article ID 421525, 16 pagesdoi:10.1155/2008/421525

Research ArticleOn Controllability and Observability of Fuzzy DynamicalMatrix Lyapunov Systems

M. S. N. Murty and G. Suresh Kumar

Department of Applied Mathematics, Acharya Nagarjuna University, Nuzvid Campus, Nuzvid 521 201, Andhra Pradesh, India

Correspondence should be addressed to M. S. N. Murty, [email protected]

Received 27 September 2007; Accepted 1 January 2008

Recommended by Hao Ying

We provide a way to combine matrix Lyapunov systems with fuzzy rules to form a new fuzzy system called fuzzy dynamical matrixLyapunov system, which can be regarded as a new approach to intelligent control. First, we study the controllability property of thefuzzy dynamical matrix Lyapunov system and provide a sufficient condition for its controllability with the use of fuzzy rule base.The significance of our result is that given a deterministic system and a fuzzy state with rule base, we can determine the rule basefor the control. Further, we discuss the concept of observability and give a sufficient condition for the system to be observable. Theadvantage of our result is that we can determine the rule base for the initial value without solving the system.

Copyright © 2008 M. S. N. Murty and G. Suresh Kumar. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. INTRODUCTION

The importance of control theory in applied mathematicsand its occurrence in several problems such as mechanics,electromagnetic theory, thermodynamics, and artificial satel-lites are well known. In general, fuzzy systems are mainlyclassified into three categories, namely pure fuzzy systems,T-S fuzzy systems, and fuzzy logic systems, using fuzzifiersand defuzzifiers. In this paper, we use fuzzy matrix Lyapunovsystem to describe fuzzy logic system. The purpose of thispaper is to provide sufficient conditions for controllabilityand observability of first-order fuzzy matrix Lyapunovsystem modeled by

X ′(t) = A(t)X(t) + X(t)B(t) + F(t)U(t),

X(0) = X0, t > 0,(1)

Y(t) = C(t)X(t) +D(t)U(t), (2)

where U(t) is an n × n fuzzy input matrix called fuzzycontrol and Y(t) is an n × n fuzzy output matrix. HereA(t),B(t),F(t),C(t), and D(t) are matrices of order n × n,whose elements are continuous functions of t on J = [0,T] ⊂R(T > 0).

The problem of controllability and observability for asystem of ordinary differential equations was studied by

Barnett and Cameron [1] and for matrix Lyapunov systemsby Murty et al.[2]. Fuzzy control usually decomposes acomplex system into several subsystems according to thehuman expert’s understanding of the system and uses a sim-ple control law to emulate the human control strategy.Thereexist two major types of fuzzy controllers, namely Mamdanifuzzy controllers and Takagi-Sugeno (TS) fuzzy controllers.They mainly differ in the consequence of fuzzy rules: theformer uses fuzzy sets whereas the latter employs (linear)functions. Takagi and Sugeno [3, 4] propose a type of fuzzymodel in which the consequent part of the rules is definednot by the membership function but by a crisp analyticalfunction. More and more interest appears to shift towardsTS fuzzy controllers in recent years, as evidenced by theincreasing number of papers in this direction and due to theirapplications in real world problems (e.g., [5–12]).

Recently, the controllability and observability criteria forfuzzy dynamical control systems were discussed by Ding andKandel [13, 14]. In this paper, by converting the fuzzy matrixLyapunov system into a Kronecker product system we obtainsufficient conditions for controllability and observability ofthe system (1) satisfying (2).

The paper is well organized as follows. In Section 2, wepresent some basic definitions and results relating to fuzzysets [13] and Kronecker product of matrices. Further, weobtain a unique solution of the system (1), when U(t) is a

mailto:[email protected]

2 Advances in Fuzzy Systems

crisp continuous matrix. In Section 3, we generate a fuzzydynamical Lyapunov system, and also obtain its solution set.In Section 4, we present a sufficient condition for the con-trollability of the system and illustrate the results by suitableexamples. In Section 5, we obtain a sufficient condition forthe observability of the fuzzy dynamical Lyapunov system,and the theorem is highlighted by a suitable example. Finally,in Section 6, we present some conclusions and future works.

This paper extends some of the results of Ding andKandel [13, 14] developed for system of fuzzy differentialequations to fuzzy matrix Lyapunov systems and includestheir results as a particular case, when B(t) = 0,X ,U , andY are column vectors of order n.

2. PRELIMINARIES

In this section, we present some definitions and resultsrelating to fuzzy sets [13] and Kronecker product of matrices.

Let Pk(Rn) denote the family of all nonempty compactconvex subsets of Rn. Define the addition and scalar multi-plication in Pk(Rn) as usual. Radstrom [15] states that Pk(Rn)is a commutative semigroup under addition, which satisfiesthe cancellation law. Moreover, if α,β ∈ R andA,B ∈ Pk(Rn),then

α(A + B) = αA + αB, α(βA) = (αβ)A, 1·A = A,(3)

and if α,β ≥ 0, then (α + β)A = αA + βA. The distancebetween A and B is defined by the Hausdorff metric

d(A,B) = inf{ε : A ⊂ N(B, ε),B ⊂ N(A, ε)

}, (4)

where

N(A, ε) = {x ∈ Rn : ‖x − y‖ < ε, forsomey ∈ A

}. (5)

Definition 1. A set-valued function F : J → Pk(Rn) is saidto be measurable if it satisfies any one of the followingequivalent conditions:

(1) for all u ∈ Rn, t → dF(t)(u) = inf v∈F(t)‖u − v‖ ismeasurable,

(2) GrF = {(t,u) ∈ J×Rn : u ∈ F(t)} ∈ Σ×β(Rn), whereΣ,β(Rn) are Borel σ-field of J and Rn, respectively(Graph measurability),

(3) there exists a sequence { fn(·)}n≥1 of measurable

functions such that F(t) = { fn(·)}n≥1, for all t ∈ J(Castaing’s representation).

We denote by S1F the set of all selections of F(·) that

belong to the Lebesgue Bochner space L1Rn(J), that is,

S1F =

{f (·) ∈ L1

Rn(J) : f (t) ∈ F(t)a.e.}. (6)

We present the Aumann’s integral as follows:

(A)∫

JF(t)dt =

{∫

Jf (t)dt, f (·) ∈ S1

F

}. (7)

We say that F : J → Pk(Rn) is integrably bounded if it ismeasurable and there exists a function h : J → R, h ∈ L1

Rn(J),such that ‖u‖ ≤ h(t), u ∈ F(t). From [16], we know that if Fis a closed valued measurable multifunction, then

∫J F(t)dt is

convex in Rn. Furthermore, if F is integrably bounded, then∫J F(t)dt is compact in Rn.

Let

En ={u : Rn −→ [0, 1]/u satisfie(i)–(iv) below

}, (8)

where

(i) u is normal, that is, there exists an x0 ∈ Rn such thatu(x0) = 1;

(ii) u is fuzzy convex, that is, for x, y ∈ Rn and 0 ≤ λ ≤ 1,

u(λx + (1− λ)y

) ≥ min[u(x),u(y)

]; (9)

(iii) u is upper semicontinuous;

(iv) [u]0 = {x ∈ Rn/u(x) > 0} is compact.

For 0 < α ≤ 1, the α-level set is denoted and defined by[u]α = {x ∈ Rn/u(x) ≥ α}. Then, from (i)–(iv) it followsthat [u]α ∈ Pk(Rn) for all 0 ≤ α ≤ 1.

Define D : En × En → [0,∞) by

D(u, v) = sup{d([u]α, [v]α

)/α ∈ [0, 1]

}, (10)

where d is the Hausdorff metric defined in Pk(Rn). It is easyto show that D is a metric in En and using results of [15],we see that (En,D) is a complete metric space, but not locallycompact. Moreover, the distance D verifies that

D(u +w, v +w) = D(u, v), u, v,w ∈ En,

D(λu, λv) = |λ|D(u, v), u, v ∈ En, λ ∈ R,

D(u +w, v + z) ≤ D(u, v) +D(w, z), u, v,w, z ∈ En.(11)

We note that (En,D) is not a vector space. But it can beimbedded isomorphically as a cone in a Banach space [15].

Regarding fundamentals of differentiability and inte-grability of fuzzy functions, we refer to Kaleva [17] andLakshmikantham and Mohapatra [18].

In the sequel, we need the following representationtheorem.

Theorem 1 (see [19]). If u ∈ En, then

(1) [u]α ∈ Pk(Rn), for all 0 ≤ α ≤ 1;

(2) [u]α2 ⊂ [u]α1 , for all 0 ≤ α1 ≤ α2 ≤ 1;

(3) if {αk} is a nondecreasing sequence converging to α > 0,then [u]α = ⋂

k≥1[u]αk .

Conversely, if {Aα : 0 ≤ α ≤ 1} is a family of subsets of Rn

satisfying (1)–(3), then there exists a u ∈ En such that [u]α =Aα for 0 < α ≤ 1 and [u]0 = ⋃

0≤α≤1Aα ⊂ A0.

M. S. N. Murty and G. Suresh Kumar 3

A fuzzy set-valued mapping F : J → En is called fuzzyintegrably bounded if F0(t) is integrably bounded.

Definition 2. Let F : J → En be a fuzzy integrably boundedmapping. The fuzzy integral of F over J denoted by

∫J F(t)dt

is defined level-set-wise by

[∫

JF(t)dt

]α= (A)

∫

JFα(t)dt, 0 < α ≤ 1. (12)

Let F : J × En → En,and consider the fuzzy differentialequation

u′ = F(t,u), u(0) = u0. (13)

Definition 3. A mapping u : J → En is a fuzzy weak solutionto (13) if it is continuous and satisfies the integral equation

u(t) = u0 +∫ t

0F(s,u(s)

)ds, ∀t ∈ J. (14)

If F is continuous, then this weak solution also satisfies (13)and we call it fuzzy strong solution to (13).

Now, we present some properties and rules for Kroneckerproducts and basic results related to matrix Lyapunovsystems.

Definition 4 (see [2]). Let A ∈ Cm×n and B ∈ Cp×q.Then theKronecker product of A and B written A⊗ B is defined to bethe partitioned matrix

A⊗ B =

⎡

⎢⎢⎢⎢⎢⎢⎣

a11B a12B · · · a1nB

a21B a22B · · · a2nB

. . · · · .

am1B am2B · · · amnB

⎤

⎥⎥⎥⎥⎥⎥⎦

(15)

which is an mp × nq matrix and is in Cmp×nq.

Definition 5 (see [2]). Let A = [ai j] ∈ Cm×n; one denotes

A = VecA =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

A.1

A.2

...

A.n

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

, whereA.j =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

a1 j

a2 j

...

amj

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

(1 ≤ j ≤ n).

(16)

The Kronecker product has the following properties andrules [2].

(1) (A⊗ B)∗ = A∗ ⊗ B∗ (A∗ denotes transpose of A).

(2) (A⊗ B)−1 = A−1 ⊗ B−1.

(3) The mixed product rule

(A⊗ B)(C ⊗D) = (AC ⊗ BD).

This rule holds, provided the dimension of the matrices issuch that the various expressions exist.

(4) ‖A⊗ B‖ = ‖A‖‖B‖.

(5) If A(t) and B(t) are matrices, then

(A⊗ B)′ = A′ ⊗ B + A⊗ B′(′ = d/dt).

(6) Vec (AYB) = (B∗ ⊗ A)VecY .

(7) If A and B are matrices both of order n× n, then

(i) Vec (AX) = (In ⊗ A)VecX ,(ii) Vec (XA) = (A∗ ⊗ In)VecX .

Now by applying the Vec operator to the matrixLyapunov system (1) satisfying (2) and using the aboveproperties, we have

X ′(t) = G(t)X(t) +(In ⊗ F(t)

)U(t), X(0) = X0,

(17)

Y(t) = (In ⊗ C(t)

)X(t) +

(In ⊗D(t)

)U(t), (18)

where G(t) = (B∗ ⊗ In) + (In ⊗ A) is an n2 × n2 matrix andX = VecX(t), U = VecU(t) are column matrices of ordern2.

The corresponding linear homogeneous system of (17) is

X ′(t) = G(t)X(t), X(0) = X0. (19)

Lemma 1. Let φ(t) and ψ(t) be the fundamental matrices forthe systems

X ′(t) = A(t)X(t), X(0) = In, (20)

[X∗(t)]′ = B∗(t)X∗(t), X(0) = In, (21)

respectively. Then the matrix ψ(t) ⊗ φ(t) is a fundamentalmatrix of (19) and the solution of (19) is X(t) = (ψ(t) ⊗φ(t))X0.

Proof. Consider(ψ(t)⊗ φ(t)

)′

= (ψ(t)′ ⊗ φ(t)

)+(ψ(t)⊗ φ′(t))

= (B∗(t)ψ(t)⊗ φ(t)

)+(ψ(t)⊗ A(t)φ(t)

)

= (B∗(t)⊗ In

)(ψ(t)⊗ φ(t)

)+(In ⊗ A(t)

)(ψ(t)⊗ φ(t)

)

= [B∗(t)⊗ In + In ⊗ A(t)

](ψ(t)⊗ φ(t)

)

= G(t)(ψ(t)⊗ φ(t)

).

(22)

Also ψ(0)⊗ φ(0) = In ⊗ In = In2 .Hence, ψ(t) ⊗ φ(t) is a fundamental matrix of (19).

Clearly, X(t) = (ψ(t)⊗ φ(t))X0 is a solution of (19).

Theorem 2. Let φ(t) and ψ(t) be the fundamental matricesfor the systems (20) and (21). Then the unique solution of theinitial value problem (17) is given by

X(t) = (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)U(s)ds.

(23)


Proof. First we show that the solution of (17) is of the formX(t) = (ψ(t) ⊗ φ(t))X0 + X(t), where X(t) is a particularsolution of (17) and is given by

X(t) =∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)U(s)ds. (24)

Let u(t) be any other solution of (17), writew(t) = u(t)−X(t), then w satisfies (19), hence w = (ψ(t)⊗φ(t))X0, u(t) =(ψ(t)⊗ φ(t))X0 + X(t).

Consider the vector X(t) = (ψ(t)⊗ φ(t))v(t), where v(t)is an arbitrary vector to be determined so as to satisfy (17),

X ′(t) = (ψ(t)⊗ φ(t)

)′v(t) +

(ψ(t)⊗ φ(t)

)v′(t)

=⇒ G(t)X(t) +(In ⊗ F(t)

)U(t)

= G(t)(ψ(t)⊗ φ(t)

)v(t) +

(ψ(t)⊗ φ(t)

)v′(t)

=⇒ (ψ(t)⊗ φ(t)

)v′(t)

= (In ⊗ F(t)

)U(t)

=⇒ v′(t)

= (ψ−1(t)⊗ φ−1(t)

)(In ⊗ F(t)

)U(t)

=⇒ v(t)

=∫ t

0

(ψ−1(s)⊗ φ−1(s)

)(In ⊗ F(s)

)U(s)ds.

(25)

Hence, the desired expression follows immediately by notingthe fact that φ(t)φ−1(s) = φ(t − s) and ψ(t)ψ−1(s) = ψ(t −s).

3. FORMATION OF FUZZY DYNAMICALLYAPUNOV SYSTEMS

Let ui(t) ∈ E1, t ∈ J , i = 1, 2, . . . ,n2, and define

U(t) = (u1(t),u2(t), . . . ,un2 (t)

)

= u1(t)× u2(t)× · · · × un2 (t)

= {(uα1(t),uα2(t), . . . ,uαn2 (t)

): α ∈ [0, 1]

}

= {(u1(t), u2(t), . . . , un2 (t)

): ui(t) ∈ uαi (t),α ∈ [0, 1]

},

(26)

where uαi (t) is the α-level set of ui(t). From the abovedefinition of U(t) and Theorem 1, it can be easily seen thatU(t) ∈ En

2.

Now by using the fuzzy control U(t), we show that thefollowing system

X ′(t) = G(t)X(t) +(In ⊗ F(t)

)U(t), X(0) = X0, (27)

Y(t) = (In ⊗ C(t)

)X(t) +

(In ⊗D(t)

)U(t) (28)

determines a fuzzy system.

Assume that U(t) is continuous in En2. The set Uα =

u1(t) × u2(t) × · · · × un2 (t) is a convex and compact setin Rn

2. For any positive number T , consider the following

differential inclusions:

X ′(t) ∈ G(t)X(t) +(In ⊗ F(t)

)Uα(t), t ∈ [0,T], (29)

X(0) = X0. (30)

Let Xα be the solution of (29) satisfying (30).

Claim (i). [X(t)]α ∈ Pk(Rn2), for every 0 ≤ α ≤ 1, t ∈ [0,T].

First, we prove that Xα is nonempty, compact, and convexin C[[0,T],Rn

2]. Since Uα(t) has measurable selection, we

have that Xα is nonempty.Let K = maxt∈[0,T]‖φ(t)‖, L = maxt∈[0,T]‖ψ(t)‖,

M = max{‖u(t)‖ : u(t) ∈ Uα(t), t ∈ [0,T]}, N =maxt∈[0,T]‖F(t)‖.

If for any X ∈ Xα, then there is a selection u(t) ∈ Uα(t)such that

X(t) = (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)u(s)ds.

(31)

Then

∥∥X(t)∥∥ ≤ ∥∥(ψ(t)⊗ φ(t)

)X0

∥∥

+∫ t

0

∥∥(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)u(s)

∥∥ds

≤ ∥∥ψ(t)

∥∥∥∥φ(t)

∥∥∥∥X0

∥∥

+∫ t

0

∥∥ψ(t − s)∥∥∥∥φ(t − s)∥∥∥∥F(s)

∥∥∥∥u(s)

∥∥ds

≤ KL∥∥X0

∥∥ + KLNMT.

(32)

Thus Xα is bounded.For any t1, t2 ∈ [0,T],

X(t1)− X(t2)

= (ψ(t1)⊗ φ(t1)

)X0

+∫ t1

0

(ψ(t1 − s)⊗ φ(t1 − s)

)(In ⊗ F(s)

)u(s)ds

− (ψ(t2)⊗ φ(t2)

)X0

−∫ t2

0

(ψ(t2 − s)⊗ φ(t2 − s)

)(In ⊗ F(s)

)u(s)ds.

(33)


Therefore∥∥X(t1)− X(t2)

∥∥

≤ ∥∥(ψ(t1)⊗ φ(t1)

)− (ψ(t2)⊗ φ(t2)

)∥∥∥∥X0

∥∥

+∫ t1

t2

∥∥(ψ(t1 − s)⊗ φ(t1 − s))(In ⊗ F(s)

)u(s)

∥∥ds

+∫ t2

0

∥∥[(ψ(t1 − s)⊗ φ(t1 − s))

− (ψ(t2 − s)⊗ φ(t2 − s)

)](In ⊗ F(s)

)u(s)

∥∥ds

≤ ∥∥(ψ(t1)⊗ φ(t1)

)− (ψ(t2)⊗ φ(t2)

)∥∥∥∥X0

∥∥

+ KLNM|t1 − t2|

+MN∫ T

0

∥∥(ψ(t1 − s)⊗ φ(t1 − s))

− (ψ(t2 − s)⊗ φ(t2 − s)

)∥∥ds.(34)

Since φ(t) and ψ(t) are both uniformly continuous on [0,T],X is equicontinuous. Thus, Xα is relatively compact. If Xα isclosed, then it is compact.

Let Xk ∈ Xα and Xk → X . For each Xk, there is a uk ∈Uα(t) such that

Xk(t) = (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)uk(s)ds.

(35)

Since uk ∈ Uα(t) is closed, then there exists a subsequence{ukj} of {uk} converging weakly to u ∈ Uα(t). From Mazur’stheorem [20], there exists a sequence of numbers λj > 0,∑λj = 1 such that

∑λjukj converges strongly to u.

Thus, from (35) we have∑λjXkj (t)

=∑λj(ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)∑λjukj (s)ds.

(36)

From Fatou’s lemma, taking the limit as j →∞ on both sidesof (36), we have

X(t) = (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)u(s)ds.

(37)

Thus, X(t) ∈ Xα, and hence Xα is closed.Let X1, X2 ∈ Xα, then there exist u1,u2 ∈ Uα(t) such that

X ′1(t) = G(t)X1(t) +(In ⊗ F(t)

)u1(t),

X ′2(t) = G(t)X2(t) +(In ⊗ F(t)

)u2(t).

(38)

Let X = λX1(t) + (1− λ)X2(t), 0 ≤ λ ≤ 1, then

X ′ = λX ′1(t) + (1− λ)X ′2(t)

= λ(G(t)X1(t) +

(In ⊗ F(t)

)u1(t)

)

+ (1− λ)(G(t)X2(t) +

(In ⊗ F(t)

)u2(t)

)

= G(t)[λX1(t) + (1− λ)X2(t)

]

+(In ⊗ F(t)

)[λu1(t) + (1− λ)u2(t)

].

(39)

Since Uα(t) is convex, λu1(t) + (1− λ)u2(t) ∈ Uα(t), we have

X ′(t) ∈ G(t)X(t) +(In ⊗ F(t)

)Uα(t), (40)

that is X ∈ Xα. Thus Xα is convex. Therefore, Xα isnonempty, compact, and convex in C[[0,T],Rn

2]. Thus,

from Arzela-Ascoli theorem, we know that [X(t)]α

is com-pact in Rn

2for every t ∈ [0,T]. Also it is obvious that [X(t)]

α

is convex in Rn2. Thus, we have [X(t)]

α ∈ Pk(Rn2), for every

t ∈ [0,T]. Hence the claim.

Claim (ii). [X(t)]α2 ⊂ [X(t)]

α1, for all 0 ≤ α1 ≤ α2 ≤ 1.

Let 0 ≤ α1 ≤ α2 ≤ 1. Since Uα2 (t) ⊂ Uα1 (t), we have

Uα2 (t) = uα21 (t)× uα2

2 (t)× · · · × uα2n2 (t)

⊂ uα11 (t)× uα1

2 (t)× · · · × uα1n2 (t)

= Uα1(t).

(41)

Thus, we have the selection inclusion S1Uα2 (t)

⊂ S1Uα1 (t)

and the

following inclusion:

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uα2 (t)

⊂ G(t)X +(In ⊗ F(t)

)Uα1 (t).

(42)

Consider the differential inclusions

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uα2 (t), t ∈ [0,T], (43)

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uα1 (t), t ∈ [0,T]. (44)

Let Xα2 and Xα1 be the solution sets of (43) and (44), respec-tively. Clearly, the solution of (43) satisfies the followinginclusion:

X(t) ∈ (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)S1Uα2 (s)

ds

⊂ (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)S1Uα1 (s)

ds.

(45)

Thus Xα2 ⊂ Xα1 , and hence Xα2 (t) ⊂ Xα1 (t). Hence theclaim.


Claim (iii). If {αk} is a nondecreasing sequence convergingto α > 0, then Xα(t) = ⋂

k≥1Xαk (t).

Let Uαk (t) = uαk1 × uαk2 × · · · × uαkn2 , Uα(t) = uα1 × uα2 ×· · · × uαn2 and consider the inclusions

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uαk (t), (46)

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uα(t). (47)

Let Xαk and Xα be the solution sets of (46) and (47),respectively. Since ui(t) is a fuzzy set and from Theorem 1,we have

uαi =⋂

k≥1

uαki . (48)

Consider

Uα(t) = uα1 × uα2 × · · · × uαn2

=⋂

k≥1

uαk1 ×⋂

k≥1

uαk2 × · · · ×⋂

k≥1

uαkn2

=⋂

k≥1

(uαk1 × uαk2 × · · · × uαkn2

)

=⋂

k≥1

Uαk (t)

(49)

and then S1Uα(t)

= S1⋂k≥1U

αk (t). Therefore

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uα(t)

= G(t)X +(In ⊗ F(t)

)⋂

k≥1

Uαk (t)

⊂ G(t)X +(In ⊗ F(t)

)Uαk (t), k = 1, 2, . . . .

(50)

Thus, we have Xα ⊂ Xαk , k = 1, 2, . . ., which implies that

Xα ⊂⋂

k≥1

Xαk . (51)

Let X be the solution set to the inclusion

X ′(t) ∈ G(t)X +(In ⊗ F(t)

)Uαk (t), k ≥ 1. (52)

Then,

X(t) ∈ (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)S1Uαk (t)

ds.

(53)

It follows that

X(t) ∈ (ψ(t)⊗ φ(t)

)X0

+⋂

k≥1

∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)S1Uαk

ds

⊂ (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)S1⋂

k≥1Uαkds

= (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)S1Uαds.

(54)

This implies that X ∈ Xα. Therefore,

⋂

k≥1

Xαk ⊂ Xα. (55)

From (51) and (55), we have

Xα =⋂

k≥1

Xαk , (56)

and hence,

Xα(t) =⋂

k≥1

Xαk (t). (57)

From Claims 3–3 and applying Theorem 1, there existsX(t) ∈ En

2on [0,T] such that Xα(t) is a solution set to

the differential inclusions (29) and (30). Hence, the system(27), (28) is a fuzzy dynamical Lyapunov system, and it canbe expressed as

X ′(t) = G(t)X(t) +(In ⊗ F(t)

)U(t), X(0) = {

X0}

,(58)

Y(t) = (In ⊗ C(t)

)X(t) +

(In ⊗D(t)

)U(t). (59)

The solution set of the fuzzy dynamical system (58), (59) isgiven by

X(t) ∈ (ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)U(s)ds.

(60)

Remark 1. Consider a special case. If the input is in the form

U(t) = u1(t)× u2(t)× · · · × ui(t)× · · · × un2 (t), (61)

where uk(t) ∈ R1, k /= i, are crisp numbers, then the ithcomponent of the solution set of (27) is a fuzzy set in E1.

Proof. The proof follows along similar lines as in the abovediscussion.


4. CONTROLLABILITY OF FUZZY DYNAMICALLYAPUNOV SYSTEMS

In this section, we discuss the concept of controllability of thefuzzy system (58) satisfying (59).

Definition 6. The fuzzy system (58), (59) is said to becompletely controllable if for any initial state X(0) = X0 andany given final state X f there exists a finite time t1 > 0 and a

control U(t), 0 ≤ t ≤ t1, such that X(t1) = X f .

Lemma 2. If F is a fuzzy set, then∫ T

0 F dt = TF.

Proof. Let [F]α be the α-level set of F. Since[∫ T

0F dt

]α=∫ T

0[F]αdt = T[F]α. (62)

From the definition of fuzzy set, we have∫ T

0 F dt = TF.

Lemma 3. Let P,Q be two fuzzy sets and let h(t) be a nonzerocontinuous function on [0,T], satisfying

∫ T

0h(t)P dt =

∫ T

0h(t)Qdt (63)

then P = Q.

Proof. For each α-level, we have∫ T

0h(t)[P]αdt =

[∫ T

0h(t)P dt

]α

=[∫ T

0h(t)Qdt

]α

=∫ T

0h(t)[Q]αdt.

(64)

Suppose that P /= Q, then for some α ∈ [0, 1], wehave [P]α /= [Q]α. Without loss of generality, we assumethat P,Q ∈ E1. Let Pα = [Pmin(α),Pmax(α)] andQα = [Qmin(α),Qmax(α)]. Then, we have either (i)Pmin(α) /= Qmin(α) or (ii) Pmax(α) /= Qmax(α) holds.

If (i) holds, then∫ T

0h(t)Pmin(α)dt /=

∫ T

0h(t)Qmin(α)dt. (65)

If (ii) holds, then∫ T

0h(t)Pmax(α)dt /=

∫ T

0h(t)Qmax(α)dt. (66)

Thus, in both cases (i) and (ii), we have∫ T

0h(t)

[Pmin(α),Pmax(α)

]dt /=

∫ T

0h(t)

[Qmin(α),Qmax(α)

]dt.

(67)

This implies that∫ T

0h(t)Pαdt /=

∫ T

0h(t)Qαdt, (68)

which is a contradiction to (64). Hence P = Q.

Definition 7 (see [20]). Let u, v ∈ E1, k ∈ R1, and let [u]α bethe α-level set of u. One defines the sum of u and v by

[u + v]α = [u]α + [v]α = {a + b : a ∈ [u]α, b ∈ [v]α

},

(69)

the difference between u and v by

[u− v]α = [u]α − [v]α = {a− b : a ∈ [u]α, b ∈ [v]α

}, (70)

and the scalar product by

[ku]α = k[u]α = {ka : a ∈ [u]α

}. (71)

Definition 8 (see [20]). Let x, y ∈ En2

and x = x1 × x2 ×· · ·×xn2 , y = y1× y2×· · ·× yn2 , xi, yi ∈ E1, i = 1, 2, . . . ,n2.If y = z + x, then z = y − x which is defined by

[z]α = [y − x]α = [y]α − [x]α =

⎡

⎢⎢⎢⎣

[y1]α − [x1]α

· · ·[yn2 ]α − [xn2 ]α

⎤

⎥⎥⎥⎦.

(72)

If y = w − x, then w = y + x which is defined by

[w]α = [y + x]α = [y]α + [x]α =

⎡

⎢⎢⎢⎣

[y1]α + [x1]α

· · ·[yn2 ]α + [xn2 ]α

⎤

⎥⎥⎥⎦.

(73)

Definition 9 ([20]). Let

C =

⎡

⎢⎢⎢⎢⎢⎢⎣

c11 c12 · · · c1n2

c21 c22 · · · c2n2

· · · · · · · · · · · ·cn21 cn22 · · · cn2n2

⎤

⎥⎥⎥⎥⎥⎥⎦

(74)

be an n2 × n2 matrix, p = p1 × p2 × · · · × pn2 , let pi ∈ E1,i = 1, 2, . . . ,n2, be a fuzzy set in En

2, and let [pi]

α be α-levelsets of pi. Define the product Cp of C and p as

[Cp]α = C[p]α =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

c11 c12 · · · c1n2

c21 c22 · · · c2n2

· · · · · · · · · · · ·cn21 cn22 · · · cn2n2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

[p1]α

[p2]α

· · ·[pn2 ]α

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

c11[p1]α + · · · + c1n2 [pn2 ]α

c21[p1]α + · · · + c2n2 [pn2 ]α

· · ·cn21[p1]α + · · · + cn2n2 [pn2 ]α

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(75)

All these definitions yield the following lemma.

Lemma 4. Cp is a fuzzy set in En2.

Proof. The proof is similar to proof of Lemma 3.1 [13].


Theorem 3. The fuzzy system (58),(59) is completely control-lable if the n2 × n2 symmetric controllability matrix

W(0,T) =∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

×(In ⊗ F(t))∗(

ψ(T − t)⊗ φ(T − t))∗dt(76)

is nonsingular. Furthermore,the fuzzy control U(t) whichtransfers the state of the system from X(0) = X0 to a fuzzystate X(T) = X f = (x f1 , x f2 , . . . , x fn2 ) can be determined bythe following fuzzy rule base:

R : IF x1 is in x f1 , . . . , xn2 is in x fn2 ,

THEN u1 is in u1, . . . , un2 is in un2 ,(77)

where

(u1(t), u2(t), . . . ,ui(t), . . . , un2 (t)

)

= 1T

(In ⊗ F(t)

)−1(ψ(T − t)⊗ φ(T − t))−1

× (x1(T), x2(T), . . . , x fi , . . . , xn2 (T)

)

− (In ⊗ F(t)

)∗(ψ(T − t)⊗ φ(T − t))∗

×W−1(0,T)(ψ(T)⊗ φ(T)

)X0, i = 1, 2, . . . ,n2.

(78)

Proof. Suppose that the symmetric controllability matrixW(0,T) is nonsingular. Therefore W−1(0,T) exists. Multi-plying W−1(0,T)(ψ(T)⊗ φ(T))X0 on both sides of (76), wehave

(ψ(T)⊗ φ(T)

)X0 =

∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

× (In ⊗ F(t)

)∗(ψ(T − t)⊗ φ(T − t))∗

×W−1(0,T)(ψ(T)⊗ φ(T)

)X0dt.

(79)

Now our problem is to find the control U(t) such that

X(T) = X f

= (ψ(T)⊗ φ(T)

)X0

+∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)U(t)dt.

(80)

Since X is fuzzy and from Lemma 4, U(t) must be fuzzy,otherwise the fuzzy left side of (80) cannot be equal to thecrisp right side. By Lemma 2, X f can be written as

X f = 1T

∫ T

0X f dt

= 1T

∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

×(In ⊗ F(t))−1(

ψ(T − t)⊗ φ(T − t))−1X f dt.

(81)

From (80) and (81), we have

1T

∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

×(In ⊗ F(t))−1(

ψ(T − t)⊗ φ(T − t))−1X f dt

= (ψ(T)⊗ φ(T)

)X0

+∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)U(t)dt.

(82)

Combining (79) and (82), we have

1T

∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

× (In ⊗ F(t)

)−1(ψ(T − t)⊗ φ(T − t))−1

X f dt

=∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

× (In ⊗ F(t)

)∗(ψ(T − t)⊗ φ(T − t))∗

×W−1(0,T)(ψ(T)⊗ φ(T)

)X0dt

+∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)U(t)dt.

(83)

It follows that

∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)U(t)dt

=∫ T

0

(ψ(T − t)⊗ φ(T − t))(In ⊗ F(t)

)

×{

1T

(In ⊗ F(t)

)−1(ψ(T − t)⊗ φ(T − t))−1

X f

− (In ⊗ F(t)

)∗(ψ(T − t)⊗ φ(T − t))∗

×W−1(0,T)(ψ(T)⊗ φ(T)

)X0

}dt.

(84)


By using Lemma 3, we get

U(t) = 1T

(In ⊗ F(t)

)−1(ψ(T − t)⊗ φ(T − t))−1

X f

− (In ⊗ F(t)

)∗(ψ(T − t)⊗ φ(T − t))∗

×W−1(0,T)(ψ(T)⊗ φ(T)

)X0.

(85)

Now we have two special cases for (85). First, let X(T) =X f = (x1(T), x2(T), . . . , xn2 (T)) be a crisp point, then we

will get a corresponding control U(t) = (u1, u2, . . . , un2 ),satisfying (85).

Second, let X(T) = (x1(T), x2(T), . . . , x fi , . . . , xn2 (T)),

then the corresponding control U(t) will take the formU(t) = (u1, u2, . . . ,ui, . . . , un2 ) in wich the ith componentof U(t) is a fuzzy set in E1. Obviously, ui(t) is in ui(t), thegrade of the membership can be determined by μxfi (xi(T)),the grade of the membership of xi(T) in x fi . Thus, based onthe above discussion, we have a fuzzy rule base for the controlU ,and is given by (77) and (78).

Remark 2. The nonsingularity of the symmetric control-lability matrix W(0,T) in Theorem 3 is only a sufficientcondition but not necessary because the fuzzy rule basecannot guarantee the nonsingularity of the controllabilitymatrix.

Example 1. Consider the fuzzy dynamical matrix Lyapunovsystem (1) satisfying (2) with

A(t) =⎡

⎣0 −1

1 0

⎤

⎦ , B(t) =⎡

⎣1 0

0 1

⎤

⎦ ,

F(t) =⎡

⎣et 0

0 et

⎤

⎦ , C(t) =⎡

⎣0 1

1 0

⎤

⎦ ,

D(t) =⎡

⎣0 0

0 0

⎤

⎦ , T = π

2, X0 =

⎡

⎣1 1

1 1

⎤

⎦ .

(86)

Let X f = (x f1 , x f2 , x f3 , x f4 ) ∈ E4, where

[X f

]α =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[x f1

]α

[x f2

]α

[x f3

]α

[x f4

]α

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[α− 1, 1− α]

[α− 1, 1− α]

[0.1(α− 1), 0.1(1− α)

]

[0.1(α− 1), 0.1(1− α)

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (87)

We select the points x1 = 0.5, x2 = 0.25, x3 = 0.05, andx4 = 0.025 which are in x f1 , x f2 , x f3 , and x f4 with grades of

the membership are 0.5, 0.75, 0.5, and 0.75, respectively. Thefundamental matrices of (20), (21) are

φ(t) =⎡

⎣cos t − sint

sint cos t

⎤

⎦ , ψ(t) =⎡

⎣et 0

0 et

⎤

⎦ . (88)

Now the fundamental matrix of (19) is

ψ(t)⊗ φ(t) =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

etcos t −et sint 0 0

et sint etcos t 0 0

0 0 etcos t −et sint

0 0 et sint etcos t

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

. (89)

Consider

(ψ(θ)⊗ φ(θ)

)(In ⊗ F(t)

)(In ⊗ F(t)

)∗(ψ(θ)⊗ φ(θ)

)∗

= eθ

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

cos θ − sin θ 0 0

sin θ cos θ 0 0

0 0 cos θ − sin θ

0 0 sin θ cos θ

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

et

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

× et

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

eθ

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

cos θ sin θ 0 0

− sin θ cos θ 0 0

0 0 cos θ sin θ

0 0 − sin θ cos θ

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

eπ 0 0 0

0 eπ 0 0

0 0 eπ 0

0 0 0 eπ

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

,

(90)

where θ = π/2− t.Therefore,

W(

0,π

2

)=∫ π/2

0

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

eπ 0 0 0

0 eπ 0 0

0 0 eπ 0

0 0 0 eπ

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

dt = π

2eπ

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

.

(91)

Clearly, it is nonsingular.


Thus, from Theorem 3, the input U can be chosen by thefollowing α-level sets:

Uα(t)

= 2e−t

π

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

e−θ

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

cos θ sin θ 0 0


0 0 cos θ sin θ


⎤

⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[α− 1, 1− α]

[α− 1, 1− α]

[0.1(α− 1), 0.1(1− α)

]

[0.1(α− 1), 0.1(1− α)

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

− et

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

× eθ

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

cos θ sin θ 0 0


0 0 cos θ sin θ


⎤

⎥⎥⎥⎥⎥⎥⎥⎦

2e−π

π

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

×eπ/2

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(π

2

)− sin

(π

2

)0 0

sin(π

2

)cos

(π

2

)0 0

0 0 cos(π

2

)− sin

(π

2

)

0 0 sin(π

2

)cos

(π

2

)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1

1

1

1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

.

(92)

Hence, the α-level sets of fuzzy control are given by

Uα(t) = 2e−π/2

π

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(sint +cos t)[α− 1, 1− α]

(sint−cos t)[α− 1, 1− α]

(sint +cos t)[0.1(α− 1), 0.1(1− α)

]

(sint−cos t)[0.1(α− 1), 0.1(1− α)

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

− 2π

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

cos t − sint

sint +cos t

cos t − sint

sint +cos t

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

(93)

and the corresponding control function to the point(0.5, 0.25, 0.05, 0.025)∗ is

U(t)=

⎡

⎢⎢⎢⎢⎢⎢⎣

u1

u2

u3

u4

⎤

⎥⎥⎥⎥⎥⎥⎦

= 2e−π/2

π

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0.5(sint +cos t)

0.75(sint−cos t)

0.5(sint +cos t)

0.75(sint−cos t)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

− 2π

⎡

⎢⎢⎢⎢⎢⎢⎣

cos t−sint

sint +cos t

cos t−sint

sint +cos t

⎤

⎥⎥⎥⎥⎥⎥⎦

.

(94)

Example 2. Consider the fuzzy dynamical matrix Lyapunovsystem (1) satisfying (2) with

A(t) =

⎡

⎢⎢⎢⎣

1t + 1

0

0−1t + 1

⎤

⎥⎥⎥⎦

, B(t) =⎡

⎢⎣

1t + 1

e−t

t + 1

0 1

⎤

⎥⎦ ,

F(t) =⎡

⎢⎣

12− t 0

0 (2− t)et−1

⎤

⎥⎦ , C(t) =

⎡

⎣0 et

et 0

⎤

⎦ ,

D(t) =⎡

⎣0 0

0 0

⎤

⎦ , T = 1, X0 =⎡

⎣1 1

0 0

⎤

⎦ .

(95)

Let X f = (x f1 , x f2 , x f3 , x f4 ) ∈ E4, where

[X f

]α =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[x f1

]α

[x f2

]α

[x f3

]α

[x f4

]α

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

[0.5α + 0.5, 1]

[0.8α + 0.2, 1]

[α− 1, 1− α][0.2(α− 1), 0.2(1− α)

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (96)

We select the points x1 = 0.75, x2 = 0.8, x3 = 0.75, andx4 = 0.1 which are in x f1 , x f2 , x f3 , and x f4 with grades of themembership being 0.5, 0.75, 0.25, and 0.5, respectively. Thefundamental matrices of (20), (21) are

φ(t) =

⎡

⎢⎢⎣

t + 1 0

01

t + 1

⎤

⎥⎥⎦ , ψ(t) =

⎡

⎣t + 1 t

0 et

⎤

⎦ . (97)

Now the fundamental matrix of (19) is

ψ(t)⊗ φ(t) =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(t + 1)2 0 t(t + 1) 0

0 1 0t

t + 1

0 0 et(t + 1) 0

0 0 0et

t + 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (98)


It is easily seen that

(ψ(1− t)⊗ φ(1− t))(In ⊗ F(t)

)(In ⊗ F(t)

)∗

×(ψ(1− t)⊗ φ(1− t))∗

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2t2−6t+5 0 (1−t)e1−t 0

0(2t2−6t+5

)e2(t−1) 0 (1−t)et−1

(1−t)e1−t 0 e2(1−t) 0

0 (1−t)et−1 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(99)

Therefore,

W(

0,π

2

)

=∫ 1

0

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2t2−6t+5 0 (1−t)e1−t 0

0(2t2−6t+5

)e2(t−1) 0 (1−t)et−1

(1−t)e1−t 0 e2(1−t) 0

0 (1−t)et−1 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dt

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

83

0 1 0

032

(1− 3e−2

)0 1− 2e−1

1 0e2 − 1

20

0 1− 2e−1 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(100)

Clearly, it is nonsingular.Thus, from Theorem 3, the input U can be chosen by the

following α-level sets, given by

Uα(t) =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

12− t 0

(t − 12− t

)et−1 0

0e1−t

2− t 0t − 12− t

0 0 et−1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[0.5α + 0.5, 1]

[0.8α + 0.2, 1]

[α− 1, 1− α][0.2(α− 1),

0.2(1− α)]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(9(e2 − 1

)− 6e)(2− t)

4e2 − 7

0(9(e2 − 1

)− 6e)(1− t) + 2(8e − 9)e1−t

4e2 − 7

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

(101)

and the corresponding control function to the point(0.75, 0.8, 0.75, 0.1)∗ is

U(t) =

⎡

⎢⎢⎢⎢⎢⎢⎣

u1

u2

u3

u4

⎤

⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 + 3(t − 1)et−1

4(2− t)3e1−t + 2(t − 1)

4(2− t)et−1

4

0.5

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(9(e2 − 1

)− 6e)(2− t)

4e2 − 7

0(9(e2 − 1

)− 6e)(1− t) + 2(8e − 9)e1−t

4e2 − 7

0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(102)

5. OBSERVABILITY OF FUZZY DYNAMICALLYAPUNOV SYSTEMS

In this section, we discuss the concept of observability of thefuzzy system (58), (59).

Definition 10. The fuzzy system (58), (59) is said to be com-pletely observable over the interval [0,T] if the knowledgeof rule base of input U and output Y over [0,T] suffices todetermine a rule base of initial state X0.

Let ui , yi , i = 1, 2, . . . ,n2, = 1, 2, . . . ,m, be fuzzy sets in

E1. We assume that the rule base for the input and output is

R : IF u1(t) is in u1(t), . . . , un2 (t) is in un2 (t),

THEN y1(t) is in y1(t), . . . , yn2 (t) is in yn2 (t),

= 1, 2, . . . ,m,

(103)

and the relation between input and output is

Y(t) = (In ⊗ C(t)

)X(t) +

(In ⊗D(t)

)U(t). (104)

Theorem 4. Assume that the fuzzy rule base (103) holds, thenthe system (58), (59) is completely observable over the interval[0,T] if (In⊗C(T))(ψ(T)⊗φ(T)) is nonsingular. Furthermore,if X0 = (x1

0, x20, . . . , xn

2

0 ), then one has the following rule base forthe initial value X0:

R : IF u1(T) is in u1(T), . . . , un2 (T) is in un2 (T),

IF y1(T) is in y1(T), . . . , yn2 (T) is inyn2 (T),

THEN x10 is in x0(1), . . . , xn

2

0 (t) is in x0(n2),

= 1, 2, . . . ,m,

(105)


where

x0(i) = [(In ⊗ C(T)

)(ψ(T)⊗ φ(T)

)]−1

×{Vi (T)− (

In ⊗D(T))U(T)− (

In ⊗ C(T))

×∫ T

0

(ψ(T − s)⊗ φ(T − s))(In ⊗ F(s)

)Hi (s)ds

},

(106)

X0 =[(In ⊗ C(T)

)(ψ(T)⊗ φ(T)

)]−1

×{y(T)− (

In ⊗D(T))u(T)− (

In ⊗ C(T))

×∫ T

0

(ψ(T − s)⊗ φ(T − s))(In ⊗ F(s)

)u(s)ds

},

(107)

Hi (t) = u1(t)× · · · × ui (t)× · · · × un2 (t),

Vi (t) = y1(t)× · · · × yi (t)× · · · × yn2 (t),

i = 1, 2, . . . ,n2, = 1, 2, . . . ,m.

(108)

Proof. Without loss of generality, we prove this theorem byconsidering = 1. Let

u(t) = (u1(t), u2(t), . . . , un2 (t)

),

y(t) = (y1(t), y2(t), . . . , yn2 (t)

).

(109)

Let μu1i (t)(ui(t)) be the grade of the membership of ui(t) in

u1i (t), and let μy1

i (t)( yi(t)) be the grade of the membership ofyi(t) in y1

i (t). Since (In⊗C(T))(ψ(T)⊗φ(T)) is nonsingularand from (60), we have

X0 =[(In ⊗ C(T)

)(ψ(T)⊗ φ(T)

)]−1

×{y(T)− (

In ⊗D(T))u(T)− (

In ⊗ C(T))

×∫ T

0

(ψ(T − s)⊗ φ(T − s))(In ⊗ F(s)

)u(s)ds

}.

(110)

Observing (104), when the input and output are both fuzzysets it follows from Definition 8 that

(In ⊗ C(t)

)X(t) = Y(t)− (

In ⊗D(t))U(t) (111)

is a fuzzy set. Substituting (60) in (111), we have

(In ⊗ C(t)

)((ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)U(s)ds

)

= Y(t)− (In ⊗D(t)

)U(t).

(112)

Using Definition 8,(In ⊗ C(t)

)(ψ(t)⊗ φ(t)

)X0

= Y(t)− (In ⊗D(t)

)U(t)− (

In ⊗ C(t))

×∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)U(s)ds.

(113)

Since (In ⊗ C(T))(ψ(T)⊗ φ(T)) is nonsingular, we have

X0 =[(In ⊗ C(T)

)(ψ(T)⊗ φ(T)

)]−1

×{Y(T)− (

In ⊗D(T))U(T)− (

In ⊗ C(T))

×∫ T

0

(ψ(T − s)⊗ φ(T − s))(In ⊗ F(s)

)U(s)ds

}.

(114)

Now, the initial value X0 is no more a crisp value, but shouldbe a fuzzy set. In order to determine each component of X0,let us assume

H1i (t) = u1(t)× · · · × u1

i (t)× · · · × un2 (t),

V 1i (t) = y1(t)× · · · × y1

i (t)× · · · × yn2 (t),

i = 1, 2, . . . ,n2.

(115)

From Remark 1, we know that the ith component of the set(ψ(t)⊗ φ(t)

)X0

+∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)H1i (s)ds

(116)

is a fuzzy set in E1. From Lemma 4, we know that the product

(In ⊗ C(t)

)∫ t

0

(ψ(t − s)⊗ φ(t − s))(In ⊗ F(s)

)H1i (s)ds

(117)

is a fuzzy set in En2. Hence, X0 is a fuzzy set in En

2, and the

ith component of it denoted by x10(i) is a fuzzy set in E1. The

grade of the membership of xi0 in x10(i) is defined by

μx10(i)

(xi0) = min

{μu1

i (t)

(ui(t)

),μy1

i (t)

(yi(t)

)}. (118)

Now, we are in a position to determine the rule base for theinitial value and it is given by (105), (106), (107), and (108).

In general, it is difficult to compute x0(i), but to solve thereal problems, we choose the following approximation. Now,

we take the point (xi0,μx0(i)(xi0)) and the zero-level set [x0(i)]

0

to determine a triangle as the new fuzzy set x0(i).We can use the center average defuzzifier

xi0 =∑m

=1

(xi0)μx0(i)

((xi0))

∑m=1μx0(i)

((xi0)) (119)

to determine the initial value X0 = (x10, x2

0, . . . , xn2

0 ). To obtainmore accurate value for the initial state, more rule bases maybe provided.


Example 3. Consider the fuzzy matrix Lyapunov system

X ′(t) =⎡

⎣0 −1

1 0

⎤

⎦X(t) + X(t)

⎡

⎣1 0

0 1

⎤

⎦ +

⎡

⎣et 0

0 et

⎤

⎦U ,

0 ≤ t ≤ π

2,

Y(t) =⎡

⎣0 1

1 0

⎤

⎦X(t).

(120)

The α-level sets of fuzzy input U(t) and fuzzy output Y(t) byrule base 1 and rule base 2 are given as follows.

Rule 1:

[U (1)]α =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[0,−0.75(α− 1)

]

[0.75(α− 1) + 1, 1

]

[0,−0.5(α− 1)

]

[0.5(α− 1) + 1, 1

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

[Y (1)]α =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[0,−2(α + 1)

]

[0.5α + 2.5, 3][0,−1.5(α− 1)

]

[0.5(α− 1) + 3, 3

]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(121)

Rule 2:

[U (2)]α=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[0,−0.8(α−1)

]

[0.8α + 0.2, 1][0,−0.5(α−1)

]

[0.5α + 0.5, 1]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,[Y (2)]α=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[0,−1.5(α−1)

]

[α + 1, 2][0,−2.5(α−1)

]

[2α + 1, 3]

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(122)

From rule base 1, we select

u1 = (u1, u2, u3, u4

) = (0.5, 0.85, 0.4, 0.75), (123)

the grades of the membership of u1, u2, u3, and u4 are 1/3,0.8, 0.2, and 1/2, respectively. Also

y1 = (y1, y2, y3, y4

) = (1, 2.8, 0.5, 2.9), (124)

the grades of the membership of y1, y2, y3, and y4 are 1/2,0.6, 2/3, and 0.8, respectively.

From rule base 2, we select

u2 = (u1, u2, u3, u4

) = (0.5, 0.8, 0.25, 0.75), (125)

the grades of the membership of u1, u2, u3, and u4 are 3/8,3/4, 1/2, and 1/2, respectively. Also

y2 = (y1, y2, y3, y4

) = (1, 1.75, 2, 1.5), (126)

the grades of the membership of y1, y2, y3, and y4 are 1/3,3/4, 0.2, and 0.25, respectively.

For rule base 1, by formula (106), (107),we have

X0 = e−π/2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎤

⎥⎥⎥⎥⎥⎥⎦

·

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡

⎢⎢⎢⎢⎢⎢⎣

1

2.8

0.5

2.9

⎤

⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎣

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

⎤

⎥⎥⎥⎥⎥⎥⎦

∫ π/2

0eπ/2−s

·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(π

2−s)−sin

(π

2−s)

0 0

sin(π

2−s)

cos(π

2−s)

0 0

0 0 cos(π

2−s)−sin

(π

2−s)

0 0 ,sin(π

2−s)

cos(π

2−s)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·es

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

0.5

0.85

0.4

0.75

⎤

⎥⎥⎥⎥⎥⎥⎦

ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡

⎢⎢⎢⎢⎢⎢⎣

−1.142

−0.9324

−1.046

−0.9532

⎤

⎥⎥⎥⎥⎥⎥⎦

,

x10(1) = e−π/2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎤

⎥⎥⎥⎥⎥⎥⎦

·

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡

⎢⎢⎢⎢⎢⎢⎣

[0,−2(α− 1)

]

2.8

0.5

2.9

⎤

⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎣

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

⎤

⎥⎥⎥⎥⎥⎥⎦

∫ π/2

0eπ/2−s


·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(π

2−s)−sin

(π

2−s)

0 0

sin(π

2−s)

cos(π

2−s)

0 0

0 0 cos(π

2−s)−sin

(π

2−s)

0 0 sin(π

2−s)

cos(π

2−s)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·es

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

[0,−0.75(α− 1)

]

0.85

0.4

0.75

⎤

⎥⎥⎥⎥⎥⎥⎦

ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡

⎢⎢⎢⎢⎢⎢⎣

[−1.6 + 0.75α,−0.434− 0.416α]

[−1.4324,−0.6824− 0.75α]

−1.046

−0.9532

⎤

⎥⎥⎥⎥⎥⎥⎦

(127)

when α = 0, we get the biggest interval [−1.6, −0.434]and x1

0 = −1.142 is located in this interval. We choose itsmembership grade in x1

0(1) as

μx10(1)

(x1

0

) = min{

13

,12

}= 1

3= 0.333,

x10(2) = e−π/2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎤

⎥⎥⎥⎥⎥⎥⎦

·

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡

⎢⎢⎢⎢⎢⎢⎣

1

[0.5α + 2.5, 3]

0.5

2.9

⎤

⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎣

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

⎤

⎥⎥⎥⎥⎥⎥⎦

∫ π/2

0eπ/2−s

·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(π

2−s)−sin

(π

2−s)

0 0

sin(π

2−s)

cos(π

2−s)

0 0

0 0 cos(π

2−s)−sin

(π

2−s)

0 0 sin(π

2−s)

cos(π

2−s)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·es

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

0.5[0.75(α− 1) + 1, 1

]

0.4

0.75

⎤

⎥⎥⎥⎥⎥⎥⎦

ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡

⎢⎢⎢⎢⎢⎢⎣

[−1.292,−0.542− 0.75α]

[−1.124,−0.27− 0.854α]

−1.046

−0.9532

⎤

⎥⎥⎥⎥⎥⎥⎦

,

(128)



0(2) as

μx10(2)

(x2

0

) = min{0.8, 0.6} = 0.6,

x10(3) = e−π/2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎤

⎥⎥⎥⎥⎥⎥⎦

·

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡

⎢⎢⎢⎢⎢⎢⎣

1

2.8[0,−1.5(α− 1)

]

2.9

⎤

⎥⎥⎥⎥⎥⎥⎦

−

⎡

⎢⎢⎢⎢⎢⎢⎣

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

⎤

⎥⎥⎥⎥⎥⎥⎦

∫ π/2

0eπ/2−s

·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(π

2−s)−sin

(π

2−s)

0 0

sin(π

2−s)

cos(π

2−s)

0 0

0 0 cos(π

2−s)−sin

(π

2−s)

0 0 sin(π

2−s)

cos(π

2−s)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·es

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

0.5

0.85[0,−0.5(α− 1)

]

0.75

⎤

⎥⎥⎥⎥⎥⎥⎦

ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


=

⎡

⎢⎢⎢⎢⎢⎢⎣

−1.142

−0.9324

[−1.25 + 0.5α,−0.438− 0.312α]

[−1.3532,−0.8532− 0.5α]

⎤

⎥⎥⎥⎥⎥⎥⎦

,

(129)



0(3) as

μx10(3)

(x3

0

) = min{

0.2,23

}= 0.2,

x10(4) = e−π/2

⎡

⎢⎢⎢⎢⎣

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1

⎤

⎥⎥⎥⎥⎦

·

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡

⎢⎢⎢⎢⎣

1

2.8

0.5[0.5(α− 1) + 3, 3

]

⎤

⎥⎥⎥⎥⎦−

⎡

⎢⎢⎢⎢⎣

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

⎤

⎥⎥⎥⎥⎦

∫ π/2

0eπ/2−s

·

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(π

2−s)−sin

(π

2−s)

0 0

sin(π

2−s)

cos(π

2−s)

0 0

0 0 cos(π

2−s)−sin

(π

2−s)

0 0 sin(π

2−s)

cos(π

2−s)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

·es

⎡

⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎣

0.5

0.85

0.4[0.5(α− 1) + 1, 1

]

⎤

⎥⎥⎥⎥⎦ds

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎡

⎢⎢⎢⎢⎣

−1.142

−0.9324

[−1.296,−0.796− 0.5α]

[−1.224,−0.62− 0.604α]

⎤

⎥⎥⎥⎥⎦

,

(130)



0(4) as

μx10(4)

(x4

0

) = min{

12

, 0.8}= 1

2= 0.5. (131)

Similarly for rule base 2, by the use of formula (106), (107),we obtain the values of X0, x2

0(i), i = 1, 2, 3, 4 and given as

follows:

X0 =

⎡

⎢⎢⎢⎢⎢⎢⎣

−1.092

−0.664

−0.584

−0.812

⎤

⎥⎥⎥⎥⎥⎥⎦

,

x20(1) =

⎡

⎢⎢⎢⎢⎢⎢⎣

[−1.6 + 0.8α,−0.488− 0.312α]

[−1.164,−0.364− 0.8α]

−0.584

−0.812

⎤

⎥⎥⎥⎥⎥⎥⎦

,

x20(2) =

⎡

⎢⎢⎢⎢⎢⎢⎣

[−1.292,−0.492− 0.8α]

[−0.916, 0.092− 1.008α]

−0.584

−0.812

⎤

⎥⎥⎥⎥⎥⎥⎦

,

x20(3) =

⎡

⎢⎢⎢⎢⎢⎢⎣

−1.092

−0.664

[0.5α− 1.25,−0.23− 0.52α]

[−1.062,−0.562− 0.5α]

⎤

⎥⎥⎥⎥⎥⎥⎦

,

x20(4) =

⎡

⎢⎢⎢⎢⎢⎢⎣

−1.092

−0.664

[−0.834,−0.334− 0.5α]

[−1.374,−0.458− 0.916α]

⎤

⎥⎥⎥⎥⎥⎥⎦

.

(132)

Also the grades of the membership of x10 = −1.092,

x20 = −0.664, x3

0 = −0.584, x40 = −0.812 in x2

0(1), x20(2),

x20(3), x2

0(4) are 0.333, 0.75, 0.2, 0.25, respectively. We canuse the center average defuzzifier to determine X0 = (x1

0, x20,

x30, x4

0), where

x10 =

∑2=1x

0μx0(1)

(x0)

∑2=1μx0(1)

(x0)

= −1.142× 0.333 + (−1.092)× 0.3330.333 + 0.333

= −1.117,

x20 =

∑2=1x

0μx0(2)

(x0)

∑2=1μx0(2)

(x0)

= −0.9324× 0.6 + (−0.664)× 0.750.6 + 0.75

= −0.7833,

x30 =

∑2=1x

0μx0(3)

(x0)

∑2=1μx0(3)

(x0)

= −1.046× 0.2 + (−0.584)× 0.20.2 + 0.2

= −0.815,

x40 =

∑2=1x

0μx0(4)

(x0)

∑2=1μx0(4)

(x0)

= −0.9532× 0.5 + (−0.812)× 0.250.5 + 0.25

= −0.9061.

(133)


6. CONCLUSIONS

In this paper, we have investigated a way to incorporatematrix Lyapunov systems with a set of fuzzy rules. Here,a deterministic matrix Lyapunov system with fuzzy inputsand fuzzy outputs can generate a fuzzy dynamical matrixLyapunov system (FDMLS). Based on this result, we canstudy both controllability and observability properties of theFDMLS. First, we have provided a sufficient condition forthe controllability of the FDMLS, that is, for a given fuzzystate with a fuzzy rule base, we can determine a controlwhich transfers the initial state to the given state in a finitetime. The advantage of our approach is that all levels arerepresented by mathematical formulas. Example 1 showshow to determine the control by our formula. Next, wehave studied the observability property which concerns thefollowing problem, that is, given the input and output rulebases we can determine a rule base for the initial state with aformula. Example 3 illustrates the significance of our methodby which we can determine the rule base for initial valuewithout solving the FDMLS. Our future research works willconcentrate on the applications of these systems (FDMLS) toreal world problems.

ACKNOWLEDGMENTS

The authors would like to thank Professor H. Ying(Associate-Editor) and the anonymous referees for theirsuggestions which helped to improve the quality of thepresentation.

REFERENCES

[1] S. Barnett and R. G. Cameron, Introduction to MathematicalControl Theory, Clarendon Press, Oxford, UK, 2nd edition,1985.

[2] M. S. N. Murty, B. V. Appa Rao, and G. S. Kumar, “Con-trollability, observability, and realizability of matrix Lyapunovsystems,” Bulletin of the Korean Mathematical Society, vol. 43,no. 1, pp. 149–159, 2006.

[3] M. Sugeno, “On stability of fuzzy systems expressed by fuzzyrules with singleton consequents,” IEEE Transactions on FuzzySystems, vol. 7, no. 2, pp. 201–224, 1999.

[4] T. Takagi and M. Sugeno, “Fuzzy identification of systems andits applications to modeling and control,” IEEE Transactionson Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132,1985.

[5] A. Alwadie, H. Ying, and H. Shah, “A practical two-input two-output Takagi-Sugeno fuzzy controllers,” International Journalof Fuzzy Systems, vol. 5, no. 2, pp. 123–130, 2003.

[6] Y. S. Ding, H. Ying, and S. H. Shao, “Structure and stabilityanalysis of a Takagi-Sugeno fuzzy PI controller with applica-tion to tissue hyperthermia therapy,” Soft Computing, vol. 2,no. 4, pp. 183–190, 1999.

[7] Y. S. Ding, H. Ying, and S. H. Shao, “Typical Takagi-SugenoPI and PD fuzzy controllers: analytical structures and stabilityanalysis,” Information Sciences, vol. 151, pp. 245–262, 2003.

[8] T. A. Johansen, R. Shorten, and R. Murray-Smith, “On theinterpretation and identification of dynamic Takagi-Sugenofuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 8,no. 3, pp. 297–313, 2000.

[9] H. Ying, “Analytical analysis and feedback linearizationtracking control of the general Takagi-Sugeno fuzzy dynamicsystems,” IEEE Transactions on Systems, Man and Cybernetics,Part C, vol. 29, no. 2, pp. 290–298, 1999.

[10] H. Ying, Fuzzy Control and Modeling: Analytical Foundationsand Applications, IEEE Press, New York, NY, USA, 2000.

[11] H. Ying, “Deriving analytical input-output relationship forfuzzy controllers using arbitrary input fuzzy sets and Zadehfuzzy AND operator,” IEEE Transactions on Fuzzy Systems,vol. 14, no. 5, pp. 654–662, 2006.

[12] Y. Chen, B. Yang, A. Abraham, and L. Peng, “Automaticdesign of hierarchical Takagi-Sugeno type fuzzy systems usingevolutionary algorithms,” IEEE Transactions on Fuzzy Systems,vol. 15, no. 3, pp. 385–397, 2007.

[13] Z. Ding and A. Kandel, “On the observability of fuzzydynamical control systems (II),” Fuzzy Sets and Systems,vol. 115, no. 2, pp. 261–277, 2000.

[14] Z. Ding and A. Kandel, “On the controllability of fuzzydynamical systems (II),” Journal of Fuzzy Mathematics, vol. 18,no. 2, pp. 295–306, 2000.

[15] H. Radstrom, “An embedding theorem for space of convexsets,” Proceedings of the American Mathematical Society, vol. 3,no. 1, pp. 165–169, 1952.

[16] R. J. Aumann, “Integrals of set-valued functions,” Journal ofMathematical Analysis and Applications, vol. 12, no. 1, pp. 1–12, 1965.

[17] O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets andSystems, vol. 24, no. 3, pp. 301–317, 1987.

[18] V. Lakshmikantham and R. Mohapatra, Theory of Fuzzy Dif-ferential Equations and Inclusions, Taylor & Francis, London,UK, 2003.

[19] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets toSystems Analysis, John Willey & Sons, New York, NY, USA,1975.

[20] J. B. Conway, A Course in Functional Analysis, Springer, NewYork, NY, USA, 1990.

Submit your manuscripts athttp://www.hindawi.com

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014


Applied Computational Intelligence and Soft Computing

Advances in

Artificial Intelligence


Advances inSoftware EngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in


Documents

On Controllability and Observability of Fuzzy Dynamical ...downloads.hindawi.com/journals/afs/2008/421525.pdf · lites are well known. In general, fuzzy systems are mainly classiﬁed