# On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems

Authors: M. S. N. Murty; G. Suresh Kumar
Journal: Advances in Fuzzy Systems
 (2008)

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## Abstract

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


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## Body

## 1. Introduction

The importance of control theory in applied
mathematics and its occurrence in several problems such as mechanics,
electromagnetic theory, thermodynamics, and artificial satellites are well
known. In general, fuzzy systems are mainly classified into three categories,
namely pure fuzzy systems, T-S fuzzy systems, and fuzzy logic systems, using
fuzzifiers and defuzzifiers. In this paper, we use fuzzy matrix Lyapunov system
to describe fuzzy logic system. The purpose of this paper is to provide
sufficient conditions for controllability and observability of first-order
fuzzy matrix Lyapunov system modeled by(1)X′(t)=A(t)X(t)+X(t)B(t)+F(t)U(t),X(0)=X0,t>0,(2)Y(t)=C(t)X(t)+D(t)U(t),
where U(t) is an n×n fuzzy input
matrix called fuzzy control and Y(t) is an n×n fuzzy output
matrix. Here A(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 a
system of ordinary differential equations was studied by Barnett and Cameron [1] and for
matrix Lyapunov systems by Murty et al.[2]. Fuzzy control usually decomposes
a complex system into several subsystems according to the human expert's
understanding of the system and uses a simple control law to emulate the human
control strategy.There exist two major types of fuzzy controllers, namely Mamdani fuzzy
controllers and Takagi-Sugeno (TS) fuzzy controllers. They mainly differ in the
consequence of fuzzy rules: the former uses fuzzy
sets whereas the latter employs (linear) functions. Takagi and Sugeno [3, 4] propose a type of fuzzy model in which the consequent part of the rules is
defined not by the membership function but by a crisp analytical function. More
and more interest appears to shift towards TS fuzzy controllers in recent
years, as evidenced by the increasing number of papers in this direction and
due to their applications in real world problems (e.g., [5–12]).Recently, the controllability and observability criteria
for fuzzy dynamical control systems were discussed by Ding and Kandel [13, 14].
In this paper, by converting the fuzzy matrix Lyapunov system into a Kronecker
product system we obtain sufficient conditions for controllability and
observability of the system (1) satisfying (2).The paper is well organized as follows. In Section2,
we present some basic definitions and results relating to fuzzy sets [13] and
Kronecker product of matrices. Further, we obtain a unique solution of the
system (1), when U(t) is a crisp
continuous matrix. In Section 3, we generate a fuzzy dynamical Lyapunov system,
and also obtain its solution set. In Section 4, we present a sufficient
condition for the controllability of the system and illustrate the results by
suitable examples. In Section 5, we obtain a sufficient condition for the
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 and
Kandel [13, 14] developed for system of fuzzy differential equations to fuzzy
matrix Lyapunov systems and includes their results as a particular case, when B(t)=0,X,U, and Y are column
vectors of order n.


## 2. Preliminaries

In this section, we present some definitions and
results relating to fuzzy sets [13] and Kronecker product of matrices.LetPk(Rn) denote the
family of all nonempty compact convex subsets of Rn. Define the addition and scalar multiplication in Pk(Rn) as usual.
Radstrom [15] states that Pk(Rn) is a
commutative semigroup under addition, which satisfies the cancellation law.
Moreover, if α,β∈R and A,B∈Pk(Rn), then
(3)α(A+B)=αA+αB,α(βA)=(αβ)A,1⋅A=A,
and if α,β≥0, then (α+β)A=αA+βA. The distance between A and B is defined by
the Hausdorff metric
(4)d(A,B)=inf⁡{ϵ:A⊂N(B,ϵ),B⊂N(A,ϵ)},
where
(5)N(A,ϵ)={x∈Rn:∥x−y∥<ϵ,forsomey∈A}.Definition 1.
A set-valued functionF:J→Pk(Rn) is said to be
measurable if it satisfies any one of the following equivalent conditions:
(1)
for allu∈Rn, t→dF(t)(u)=infv∈F(t)∥u−v∥ is measurable,(2)
GrF={(t,u)∈J×Rn:u∈F(t)}∈Σ×β(Rn), where Σ,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 bySF1 the set of all
selections of F(⋅) that belong to
the Lebesgue Bochner space LRn1(J), that is,
(6)SF1={f(⋅)∈LRn1(J):f(t)∈F(t)a.e.}.
We present the Aumann's integral as follows:
(7)(A)∫JF(t)dt={∫Jf(t)dt,f(⋅)∈SF1}.We say thatF:J→Pk(Rn) is integrably
bounded if it is measurable and there exists a function h:J→R, h∈LRn1(J), such that ∥u∥≤h(t), u∈F(t). From [16], we know that if F is a closed
valued measurable multifunction, then ∫JF(t)dt is convex in Rn. Furthermore, if F is integrably
bounded, then ∫JF(t)dt is
compact in Rn.Let(8)En={u:Rn[0,1]/usatisfie(i)–(iv) below},
where
(i)
u is normal, that
is, there exists an x0∈Rn such that u(x0)=1;(ii)
u is fuzzy
convex, that is, for x,y∈Rn and 0≤λ≤1,
(9)u(λx+(1−λ)y)≥min⁡[u(x),u(y)];(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 follows that [u]α∈Pk(Rn) for all 0≤α≤1.DefineD:En×En→[0,∞) by (10)D(u,v)=sup⁡{d([u]α,[v]α)/α⁢∈[0,1]},
where d is the
Hausdorff metric defined in Pk(Rn). It is easy to 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 locally compact. Moreover, the distance D verifies that
(11)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.
We note that (En,D) is not a vector
space. But it can be imbedded isomorphically as a cone in a Banach space [15].Regarding fundamentals of differentiability and
integrability of fuzzy functions, we refer to Kaleva [17] and Lakshmikantham
and Mohapatra [18].In the sequel, we need the following representation
theorem.Theorem 1 (see [19]).
Ifu∈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.A fuzzy
set-valued mappingF:J→En is called fuzzy
integrably bounded if F0(t) is integrably
bounded.Definition 2.
LetF:J→En be a fuzzy
integrably bounded mapping. The fuzzy integral of F over J denoted by ∫JF(t)dt is defined
level-set-wise by
(12)[∫JF(t)dt]α=(A)∫JFα(t)dt,0<α≤1.
LetF:J×En→En,and consider the fuzzy differential equation
(13)u′=F(t,u),u(0)=u0.Definition 3.
A mappingu:J→En is a fuzzy weak
solution to (13) if it is continuous and satisfies the integral equation
(14)u(t)=u0+∫0tF(s,u(s))ds,∀t∈J.
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
Kronecker products and basic results related to matrix Lyapunov systems.Definition 4 (see [2]).
LetA∈Cm×n and B∈Cp×q.Then the Kronecker product of A and B written A⊗B is defined to
be the partitioned matrix
(15)A⊗B=[a11Ba12B⋯a1nBa21Ba22B⋯a2nB..⋯.am1Bam2B⋯amnB]
which is an mp×nq matrix and is
in Cmp×nq.Definition 5 (see [2]).
LetA=[aij]∈Cm×n; one denotes
(16)A^=VecA=[A.1A.2⋮A.n],whereA.j=[a1ja2j⋮amj](1≤j≤n).
The Kronecker product has the following properties and rules [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 is such that the various expressions exist.
(4)
∥A⊗B∥=∥A∥∥B∥.(5)
IfA(t) and B(t) are matrices,
then
(A⊗B)′=A′⊗B+A⊗B′(′=d/dt).(6)
Vec(AYB)=(B∗⊗A)VecY.(7)
IfA 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 matrix
Lyapunov system (1) satisfying (2) and using the above properties, we have
(17)X^'(t)=G(t)X^(t)+(In⊗F(t))U^(t),X^(0)=X^0,(18)Y^(t)=(In⊗C(t))X^(t)+(In⊗D(t))U^(t),
where G(t)=(B∗⊗In)+(In⊗A) is an n2×n2 matrix and X^=VecX(t), U^=VecU(t) are column
matrices of order n2.The
corresponding linear homogeneous system of (17) is
(19)X^'(t)=G(t)X^(t),X^(0)=X^0.Lemma 1.
Letϕ(t) and ψ(t) be the fundamental matrices for the systems
(20)X′(t)=A(t)X(t),X(0)=In,(21)[X*(t)]′=B*(t)X*(t),X(0)=In,
respectively.
Then the matrix ψ(t)⊗ϕ(t) is a
fundamental matrix of (19) and the solution of (19) is X^(t)=(ψ(t)⊗ϕ(t))X^0.Proof.
Consider(22)(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)).
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))X^0 is a solution
of (19).Theorem 2.
Letϕ(t) and ψ(t)
be the
fundamental matrices for the systems (20) and (21). Then the unique solution
of the initial value problem (17) is given by
(23)X^(t)=(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F^(s))U^(s)ds.Proof.
First
we show that the solution of (17) is of the form X^(t)=(ψ(t)⊗ϕ(t))X^0+X˜(t), where X˜(t) is a particular
solution of (17) and is given by
(24)X˜(t)=∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F^(s))U^(s)ds.
Letu(t) be any other
solution of (17), write w(t)=u(t)−X˜(t), then w satisfies
(19), hence w=(ψ(t)⊗ϕ(t))X^0, u(t)=(ψ(t)⊗ϕ(t))X^0+X˜(t).
Consider the vectorX˜(t)=(ψ(t)⊗ϕ(t))v(t), where v(t) is an arbitrary
vector to be determined so as to satisfy (17),
(25)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)=∫0t(ψ−1(s)⊗ϕ−1(s))(In⊗F^(s))U^(s)ds.
Hence, the
desired expression follows immediately by noting the fact that ϕ(t)ϕ−1(s)=ϕ(t−s) and ψ(t)ψ−1(s)=ψ(t−s).


## 3. Formation of Fuzzy Dynamical Lyapunov Systems

Letui(t)∈E1, t∈J, i=1,2,…,n2, and define
(26)U^(t)=(u1(t),u2(t),…,un2(t))=u1(t)×u2(t)×⋯×un2(t)={(u1α(t),u2α(t),…,un2α(t)):α∈[0,1]}={(u˜1(t),u˜2(t),…,u˜n2(t)):u˜i(t)∈uiα(t),α∈[0,1]},
where uiα(t) is the α-level set of ui(t). From the above definition of U^(t) and Theorem 1, it can be easily seen that U^(t)∈En2.Now by using the fuzzy controlU^(t), we show that the following system
(27)X^'(t)=G(t)X^(t)+(In⊗F(t))U^(t),X^(0)=X^0,(28)Y^(t)=(In⊗C(t))X^(t)+(In⊗D(t))U^(t)
determines a fuzzy system.Assume thatU^(t) is continuous
in En2. The set U^α=u1(t)×u2(t)×⋯×un2(t) is a convex and
compact set in Rn2. For any positive number T, consider the following differential inclusions:
(29)X^'(t)∈G(t)X^(t)+(In⊗F(t))U^α(t),t∈[0,T],(30)X^(0)=X^0.
Let X^α 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 thatX^α is nonempty,
compact, and convex in C[[0,T],Rn2]. Since U^α(t) has measurable
selection, we have that X^α is nonempty.
LetK=maxt∈[0,T]∥ϕ(t)∥, L=maxt∈[0,T]∥ψ(t)∥, M=max{∥u(t)∥:u(t)∈U^α(t),t∈[0,T]}, N=maxt∈[0,T]∥F(t)∥.
If for anyX^∈X^α, then there is a selection u(t)∈U^α(t) such that
(31)X^(t)=(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))u(s)ds.
Then
(32)∥X^(t)∥≤∥(ψ(t)⊗ϕ(t))X^0∥+∫0t∥(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))u(s)∥ds≤∥ψ(t)∥∥ϕ(t)∥∥X^0∥+∫0t∥ψ(t−s)∥∥ϕ(t−s)∥∥F(s)∥∥u(s)∥ds≤KL∥X^0∥+KLNMT.
Thus X^α is bounded.
For anyt1,t2∈[0,T],
(33)X^(t1)−X^(t2)=(ψ(t1)⊗ϕ(t1))X^0+∫0t1(ψ(t1−s)⊗ϕ(t1−s))(In⊗F(s))u(s)ds−(ψ(t2)⊗ϕ(t2))X^0−∫0t2(ψ(t2−s)⊗ϕ(t2−s))(In⊗F(s))u(s)ds.
Therefore
(34)∥X^(t1)−X^(t2)∥≤∥(ψ(t1)⊗ϕ(t1))−(ψ(t2)⊗ϕ(t2))∥∥X^0∥+∫t2t1∥(ψ(t1−s)⊗ϕ(t1−s))(In⊗F(s))u(s)∥ds+∫0t2∥[(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))∥∥X^0∥+KLNM|t1−t2|+MN∫0T∥(ψ(t1−s)⊗ϕ(t1−s))−(ψ(t2−s)⊗ϕ(t2−s))∥ds.
Since ϕ(t) and ψ(t) are both
uniformly continuous on [0,T], X^ is
equicontinuous. Thus, X^α is relatively
compact. If X^α is closed, then
it is compact.
LetX^k∈X^α and X^k→X^. For each X^k, there is a uk∈U^α(t) such that
(35)X^k(t)=(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))uk(s)ds.
Since uk∈U^α(t) is closed, then
there exists a subsequence {ukj} of {uk} converging
weakly to u∈U^α(t). From Mazur's theorem [20], there exists a sequence of
numbers λj>0, ∑λj=1 such that ∑λjukj converges
strongly to u.
Thus, from (35) we have
(36)∑λjX^kj(t)=∑λj(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))×(In⊗F(s))∑λjukj(s)ds.
From Fatou's
lemma, taking the limit as j→∞ on both sides
of (36), we have
(37)X^(t)=(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))u(s)ds.
Thus, X^(t)∈X^α, and hence X^α is closed.
LetX^1, X^2∈X^α, then there exist u1,u2∈U^α(t) such that
(38)X^1'(t)=G(t)X^1(t)+(In⊗F(t))u1(t),X^2'(t)=G(t)X^2(t)+(In⊗F(t))u2(t).
Let X^=λX^1(t)+(1−λ)X^2(t), 0≤λ≤1, then
(39)X^'=λX^1'(t)+(1−λ)X^2'(t)=λ(G(t)X^1(t)+(In⊗F(t))u1(t))+(1−λ)(G(t)X^2(t)+(In⊗F(t))u2(t))=G(t)[λX^1(t)+(1−λ)X^2(t)]+(In⊗F(t))[λu1(t)+(1−λ)u2(t)].
Since U^α(t) is convex, λu1(t)+(1−λ)u2(t)∈U^α(t), we have
(40)X^'(t)∈G(t)X^(t)+(In⊗F(t))U^α(t),
that is X^∈X^α. Thus X^α is convex.
Therefore, X^α is nonempty,
compact, and convex in C[[0,T],Rn2]. Thus, from Arzela-Ascoli theorem, we know that [X^(t)]α is compact in Rn2 for 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.
Let0≤α1≤α2≤1. Since U^α2(t)⊂U^α1(t), we have
(41)U^α2(t)=u1α2(t)×u2α2(t)×⋯×un2α2(t)⊂u1α1(t)×u2α1(t)×⋯×un2α1(t)=U^α1(t).
Thus, we have
the selection inclusion SU^α2(t)1⊂SU^α1(t)1 and the
following inclusion:
(42)X^'(t)∈G(t)X^+(In⊗F(t))U^α2(t)⊂G(t)X^+(In⊗F(t))U^α1(t).
Consider the differential inclusions
(43)X^'(t)∈G(t)X^+(In⊗F(t))U^α2(t),t∈[0,T],(44)X^'(t)∈G(t)X^+(In⊗F(t))U^α1(t),t∈[0,T].
Let X^α2 and X^α1 be the solution
sets of (43) and (44), respectively. Clearly, the solution of (43) satisfies
the following inclusion:
(45)X^(t)∈(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))SU^α2(s)1ds⊂(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))SU^α1(s)1ds.
Thus X^α2⊂X^α1, and hence X^α2(t)⊂X^α1(t). Hence the claim.Claim (iii).
If{αk} is a
nondecreasing sequence converging to α>0, then X^α(t)=⋂k≥1X^αk(t).
LetU^αk(t)=u1αk×u2αk×⋯×un2αk, U^α(t)=u1α×u2α×⋯×un2α and consider
the inclusions
(46)X^'(t)∈G(t)X^+(In⊗F(t))U^αk(t),(47)X^'(t)∈G(t)X^+(In⊗F(t))U^α(t).
Let X^αk and X^α be the solution
sets of (46) and (47), respectively. Since ui(t) is a fuzzy set
and from Theorem 1, we have
(48)uiα=⋂k≥1uiαk.
Consider
(49)U^α(t)=u1α×u2α×⋯×un2α=⋂k≥1u1αk×⋂k≥1u2αk×⋯×⋂k≥1un2αk=⋂k≥1(u1αk×u2αk×⋯×un2αk)=⋂k≥1U^αk(t)
and then SU^α(t)1=S⋂k≥1U^αk(t)1. Therefore
(50)X^'(t)∈G(t)X^+(In⊗F(t))U^α(t)=G(t)X^+(In⊗F(t))⋂k≥1U^αk(t)⊂G(t)X^+(In⊗F(t))U^αk(t),k=1,2,….
Thus, we have X^α⊂X^αk, k=1,2,…, which implies that
(51)X^α⊂⋂k≥1X^αk.
Let X^ be the solution
set to the inclusion
(52)X^'(t)∈G(t)X^+(In⊗F(t))U^αk(t),k≥1.
Then,
(53)X^(t)∈(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))SU^αk(t)1ds.
It follows that
(54)X^(t)∈(ψ(t)⊗ϕ(t))X^0+⋂k≥1∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))SU^αk1ds⊂(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))S⋂k≥1U^αk1ds=(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))SU^α1ds.
This implies
that X^∈X^α. Therefore,
(55)⋂k≥1X^αk⊂X^α.
From (51) and (55), we have
(56)X^α=⋂k≥1X^αk,
and hence,
(57)X^α(t)=⋂k≥1X^αk(t).
From Claims (i)–(iii) and applying Theorem 1, there exists X^(t)∈En2 on [0,T] such that X^α(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 can be expressed as
(58)X^'(t)=G(t)X^(t)+(In⊗F(t))U^(t),X^(0)={X^0},(59)Y^(t)=(In⊗C(t))X^(t)+(In⊗D(t))U^(t).
The solution
set of the fuzzy dynamical system (58), (59) is given by
(60)X^(t)∈(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))U^(s)ds.Remark 1.
Consider a special case. If the input is in the form(61)U^(t)=u˜1(t)×u˜2(t)×⋯×ui(t)×⋯×u˜n2(t),
where u˜k(t)∈R1, k≠i, are crisp
numbers, then the ith component of
the solution set of (27) is a fuzzy set in E1.Proof.
The proof follows along similar lines as in the above discussion.


## 4. Controllability of Fuzzy Dynamical Lyapunov Systems

In this section, we discuss the concept of
controllability of the fuzzy system (58) satisfying (59).Definition 6.
The fuzzy system (58), (59) is said to be
completely controllable if for any initial state X^(0)=X^0 and any 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.
IfF
is a fuzzy set,
then ∫0TFdt=TF.Proof.
Let[F]α be the α-level set of F. Since
(62)[∫0TFdt]α=∫0T[F]αdt=T[F]α.
From the
definition of fuzzy set, we have ∫0TFdt=TF.Lemma 3.
LetP,Q
be two fuzzy
sets and let h(t)
be a nonzero
continuous function on [0,T], satisfying
(63)∫0Th(t)Pdt=∫0Th(t)Qdt
then P=Q.Proof.
For
eachα-level, we have
(64)∫0Th(t)[P]αdt=[∫0Th(t)Pdt]α=[∫0Th(t)Qdt]α=∫0Th(t)[Q]αdt.
Suppose that P≠Q, then for some α∈[0,1], we have [P]α≠[Q]α. Without loss of generality, we assume that P,Q∈E1. Let Pα=[Pmin(α),Pmax(α)] and Qα=[Qmin(α),Qmax(α)]. Then, we have either (i) Pmin(α)≠Qmin(α) or (ii) Pmax(α)≠Qmax(α) holds.
If (i) holds, then(65)∫0Th(t)Pmin⁡(α)dt≠∫0Th(t)Qmin⁡(α)dt.
If (ii) holds, then(66)∫0Th(t)Pmax⁡(α)dt≠∫0Th(t)Qmax⁡(α)dt.
Thus, in both cases (i) and (ii), we have
(67)∫0Th(t)[Pmin⁡(α),Pmax⁡(α)]dt≠∫0Th(t)[Qmin⁡(α),Qmax⁡(α)]dt.
This implies that
(68)∫0Th(t)Pαdt≠∫0Th(t)Qαdt,
which is a
contradiction to (64). Hence P=Q.Definition 7 (see [20]).
Letu,v∈E1, k∈R1, and let [u]α be the α-level set of u. One defines the sum of u and v by
(69)[u+v]α=[u]α+[v]α={a+b:a∈[u]α,b∈[v]α},
the difference
between u and v by
(70)[u−v]α=[u]α−[v]α={a−b:a∈[u]α,b∈[v]α},
and the scalar product by
(71)[ku]α=k[u]α={ka:a∈[u]α}.Definition 8 (see [20]).
Letx,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
(72)[z]α=[y−x]α=[y]α−[x]α=[[y1]α−[x1]α⋯[yn2]α−[xn2]α].
If y=w−x, then w=y+x which is defined by
(73)[w]α=[y+x]α=[y]α+[x]α=[[y1]α+[x1]α⋯[yn2]α+[xn2]α].Definition 9 (see [20]).
Let(74)C=[c11c12⋯c1n2c21c22⋯c2n2⋯⋯⋯⋯cn21cn22⋯cn2n2]
be an n2×n2 matrix, p=p1×p2×⋯×pn2, let pi∈E1, i=1,2,…,n2, be a fuzzy set in En2, and let [pi]α be α-level sets of pi. Define the product Cp of C and p as
(75)[Cp]α=C[p]α=[c11c12⋯c1n2c21c22⋯c2n2⋯⋯⋯⋯cn21cn22⋯cn2n2][[p1]α[p2]α⋯[pn2]α]=[c11[p1]α+⋯+c1n2[pn2]αc21[p1]α+⋯+c2n2[pn2]α⋯cn21[p1]α+⋯+cn2n2[pn2]α].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 controllable if the n2×n2 symmetric
controllability matrix
(76)W(0,T)=∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))×(In⊗F(t))*(ψ(T−t)⊗ϕ(T−t))*dt
is nonsingular. Furthermore,the fuzzy control U^(t) which transfers
the state of the system from
X^(0)=X^0
to a fuzzy
state X^(T)=X^f=(xf1,xf2,…,xfn2) can be
determined by the following fuzzy rule base:
(77)R:Ix˜1isinxf1,…,x˜n2isinxfn2,THENu˜1isinu1,…,u˜n2isinun2,
where
(78)(u˜1(t),u˜2(t),…,ui(t),…,u˜n2(t))=1T(In⊗F(t))−1(ψ(T−t)⊗ϕ(T−t))−1×(x˜1(T),x˜2(T),…,xfi,…,x˜n2(T))−(In⊗F(t))*(ψ(T−t)⊗ϕ(T−t))*×W−1(0,T)(ψ(T)⊗ϕ(T))X^0,i=1,2,…,n2.Proof.
Suppose that the symmetric controllability matrixW(0,T) is nonsingular. Therefore W−1(0,T) exists.
Multiplying W−1(0,T)(ψ(T)⊗ϕ(T))X^0 on both sides
of (76), we have
(79)(ψ(T)⊗ϕ(T))X^0=∫0T(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))X^0dt.
Now our problem is to find the control U^(t) such that
(80)X^(T)=X^f=(ψ(T)⊗ϕ(T))X^0+∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))U^(t)dt.
Since X^ is fuzzy and
from Lemma 4, U^(t) must be fuzzy,
otherwise the fuzzy left side of (80) cannot be equal to the crisp right
side. By Lemma 2, X^f can be written
as
(81)X^f=1T∫0TX^fdt=1T∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))×(In⊗F(t))−1(ψ(T−t)⊗ϕ(T−t))−1X^fdt.
From (80) and (81), we have
(82)1T∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))×(In⊗F(t))−1(ψ(T−t)⊗ϕ(T−t))−1X^fdt=(ψ(T)⊗ϕ(T))X^0+∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))U^(t)dt.
Combining (79) and (82), we have
(83)1T∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))×(In⊗F(t))−1(ψ(T−t)⊗ϕ(T−t))−1X^fdt=∫0T(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))X^0dt+∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))U^(t)dt.
It follows that
(84)∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))U^(t)dt=∫0T(ψ(T−t)⊗ϕ(T−t))(In⊗F(t))×{1T(In⊗F(t))−1(ψ(T−t)⊗ϕ(T−t))−1X^f−(In⊗F(t))*(ψ(T−t)⊗ϕ(T−t))*×W−1(0,T)(ψ(T)⊗ϕ(T))X^0}dt.
By using Lemma 3, we get
(85)U^(t)=1T(In⊗F(t))−1(ψ(T−t)⊗ϕ(T−t))−1X^f−(In⊗F(t))*(ψ(T−t)⊗ϕ(T−t))*×W−1(0,T)(ψ(T)⊗ϕ(T))X^0.
Now we have two
special cases for (85). First, let X^(T)=X^f=(x˜1(T),x˜2(T),…,x˜n2(T)) be a crisp
point, then we will get a corresponding control U^(t)=(u˜1,u˜2,…,u˜n2), satisfying (85).
Second, letX^(T)=(x˜1(T),x˜2(T),…,xfi,…,x˜n2(T)), then the corresponding control U^(t) will take the
form U^(t)=(u˜1,u˜2,…,ui,…,u˜n2) in wich the ith component of U^(t) is a fuzzy set
in E1. Obviously, u˜i(t) is in ui(t), the grade of the membership can be determined by μxfi(x˜i(T)), the grade of the membership of x˜i(T) in xfi. Thus, based on the above discussion, we have a fuzzy
rule base for the control U^,and is given by (77) and (78).Remark 2.
The nonsingularity of the symmetric controllability
matrixW(0,T) in Theorem 3
is only a sufficient condition but not necessary because the fuzzy rule base cannot guarantee the nonsingularity of the controllability matrix.Example 1.
Consider the fuzzy dynamical matrix Lyapunov system
(1) satisfying (2) with
(86)A(t)=[0−110],B(t)=[1001],F(t)=[et00et],C(t)=[0110],D(t)=[0000],T=π2,X0=[1111].
Let X^f=(xf1,xf2,xf3,xf4)∈E4, where
(87)[X^f]α=[[xf1]α[xf2]α[xf3]α[xf4]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)]].
We select the points x˜1=0.5, x˜2=0.25, x˜3=0.05, and x˜4=0.025 which are in xf1, xf2, xf3, and xf4 with grades of
the membership are 0.5, 0.75, 0.5, and 0.75, respectively. The fundamental
matrices of (20), (21) are
(88)ϕ(t)=[cos⁡t−sin⁡tsin⁡tcos⁡t],ψ(t)=[et00et].
Now the
fundamental matrix of (19) is
(89)ψ(t)⊗ϕ(t)=[etcos⁡t−etsin⁡t00etsin⁡tetcos⁡t0000etcos⁡t−etsin⁡t00etsin⁡tetcos⁡t].
Consider
(90)(ψ(θ)⊗ϕ(θ))(In⊗F(t))(In⊗F(t))*(ψ(θ)⊗ϕ(θ))*=eθ[cos⁡θ−sin⁡θ00sin⁡θcos⁡θ0000cos⁡θ−sin⁡θ00sin⁡θcos⁡θ]et[1000010000100001]×et[1000010000100001]eθ[cos⁡θsin⁡θ00−sin⁡θcos⁡θ0000cos⁡θsin⁡θ00−sin⁡θcos⁡θ]=[eπ0000eπ0000eπ0000eπ],
where θ=π/2−t. Therefore,
(91)W(0,π2)=∫0π/2[eπ0000eπ0000eπ0000eπ]dt=π2eπ[1000010000100001].
Clearly, it is nonsingular.
Thus, from Theorem3, the input U^ can be chosen
by the following α-level sets:
(92)U^α(t)=2e−tπ[1000010000100001]e−θ[cos⁡θsin⁡θ00−sin⁡θcos⁡θ0000cos⁡θsin⁡θ00−sin⁡θcos⁡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[1000010000100001]×eθ[cos⁡θsin⁡θ00−sin⁡θcos⁡θ0000cos⁡θsin⁡θ00−sin⁡θcos⁡θ]2e−ππ[1000010000100001]×eπ/2[cos⁡(π2)−sin⁡(π2)00sin⁡(π2)cos⁡(π2)0000cos⁡(π2)−sin⁡(π2)00sin⁡(π2)cos⁡(π2)][1111].Hence, the α-level sets of
fuzzy control are given by
(93)U^α(t)=2e−π/2π[(sin⁡t+cos⁡t)[α−1,1−α](sin⁡t−cos⁡t)[α−1,1−α](sin⁡t+cos⁡t)[0.1(α−1),0.1(1−α)](sin⁡t−cos⁡t)[0.1(α−1),0.1(1−α)]]−2π[cos⁡t−sin⁡tsin⁡t+cos⁡tcos⁡t−sin⁡tsin⁡t+cos⁡t]
and the
corresponding control function to the point (0.5,0.25,0.05,0.025)* is
(94)U^(t)=[u˜1u˜2u˜3u˜4]=2e−π/2π[0.5(sin⁡t+cos⁡t)0.75(sin⁡t−cos⁡t)0.5(sin⁡t+cos⁡t)0.75(sin⁡t−cos⁡t)]−2π[cos⁡t−sin⁡tsin⁡t+cos⁡tcos⁡t−sin⁡tsin⁡t+cos⁡t].Example 2.
Consider the fuzzy dynamical matrix Lyapunov system
(1) satisfying (2) with
(95)A(t)=[1t+100−1t+1],B(t)=[1t+1e−tt+101],F(t)=[12−t00(2−t)et−1],C(t)=[0etet0],D(t)=[0000],T=1,X0=[1100].
Let X^f=(xf1,xf2,xf3,xf4)∈E4, where
(96)[X^f]α=[[xf1]α[xf2]α[xf3]α[xf4]α]=[[0.5α+0.5,1][0.8α+0.2,1][α−1,1−α][0.2(α−1),0.2(1−α)]].
We select the points x˜1=0.75, x˜2=0.8, x˜3=0.75, and x˜4=0.1 which are in xf1, xf2, xf3, and xf4 with grades of
the membership being 0.5, 0.75, 0.25, and 0.5, respectively. The fundamental
matrices of (20), (21) are
(97)ϕ(t)=[t+1001t+1],ψ(t)=[t+1t0et].
Now the
fundamental matrix of (19) is
(98)ψ(t)⊗ϕ(t)=[(t+1)20t(t+1)0010tt+100et(t+1)0000ett+1].
It is easily seen that
(99)(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00(2t2−6t+5)e2(t−1)0(1−t)et−1(1−t)e1−t0e2(1−t)00(1−t)et−101].
Therefore,
(100)W(0,π2)=∫01[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=[83010032(1−3e−2)01−2e−110e2−12001−2e−101].
Clearly, it is nonsingular.
Thus, from Theorem3, the input U^ can be chosen
by the following α-level sets,
given by
(101)U^α(t)=[12−t0(t−12−t)et−100e1−t2−t0t−12−t00et−100001]×[[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)4e2−70(9(e2−1)−6e)(1−t)+2(8e−9)e1−t4e2−70],
and the corresponding control function to the point (0.75,0.8,0.75,0.1)* is
(102)U^(t)=[u˜1u˜2u˜3u˜4]=[2+3(t−1)et−14(2−t)3e1−t+2(t−1)4(2−t)et−140.5]−[(9(e2−1)−6e)(2−t)4e2−70(9(e2−1)−6e)(1−t)+2(8e−9)e1−t4e2−70].


## 5. Observability of Fuzzy Dynamical Lyapunov Systems

In this
section, we discuss the concept of observability of the fuzzy system (58),
(59).Definition 10.
The fuzzy system (58), (59) is said to be
completely observable over the interval [0,T] if the
knowledge of rule base of input U^ and output Y^ over [0,T] suffices to
determine a rule base of initial state X^0.Letuiℓ, yiℓ, i=1,2,…,n2, ℓ=1,2,…,m, be fuzzy sets
in E1. We assume that the rule base for the input and output
is
(103)Rℓ:IFu˜1(t)isinu1ℓ(t),…,u˜n2(t)isinun2ℓ(t),THENy˜1(t)isiny1ℓ(t),…,y˜n2(t)isinyn2ℓ(t),ℓ=1,2,…,m,
and the relation between input and output is
(104)Y^(t)=(In⊗C(t))X^(t)+(In⊗D(t))U^(t).Theorem 4.
Assume that the fuzzy rule base (103) holds,
then the system (58), (59) is completely observable over the interval [0,T] if (In⊗C(T))(ψ(T)⊗ϕ(T))
is nonsingular.
Furthermore, if X^0=(x˜01,x˜02,…,x˜0n2), then one has the
following rule base for the initial value X^0:
(105)Rℓ:IFu˜1(T)isinu1ℓ(T),…,u˜n2(T)isinun2ℓ(T),IFy˜1(T)isiny1ℓ(T),…,y˜n2(T)isinyn2ℓ(T),THENx˜01isinx0ℓ(1),…,x˜0n2(t)isinx0ℓ(n2),ℓ=1,2,…,m,
where
(106)x0ℓ(i)=[(In⊗C(T))(ψ(T)⊗ϕ(T))]−1×{Viℓ(T)−(In⊗D(T))U^(T)−(In⊗C(T))×∫0T(ψ(T−s)⊗ϕ(T−s))(In⊗F(s))Hiℓ(s)ds},(107)X^0=[(In⊗C(T))(ψ(T)⊗ϕ(T))]−1×{y˜(T)−(In⊗D(T))u˜(T)−(In⊗C(T))×∫0T(ψ(T−s)⊗ϕ(T−s))(In⊗F(s))u˜(s)ds},(108)Hiℓ(t)=u˜1(t)×⋯×uiℓ(t)×⋯×u˜n2(t),Viℓ(t)=y˜1(t)×⋯×yiℓ(t)×⋯×y˜n2(t),i=1,2,…,n2,ℓ=1,2,…,m.Proof.
Without loss of generality, we prove this theorem by consideringℓ=1. Let
(109)u˜(t)=(u˜1(t),u˜2(t),…,u˜n2(t)),y˜(t)=(y˜1(t),y˜2(t),…,y˜n2(t)).
Let μui1(t)(u˜i(t)) be the grade of
the membership of u˜i(t) in ui1(t), and let μyi1(t)(y˜i(t)) be the grade of
the membership of y˜i(t) in yi1(t). Since (In⊗C(T))(ψ(T)⊗ϕ(T)) is nonsingular
and from (60), we have
(110)X^0=[(In⊗C(T))(ψ(T)⊗ϕ(T))]−1×{y˜(T)−(In⊗D(T))u˜(T)−(In⊗C(T))×∫0T(ψ(T−s)⊗ϕ(T−s))(In⊗F(s))u˜(s)ds}.
Observing
(104), when the input and output are both fuzzy sets it follows from Definition 8 that
(111)(In⊗C(t))X^(t)=Y^(t)−(In⊗D(t))U^(t)
is a fuzzy set. Substituting (60) in (111), we have
(112)(In⊗C(t))((ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))U^(s)ds)=Y^(t)−(In⊗D(t))U^(t).
Using
Definition 8
,
(113)(In⊗C(t))(ψ(t)⊗ϕ(t))X^0=Y^(t)−(In⊗D(t))U^(t)−(In⊗C(t))×∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))U^(s)ds.
Since (In⊗C(T))(ψ(T)⊗ϕ(T)) is nonsingular, we have
(114)X^0=[(In⊗C(T))(ψ(T)⊗ϕ(T))]−1×{Y^(T)−(In⊗D(T))U^(T)−(In⊗C(T))×∫0T(ψ(T−s)⊗ϕ(T−s))(In⊗F(s))U^(s)ds}.
Now, the
initial value X^0 is no more a
crisp value, but should be a fuzzy set. In order to determine each component of X^0, let us assume
(115)Hi1(t)=u˜1(t)×⋯×ui1(t)×⋯×u˜n2(t),Vi1(t)=y˜1(t)×⋯×yi1(t)×⋯×y˜n2(t),i=1,2,…,n2.
From Remark 1, we know that the ith component of
the set
(116)(ψ(t)⊗ϕ(t))X^0+∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))Hi1(s)ds
is a fuzzy set in E1. From Lemma 4, we know that the product
(117)(In⊗C(t))∫0t(ψ(t−s)⊗ϕ(t−s))(In⊗F(s))Hi1(s)ds
is a fuzzy set in En2. Hence, X^0 is a fuzzy set
in En2, and the ith component of
it denoted by x01(i) is a fuzzy set
in E1. The grade of the membership of x˜0i in x01(i) is defined by
(118)μx01(i)(x˜0i)=min⁡{μui1(t)(u˜i(t)),μyi1(t)(y˜i(t))}.
Now, we are in
a position to determine the rule base for the initial value and it is given by
(105), (106), (107)), and (108).
In general, it
is difficult to computex0ℓ(i), but to solve the real problems, we choose the
following approximation. Now, we take the point (x˜0i,μx0ℓ(i)(x˜0i)) and the
zero-level set [x0ℓ(i)]0 to determine a
triangle as the new fuzzy set x0ℓ(i).
We can use the center average defuzzifier(119)x˜0i=∑ℓ=1m(x˜0i)ℓμx0ℓ(i)((x˜0i)ℓ)∑ℓ=1mμx0ℓ(i)((x˜0i)ℓ)
to determine the initial value X^0=(x˜01,x˜02,…,x˜0n2). To obtain more accurate value for the initial state,
more rule bases may be provided.Example 3.
Consider the fuzzy matrix
Lyapunov system(120)X′(t)=[0−110]X(t)+X(t)[1001]+[et00et]U,0≤t≤π2,Y(t)=[0110]X(t).
The α-level sets of
fuzzy input U^(t) and fuzzy
output Y^(t) by rule base 1
and rule base 2 are given as follows.
Rule 1:
(121)[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]].
Rule 2:
(122)[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]].
From rule base 1, we select(123)u˜1=(u˜1,u˜2,u˜3,u˜4)=(0.5,0.85,0.4,0.75),
the grades of the membership of u˜1,u˜2,u˜3, and u˜4 are 1/3, 0.8, 0.2, and 1/2, respectively. Also
(124)y˜1=(y˜1,y˜2,y˜3,y˜4)=(1,2.8,0.5,2.9),
the grades of the membership of y˜1,y˜2,y˜3, and y˜4 are 1/2, 0.6, 2/3, and 0.8, respectively.
From rule base
2, we select(125)u˜2=(u˜1,u˜2,u˜3,u˜4)=(0.5,0.8,0.25,0.75),
the grades of the membership of u˜1,u˜2,u˜3, and u˜4 are 3/8, 3/4, 1/2, and 1/2, respectively. Also
(126)y˜2=(y˜1,y˜2,y˜3,y˜4)=(1,1.75,2,1.5),
the grades of the membership of y˜1,y˜2,y˜3, and y˜4 are 1/3, 3/4, 0.2, and 0.25, respectively.
For rule base 1, by formula (106), (107)),we have
(127)X^0=e−π/2[10000−1000010000−1]⋅([12.80.52.9]−[0100100000010010]∫0π/2eπ/2−s.⋅[cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)0000cos⁡(π2−s)−sin⁡(π2−s)00,sin⁡(π2−s)cos⁡(π2−s)].⋅es[1000010000100001][0.50.850.40.75]ds)=[−1.142−0.9324−1.046−0.9532],x01(1)=e−π/2[10000−1000010000−1]⋅([[0,−2(α−1)]2.80.52.9]−[0100100000010010]∫0π/2eπ/2−s.⋅[cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)0000cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)].⋅es[1000010000100001][[0,−0.75(α−1)]0.850.40.75]ds)=[[−1.6+0.75α,−0.434−0.416α][−1.4324,−0.6824−0.75α]−1.046−0.9532]
when α=0, we get the biggest interval [−1.6, −0.434] and x˜01=−1.142 is
located in this interval. We choose its membership grade in x01(1) as
(128)μx01(1)(x˜01)=min⁡{13,12}=13=0.333,x01(2)=e−π/2[10000−1000010000−1]⋅([1[0.5α+2.5,3]0.52.9]−[0100100000010010]∫0π/2eπ/2−s.⋅[cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)0000cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)].⋅es[1000010000100001][0.5[0.75(α−1)+1,1]0.40.75]ds)=[[−1.292,−0.542−0.75α][−1.124,−0.27−0.854α]−1.046−0.9532],
when α=0, we get the biggest interval [−1.124, −0.27] and x˜02=−0.9324 is
located in this interval. We choose its membership grade in x01(2) as
(129)μx01(2)(x˜02)=min⁡{0.8,0.6}=0.6,x01(3)=e−π/2[10000−1000010000−1]⋅([12.8[0,−1.5(α−1)]2.9]−[0100100000010010]∫0π/2eπ/2−s.⋅[cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)0000cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)].⋅es[1000010000100001][0.50.85[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]],
when α=0, we get the biggest interval [−1.25, −0.438] and x˜03=−1.046 is
located in this interval. We choose its membership grade in x01(3) as
(130)μx01(3)(x˜03)=min⁡{0.2,23}=0.2,x01(4)=e−π/2[10000−1000010000−1]⋅([12.80.5[0.5(α−1)+3,3]]−[0100100000010010]∫0π/2eπ/2−s.⋅[cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)0000cos⁡(π2−s)−sin⁡(π2−s)00sin⁡(π2−s)cos⁡(π2−s)].⋅es[1000010000100001][0.50.850.4[0.5(α−1)+1,1]]ds)=[−1.142−0.9324[−1.296,−0.796−0.5α][−1.224,−0.62−0.604α]],
when α=0, we get the biggest interval [−1.224, −0.62] and x˜04=−0.9532 is
located in this interval. We choose its membership grade in x01(4) as
(131)μx01(4)(x˜04)=min⁡{12,0.8}=12=0.5.
Similarly for rule base 2, by the use of formula (106), (107), we obtain the values of X^0, x02(i), i=1,2,3,4 and given as follows:
(132)X^0=[−1.092−0.664−0.584−0.812],x02(1)=[[−1.6+0.8α,−0.488−0.312α][−1.164,−0.364−0.8α]−0.584−0.812],x02(2)=[[−1.292,−0.492−0.8α][−0.916,0.092−1.008α]−0.584−0.812],x02(3)=[−1.092−0.664[0.5α−1.25,−0.23−0.52α][−1.062,−0.562−0.5α]],x02(4)=[−1.092−0.664[−0.834,−0.334−0.5α][−1.374,−0.458−0.916α]].
Also the grades of the membership of x˜01=−1.092,x˜02=−0.664,x˜03=−0.584,x˜04=−0.812 in x02(1), x02(2), x02(3),x02(4) are 0.333,
0.75, 0.2, 0.25, respectively. We can use
the center average defuzzifier to determine X^0=(x˜01,x˜02,x˜03,x˜04), where
(133)x˜01=∑ℓ=12x˜0ℓμx0ℓ(1)(x˜0ℓ)∑ℓ=12μx0ℓ(1)(x˜0ℓ)=−1.142×0.333+(−1.092)×0.3330.333+0.333=−1.117,x˜02=∑ℓ=12x˜0ℓμx0ℓ(2)(x˜0ℓ)∑ℓ=12μx0ℓ(2)(x˜0ℓ)=−0.9324×0.6+(−0.664)×0.750.6+0.75=−0.7833,x˜03=∑ℓ=12x˜0ℓμx0ℓ(3)(x˜0ℓ)∑ℓ=12μx0ℓ(3)(x˜0ℓ)=−1.046×0.2+(−0.584)×0.20.2+0.2=−0.815,x˜04=∑ℓ=12x˜0ℓμx0ℓ(4)(x˜0ℓ)∑ℓ=12μx0ℓ(4)(x˜0ℓ)=−0.9532×0.5+(−0.812)×0.250.5+0.25=−0.9061.


## 6. Conclusions

In this paper,
we have investigated a way to incorporate matrix Lyapunov systems with a set of
fuzzy rules. Here, a deterministic matrix Lyapunov system with fuzzy inputs and
fuzzy outputs can generate a fuzzy dynamical matrix Lyapunov system (FDMLS).
Based on this result, we can study both controllability and observability
properties of the FDMLS. First, we have provided a sufficient condition for the
controllability of the FDMLS, that is, for a given fuzzy state with a fuzzy
rule base, we can determine a control which transfers the initial state to the
given state in a finite time. The advantage of our approach is that all levels
are represented by mathematical formulas. Example1 shows how to determine
the control by our formula. Next, we have studied the observability property
which concerns the following problem, that is, given the input and output rule
bases we can determine a rule base for the initial state with a formula.
Example 3 illustrates the significance of our method by which we can
determine the rule base for initial value without solving the FDMLS. Our future
research works will concentrate on the applications of these systems (FDMLS) to
real world problems.



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*Source: 421525-2008-04-30.xml*