On Controllability and Observability of Fuzzy Dynamical Matrix ...
Hindawi Publishing Corporation Advances in Fuzzy Systems Volume 2008, Article ID 421525, 16 pages doi:10.1155/2008/421525
Research Article On Controllability and Observability of Fuzzy Dynamical Matrix 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 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. Copyright © 2008 M. S. N. Murty and G. Suresh Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 X (t) = A(t)X(t) + X(t)B(t) + F(t)U(t), X(0) = X0 ,
t > 0,
Y (t) = C(t)X(t) + D(t)U(t),
(1) (2)
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 Section 2, 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
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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
α(βA) = (αβ)A,
1·A = A, (3)
and if α, β ≥ 0, then (α + β)A = αA + βA. The distance between A and B is defined by the Hausdorff metric
d(A, B) = inf : A ⊂ N(B, ), B ⊂ N(A, ) ,
(4)
where
N(A, ) = x ∈ Rn : x − y < , forsomey ∈ A .
(5)
Definition 1. A set-valued function F : J → Pk (Rn ) is said to be measurable if it satisfies any one of the following equivalent conditions: (1) for all u ∈ Rn , t → dF(t) (u) = inf v∈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 by S1F the set of all selections of F(·) that belong to the Lebesgue Bochner space L1Rn (J), that is,
S1F = f (·) ∈ L1Rn (J) : f (t) ∈ F(t)a.e. .
(6)
We present the Aumann’s integral as follows:
(A)
J
F(t)dt =
En = u : Rn −→ [0, 1]/u satisfie(i)–(iv) below ,
(8)
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,
In this section, we present some definitions and results relating to fuzzy sets [13] and Kronecker product of matrices. Let Pk (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 α(A + B) = αA + αB,
We say that F : J → Pk (Rn ) is integrably bounded if it is measurable and there exists a function h : J → R, h ∈ L1Rn (J), such that u ≤ h(t), u ∈ F(t). From [16], we know that if F is a closed valued measurable multifunction, then J F(t)dt is n if F is integrably bounded, then convex in R . Furthermore, n. F(t)dt is compact in R J Let
J
f (t)dt, f (·) ∈ S1F .
(7)
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 follows that [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 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 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 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]). 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≤1 Aα ⊂ A0 .
M. S. N. Murty and G. Suresh Kumar
3
A fuzzy set-valued mapping F : J → En is called fuzzy integrably bounded if F0 (t) is integrably bounded.
(4) A ⊗ B = AB. (5) If A(t) and B(t) are matrices, then (A ⊗ B) = A ⊗ B + A ⊗ B ( = d/dt). (6) Vec (AY B) = (B∗ ⊗ A)Vec Y . (7) If A and B are matrices both of order n × n, then
Definition 2. Let F : J → En be a fuzzy integrably bounded mapping. The fuzzy integral of F over J denoted by J F(t)dt is defined level-set-wise by
α
J
F(t)dt
= (A)
J
Fα (t)dt,
0 < α ≤ 1.
Let F : J × En → En ,and consider the fuzzy differential equation u = F(t, u),
u(0) = u0 .
(i) Vec (AX) = (In ⊗ A)Vec X, (ii) Vec (XA) = (A∗ ⊗ In )Vec X.
(12)
(13)
Now by applying the Vec operator to the matrix Lyapunov system (1) satisfying (2) and using the above properties, we have
Definition 3. A mapping u : J → En is a fuzzy weak solution to (13) if it is continuous and satisfies the integral equation u(t) = u0 +
t 0
F 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 Kronecker products and basic results related to matrix Lyapunov systems. Definition 4 (see [2]). Let A ∈ C m×n and B ∈ C p×q .Then the Kronecker product of A and B written A ⊗ B is defined to be the partitioned matrix ⎡
a11 B a12 B · · · a1n B
⎤
⎢ ⎥ ⎢a B a B · · · a B⎥ ⎢ 21 22 2n ⎥ ⎥ A⊗B =⎢ ⎢ ⎥ ⎢ . . ··· . ⎥ ⎣ ⎦ am1 B am2 B · · · amn B
⎤
⎢ ⎥ ⎢A ⎥ ⎢ .2 ⎥ ⎢ ⎥ ⎥, A = VecA = ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎣ ⎦
⎡
a1 j
A.n
+ In ⊗ D(t) U(t), Y (t) = In ⊗ C(t) X(t)
(18)
where G(t) = (B∗ ⊗ In ) + (In ⊗ A) is an n2 × n2 matrix and X = Vec X(t), U = Vec U(t) are column matrices of order n2 . 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 for the systems X (t) = A(t)X(t),
X(0) = In ,
[X ∗ (t)] = B∗ (t)X ∗ (t),
X(0) = In ,
(20) (21)
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))X0 .
ψ(t) ⊗ φ(t)
⎤
⎥ ⎢ ⎢a ⎥ ⎢ 2j ⎥ ⎥ ⎢ ⎥ whereA. j = ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎦ ⎣
(17)
Proof. Consider
Definition 5 (see [2]). Let A = [ai j ] ∈ C m×n ; one denotes A.1
X(0) = X0 ,
(15)
which is an mp × nq matrix and is in C mp×nq . ⎡
+ In ⊗ F(t) U(t), X (t) = G(t)X(t)
(1 ≤ j ≤ n).
= 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)
am j (16)
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.
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 matrices for the systems (20) and (21). Then the unique solution of the initial 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)
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Proof. First we show that the solution of (17) is of the form X(t) = (ψ(t) ⊗ φ(t))X0 + X(t), where X(t) is a particular solution of (17) and is given by X(t) =
t
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds.
0
(24)
Let u(t) be any other solution of (17), write w(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) + In ⊗ F(t) =⇒ G(t)X(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)
(25)
is continuous in En . The set U α = Assume that U(t) u1 (t) × u2 (t) × · · · × un2 (t) is a convex and compact set 2 in Rn . For any positive number T, consider the following differential inclusions: 2
α
∈ Pk (Rn 2 ), for every 0 ≤ α ≤ 1, t ∈ [0, T]. Claim (i). [X(t)] First, we prove that X α is nonempty, compact, and convex 2 in C[[0, T], Rn ]. Since U α (t) has measurable selection, we α have that X is nonempty. Let K = 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 any X ∈ X α , then there is a selection u(t) ∈ U α (t) such that
0
t
+
=⇒ v(t) =
0
ψ −1 (s) ⊗ φ−1 (s) In ⊗ F(s) U(s)ds.
t
+
t
Let ui (t) ∈ E1 , t ∈ J, i = 1, 2, . . . , n2 , and define
+ 0
(26) where uαi (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 2 U(t) ∈ En . we show that the Now by using the fuzzy control U(t), following system
+ In ⊗ F(t) U(t), X (t) = G(t)X(t)
X(0) = X0 , (27)
+ In ⊗ D(t) U(t) Y (t) = In ⊗ C(t) X(t)
determines a fuzzy system.
ψ(t − s)φ(t − s)F(s)u(s)ds
≤ KLX0 + KLNMT.
= u1 (t) × u2 (t) × · · · × un2 (t) = uα1 (t), uα2 (t), . . . , uαn2 (t) : α ∈ [0, 1] 2 (t), . . . , u n2 (t) : u i (t) ∈ uαi (t), α ∈ [0, 1] , = u 1 (t), u
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) u(s)ds
≤ ψ(t)φ(t)X0
FORMATION OF FUZZY DYNAMICAL LYAPUNOV SYSTEMS
(31)
X(t) ≤ ψ(t) ⊗ φ(t) X0
0
U(t) = u1 (t), u2 (t), . . . , un2 (t)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) u(s)ds.
Then
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.
(30)
X(t) = ψ(t) ⊗ φ(t) X0
= ψ −1 (t) ⊗ φ−1 (t) In ⊗ F(t) U(t) t
(29)
Let X α be the solution of (29) satisfying (30).
t ∈ [0, T],
X(0) = X0 .
=⇒ v (t)
+ In ⊗ F(t) U α (t), X (t) ∈ G(t)X(t)
(28)
(32) Thus X α is bounded. For any t1 , t2 ∈ [0, T], 2) 1 ) − X(t X(t = 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)
M. S. N. Murty and G. Suresh Kumar
5 Let X = λX1 (t) + (1 − λ)X2 (t), 0 ≤ λ ≤ 1, then
Therefore X(t 2 ) 1 ) − X(t
X = λX1 (t) + (1 − λ)X2 (t)
≤ ψ(t1 ) ⊗ φ(t1 ) − ψ(t2 ) ⊗ φ(t2 ) X0 t1
+
t2
t2
+ 0
= λ G(t)X1 (t) + In ⊗ F(t) u1 (t)
ψ(t1 − s) ⊗ φ(t1 − s) In ⊗ F(s) u(s)ds 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
+ MN
0
ψ(t1 − s) ⊗ φ(t1 − s)
(34) 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. For each Xk , there is a uk ∈ Let Xk ∈ X α and Xk → X. α U (t) such that
0
α2
U α2 (t) = uα1 2 (t) × uα2 2 (t) × · · · × uαn22 (t)
+ 0
α
(41)
Thus, we have the selection inclusion S1U α2 (t) ⊂ S1U α1 (t) and the following inclusion:
X (t) ∈ G(t)X + In ⊗ F(t) U α2 (t) α 1 (t). ⊂ G(t)X + In ⊗ F(t) U
(42)
Consider the differential inclusions
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s)
λ j uk j (s)ds. (36)
From Fatou’s lemma, taking the limit as j → ∞ on both sides of (36), we have
X(t) = ψ(t) ⊗ φ(t) X0 t
+ 0
X (t) ∈ G(t)X + In ⊗ F(t) U α2 (t),
λ j ψ(t) ⊗ φ(t) X0
t
α
1 (t). = Uα
(35)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) uk (s)ds.
λ j Xk j (t) =
α1
α
(40)
⊂ [X(t)] , for all 0 ≤ α1 ≤ α2 ≤ 1. Claim (ii). [X(t)] Let 0 ≤ α1 ≤ α2 ≤ 1. Since U α2 (t) ⊂ U α1 (t), we have
⊂ u1 1 (t) × u2 1 (t) × · · · × un21 (t)
Since uk ∈ U α (t) is closed, then there exists a subsequence α (t). From Mazur’s {uk j } of {uk } converging weakly to u ∈ U theorem [20], thereexists a sequence of numbers λ j > 0, λ j = 1 such that λ j uk j converges strongly to u. Thus, from (35) we have
that is X ∈ X α . Thus X α is convex. Therefore, X α is 2 nonempty, compact, and convex in C[[0, T], Rn ]. Thus, α from Arzela-Ascoli theorem, we know that [X(t)] is comα 2 n pact in R for every t ∈ [0, T]. Also it is obvious that [X(t)] α 2 2 is convex in Rn . Thus, we have [X(t)] ∈ Pk (Rn ), for every t ∈ [0, T]. Hence the claim.
t
(39)
Since U α (t) is convex, λu1 (t) + (1 − λ)u2 (t) ∈ U α (t), we have
Xk (t) = ψ(t) ⊗ φ(t) X0 +
= G(t) λX1 (t) + (1 − λ)X2 (t)
+ In ⊗ F(t) λu1 (t) + (1 − λ)u2 (t) .
− ψ(t2 − s) ⊗ φ(t2 − s) ds.
+ In ⊗ F(t) U α (t), X (t) ∈ G(t)X(t)
+ KLNM |t1 − t2 | T
+ (1 − λ) G(t)X2 (t) + In ⊗ F(t) u2 (t)
X (t) ∈ G(t)X + In ⊗ F(t) U α1 (t),
X2 (t) = G(t)X2 (t) + In ⊗ F(t) u2 (t).
(44)
+
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U α2 (s) ds (45)
⊂ ψ(t) ⊗ φ(t) X0
∈ X α , and hence X α is closed. Thus, X(t) Let X1 , X2 ∈ X α , then there exist u1 , u2 ∈ U α (t) such that
X1 (t) = G(t)X1 (t) + In ⊗ F(t) u1 (t),
t ∈ [0, T].
X(t) ∈ ψ(t) ⊗ φ(t) X0
0
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) u(s)ds.
(43)
Let X α2 and X α1 be the solution sets of (43) and (44), respectively. Clearly, the solution of (43) satisfies the following inclusion: t
(37)
t ∈ [0, T],
t
+ 0
(38)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U α1 (s) ds.
Thus X α2 ⊂ X α1 , and hence X α2 (t) ⊂ X α1 (t). Hence the claim.
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Claim (iii). If {αk } is a nondecreasing sequence converging to α > 0, then X α (t) = k≥1 X αk (t). Let U αk (t) = uα1 k × uα2 k × · · · × uαn2k , U α (t) = uα1 × uα2 × · · · × uαn2 and consider the inclusions
X (t) ∈ G(t)X + In ⊗ F(t) U αk (t),
X (t) ∈ G(t)X + In ⊗ F(t) U α (t).
(46) (47)
It follows that
X(t) ∈ ψ(t) ⊗ φ(t) X0
+
t k≥1 0
uαi =
k≥1
⊂ ψ(t) ⊗ φ(t) X0 t
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1
α k ≥1 U k
0
ds
= ψ(t) ⊗ φ(t) X0 t
+
uαi k .
+ 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
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U αk ds
0
(48)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U α ds. (54)
This implies that X ∈ X α . Therefore,
Consider
U α (t) = uα1 × uα2 × · · · × uαn2 =
uα1 k ×
k≥1
=
uα1 k
uα2 k × · · · ×
α × u2 k
k≥1
=
(55)
k≥1
k≥1
X αk ⊂ X α .
k≥1
uαn2k
α × · · · × un2k
From (51) and (55), we have X α =
(49)
X αk ,
(56)
X αk (t).
(57)
k≥1
and hence,
U αk (t)
X α (t) =
k≥1
k≥1
and then S1U α (t) = S1
α k ≥1 U k (t)
. Therefore
X (t) ∈ G(t)X + In ⊗ F(t) U α (t) α = G(t)X + In ⊗ F(t) U k (t)
(50)
k≥1
α k (t), ⊂ G(t)X + In ⊗ F(t) U
k = 1, 2, . . . .
X αk .
(51)
k≥1
+ In ⊗ D(t) U(t). Y (t) = In ⊗ C(t) X(t)
t
+ 0
k ≥ 1.
(52)
Then,
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds.
X(t) ∈ ψ(t) ⊗ φ(t) X0 t 0
(60)
Remark 1. Consider a special case. If the input is in the form U(t) =u 1 (t) × u 2 (t) × · · · × ui (t) × · · · × u n2 (t),
+
(59)
The solution set of the fuzzy dynamical system (58), (59) is given by
X (t) ∈ G(t)X + In ⊗ F(t) U αk (t),
X(0) = X0 , (58)
X(t) ∈ ψ(t) ⊗ φ(t) X0
Let X be the solution set to the inclusion
+ In ⊗ F(t) U(t), X (t) = G(t)X(t)
Thus, we have X α ⊂ X αk , k = 1, 2, . . ., which implies that X α ⊂
From Claims 3–3 and applying Theorem 1, there exists 2 X(t) ∈ En 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
(61)
where uk (t) ∈ R1 , k = / i, are crisp numbers, then the ith component of the solution set of (27) is a fuzzy set in E1 .
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U αk (t) ds. (53)
Proof. The proof follows along similar lines as in the above discussion.
M. S. N. Murty and G. Suresh Kumar 4.
7 Definition 7 (see [20]). Let u, v ∈ E1 , k ∈ R1 , and let [u]α be the α-level set of u. One defines the sum of u and v by
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 = X0 and completely controllable if for any initial state X(0) any given final state X f there exists a finite time t1 > 0 and a 1 ) = X f . 0 ≤ t ≤ t1 , such that X(t control U(t), Lemma 2. If F is a fuzzy set, then
T 0
F dt = TF.
α
0
=
T
[F]α dt = T[F]α .
0
From the definition of fuzzy set, we have
T 0
(69) the difference between u and v by
and the scalar product by
0
h(t)P dt =
⎡
(62)
⎢ ⎢ [z]α = [y − x]α = [y]α − [x]α = ⎢ ⎣
F dt = TF.
T
[yn2 ]α − [xn2 ]α
⎡ ⎢ ⎢ ⎣
[y1 ]α + [x1 ]α
[w]α = [y + x]α = [y]α + [x]α = ⎢ T 0
h(t)[P]α dt = = =
T 0
T 0
0
h(t)Pmin (α)dt = /
⎡
h(t)Q dt
(64)
α
T
h(t)Qmin (α)dt.
(65)
h(t)Qmax (α)dt.
(66)
If (ii) holds, then T 0
h(t)Pmax (α)dt = /
T 0
Thus, in both cases (i) and (ii), we have
h(t) Pmin (α), Pmax (α) dt = /
T 0
h(t) Qmin (α), Qmax (α) dt. (67)
This implies that T 0
h(t)P α dt = /
T 0
⎥ ⎥ ⎥. ⎦
Definition 9 ([20]). Let
h(t)[Q] dt.
0
⎤
(73)
α
T 0
⎥ ⎥ ⎥. ⎦
α
h(t)P dt
Suppose that P = / Q, then for some α ∈ [0, 1], we α have [P]α = / [Q] . Without loss of generality, we assume that P, Q ∈ E1 . Let P α = [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 T
···
[yn2 ]α + [xn2 ]α
Proof. For each α-level, we have
0
···
⎤
(72)
(63)
then P = Q.
T
[y1 ]α − [x1 ]α
If y = w − x, then w = y + x which is defined by h(t)Q dt
0
(71)
2
Lemma 3. Let P, Q be two fuzzy sets and let h(t) be a nonzero continuous function on [0, T], satisfying T
[u − v]α = [u]α − [v]α = a − b : a ∈ [u]α , b ∈ [v]α , (70)
Definition 8 (see [20]). Let x, y ∈ En 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
Proof. Let [F] be the α-level set of F. Since F dt
[ku]α = k[u]α = ka : a ∈ [u]α .
α
T
[u + v]α = [u]α + [v]α = a + b : a ∈ [u]α , b ∈ [v]α ,
c11
c12 · · · c1n2
⎤
⎢ ⎥ ⎢c ⎥ ⎢ 21 c22 · · · c2n2 ⎥ ⎥ C=⎢ ⎢ ⎥ ⎢· · · · · · · · · · · · ⎥ ⎣ ⎦ cn2 1 cn2 2 · · · cn2 n2
(74)
be an n2 × n2 matrix, p = p1 × p2 × · · · × pn2 , let pi ∈ E1 , 2 i = 1, 2, . . . , n2 , be a fuzzy set in En , and let [pi ]α be α-level sets of pi . Define the product C p of C and p as ⎡
c11
c12 · · · c1n2
⎤⎡
[p1 ]α
⎤
⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ c21 c22 · · · c2n2 ⎥ ⎢ [p2 ]α ⎥ ⎥ ⎢ ⎥ ⎢ α α ⎥⎢ ⎥ [C p] = C[p] = ⎢ ⎥⎢ ⎥ ⎢ ⎢· · · · · · · · · · · · ⎥ ⎢ · · · ⎥ ⎥⎢ ⎥ ⎢ ⎦⎣ ⎦ ⎣ cn2 1 cn2 2 · · · cn2 n2 [pn2 ]α ⎡
c11 [p1 ]α + · · · + c1n2 [pn2 ]α
⎤
⎥ ⎢ ⎥ ⎢ ⎢ c21 [p1 ]α + · · · + c2n2 [pn2 ]α ⎥ ⎥ ⎢ ⎥. =⎢ ⎥ ⎢ ⎥ ⎢ · · · ⎥ ⎢ ⎣ α α⎦ cn2 1 [p1 ] + · · · + cn2 n2 [pn2 ]
All these definitions yield the following lemma. h(t)Qα dt,
which is a contradiction to (64). Hence P = Q.
(68)
2
Lemma 4. C p is a fuzzy set in En . Proof. The proof is similar to proof of Lemma 3.1 [13].
(75)
8
Advances in Fuzzy Systems
Theorem 3. The fuzzy system (58),(59) is completely controllable if the n2 × n2 symmetric controllability matrix T
W(0, T) =
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
0
must be fuzzy, Since X is fuzzy and from Lemma 4, U(t) otherwise the fuzzy left side of (80) cannot be equal to the crisp right side. By Lemma 2, X f can be written as
∗ ∗ × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) dt
(76)
is nonsingular. Furthermore,the fuzzy control U(t) which = X0 to a fuzzy transfers the state of the system from X(0) = X f = (x f1 , x f2 , . . . , x fn2 ) can be determined by state X(T) the following fuzzy rule base:
R : IF x1 is in x f1 , . . . , xn2 is in x fn2 ,
(77)
THEN u1 is in u1 , . . . , un2 is in un2 ,
X f =
1 T
=
1 T
T 0
T
X f dt
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
0
−1 −1 × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f dt.
(81) From (80) and (81), we have 1 T
T
0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
−1 −1 × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f dt = ψ(T) ⊗ φ(T) X0
where
u1 (t), u2 (t), . . . , ui (t), . . . , un2 (t) =
T
+
× x1 (T), x2 (T), . . . , x fi , . . . , xn2 (T)
1 T
T 0
ψ(T) ⊗ φ(T) X0 =
T 0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
× In ⊗ F(t)
∗
=
T
(79) Now our problem is to find the control U(t) such that
0
T
+ 0
= ψ(T) ⊗ φ(T) X0
0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt.
(83) It follows that T
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt.
(80)
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt T 0
×
× W −1 (0, T) ψ(T) ⊗ φ(T) X0 dt
=
X(T) = X f
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
∗ ∗ × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t)
0
T
∗
× W −1 (0, T) ψ(T) ⊗ φ(T) X0 dt.
+
−1 −1 × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f dt
ψ(T − t) ⊗ φ(T − t)
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
i = 1, 2, . . . , n2 . (78)
Proof. Suppose that the symmetric controllability matrix W(0, T) is nonsingular. Therefore W −1 (0, T) exists. Multiplying W −1 (0, T)(ψ(T) ⊗ φ(T))X0 on both sides of (76), we have
Combining (79) and (82), we have
∗ ∗ − In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) × W −1 (0, T) ψ(T) ⊗ φ(T) X0 ,
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt.
(82)
−1 −1 1 In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) T
0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
−1 −1 1 In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f T ∗ ∗ − In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) × W −1 (0, T) ψ(T) ⊗ φ(T) X0 dt.
(84)
M. S. N. Murty and G. Suresh Kumar
9
By using Lemma 3, we get U(t) =
the membership are 0.5, 0.75, 0.5, and 0.75, respectively. The fundamental matrices of (20), (21) are
−1 −1 1 In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f T ∗ ∗ − In ⊗ F(t) ψ(T − t) ⊗ φ(T − t)
⎡
φ(t) = ⎣
(85)
× W −1 (0, T) ψ(T) ⊗ φ(T) X0 .
⎡
A(t) = ⎣
1 0
⎡
F(t) = ⎣ ⎡
et 0 0 et
⎦,
B(t) = ⎣ ⎡
⎤ ⎦,
C(t) = ⎣
⎦,
0 0
π T= , 2
0 et
⎤ ⎦.
(88)
⎡
0
0
⎤
(89)
et sint et cos t
0
cos θ − sin θ
⎢ ⎢ ⎢ sin θ θ⎢ =e ⎢ ⎢ 0 ⎢ ⎣
cos θ 0
0 ⎡
0 ⎤
1 0 0 0
∗
ψ(θ) ⊗ φ(θ) ⎤
0
0
0
0
sin θ
cos θ
⎡
cos θ
0
0 0 0 1
⎤
⎡
(86)
eπ 0
0
0
0 eπ
0
⎡
∗ ⎤
1 0 0 0
⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢0 1 0 0⎥ ⎥ t⎢ ⎥ ⎥e ⎢ ⎥ ⎢0 0 1 0⎥ cos θ − sin θ ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦
0 0 0 1
sin θ
⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢0 1 0 0⎥ ⎢− sin θ cos θ ⎥ θ⎢ t⎢ ⎢ ⎥e ⎢ ×e ⎢0 0 1 0⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎣
0 1
0
⎤
0
0
0
0 ⎥ ⎥
cos θ
⎥ ⎥ ⎥
sin θ ⎥ ⎥ ⎦
− sin θ cos θ
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0 eπ 0 0 ⎥ ⎢ ⎥ ⎥, =⎢ ⎢ 0 0 eπ 0 ⎥ ⎢ ⎥ ⎣ ⎦
1 0
X0 = ⎣
0
⎦,
⎦,
0
⎥ ⎥ ⎥ ⎥ ⎥. t t e cos t −e sint⎥ ⎥ ⎦
et cos t
ψ(θ) ⊗ φ(θ) In ⊗ F(t) In ⊗ F(t)
⎤
0 1
0
Consider
1 0
⎡
⎤
0 0
D(t) = ⎣
⎡
ψ(t) = ⎣
⎦,
0
Example 1. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with ⎤
et 0
et cos t −et sint
⎢ ⎢ t ⎢ e sint ⎢ ψ(t) ⊗ φ(t) = ⎢ ⎢ 0 ⎢ ⎣
Remark 2. The nonsingularity of the symmetric controllability matrix W(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.
0 −1
cos t
sint
⎡
⎤
Now the fundamental matrix of (19) is
Now we have two special cases for (85). First, let X(T) = 2 X f = (x1 (T), x2 (T), . . . , xn (T)) be a crisp point, then we = (u 1 , u 2 , . . . , u n2 ), will get a corresponding control U(t) satisfying (85). = (x1 (T), x2 (T), . . . , x fi , . . . , xn2 (T)), Second, let X(T) will take the form then the corresponding control U(t) 2 U(t) = (u1 , u2 , . . . , ui , . . . , un ) in wich the ith component of U(t) is a fuzzy set in E1 . Obviously, ui (t) is in ui (t), the grade of the membership can be determined by μx fi (xi (T)), the grade of the membership of xi (T) in x fi . Thus, based on the above discussion, we have a fuzzy rule base for the control U,and is given by (77) and (78).
⎡
cos t − sint
⎤
1 1
⎦.
1 1
0
(90) Let X f = (x f1 , x f2 , x f3 , x f4 ) ∈ E4 , where ⎡
α ⎤
⎡
where θ = π/2 − t.Therefore,
⎤
x f1 [α − 1, 1 − α] ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ α⎥ ⎢ x f2 ⎥ ⎢ ⎥ [α − 1, 1 − α] ⎥ ⎢ ⎥
α ⎢ ⎥=⎢ ⎥. X f = ⎢ ⎢ α ⎥ ⎢
⎥ ⎥ ⎢ ⎢ x ⎥ ⎢ f3 ⎥ ⎢ 0.1(α − 1), 0.1(1 − α) ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣
⎣ ⎦
α 0.1(α − 1), 0.1(1 − α) x f4
⎡
(87)
We select the points x1 = 0.5, x2 = 0.25, x3 = 0.05, and x4 = 0.025 which are in x f1 , x f2 , x f3 , and x f4 with grades of
π W 0, 2
eπ 0
0
0
0 eπ
0
⎤
⎡
1 0 0 0
⎤
⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ π/2 ⎢ ⎢ 0 eπ 0 0 ⎥ 0 1 0 0⎥ π π⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ dt = e ⎢ ⎥. = ⎢ ⎥ ⎢ ⎥ π 2 0 ⎢0 0 e 0⎥ ⎢0 0 1 0 ⎥ ⎣ ⎦ ⎣ ⎦
0
Clearly, it is nonsingular.
0 0 0 1 (91)
10
Advances in Fuzzy Systems
Thus, from Theorem 3, the input U can be chosen by the following α-level sets:
⎡
U α (t) ⎡
2e−t = π
and the corresponding control function to the point (0.5, 0.25, 0.05, 0.025)∗ is
1 0 0 0
⎤
⎡
cos θ
sin θ
⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢0 1 0 0 ⎥ ⎢− sin θ cos θ ⎢ ⎥ −θ ⎢ ⎢ ⎥e ⎢ ⎢0 0 1 0 ⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎣
0
0 0 0 1
0
0
0 ⎥ ⎥
sin θ
⎢ ⎢ ⎢− sin θ cos θ θ⎢ ×e ⎢ ⎢ 0 0 ⎢ ⎣
0 ⎡
0
0 0 cos θ
π 2
⎢ ⎢ ⎢ π ⎢ ⎢ sin ⎢ 2 ⎢ × eπ/2 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎣
0
0 0 0
⎦
π 2
π cos 2 0 0
⎥ ⎥
⎡
1 ⎢ ⎢t + 1
A(t) = ⎢
⎦
⎣
0 0 1 ⎤
1 0 0 0
⎥ ⎥ ⎦
⎢ ⎥ ⎢ ⎥ ⎢0 1 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 0 1 0⎥ ⎢ ⎥ ⎣ ⎦ ⎤
0
0
⎥ ⎥⎡ ⎤ ⎥1 ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ 0 0 ⎥⎢1⎥ ⎥⎢ ⎥ ⎢ ⎥. ⎥ ⎢1⎥ π π ⎥ ⎥ ⎢ ⎥ − sin cos ⎥⎣ ⎦ 2 2 ⎥ ⎥ ⎥1 π π ⎦
2
cos
2e−π/2 U (t) = 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)
cos t − sint
⎢ ⎥ ⎢sint +cos t ⎥ ⎢ ⎥ ⎢ ⎥. ⎢ ⎥ ⎢cos t − sint⎥ ⎣ ⎦
sint +cos t
1
0 D(t) = ⎣
⎡
0 0 0 0
1
e−t
0
1
⎤
0 (2 − t)et−1
⎡
⎥ ⎦,
C(t) = ⎣ ⎡
⎤ ⎦,
⎤
⎢ ⎥ B(t) = ⎣ t + 1 t + 1 ⎦ ,
⎥ ⎥, −1 ⎦
t+1
⎢ F(t) = ⎣ 2 − t ⎡
⎤
0 ⎥
X0 = ⎣
T = 1,
0 et et
0
⎤ ⎦,
(95)
⎤
1 1
⎦.
0 0
Let X f = (x f1 , x f2 , x f3 , x f4 ) ∈ E4 , where ⎡
α ⎤
⎡ ⎢ ⎥ ⎢ ⎢ α ⎥ ⎢ ⎢ x ⎥ ⎥ ⎢
α ⎢ ⎢ f2 ⎥ ⎢ ⎢ X f = ⎢ α ⎥ =⎢ ⎥ ⎢ ⎢ x f3 ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣
α
x f1
x f4
⎤
[0.5α + 0.5, 1] [0.8α + 0.2, 1] [α − 1, 1 − α] 0.2(α − 1), 0.2(1 − α)
⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(96)
2
Hence, the α-level sets of fuzzy control are given by
α
⎤
0 0 0 1
(92)
⎡
0
⎡ ⎡
0 ⎥ ⎥ 2e−π ⎥ sin θ ⎥ ⎥ π
sin
cos t − sint
Example 2. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with
⎤
− sin
⎡
(94)
1 0 0⎥ ⎥ ⎥ 0 1 0⎥ ⎥
⎤
− sin θ cos θ
⎢cos
0
⎤
− sin θ cos θ
0
⎤ [α − 1, 1 − α] ⎡ ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ [α − 1, 1 − α] ⎢0 ⎥ ⎢ ⎥ ⎢ t⎢ ⎥−e ⎢ × ⎢
⎢0 ⎢ 0.1(α − 1), 0.1(1 − α) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ ⎥ ⎢ ⎦ ⎣
0 0.1(α − 1), 0.1(1 − α)
cos θ
⎥
sin θ ⎥ ⎥
cos θ
⎡
⎡
⎥ ⎥
0.5(sint +cos t)
⎢ ⎥ ⎢ ⎥ ⎢0.75(sint −cos t)⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥− ⎢ 0.5(sint +cos t) ⎥ π ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.75(sint −cos t)
⎢ ⎥ ⎢u ⎥ ⎢ 2 ⎥ 2e−π/2 ⎥ =⎢ U(t) ⎢ ⎥= π ⎢u ⎥ ⎣ 3⎦ u4
⎤
0
u1
⎡
⎤
We select the points x1 = 0.75, x2 = 0.8, x3 = 0.75, and x4 = 0.1 which are in x f1 , x f2 , x f3 , and x f4 with grades of the membership being 0.5, 0.75, 0.25, and 0.5, respectively. The fundamental matrices of (20), (21) are ⎡ ⎢ φ(t) = ⎢ ⎣
t+1 0
0
sint +cos t (93)
ψ(t) = ⎣
t+1
t+1 t et
0
⎤ ⎦.
(97)
Now the fundamental matrix of (19) is ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ψ(t) ⊗ φ(t) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ sint +cos t 2⎢ ⎥ ⎥ − ⎢ ⎥ π⎢ ⎢cos t − sint⎥ ⎣ ⎦
⎡
⎥ ⎥ 1 ⎦,
⎡
⎤
⎤
(t + 1)2 0 t(t + 1) 0 0 0
0
⎤ ⎥
t ⎥ ⎥ ⎥ 1 0 t + 1⎥ ⎥ ⎥. 0 et (t + 1) 0 ⎥ ⎥ ⎥ ⎥
0
0
et ⎦ t+1
(98)
M. S. N. Murty and G. Suresh Kumar
11
It is easily seen that
ψ(1 − t) ⊗ φ(1 − t) In ⊗ F(t) In ⊗ F(t)
and the corresponding control function to the point (0.75, 0.8, 0.75, 0.1)∗ is
∗
⎡
⎡
⎤ 2t 2 − 6t+5 0 (1 − t)e1−t 0 ⎢ ⎥ ⎢ ⎥ 2 2(t−1) t −1 ⎥ ⎢ 0 2t − 6t+5 e 0 (1 − t)e ⎥ ⎢ ⎢ ⎥ ⎥. =⎢ 2(1 −t) ⎢ (1 − t)e1−t ⎥ 0 e 0 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ t −1 0 (1 − t)e 0 1
⎡
(99)
π 2
⎡
2t 2 − 6t+5
0
⎢ ⎢ 2 0 2t − 6t+5 e2(t−1) 1⎢ ⎢ ⎢ = ⎢ 1−t 0⎢ 0 ⎢ (1 − t)e ⎢ ⎣ 0 (1 − t)et−1 ⎡
(1 − t)e1−t 0 e2(1−t) 0
0
⎤
1
⎢2 − t 0 ⎢ ⎢ 1−t ⎢ ⎢ 0 e α ⎢ U (t) = ⎢ 2−t ⎢ ⎢ 0 0 ⎢ ⎣
0 ⎡
t − 1 t−1 e 2−t 0 et−1
0
0
⎤⎡[0.5α + 0.5, 1] 0 ⎥⎢ ⎥⎢ ⎥⎢ ⎢[0.8α + 0.2, 1] t − 1⎥ ⎥⎢ ⎢ 2 − t⎥ [α − 1, 1 − α] ⎥⎢ ⎥⎢ ⎢ ⎥
0 ⎥⎢ ⎦⎢ ⎣ 0.2(α − 1), 0.2(1 − α) 1
9 e2 − 1 − 6e (2 − t) 4e2 − 7 0
OBSERVABILITY OF FUZZY DYNAMICAL LYAPUNOV SYSTEMS
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 X0 .
Clearly, it is nonsingular. Thus, from Theorem 3, the input U can be chosen by the following α-level sets, given by
5.
In this section, we discuss the concept of observability of the fuzzy system (58), (59).
(100)
⎤
(102)
⎥ ⎦
0 1 0 ⎥ ⎢3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3 ⎢0 1 − 3e−2 0 1 − 2e−1 ⎥ ⎢ ⎥ 2 ⎥. =⎢ ⎢ ⎥ 2−1 ⎢ ⎥ e ⎢1 ⎥ 0 0 ⎢ ⎥ 2 ⎢ ⎥ ⎣ ⎦ 0 1 − 2e−1 0 1
⎡
⎥ ⎥
⎤
8
9 e2 − 1 − 6e (2 − t) 4e2 − 7 0
0
(1 − t)et−1⎥ ⎥ ⎥ ⎥dt 0 ⎥ ⎥ 1
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥. − ⎢ 2 ⎢ 9 e − 1 − 6e (1 − t) + 2(8e − 9)e1−t ⎥ ⎥ ⎢ ⎥ ⎢ 2−7 ⎥ ⎢ 4e ⎦ ⎣
Therefore, W 0,
⎤
2 + 3(t − 1)et−1 ⎥ ⎡ ⎤ ⎢ 4(2 − t) ⎢ ⎥ u1 ⎢ ⎥ ⎢ ⎥ ⎢ 3e1−t + 2(t − 1) ⎥ ⎢u ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ 4(2 − t) ⎥ ⎢ ⎥ U(t) =⎢ ⎢ ⎥=⎢ ⎥ t −1 ⎢u ⎥ 3 ⎥ ⎢ e ⎥ ⎣ ⎦ ⎢ ⎢ ⎥ ⎢ ⎥ 4 u4 ⎣ ⎦ 0.5
∗ × ψ(1 − t) ⊗ φ(1 − t)
E1 .
Let u i , yi , i = 1, 2, . . . , n2 , = 1, 2, . . . , m, be fuzzy sets in We assume that the rule base for the input and output is
R : IF u1 (t) is in u 1 (t), . . . , un2 (t) is in u n2 (t), THEN y1 (t) is in y1 (t), . . . , yn2 (t) is in yn 2 (t),
= 1, 2, . . . , m,
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎤
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥, − ⎢ 2 ⎢ 9 e − 1 − 6e (1 − t) + 2(8e − 9)e1−t ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 4e2 − 7 ⎦ ⎣
(103)
and the relation between input and output is
+ In ⊗ D(t) U(t). Y (t) = In ⊗ C(t) X(t)
(104)
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, 2 if X0 = (x01 , x02 , . . . , x0n ), then one has the following rule base for the initial value X0 : R : IF u1 (T) is in u 1 (T), . . . , un2 (T) is in u n2 (T), IF y1 (T) is in y1 (T), . . . , yn2 (T) is inyn 2 (T), 2
THEN x01 is in x0 (1), . . . , x0n (t) is in x0 n2 ,
0
(101)
= 1, 2, . . . , m,
(105)
12
Advances in Fuzzy Systems
where
Using Definition 8,
x0 (i) =
In ⊗ C(T) ψ(T) ⊗ φ(T)
×
Vi (T) − In
×
T
−1
⊗ D(T) U(T) − In ⊗ C(T)
ψ(T − s) ⊗ φ(T − s) In ⊗ F(s)
Hi (s)ds
×
X0 =
In ⊗ C(T) ψ(T) ⊗ φ(T)
(T) − In ⊗ C(T) × y(T) − In ⊗ D(T) u ×
T
0
In ⊗ C(T) ψ(T) ⊗ φ(T)
T
×
ψ(T − s) ⊗ φ(T − s) In ⊗ F(s) u(s)ds ,
0
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds.
−1
× Y (T) − In ⊗ D(T) U(T) − In ⊗ C(T)
(113)
Since (In ⊗ C(T))(ψ(T) ⊗ φ(T)) is nonsingular, we have X0 =
−1
t
,
(106)
= Y (t) − In ⊗ D(t) U(t) − In ⊗ C(t)
0
In ⊗ C(t) ψ(t) ⊗ φ(t) X0
ψ(T − s) ⊗ φ(T − s) In ⊗ F(s) U(s)ds .
0
(114)
(107) Hi (t) = u1 (t) × · · · × u i (t) × · · · × un2 (t), Vi (t) = y1 (t) × · · · × yi (t) × · · · × yn2 (t), i = 1, 2, . . . , n2 ,
(108)
Hi1 (t) = u1 (t) × · · · × u1i (t) × · · · × un2 (t),
= 1, 2, . . . , m.
Proof. Without loss of generality, we prove this theorem by considering = 1. Let
(109)
y(t) = y1 (t), y2 (t), . . . , yn2 (t) .
Let μu1i (t) (ui (t)) be the grade of the membership of ui (t) in u1i (t), and let μ yi1 (t) ( yi (t)) be the grade of the membership of yi (t) in yi1 (t). Since (In ⊗ C(T))(ψ(T) ⊗ φ(T)) is nonsingular and from (60), we have
X0 =
In ⊗ C(T) ψ(T) ⊗ φ(T)
−1
T 0
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
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) ⊗ φ(t) X0 t
+ 0
(116)
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
t 0
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) Hi1 (s)ds
is a fuzzy set in En . Hence, X0 is a fuzzy set in En , and the ith component of it denoted by x01 (i) is a fuzzy set in E1 . The grade of the membership of x0i in x01 (i) is defined by 2
(110)
(117)
× y(T) − In ⊗ D(T) u (T) − In ⊗ C(T) ×
(115)
i = 1, 2, . . . , n2 .
In ⊗ C(t)
Vi1 (t) = y1 (t) × · · · × yi1 (t) × · · · × yn2 (t),
From Remark 1, we know that the ith component of the set
u(t) = u1 (t), u2 (t), . . . , un2 (t) ,
Now, the initial value X0 is no more a crisp value, but should be a fuzzy set. In order to determine each component of X0 , let us assume
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds
2
μx01 (i) x0i = min μu1i (t) ui (t) , μ yi1 (t) yi (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 compute x0 (i), but to solve the real problems, we choose the following approximation. Now, 0 we take the point (x0i , μx0 (i) (x0i )) and the zero-level set [x0 (i)] to determine a triangle as the new fuzzy set x0 (i). We can use the center average defuzzifier ! m i
0 μx0 (i) x0i
=1 x x0i = m i ! 0
=1 μx0 (i) x
(119)
to determine the initial value X0 = (x01 , x02 , . . . , x0n ). To obtain more accurate value for the initial state, more rule bases may be provided. 2
= Y (t) − In ⊗ D(t) U(t).
(112)
(118)
M. S. N. Murty and G. Suresh Kumar
13 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
Example 3. Consider the fuzzy matrix Lyapunov system ⎡
⎤
⎡
⎤
⎡
⎤
0 −1 1 0 et 0 ⎦ X(t) + X(t) ⎣ ⎦+⎣ ⎦ U, X (t) = ⎣ 1 0 0 1 0 et 0≤t≤ ⎡
⎤
π , 2
⎡
(120)
X0 = e
0 1 ⎦ X(t). Y (t) = ⎣ 1 0
⎡
0, −0.75(α − 1)
⎜⎡ ⎤ ⎡ ⎜ ⎜ 1 ⎜ ⎢ ⎥ ⎢0 ⎜⎢ ⎥ ⎢ ⎜ ⎢2.8⎥ ⎢1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢ ⎜⎢ ⎢0 ⎢ 0.5 ⎜⎣ ⎥ ⎦ ⎣ ⎜ ⎜ 2.9 0 ⎜ ⎝
⎤
⎢ ⎥ ⎢
⎥ ⎢ 0.75(α − 1) + 1, 1 ⎥ ⎢ ⎥
(1) α ⎢ ⎥ U =⎢
⎥, ⎢ 0, −0.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣
⎦ 0.5(α − 1) + 1, 1 ⎡
0, −2(α + 1)
⎤
⎡
⎤
(121)
·⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
From rule base 1, we select (123)
the grades of the membership of u1 , u2 , u3 , and u4 are 1/3, 0.8, 0.2, and 1/2, respectively. Also
(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),
0
0
0
0
(126)
1 0 0 0
⎤⎡
eπ/2−s
0.5
1 0 0 0
x01 (1)
=e
(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),
0
0 1 0
⎤ −1.142 ⎢ ⎥ ⎢−0.9324⎥ ⎢ ⎥ ⎥, =⎢ ⎢ ⎥ ⎢ −1.046 ⎥ ⎣ ⎦ −0.9532 ⎡
(122)
y1 = y1 , y2 , y3 , y4 = (1, 2.8, 0.5, 2.9),
⎥ ⎥ ⎦
0 0 1⎥
⎤
0
0
2
2
⎥ ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ π π ⎥ cos − s −sin − s ⎥ 2 2 ⎥ ⎥ ⎥ ⎦ π π ,sin − s cos − s
⎤
⎡
[2α + 1, 3]
⎥
0 0 0⎥ ⎥ π/2
⎟ ⎟ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎢0 1 0 0⎥ ⎢0.85⎥ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎥⎢ ⎥ ds⎟ ·e s ⎢ ⎢ ⎥⎢ ⎥ ⎟ ⎢0 0 1 0⎥ ⎢ 0.4 ⎥ ⎟ ⎣ ⎦⎣ ⎦ ⎟ ⎟ ⎟ 0 0 0 1 0.75 ⎟ ⎠ ⎡
⎤
u1 = u1 , u2 , u3 , u4 = (0.5, 0.85, 0.4, 0.75),
⎤
⎞
⎢ ⎥ ⎢ ⎥ ⎢ [α + 1, 2] ⎥ ⎥
(2) α ⎢ ⎢ ⎥ Y = ⎢
⎥. ⎢ 0,−2.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
[0.5α + 0.5, 1]
1 0 0
π π − s −sin −s 2 2 ⎢ ⎢ ⎢ ⎢sin π − s cos π − s ⎢ 2 2 ⎢
0,−1.5(α − 1)
⎢ ⎥ ⎢ ⎥ ⎢[0.8α + 0.2, 1]⎥ ⎥
(2) α ⎢ ⎢ ⎥ U = ⎢
⎥ , ⎢ 0,−0.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
⎥ ⎥ ⎦
0⎥
⎢cos
Rule 2: 0,−0.8(α − 1)
⎡
⎢ ⎥ ⎢ ⎥ ⎢ [0.5α + 2.5, 3] ⎥ ⎢ ⎥
(1) α ⎢ ⎥ Y =⎢
⎥. ⎢ 0, −1.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣
⎦ 0.5(α − 1) + 3, 3
⎥
0⎥ ⎥
⎛
Rule 1:
⎢ ⎢0 −1 0 ⎢ ⎢ ⎢ ⎢0 0 1 ⎣
⎤
0 0 0 −1
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.
⎡
−π/2
1 0 0 0
−π/2
⎢ ⎢0 −1 0 ⎢ ⎢ ⎢ ⎢0 0 1 ⎣
⎤ ⎥
0⎥ ⎥ ⎥ ⎥ ⎦
0⎥
0 0 0 −1
⎛ ⎜ ⎡
⎤ ⎡ ⎜ ⎜ 0, −2(α − 1) 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ ⎜⎢ 2.8 ⎥ ⎢1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢0 ⎢ 0.5 ⎜⎣ ⎦ ⎣ ⎜ ⎜ 0 2.9 ⎜ ⎝
⎤
1 0 0
⎥
0 0 0⎥ ⎥ π/2 ⎥ ⎥ ⎦
0 0 1⎥ 0 1 0
0
eπ/2−s
14
Advances in Fuzzy Systems ⎤ π π 0 0 ⎥ ⎢cos 2 − s −sin 2 − s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢sin π − s cos π − s 0 0 ⎥ ⎢ 2 2 ⎥ ⎢ ·⎢ ⎥ ⎥ ⎢ π π ⎢ 0 0 cos − s −sin − s ⎥ ⎢ 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ π π 0 0 sin − s cos − s
⎞
⎡
2
⎟ ⎟ ⎟ ⎢ ⎥ ⎢
⎥ ⎟ ⎢0 1 0 0⎥ ⎢ 0.75(α − 1) + 1, 1 ⎥ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎥⎢ ⎥ ds⎟ ·e s ⎢ ⎢ ⎥⎢ ⎥ ⎟ ⎢0 0 1 0 ⎥ ⎢ ⎥ ⎟ 0.4 ⎣ ⎦⎣ ⎦ ⎟ ⎟ ⎟ 0 0 0 1 0.75 ⎟ ⎠ ⎡
2
⎡
⎞ ⎡
1 0 0 0
⎤ ⎡
⎢ ⎥⎢ ⎢ 0 1 0 0⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ ·es ⎢ ⎢ ⎥⎢ ⎢ 0 0 1 0⎥ ⎢ ⎣ ⎦⎣
0, −0.75(α − 1) 0.4
0 0 0 1 ⎡
0.75
[−1.6 + 0.75α, −0.434 − 0.416α]
⎢ ⎢ [−1.4324, −0.6824 − 0.75α] ⎢ =⎢ ⎢ ⎢ −1.046 ⎣ −0.9532
⎤
⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢0 −1 0 ⎢ 1 −π/2 ⎢ x0 (2) = e ⎢ ⎢0 0 1 ⎣
μx01 (2) x02 = min{0.8, 0.6} = 0.6, ⎡
1 0 0 0
⎢ ⎢0 −1 0 ⎢ 1 −π/2 ⎢ x0 (3) = e ⎢ ⎢0 0 1 ⎣
⎤ ⎥
0⎥ ⎥ ⎥ ⎥ ⎦
0⎥
0 0 0 −1
⎛
⎜⎡ ⎤ ⎡ ⎜ ⎜ 1 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ ⎜⎢ 2.8 ⎥ ⎢1 ⎜ −⎢ ·⎜ ⎢
⎥ ⎢ ⎢ ⎜ ⎢ 0, −1.5(α − 1) ⎥ ⎥ ⎢0 ⎜⎣ ⎦ ⎣ ⎜ ⎜ 0 2.9 ⎜ ⎝
⎤ ⎥
0⎥ ⎥ ⎥ ⎥ ⎦
0⎥
0 0 0 −1
⎤
1 1 1 , = = 0.333, 3 2 3
1 0 0 0
[−1.292, −0.542 − 0.75α]
when α = 0, we get the biggest interval [−1.124, −0.27] and x02 = −0.9324 is located in this interval. We choose its membership grade in x01 (2) as
when α = 0, we get the biggest interval [−1.6, −0.434] and x01 = −1.142 is located in this interval. We choose its membership grade in x01 (1) as
⎤
0.5
(128)
(127)
μx01 (1) x01 = min
⎤⎡
⎢ ⎥ ⎢[−1.124, −0.27 − 0.854α]⎥ ⎢ ⎥ ⎥, =⎢ ⎢ ⎥ ⎢ ⎥ −1.046 ⎣ ⎦ −0.9532
⎟ ⎟ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ds⎟ ⎥ ⎟ ⎥ ⎟ ⎦ ⎟ ⎟ ⎟ ⎟ ⎠
⎤
0.85
1 0 0 0
1 0 0
⎤ ⎥
0 0 0⎥ ⎥ π/2 ⎥ ⎥ ⎦
0 0 1⎥
0
eπ/2−s
0 1 0
⎤ π π 0 0 ⎥ ⎢cos 2 − s −sin 2 − s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢sin π − s cos π − s 0 0 ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ·⎢ ⎥ ⎢ π π ⎢ 0 0 cos − s −sin − s ⎥ ⎢ 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ π π 0 0 sin − s cos − s
⎛
⎡
⎜⎡ ⎤ ⎡ ⎜ ⎜ 1 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ ⎜ ⎢[0.5α + 2.5, 3]⎥ ⎢1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢0 0.5 ⎜⎢ ⎦ ⎣ ⎜⎣ ⎜ 2.9 0 ⎜ ⎝
1 0 0
⎤ ⎥
0 0 0⎥ ⎥ π/2 ⎥ ⎥ ⎦
0 0 1⎥
0
eπ/2−s
0 1 0
2
2
⎤
⎞
0 0 ⎥ ⎢cos 2 − s −sin 2 − s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ π π ⎥ ⎢sin − s cos −s 0 0 ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ·⎢ ⎥ ⎢ π π ⎥ ⎢ 0 0 cos − s sin − s − ⎢ 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ π π 0 0 sin − s cos − s
⎟ ⎟ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎟ ⎢0 1 0 0 ⎥ ⎢ ⎥ 0.85 ⎢ ⎥⎢ ⎥ ⎟ ⎟ s⎢ ⎥ ⎢ ⎥ ·e ⎢ ⎥ ds⎟ ⎥ ⎢
⎢0 0 1 0⎥ ⎢ 0, −0.5(α − 1) ⎥ ⎟ ⎣ ⎦⎣ ⎦ ⎟ ⎟ ⎟ 0 0 0 1 0.75 ⎟ ⎠
⎡
π
π
2
2
⎡
1 0 0 0
⎤⎡
0.5
⎤
M. S. N. Murty and G. Suresh Kumar ⎡
15 ⎤
−1.142
follows:
⎢ ⎥ ⎢ ⎥ −0.9324 ⎢ ⎥ ⎢ ⎥, =⎢ ⎥ ⎢[−1.25 + 0.5α, −0.438 − 0.312α]⎥ ⎣ ⎦ [−1.3532, −0.8532 − 0.5α]
⎤ −1.092 ⎢ ⎥ ⎢−0.664⎥ ⎢ ⎥ ⎥, X0 = ⎢ ⎢ ⎥ ⎢−0.584⎥ ⎣ ⎦ −0.812 ⎡ ⎤ [−1.6 + 0.8α, −0.488 − 0.312α] ⎢ ⎥ ⎢ [−1.164, −0.364 − 0.8α] ⎥ ⎢ ⎥ 2 ⎢ ⎥, x0 (1) = ⎢ ⎥ ⎢ ⎥ −0.584 ⎣ ⎦ −0.812 ⎡ ⎤ [−1.292, −0.492 − 0.8α] ⎢ ⎥ ⎢[−0.916, 0.092 − 1.008α]⎥ ⎢ ⎥ 2 ⎥, x0 (2) = ⎢ ⎢ ⎥ ⎢ ⎥ −0.584 ⎣ ⎦ −0.812 ⎡ ⎤ −1.092 ⎢ ⎥ ⎢ ⎥ −0.664 ⎢ ⎥ ⎥, x02 (3) = ⎢ ⎢ ⎥ ⎢[0.5α − 1.25, −0.23 − 0.52α]⎥ ⎣ ⎦ [−1.062, −0.562 − 0.5α] ⎡ ⎤ −1.092 ⎢ ⎥ ⎢ ⎥ −0.664 ⎢ ⎥ 2 ⎢ ⎥. x0 (4) = ⎢ ⎥ ⎢ [−0.834, −0.334 − 0.5α] ⎥ ⎣ ⎦ [−1.374, −0.458 − 0.916α]
(129) when α = 0, we get the biggest interval [−1.25, −0.438] and x03 = −1.046 is located in this interval. We choose its membership grade in x01 (3) as
μx01 (3) x03 = min 0.2, ⎡
1 ⎢ ⎢ 0 x01 (4) = e−π/2 ⎢ ⎢0 ⎣ 0 ⎛
0 −1 0 0
0 0 1 0
2 = 0.2, 3
⎤
0 ⎥ 0⎥ ⎥ 0⎥ ⎦ −1
⎜ ⎜⎡ ⎤ ⎡ ⎜ 1 ⎜ 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ 2.8 1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢0 ⎜⎢ 0.5 ⎦ ⎣ ⎣ ⎜
⎜ 0.5(α − 1) + 3, 3 0 ⎜ ⎜ ⎝ ⎡
⎡
0
0
0
0
1 0 0
⎢ ⎢ s ⎢0 1 0 ·e ⎢ ⎣0 0 1
0 0 0
0 0 0 1
⎤
0 ⎥ π/2 0⎥ ⎥ eπ/2−s 1⎥ ⎦ 0 0 ⎤
π π − s −sin −s ⎢cos 2 2 ⎢ ⎢ ⎢ ⎢sin π − s cos π − s ⎢ 2 2 ⎢
·⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
1 0 0 0
0
0
⎥ ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ π π ⎥ cos − s −sin − s ⎥ 2 2 ⎥ ⎥ ⎥ ⎦ π π sin − s cos − s
2
⎞
2
⎟ ⎟ ⎟ 0.5 ⎟ 0 ⎥⎢ ⎥ ⎟ ⎢ ⎥ ⎟ 0.85 0⎥ ⎥⎢ ⎥ ds⎟ ⎢ ⎥ ⎟ ⎟ 0.4 0⎥ ⎦ ⎣
⎦ ⎟ ⎟ 1 0.5(α − 1) + 1, 1 ⎟ ⎟ ⎠ ⎤⎡
0 μx0 (1) x0
=1 x 2 0
=1 μx0 (1) x
−1.142 × 0.333 + (−1.092) × 0.333
0.333 + 0.333
= −1.117,
0 μx0 (2) x0
=1 x 2 x0 = 2 0
=1 μx0 (2) x 2
=
−0.9324 × 0.6 + (−0.664) × 0.75
1 1 (131) , 0.8 = = 0.5. 2 2 Similarly for rule base 2, by the use of formula (106), (107), we obtain the values of X0 , x02 (i), i = 1, 2, 3, 4 and given as
0.6 + 0.75
= −0.7833,
0 μx0 (3) x0
=1 x 3 x0 = 2 0
=1 μx0 (3) x 2
when α = 0, we get the biggest interval [−1.224, −0.62] and x04 = −0.9532 is located in this interval. We choose its membership grade in x01 (4) as
= =
(130)
Also the grades of the membership of x01 = −1.092, x02 = −0.664, x03 = −0.584, x04 = −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 X0 = (x01 , x02 , x03 , x04 ), where x01
⎤ −1.142 ⎢ ⎥ ⎢ ⎥ −0.9324 ⎥ =⎢ ⎢ [−1.296, −0.796 − 0.5α] ⎥ , ⎣ ⎦ [−1.224, −0.62 − 0.604α]
(132)
2
⎤
⎡
μx01 (4) x04 = min
⎡
=
−1.046 × 0.2 + (−0.584) × 0.2
0.2 + 0.2
= −0.815,
0 μx0 (4) x0
=1 x 4 x0 = 2 0
=1 μx0 (4) x 2
=
−0.9532 × 0.5 + (−0.812) × 0.25
0.5 + 0.25
= −0.9061.
(133)
16 6.
Advances in Fuzzy Systems 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. Example 1 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. ACKNOWLEDGMENTS The authors would like to thank Professor H. Ying (Associate-Editor) and the anonymous referees for their suggestions which helped to improve the quality of the presentation. REFERENCES [1] S. Barnett and R. G. Cameron, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, UK, 2nd edition, 1985. [2] M. S. N. Murty, B. V. Appa Rao, and G. S. Kumar, “Controllability, observability, and realizability of matrix Lyapunov systems,” Bulletin of the Korean Mathematical Society, vol. 43, no. 1, pp. 149–159, 2006. [3] M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 2, pp. 201–224, 1999. [4] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985. [5] A. Alwadie, H. Ying, and H. Shah, “A practical two-input twooutput Takagi-Sugeno fuzzy controllers,” International Journal of Fuzzy Systems, vol. 5, no. 2, pp. 123–130, 2003. [6] Y. S. Ding, H. Ying, and S. H. Shao, “Structure and stability analysis of a Takagi-Sugeno fuzzy PI controller with application 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-Sugeno PI and PD fuzzy controllers: analytical structures and stability analysis,” Information Sciences, vol. 151, pp. 245–262, 2003. [8] T. A. Johansen, R. Shorten, and R. Murray-Smith, “On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 3, pp. 297–313, 2000.
[9] H. Ying, “Analytical analysis and feedback linearization tracking control of the general Takagi-Sugeno fuzzy dynamic systems,” 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 Foundations and Applications, IEEE Press, New York, NY, USA, 2000. [11] H. Ying, “Deriving analytical input-output relationship for fuzzy controllers using arbitrary input fuzzy sets and Zadeh fuzzy 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, “Automatic design of hierarchical Takagi-Sugeno type fuzzy systems using evolutionary algorithms,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 3, pp. 385–397, 2007. [13] Z. Ding and A. Kandel, “On the observability of fuzzy dynamical 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 fuzzy dynamical systems (II),” Journal of Fuzzy Mathematics, vol. 18, no. 2, pp. 295–306, 2000. [15] H. Radstrom, “An embedding theorem for space of convex sets,” Proceedings of the American Mathematical Society, vol. 3, no. 1, pp. 165–169, 1952. [16] R. J. Aumann, “Integrals of set-valued functions,” Journal of Mathematical Analysis and Applications, vol. 12, no. 1, pp. 1– 12, 1965. [17] O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987. [18] V. Lakshmikantham and R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London, UK, 2003. [19] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, John Willey & Sons, New York, NY, USA, 1975. [20] J. B. Conway, A Course in Functional Analysis, Springer, New York, NY, USA, 1990.
Research Article On Controllability and Observability of Fuzzy Dynamical Matrix 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 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. Copyright © 2008 M. S. N. Murty and G. Suresh Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 X (t) = A(t)X(t) + X(t)B(t) + F(t)U(t), X(0) = X0 ,
t > 0,
Y (t) = C(t)X(t) + D(t)U(t),
(1) (2)
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 Section 2, 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
2
Advances in Fuzzy Systems
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
α(βA) = (αβ)A,
1·A = A, (3)
and if α, β ≥ 0, then (α + β)A = αA + βA. The distance between A and B is defined by the Hausdorff metric
d(A, B) = inf : A ⊂ N(B, ), B ⊂ N(A, ) ,
(4)
where
N(A, ) = x ∈ Rn : x − y < , forsomey ∈ A .
(5)
Definition 1. A set-valued function F : J → Pk (Rn ) is said to be measurable if it satisfies any one of the following equivalent conditions: (1) for all u ∈ Rn , t → dF(t) (u) = inf v∈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 by S1F the set of all selections of F(·) that belong to the Lebesgue Bochner space L1Rn (J), that is,
S1F = f (·) ∈ L1Rn (J) : f (t) ∈ F(t)a.e. .
(6)
We present the Aumann’s integral as follows:
(A)
J
F(t)dt =
En = u : Rn −→ [0, 1]/u satisfie(i)–(iv) below ,
(8)
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,
In this section, we present some definitions and results relating to fuzzy sets [13] and Kronecker product of matrices. Let Pk (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 α(A + B) = αA + αB,
We say that F : J → Pk (Rn ) is integrably bounded if it is measurable and there exists a function h : J → R, h ∈ L1Rn (J), such that u ≤ h(t), u ∈ F(t). From [16], we know that if F is a closed valued measurable multifunction, then J F(t)dt is n if F is integrably bounded, then convex in R . Furthermore, n. F(t)dt is compact in R J Let
J
f (t)dt, f (·) ∈ S1F .
(7)
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 follows that [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 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 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 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]). 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≤1 Aα ⊂ A0 .
M. S. N. Murty and G. Suresh Kumar
3
A fuzzy set-valued mapping F : J → En is called fuzzy integrably bounded if F0 (t) is integrably bounded.
(4) A ⊗ B = AB. (5) If A(t) and B(t) are matrices, then (A ⊗ B) = A ⊗ B + A ⊗ B ( = d/dt). (6) Vec (AY B) = (B∗ ⊗ A)Vec Y . (7) If A and B are matrices both of order n × n, then
Definition 2. Let F : J → En be a fuzzy integrably bounded mapping. The fuzzy integral of F over J denoted by J F(t)dt is defined level-set-wise by
α
J
F(t)dt
= (A)
J
Fα (t)dt,
0 < α ≤ 1.
Let F : J × En → En ,and consider the fuzzy differential equation u = F(t, u),
u(0) = u0 .
(i) Vec (AX) = (In ⊗ A)Vec X, (ii) Vec (XA) = (A∗ ⊗ In )Vec X.
(12)
(13)
Now by applying the Vec operator to the matrix Lyapunov system (1) satisfying (2) and using the above properties, we have
Definition 3. A mapping u : J → En is a fuzzy weak solution to (13) if it is continuous and satisfies the integral equation u(t) = u0 +
t 0
F 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 Kronecker products and basic results related to matrix Lyapunov systems. Definition 4 (see [2]). Let A ∈ C m×n and B ∈ C p×q .Then the Kronecker product of A and B written A ⊗ B is defined to be the partitioned matrix ⎡
a11 B a12 B · · · a1n B
⎤
⎢ ⎥ ⎢a B a B · · · a B⎥ ⎢ 21 22 2n ⎥ ⎥ A⊗B =⎢ ⎢ ⎥ ⎢ . . ··· . ⎥ ⎣ ⎦ am1 B am2 B · · · amn B
⎤
⎢ ⎥ ⎢A ⎥ ⎢ .2 ⎥ ⎢ ⎥ ⎥, A = VecA = ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎣ ⎦
⎡
a1 j
A.n
+ In ⊗ D(t) U(t), Y (t) = In ⊗ C(t) X(t)
(18)
where G(t) = (B∗ ⊗ In ) + (In ⊗ A) is an n2 × n2 matrix and X = Vec X(t), U = Vec U(t) are column matrices of order n2 . 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 for the systems X (t) = A(t)X(t),
X(0) = In ,
[X ∗ (t)] = B∗ (t)X ∗ (t),
X(0) = In ,
(20) (21)
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))X0 .
ψ(t) ⊗ φ(t)
⎤
⎥ ⎢ ⎢a ⎥ ⎢ 2j ⎥ ⎥ ⎢ ⎥ whereA. j = ⎢ ⎢ .. ⎥ ⎢ . ⎥ ⎦ ⎣
(17)
Proof. Consider
Definition 5 (see [2]). Let A = [ai j ] ∈ C m×n ; one denotes A.1
X(0) = X0 ,
(15)
which is an mp × nq matrix and is in C mp×nq . ⎡
+ In ⊗ F(t) U(t), X (t) = G(t)X(t)
(1 ≤ j ≤ n).
= 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)
am j (16)
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.
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 matrices for the systems (20) and (21). Then the unique solution of the initial 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)
4
Advances in Fuzzy Systems
Proof. First we show that the solution of (17) is of the form X(t) = (ψ(t) ⊗ φ(t))X0 + X(t), where X(t) is a particular solution of (17) and is given by X(t) =
t
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds.
0
(24)
Let u(t) be any other solution of (17), write w(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) + In ⊗ F(t) =⇒ G(t)X(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)
(25)
is continuous in En . The set U α = Assume that U(t) u1 (t) × u2 (t) × · · · × un2 (t) is a convex and compact set 2 in Rn . For any positive number T, consider the following differential inclusions: 2
α
∈ Pk (Rn 2 ), for every 0 ≤ α ≤ 1, t ∈ [0, T]. Claim (i). [X(t)] First, we prove that X α is nonempty, compact, and convex 2 in C[[0, T], Rn ]. Since U α (t) has measurable selection, we α have that X is nonempty. Let K = 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 any X ∈ X α , then there is a selection u(t) ∈ U α (t) such that
0
t
+
=⇒ v(t) =
0
ψ −1 (s) ⊗ φ−1 (s) In ⊗ F(s) U(s)ds.
t
+
t
Let ui (t) ∈ E1 , t ∈ J, i = 1, 2, . . . , n2 , and define
+ 0
(26) where uαi (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 2 U(t) ∈ En . we show that the Now by using the fuzzy control U(t), following system
+ In ⊗ F(t) U(t), X (t) = G(t)X(t)
X(0) = X0 , (27)
+ In ⊗ D(t) U(t) Y (t) = In ⊗ C(t) X(t)
determines a fuzzy system.
ψ(t − s)φ(t − s)F(s)u(s)ds
≤ KLX0 + KLNMT.
= u1 (t) × u2 (t) × · · · × un2 (t) = uα1 (t), uα2 (t), . . . , uαn2 (t) : α ∈ [0, 1] 2 (t), . . . , u n2 (t) : u i (t) ∈ uαi (t), α ∈ [0, 1] , = u 1 (t), u
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) u(s)ds
≤ ψ(t)φ(t)X0
FORMATION OF FUZZY DYNAMICAL LYAPUNOV SYSTEMS
(31)
X(t) ≤ ψ(t) ⊗ φ(t) X0
0
U(t) = u1 (t), u2 (t), . . . , un2 (t)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) u(s)ds.
Then
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.
(30)
X(t) = ψ(t) ⊗ φ(t) X0
= ψ −1 (t) ⊗ φ−1 (t) In ⊗ F(t) U(t) t
(29)
Let X α be the solution of (29) satisfying (30).
t ∈ [0, T],
X(0) = X0 .
=⇒ v (t)
+ In ⊗ F(t) U α (t), X (t) ∈ G(t)X(t)
(28)
(32) Thus X α is bounded. For any t1 , t2 ∈ [0, T], 2) 1 ) − X(t X(t = 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)
M. S. N. Murty and G. Suresh Kumar
5 Let X = λX1 (t) + (1 − λ)X2 (t), 0 ≤ λ ≤ 1, then
Therefore X(t 2 ) 1 ) − X(t
X = λX1 (t) + (1 − λ)X2 (t)
≤ ψ(t1 ) ⊗ φ(t1 ) − ψ(t2 ) ⊗ φ(t2 ) X0 t1
+
t2
t2
+ 0
= λ G(t)X1 (t) + In ⊗ F(t) u1 (t)
ψ(t1 − s) ⊗ φ(t1 − s) In ⊗ F(s) u(s)ds 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
+ MN
0
ψ(t1 − s) ⊗ φ(t1 − s)
(34) 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. For each Xk , there is a uk ∈ Let Xk ∈ X α and Xk → X. α U (t) such that
0
α2
U α2 (t) = uα1 2 (t) × uα2 2 (t) × · · · × uαn22 (t)
+ 0
α
(41)
Thus, we have the selection inclusion S1U α2 (t) ⊂ S1U α1 (t) and the following inclusion:
X (t) ∈ G(t)X + In ⊗ F(t) U α2 (t) α 1 (t). ⊂ G(t)X + In ⊗ F(t) U
(42)
Consider the differential inclusions
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s)
λ j uk j (s)ds. (36)
From Fatou’s lemma, taking the limit as j → ∞ on both sides of (36), we have
X(t) = ψ(t) ⊗ φ(t) X0 t
+ 0
X (t) ∈ G(t)X + In ⊗ F(t) U α2 (t),
λ j ψ(t) ⊗ φ(t) X0
t
α
1 (t). = Uα
(35)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) uk (s)ds.
λ j Xk j (t) =
α1
α
(40)
⊂ [X(t)] , for all 0 ≤ α1 ≤ α2 ≤ 1. Claim (ii). [X(t)] Let 0 ≤ α1 ≤ α2 ≤ 1. Since U α2 (t) ⊂ U α1 (t), we have
⊂ u1 1 (t) × u2 1 (t) × · · · × un21 (t)
Since uk ∈ U α (t) is closed, then there exists a subsequence α (t). From Mazur’s {uk j } of {uk } converging weakly to u ∈ U theorem [20], thereexists a sequence of numbers λ j > 0, λ j = 1 such that λ j uk j converges strongly to u. Thus, from (35) we have
that is X ∈ X α . Thus X α is convex. Therefore, X α is 2 nonempty, compact, and convex in C[[0, T], Rn ]. Thus, α from Arzela-Ascoli theorem, we know that [X(t)] is comα 2 n pact in R for every t ∈ [0, T]. Also it is obvious that [X(t)] α 2 2 is convex in Rn . Thus, we have [X(t)] ∈ Pk (Rn ), for every t ∈ [0, T]. Hence the claim.
t
(39)
Since U α (t) is convex, λu1 (t) + (1 − λ)u2 (t) ∈ U α (t), we have
Xk (t) = ψ(t) ⊗ φ(t) X0 +
= G(t) λX1 (t) + (1 − λ)X2 (t)
+ In ⊗ F(t) λu1 (t) + (1 − λ)u2 (t) .
− ψ(t2 − s) ⊗ φ(t2 − s) ds.
+ In ⊗ F(t) U α (t), X (t) ∈ G(t)X(t)
+ KLNM |t1 − t2 | T
+ (1 − λ) G(t)X2 (t) + In ⊗ F(t) u2 (t)
X (t) ∈ G(t)X + In ⊗ F(t) U α1 (t),
X2 (t) = G(t)X2 (t) + In ⊗ F(t) u2 (t).
(44)
+
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U α2 (s) ds (45)
⊂ ψ(t) ⊗ φ(t) X0
∈ X α , and hence X α is closed. Thus, X(t) Let X1 , X2 ∈ X α , then there exist u1 , u2 ∈ U α (t) such that
X1 (t) = G(t)X1 (t) + In ⊗ F(t) u1 (t),
t ∈ [0, T].
X(t) ∈ ψ(t) ⊗ φ(t) X0
0
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) u(s)ds.
(43)
Let X α2 and X α1 be the solution sets of (43) and (44), respectively. Clearly, the solution of (43) satisfies the following inclusion: t
(37)
t ∈ [0, T],
t
+ 0
(38)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U α1 (s) ds.
Thus X α2 ⊂ X α1 , and hence X α2 (t) ⊂ X α1 (t). Hence the claim.
6
Advances in Fuzzy Systems
Claim (iii). If {αk } is a nondecreasing sequence converging to α > 0, then X α (t) = k≥1 X αk (t). Let U αk (t) = uα1 k × uα2 k × · · · × uαn2k , U α (t) = uα1 × uα2 × · · · × uαn2 and consider the inclusions
X (t) ∈ G(t)X + In ⊗ F(t) U αk (t),
X (t) ∈ G(t)X + In ⊗ F(t) U α (t).
(46) (47)
It follows that
X(t) ∈ ψ(t) ⊗ φ(t) X0
+
t k≥1 0
uαi =
k≥1
⊂ ψ(t) ⊗ φ(t) X0 t
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1
α k ≥1 U k
0
ds
= ψ(t) ⊗ φ(t) X0 t
+
uαi k .
+ 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
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U αk ds
0
(48)
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U α ds. (54)
This implies that X ∈ X α . Therefore,
Consider
U α (t) = uα1 × uα2 × · · · × uαn2 =
uα1 k ×
k≥1
=
uα1 k
uα2 k × · · · ×
α × u2 k
k≥1
=
(55)
k≥1
k≥1
X αk ⊂ X α .
k≥1
uαn2k
α × · · · × un2k
From (51) and (55), we have X α =
(49)
X αk ,
(56)
X αk (t).
(57)
k≥1
and hence,
U αk (t)
X α (t) =
k≥1
k≥1
and then S1U α (t) = S1
α k ≥1 U k (t)
. Therefore
X (t) ∈ G(t)X + In ⊗ F(t) U α (t) α = G(t)X + In ⊗ F(t) U k (t)
(50)
k≥1
α k (t), ⊂ G(t)X + In ⊗ F(t) U
k = 1, 2, . . . .
X αk .
(51)
k≥1
+ In ⊗ D(t) U(t). Y (t) = In ⊗ C(t) X(t)
t
+ 0
k ≥ 1.
(52)
Then,
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds.
X(t) ∈ ψ(t) ⊗ φ(t) X0 t 0
(60)
Remark 1. Consider a special case. If the input is in the form U(t) =u 1 (t) × u 2 (t) × · · · × ui (t) × · · · × u n2 (t),
+
(59)
The solution set of the fuzzy dynamical system (58), (59) is given by
X (t) ∈ G(t)X + In ⊗ F(t) U αk (t),
X(0) = X0 , (58)
X(t) ∈ ψ(t) ⊗ φ(t) X0
Let X be the solution set to the inclusion
+ In ⊗ F(t) U(t), X (t) = G(t)X(t)
Thus, we have X α ⊂ X αk , k = 1, 2, . . ., which implies that X α ⊂
From Claims 3–3 and applying Theorem 1, there exists 2 X(t) ∈ En 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
(61)
where uk (t) ∈ R1 , k = / i, are crisp numbers, then the ith component of the solution set of (27) is a fuzzy set in E1 .
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) S1U αk (t) ds. (53)
Proof. The proof follows along similar lines as in the above discussion.
M. S. N. Murty and G. Suresh Kumar 4.
7 Definition 7 (see [20]). Let u, v ∈ E1 , k ∈ R1 , and let [u]α be the α-level set of u. One defines the sum of u and v by
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 = X0 and completely controllable if for any initial state X(0) any given final state X f there exists a finite time t1 > 0 and a 1 ) = X f . 0 ≤ t ≤ t1 , such that X(t control U(t), Lemma 2. If F is a fuzzy set, then
T 0
F dt = TF.
α
0
=
T
[F]α dt = T[F]α .
0
From the definition of fuzzy set, we have
T 0
(69) the difference between u and v by
and the scalar product by
0
h(t)P dt =
⎡
(62)
⎢ ⎢ [z]α = [y − x]α = [y]α − [x]α = ⎢ ⎣
F dt = TF.
T
[yn2 ]α − [xn2 ]α
⎡ ⎢ ⎢ ⎣
[y1 ]α + [x1 ]α
[w]α = [y + x]α = [y]α + [x]α = ⎢ T 0
h(t)[P]α dt = = =
T 0
T 0
0
h(t)Pmin (α)dt = /
⎡
h(t)Q dt
(64)
α
T
h(t)Qmin (α)dt.
(65)
h(t)Qmax (α)dt.
(66)
If (ii) holds, then T 0
h(t)Pmax (α)dt = /
T 0
Thus, in both cases (i) and (ii), we have
h(t) Pmin (α), Pmax (α) dt = /
T 0
h(t) Qmin (α), Qmax (α) dt. (67)
This implies that T 0
h(t)P α dt = /
T 0
⎥ ⎥ ⎥. ⎦
Definition 9 ([20]). Let
h(t)[Q] dt.
0
⎤
(73)
α
T 0
⎥ ⎥ ⎥. ⎦
α
h(t)P dt
Suppose that P = / Q, then for some α ∈ [0, 1], we α have [P]α = / [Q] . Without loss of generality, we assume that P, Q ∈ E1 . Let P α = [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 T
···
[yn2 ]α + [xn2 ]α
Proof. For each α-level, we have
0
···
⎤
(72)
(63)
then P = Q.
T
[y1 ]α − [x1 ]α
If y = w − x, then w = y + x which is defined by h(t)Q dt
0
(71)
2
Lemma 3. Let P, Q be two fuzzy sets and let h(t) be a nonzero continuous function on [0, T], satisfying T
[u − v]α = [u]α − [v]α = a − b : a ∈ [u]α , b ∈ [v]α , (70)
Definition 8 (see [20]). Let x, y ∈ En 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
Proof. Let [F] be the α-level set of F. Since F dt
[ku]α = k[u]α = ka : a ∈ [u]α .
α
T
[u + v]α = [u]α + [v]α = a + b : a ∈ [u]α , b ∈ [v]α ,
c11
c12 · · · c1n2
⎤
⎢ ⎥ ⎢c ⎥ ⎢ 21 c22 · · · c2n2 ⎥ ⎥ C=⎢ ⎢ ⎥ ⎢· · · · · · · · · · · · ⎥ ⎣ ⎦ cn2 1 cn2 2 · · · cn2 n2
(74)
be an n2 × n2 matrix, p = p1 × p2 × · · · × pn2 , let pi ∈ E1 , 2 i = 1, 2, . . . , n2 , be a fuzzy set in En , and let [pi ]α be α-level sets of pi . Define the product C p of C and p as ⎡
c11
c12 · · · c1n2
⎤⎡
[p1 ]α
⎤
⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ c21 c22 · · · c2n2 ⎥ ⎢ [p2 ]α ⎥ ⎥ ⎢ ⎥ ⎢ α α ⎥⎢ ⎥ [C p] = C[p] = ⎢ ⎥⎢ ⎥ ⎢ ⎢· · · · · · · · · · · · ⎥ ⎢ · · · ⎥ ⎥⎢ ⎥ ⎢ ⎦⎣ ⎦ ⎣ cn2 1 cn2 2 · · · cn2 n2 [pn2 ]α ⎡
c11 [p1 ]α + · · · + c1n2 [pn2 ]α
⎤
⎥ ⎢ ⎥ ⎢ ⎢ c21 [p1 ]α + · · · + c2n2 [pn2 ]α ⎥ ⎥ ⎢ ⎥. =⎢ ⎥ ⎢ ⎥ ⎢ · · · ⎥ ⎢ ⎣ α α⎦ cn2 1 [p1 ] + · · · + cn2 n2 [pn2 ]
All these definitions yield the following lemma. h(t)Qα dt,
which is a contradiction to (64). Hence P = Q.
(68)
2
Lemma 4. C p is a fuzzy set in En . Proof. The proof is similar to proof of Lemma 3.1 [13].
(75)
8
Advances in Fuzzy Systems
Theorem 3. The fuzzy system (58),(59) is completely controllable if the n2 × n2 symmetric controllability matrix T
W(0, T) =
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
0
must be fuzzy, Since X is fuzzy and from Lemma 4, U(t) otherwise the fuzzy left side of (80) cannot be equal to the crisp right side. By Lemma 2, X f can be written as
∗ ∗ × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) dt
(76)
is nonsingular. Furthermore,the fuzzy control U(t) which = X0 to a fuzzy transfers the state of the system from X(0) = X f = (x f1 , x f2 , . . . , x fn2 ) can be determined by state X(T) the following fuzzy rule base:
R : IF x1 is in x f1 , . . . , xn2 is in x fn2 ,
(77)
THEN u1 is in u1 , . . . , un2 is in un2 ,
X f =
1 T
=
1 T
T 0
T
X f dt
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
0
−1 −1 × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f dt.
(81) From (80) and (81), we have 1 T
T
0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
−1 −1 × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f dt = ψ(T) ⊗ φ(T) X0
where
u1 (t), u2 (t), . . . , ui (t), . . . , un2 (t) =
T
+
× x1 (T), x2 (T), . . . , x fi , . . . , xn2 (T)
1 T
T 0
ψ(T) ⊗ φ(T) X0 =
T 0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
× In ⊗ F(t)
∗
=
T
(79) Now our problem is to find the control U(t) such that
0
T
+ 0
= ψ(T) ⊗ φ(T) X0
0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt.
(83) It follows that T
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt.
(80)
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt T 0
×
× W −1 (0, T) ψ(T) ⊗ φ(T) X0 dt
=
X(T) = X f
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
∗ ∗ × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t)
0
T
∗
× W −1 (0, T) ψ(T) ⊗ φ(T) X0 dt.
+
−1 −1 × In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f dt
ψ(T − t) ⊗ φ(T − t)
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
i = 1, 2, . . . , n2 . (78)
Proof. Suppose that the symmetric controllability matrix W(0, T) is nonsingular. Therefore W −1 (0, T) exists. Multiplying W −1 (0, T)(ψ(T) ⊗ φ(T))X0 on both sides of (76), we have
Combining (79) and (82), we have
∗ ∗ − In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) × W −1 (0, T) ψ(T) ⊗ φ(T) X0 ,
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t) U(t)dt.
(82)
−1 −1 1 In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) T
0
ψ(T − t) ⊗ φ(T − t) In ⊗ F(t)
−1 −1 1 In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f T ∗ ∗ − In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) × W −1 (0, T) ψ(T) ⊗ φ(T) X0 dt.
(84)
M. S. N. Murty and G. Suresh Kumar
9
By using Lemma 3, we get U(t) =
the membership are 0.5, 0.75, 0.5, and 0.75, respectively. The fundamental matrices of (20), (21) are
−1 −1 1 In ⊗ F(t) ψ(T − t) ⊗ φ(T − t) X f T ∗ ∗ − In ⊗ F(t) ψ(T − t) ⊗ φ(T − t)
⎡
φ(t) = ⎣
(85)
× W −1 (0, T) ψ(T) ⊗ φ(T) X0 .
⎡
A(t) = ⎣
1 0
⎡
F(t) = ⎣ ⎡
et 0 0 et
⎦,
B(t) = ⎣ ⎡
⎤ ⎦,
C(t) = ⎣
⎦,
0 0
π T= , 2
0 et
⎤ ⎦.
(88)
⎡
0
0
⎤
(89)
et sint et cos t
0
cos θ − sin θ
⎢ ⎢ ⎢ sin θ θ⎢ =e ⎢ ⎢ 0 ⎢ ⎣
cos θ 0
0 ⎡
0 ⎤
1 0 0 0
∗
ψ(θ) ⊗ φ(θ) ⎤
0
0
0
0
sin θ
cos θ
⎡
cos θ
0
0 0 0 1
⎤
⎡
(86)
eπ 0
0
0
0 eπ
0
⎡
∗ ⎤
1 0 0 0
⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢0 1 0 0⎥ ⎥ t⎢ ⎥ ⎥e ⎢ ⎥ ⎢0 0 1 0⎥ cos θ − sin θ ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦
0 0 0 1
sin θ
⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢0 1 0 0⎥ ⎢− sin θ cos θ ⎥ θ⎢ t⎢ ⎢ ⎥e ⎢ ×e ⎢0 0 1 0⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎣
0 1
0
⎤
0
0
0
0 ⎥ ⎥
cos θ
⎥ ⎥ ⎥
sin θ ⎥ ⎥ ⎦
− sin θ cos θ
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0 eπ 0 0 ⎥ ⎢ ⎥ ⎥, =⎢ ⎢ 0 0 eπ 0 ⎥ ⎢ ⎥ ⎣ ⎦
1 0
X0 = ⎣
0
⎦,
⎦,
0
⎥ ⎥ ⎥ ⎥ ⎥. t t e cos t −e sint⎥ ⎥ ⎦
et cos t
ψ(θ) ⊗ φ(θ) In ⊗ F(t) In ⊗ F(t)
⎤
0 1
0
Consider
1 0
⎡
⎤
0 0
D(t) = ⎣
⎡
ψ(t) = ⎣
⎦,
0
Example 1. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with ⎤
et 0
et cos t −et sint
⎢ ⎢ t ⎢ e sint ⎢ ψ(t) ⊗ φ(t) = ⎢ ⎢ 0 ⎢ ⎣
Remark 2. The nonsingularity of the symmetric controllability matrix W(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.
0 −1
cos t
sint
⎡
⎤
Now the fundamental matrix of (19) is
Now we have two special cases for (85). First, let X(T) = 2 X f = (x1 (T), x2 (T), . . . , xn (T)) be a crisp point, then we = (u 1 , u 2 , . . . , u n2 ), will get a corresponding control U(t) satisfying (85). = (x1 (T), x2 (T), . . . , x fi , . . . , xn2 (T)), Second, let X(T) will take the form then the corresponding control U(t) 2 U(t) = (u1 , u2 , . . . , ui , . . . , un ) in wich the ith component of U(t) is a fuzzy set in E1 . Obviously, ui (t) is in ui (t), the grade of the membership can be determined by μx fi (xi (T)), the grade of the membership of xi (T) in x fi . Thus, based on the above discussion, we have a fuzzy rule base for the control U,and is given by (77) and (78).
⎡
cos t − sint
⎤
1 1
⎦.
1 1
0
(90) Let X f = (x f1 , x f2 , x f3 , x f4 ) ∈ E4 , where ⎡
α ⎤
⎡
where θ = π/2 − t.Therefore,
⎤
x f1 [α − 1, 1 − α] ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ α⎥ ⎢ x f2 ⎥ ⎢ ⎥ [α − 1, 1 − α] ⎥ ⎢ ⎥
α ⎢ ⎥=⎢ ⎥. X f = ⎢ ⎢ α ⎥ ⎢
⎥ ⎥ ⎢ ⎢ x ⎥ ⎢ f3 ⎥ ⎢ 0.1(α − 1), 0.1(1 − α) ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣
⎣ ⎦
α 0.1(α − 1), 0.1(1 − α) x f4
⎡
(87)
We select the points x1 = 0.5, x2 = 0.25, x3 = 0.05, and x4 = 0.025 which are in x f1 , x f2 , x f3 , and x f4 with grades of
π W 0, 2
eπ 0
0
0
0 eπ
0
⎤
⎡
1 0 0 0
⎤
⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ π/2 ⎢ ⎢ 0 eπ 0 0 ⎥ 0 1 0 0⎥ π π⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ dt = e ⎢ ⎥. = ⎢ ⎥ ⎢ ⎥ π 2 0 ⎢0 0 e 0⎥ ⎢0 0 1 0 ⎥ ⎣ ⎦ ⎣ ⎦
0
Clearly, it is nonsingular.
0 0 0 1 (91)
10
Advances in Fuzzy Systems
Thus, from Theorem 3, the input U can be chosen by the following α-level sets:
⎡
U α (t) ⎡
2e−t = π
and the corresponding control function to the point (0.5, 0.25, 0.05, 0.025)∗ is
1 0 0 0
⎤
⎡
cos θ
sin θ
⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢0 1 0 0 ⎥ ⎢− sin θ cos θ ⎢ ⎥ −θ ⎢ ⎢ ⎥e ⎢ ⎢0 0 1 0 ⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎣
0
0 0 0 1
0
0
0 ⎥ ⎥
sin θ
⎢ ⎢ ⎢− sin θ cos θ θ⎢ ×e ⎢ ⎢ 0 0 ⎢ ⎣
0 ⎡
0
0 0 cos θ
π 2
⎢ ⎢ ⎢ π ⎢ ⎢ sin ⎢ 2 ⎢ × eπ/2 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎣
0
0 0 0
⎦
π 2
π cos 2 0 0
⎥ ⎥
⎡
1 ⎢ ⎢t + 1
A(t) = ⎢
⎦
⎣
0 0 1 ⎤
1 0 0 0
⎥ ⎥ ⎦
⎢ ⎥ ⎢ ⎥ ⎢0 1 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 0 1 0⎥ ⎢ ⎥ ⎣ ⎦ ⎤
0
0
⎥ ⎥⎡ ⎤ ⎥1 ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ 0 0 ⎥⎢1⎥ ⎥⎢ ⎥ ⎢ ⎥. ⎥ ⎢1⎥ π π ⎥ ⎥ ⎢ ⎥ − sin cos ⎥⎣ ⎦ 2 2 ⎥ ⎥ ⎥1 π π ⎦
2
cos
2e−π/2 U (t) = 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)
cos t − sint
⎢ ⎥ ⎢sint +cos t ⎥ ⎢ ⎥ ⎢ ⎥. ⎢ ⎥ ⎢cos t − sint⎥ ⎣ ⎦
sint +cos t
1
0 D(t) = ⎣
⎡
0 0 0 0
1
e−t
0
1
⎤
0 (2 − t)et−1
⎡
⎥ ⎦,
C(t) = ⎣ ⎡
⎤ ⎦,
⎤
⎢ ⎥ B(t) = ⎣ t + 1 t + 1 ⎦ ,
⎥ ⎥, −1 ⎦
t+1
⎢ F(t) = ⎣ 2 − t ⎡
⎤
0 ⎥
X0 = ⎣
T = 1,
0 et et
0
⎤ ⎦,
(95)
⎤
1 1
⎦.
0 0
Let X f = (x f1 , x f2 , x f3 , x f4 ) ∈ E4 , where ⎡
α ⎤
⎡ ⎢ ⎥ ⎢ ⎢ α ⎥ ⎢ ⎢ x ⎥ ⎥ ⎢
α ⎢ ⎢ f2 ⎥ ⎢ ⎢ X f = ⎢ α ⎥ =⎢ ⎥ ⎢ ⎢ x f3 ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣
α
x f1
x f4
⎤
[0.5α + 0.5, 1] [0.8α + 0.2, 1] [α − 1, 1 − α] 0.2(α − 1), 0.2(1 − α)
⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(96)
2
Hence, the α-level sets of fuzzy control are given by
α
⎤
0 0 0 1
(92)
⎡
0
⎡ ⎡
0 ⎥ ⎥ 2e−π ⎥ sin θ ⎥ ⎥ π
sin
cos t − sint
Example 2. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with
⎤
− sin
⎡
(94)
1 0 0⎥ ⎥ ⎥ 0 1 0⎥ ⎥
⎤
− sin θ cos θ
⎢cos
0
⎤
− sin θ cos θ
0
⎤ [α − 1, 1 − α] ⎡ ⎥ ⎢ 1 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ [α − 1, 1 − α] ⎢0 ⎥ ⎢ ⎥ ⎢ t⎢ ⎥−e ⎢ × ⎢
⎢0 ⎢ 0.1(α − 1), 0.1(1 − α) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ ⎥ ⎢ ⎦ ⎣
0 0.1(α − 1), 0.1(1 − α)
cos θ
⎥
sin θ ⎥ ⎥
cos θ
⎡
⎡
⎥ ⎥
0.5(sint +cos t)
⎢ ⎥ ⎢ ⎥ ⎢0.75(sint −cos t)⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥− ⎢ 0.5(sint +cos t) ⎥ π ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.75(sint −cos t)
⎢ ⎥ ⎢u ⎥ ⎢ 2 ⎥ 2e−π/2 ⎥ =⎢ U(t) ⎢ ⎥= π ⎢u ⎥ ⎣ 3⎦ u4
⎤
0
u1
⎡
⎤
We select the points x1 = 0.75, x2 = 0.8, x3 = 0.75, and x4 = 0.1 which are in x f1 , x f2 , x f3 , and x f4 with grades of the membership being 0.5, 0.75, 0.25, and 0.5, respectively. The fundamental matrices of (20), (21) are ⎡ ⎢ φ(t) = ⎢ ⎣
t+1 0
0
sint +cos t (93)
ψ(t) = ⎣
t+1
t+1 t et
0
⎤ ⎦.
(97)
Now the fundamental matrix of (19) is ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ψ(t) ⊗ φ(t) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ sint +cos t 2⎢ ⎥ ⎥ − ⎢ ⎥ π⎢ ⎢cos t − sint⎥ ⎣ ⎦
⎡
⎥ ⎥ 1 ⎦,
⎡
⎤
⎤
(t + 1)2 0 t(t + 1) 0 0 0
0
⎤ ⎥
t ⎥ ⎥ ⎥ 1 0 t + 1⎥ ⎥ ⎥. 0 et (t + 1) 0 ⎥ ⎥ ⎥ ⎥
0
0
et ⎦ t+1
(98)
M. S. N. Murty and G. Suresh Kumar
11
It is easily seen that
ψ(1 − t) ⊗ φ(1 − t) In ⊗ F(t) In ⊗ F(t)
and the corresponding control function to the point (0.75, 0.8, 0.75, 0.1)∗ is
∗
⎡
⎡
⎤ 2t 2 − 6t+5 0 (1 − t)e1−t 0 ⎢ ⎥ ⎢ ⎥ 2 2(t−1) t −1 ⎥ ⎢ 0 2t − 6t+5 e 0 (1 − t)e ⎥ ⎢ ⎢ ⎥ ⎥. =⎢ 2(1 −t) ⎢ (1 − t)e1−t ⎥ 0 e 0 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ t −1 0 (1 − t)e 0 1
⎡
(99)
π 2
⎡
2t 2 − 6t+5
0
⎢ ⎢ 2 0 2t − 6t+5 e2(t−1) 1⎢ ⎢ ⎢ = ⎢ 1−t 0⎢ 0 ⎢ (1 − t)e ⎢ ⎣ 0 (1 − t)et−1 ⎡
(1 − t)e1−t 0 e2(1−t) 0
0
⎤
1
⎢2 − t 0 ⎢ ⎢ 1−t ⎢ ⎢ 0 e α ⎢ U (t) = ⎢ 2−t ⎢ ⎢ 0 0 ⎢ ⎣
0 ⎡
t − 1 t−1 e 2−t 0 et−1
0
0
⎤⎡[0.5α + 0.5, 1] 0 ⎥⎢ ⎥⎢ ⎥⎢ ⎢[0.8α + 0.2, 1] t − 1⎥ ⎥⎢ ⎢ 2 − t⎥ [α − 1, 1 − α] ⎥⎢ ⎥⎢ ⎢ ⎥
0 ⎥⎢ ⎦⎢ ⎣ 0.2(α − 1), 0.2(1 − α) 1
9 e2 − 1 − 6e (2 − t) 4e2 − 7 0
OBSERVABILITY OF FUZZY DYNAMICAL LYAPUNOV SYSTEMS
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 X0 .
Clearly, it is nonsingular. Thus, from Theorem 3, the input U can be chosen by the following α-level sets, given by
5.
In this section, we discuss the concept of observability of the fuzzy system (58), (59).
(100)
⎤
(102)
⎥ ⎦
0 1 0 ⎥ ⎢3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3 ⎢0 1 − 3e−2 0 1 − 2e−1 ⎥ ⎢ ⎥ 2 ⎥. =⎢ ⎢ ⎥ 2−1 ⎢ ⎥ e ⎢1 ⎥ 0 0 ⎢ ⎥ 2 ⎢ ⎥ ⎣ ⎦ 0 1 − 2e−1 0 1
⎡
⎥ ⎥
⎤
8
9 e2 − 1 − 6e (2 − t) 4e2 − 7 0
0
(1 − t)et−1⎥ ⎥ ⎥ ⎥dt 0 ⎥ ⎥ 1
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥. − ⎢ 2 ⎢ 9 e − 1 − 6e (1 − t) + 2(8e − 9)e1−t ⎥ ⎥ ⎢ ⎥ ⎢ 2−7 ⎥ ⎢ 4e ⎦ ⎣
Therefore, W 0,
⎤
2 + 3(t − 1)et−1 ⎥ ⎡ ⎤ ⎢ 4(2 − t) ⎢ ⎥ u1 ⎢ ⎥ ⎢ ⎥ ⎢ 3e1−t + 2(t − 1) ⎥ ⎢u ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ 4(2 − t) ⎥ ⎢ ⎥ U(t) =⎢ ⎢ ⎥=⎢ ⎥ t −1 ⎢u ⎥ 3 ⎥ ⎢ e ⎥ ⎣ ⎦ ⎢ ⎢ ⎥ ⎢ ⎥ 4 u4 ⎣ ⎦ 0.5
∗ × ψ(1 − t) ⊗ φ(1 − t)
E1 .
Let u i , yi , i = 1, 2, . . . , n2 , = 1, 2, . . . , m, be fuzzy sets in We assume that the rule base for the input and output is
R : IF u1 (t) is in u 1 (t), . . . , un2 (t) is in u n2 (t), THEN y1 (t) is in y1 (t), . . . , yn2 (t) is in yn 2 (t),
= 1, 2, . . . , m,
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎤
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥, − ⎢ 2 ⎢ 9 e − 1 − 6e (1 − t) + 2(8e − 9)e1−t ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 4e2 − 7 ⎦ ⎣
(103)
and the relation between input and output is
+ In ⊗ D(t) U(t). Y (t) = In ⊗ C(t) X(t)
(104)
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, 2 if X0 = (x01 , x02 , . . . , x0n ), then one has the following rule base for the initial value X0 : R : IF u1 (T) is in u 1 (T), . . . , un2 (T) is in u n2 (T), IF y1 (T) is in y1 (T), . . . , yn2 (T) is inyn 2 (T), 2
THEN x01 is in x0 (1), . . . , x0n (t) is in x0 n2 ,
0
(101)
= 1, 2, . . . , m,
(105)
12
Advances in Fuzzy Systems
where
Using Definition 8,
x0 (i) =
In ⊗ C(T) ψ(T) ⊗ φ(T)
×
Vi (T) − In
×
T
−1
⊗ D(T) U(T) − In ⊗ C(T)
ψ(T − s) ⊗ φ(T − s) In ⊗ F(s)
Hi (s)ds
×
X0 =
In ⊗ C(T) ψ(T) ⊗ φ(T)
(T) − In ⊗ C(T) × y(T) − In ⊗ D(T) u ×
T
0
In ⊗ C(T) ψ(T) ⊗ φ(T)
T
×
ψ(T − s) ⊗ φ(T − s) In ⊗ F(s) u(s)ds ,
0
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds.
−1
× Y (T) − In ⊗ D(T) U(T) − In ⊗ C(T)
(113)
Since (In ⊗ C(T))(ψ(T) ⊗ φ(T)) is nonsingular, we have X0 =
−1
t
,
(106)
= Y (t) − In ⊗ D(t) U(t) − In ⊗ C(t)
0
In ⊗ C(t) ψ(t) ⊗ φ(t) X0
ψ(T − s) ⊗ φ(T − s) In ⊗ F(s) U(s)ds .
0
(114)
(107) Hi (t) = u1 (t) × · · · × u i (t) × · · · × un2 (t), Vi (t) = y1 (t) × · · · × yi (t) × · · · × yn2 (t), i = 1, 2, . . . , n2 ,
(108)
Hi1 (t) = u1 (t) × · · · × u1i (t) × · · · × un2 (t),
= 1, 2, . . . , m.
Proof. Without loss of generality, we prove this theorem by considering = 1. Let
(109)
y(t) = y1 (t), y2 (t), . . . , yn2 (t) .
Let μu1i (t) (ui (t)) be the grade of the membership of ui (t) in u1i (t), and let μ yi1 (t) ( yi (t)) be the grade of the membership of yi (t) in yi1 (t). Since (In ⊗ C(T))(ψ(T) ⊗ φ(T)) is nonsingular and from (60), we have
X0 =
In ⊗ C(T) ψ(T) ⊗ φ(T)
−1
T 0
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
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) ⊗ φ(t) X0 t
+ 0
(116)
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
t 0
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) Hi1 (s)ds
is a fuzzy set in En . Hence, X0 is a fuzzy set in En , and the ith component of it denoted by x01 (i) is a fuzzy set in E1 . The grade of the membership of x0i in x01 (i) is defined by 2
(110)
(117)
× y(T) − In ⊗ D(T) u (T) − In ⊗ C(T) ×
(115)
i = 1, 2, . . . , n2 .
In ⊗ C(t)
Vi1 (t) = y1 (t) × · · · × yi1 (t) × · · · × yn2 (t),
From Remark 1, we know that the ith component of the set
u(t) = u1 (t), u2 (t), . . . , un2 (t) ,
Now, the initial value X0 is no more a crisp value, but should be a fuzzy set. In order to determine each component of X0 , let us assume
ψ(t − s) ⊗ φ(t − s) In ⊗ F(s) U(s)ds
2
μx01 (i) x0i = min μu1i (t) ui (t) , μ yi1 (t) yi (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 compute x0 (i), but to solve the real problems, we choose the following approximation. Now, 0 we take the point (x0i , μx0 (i) (x0i )) and the zero-level set [x0 (i)] to determine a triangle as the new fuzzy set x0 (i). We can use the center average defuzzifier ! m i
0 μx0 (i) x0i
=1 x x0i = m i ! 0
=1 μx0 (i) x
(119)
to determine the initial value X0 = (x01 , x02 , . . . , x0n ). To obtain more accurate value for the initial state, more rule bases may be provided. 2
= Y (t) − In ⊗ D(t) U(t).
(112)
(118)
M. S. N. Murty and G. Suresh Kumar
13 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
Example 3. Consider the fuzzy matrix Lyapunov system ⎡
⎤
⎡
⎤
⎡
⎤
0 −1 1 0 et 0 ⎦ X(t) + X(t) ⎣ ⎦+⎣ ⎦ U, X (t) = ⎣ 1 0 0 1 0 et 0≤t≤ ⎡
⎤
π , 2
⎡
(120)
X0 = e
0 1 ⎦ X(t). Y (t) = ⎣ 1 0
⎡
0, −0.75(α − 1)
⎜⎡ ⎤ ⎡ ⎜ ⎜ 1 ⎜ ⎢ ⎥ ⎢0 ⎜⎢ ⎥ ⎢ ⎜ ⎢2.8⎥ ⎢1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢ ⎜⎢ ⎢0 ⎢ 0.5 ⎜⎣ ⎥ ⎦ ⎣ ⎜ ⎜ 2.9 0 ⎜ ⎝
⎤
⎢ ⎥ ⎢
⎥ ⎢ 0.75(α − 1) + 1, 1 ⎥ ⎢ ⎥
(1) α ⎢ ⎥ U =⎢
⎥, ⎢ 0, −0.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣
⎦ 0.5(α − 1) + 1, 1 ⎡
0, −2(α + 1)
⎤
⎡
⎤
(121)
·⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
From rule base 1, we select (123)
the grades of the membership of u1 , u2 , u3 , and u4 are 1/3, 0.8, 0.2, and 1/2, respectively. Also
(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),
0
0
0
0
(126)
1 0 0 0
⎤⎡
eπ/2−s
0.5
1 0 0 0
x01 (1)
=e
(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),
0
0 1 0
⎤ −1.142 ⎢ ⎥ ⎢−0.9324⎥ ⎢ ⎥ ⎥, =⎢ ⎢ ⎥ ⎢ −1.046 ⎥ ⎣ ⎦ −0.9532 ⎡
(122)
y1 = y1 , y2 , y3 , y4 = (1, 2.8, 0.5, 2.9),
⎥ ⎥ ⎦
0 0 1⎥
⎤
0
0
2
2
⎥ ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ π π ⎥ cos − s −sin − s ⎥ 2 2 ⎥ ⎥ ⎥ ⎦ π π ,sin − s cos − s
⎤
⎡
[2α + 1, 3]
⎥
0 0 0⎥ ⎥ π/2
⎟ ⎟ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎢0 1 0 0⎥ ⎢0.85⎥ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎥⎢ ⎥ ds⎟ ·e s ⎢ ⎢ ⎥⎢ ⎥ ⎟ ⎢0 0 1 0⎥ ⎢ 0.4 ⎥ ⎟ ⎣ ⎦⎣ ⎦ ⎟ ⎟ ⎟ 0 0 0 1 0.75 ⎟ ⎠ ⎡
⎤
u1 = u1 , u2 , u3 , u4 = (0.5, 0.85, 0.4, 0.75),
⎤
⎞
⎢ ⎥ ⎢ ⎥ ⎢ [α + 1, 2] ⎥ ⎥
(2) α ⎢ ⎢ ⎥ Y = ⎢
⎥. ⎢ 0,−2.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
[0.5α + 0.5, 1]
1 0 0
π π − s −sin −s 2 2 ⎢ ⎢ ⎢ ⎢sin π − s cos π − s ⎢ 2 2 ⎢
0,−1.5(α − 1)
⎢ ⎥ ⎢ ⎥ ⎢[0.8α + 0.2, 1]⎥ ⎥
(2) α ⎢ ⎢ ⎥ U = ⎢
⎥ , ⎢ 0,−0.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
⎥ ⎥ ⎦
0⎥
⎢cos
Rule 2: 0,−0.8(α − 1)
⎡
⎢ ⎥ ⎢ ⎥ ⎢ [0.5α + 2.5, 3] ⎥ ⎢ ⎥
(1) α ⎢ ⎥ Y =⎢
⎥. ⎢ 0, −1.5(α − 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣
⎦ 0.5(α − 1) + 3, 3
⎥
0⎥ ⎥
⎛
Rule 1:
⎢ ⎢0 −1 0 ⎢ ⎢ ⎢ ⎢0 0 1 ⎣
⎤
0 0 0 −1
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.
⎡
−π/2
1 0 0 0
−π/2
⎢ ⎢0 −1 0 ⎢ ⎢ ⎢ ⎢0 0 1 ⎣
⎤ ⎥
0⎥ ⎥ ⎥ ⎥ ⎦
0⎥
0 0 0 −1
⎛ ⎜ ⎡
⎤ ⎡ ⎜ ⎜ 0, −2(α − 1) 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ ⎜⎢ 2.8 ⎥ ⎢1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢0 ⎢ 0.5 ⎜⎣ ⎦ ⎣ ⎜ ⎜ 0 2.9 ⎜ ⎝
⎤
1 0 0
⎥
0 0 0⎥ ⎥ π/2 ⎥ ⎥ ⎦
0 0 1⎥ 0 1 0
0
eπ/2−s
14
Advances in Fuzzy Systems ⎤ π π 0 0 ⎥ ⎢cos 2 − s −sin 2 − s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢sin π − s cos π − s 0 0 ⎥ ⎢ 2 2 ⎥ ⎢ ·⎢ ⎥ ⎥ ⎢ π π ⎢ 0 0 cos − s −sin − s ⎥ ⎢ 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ π π 0 0 sin − s cos − s
⎞
⎡
2
⎟ ⎟ ⎟ ⎢ ⎥ ⎢
⎥ ⎟ ⎢0 1 0 0⎥ ⎢ 0.75(α − 1) + 1, 1 ⎥ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎥⎢ ⎥ ds⎟ ·e s ⎢ ⎢ ⎥⎢ ⎥ ⎟ ⎢0 0 1 0 ⎥ ⎢ ⎥ ⎟ 0.4 ⎣ ⎦⎣ ⎦ ⎟ ⎟ ⎟ 0 0 0 1 0.75 ⎟ ⎠ ⎡
2
⎡
⎞ ⎡
1 0 0 0
⎤ ⎡
⎢ ⎥⎢ ⎢ 0 1 0 0⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ ·es ⎢ ⎢ ⎥⎢ ⎢ 0 0 1 0⎥ ⎢ ⎣ ⎦⎣
0, −0.75(α − 1) 0.4
0 0 0 1 ⎡
0.75
[−1.6 + 0.75α, −0.434 − 0.416α]
⎢ ⎢ [−1.4324, −0.6824 − 0.75α] ⎢ =⎢ ⎢ ⎢ −1.046 ⎣ −0.9532
⎤
⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎢ ⎢0 −1 0 ⎢ 1 −π/2 ⎢ x0 (2) = e ⎢ ⎢0 0 1 ⎣
μx01 (2) x02 = min{0.8, 0.6} = 0.6, ⎡
1 0 0 0
⎢ ⎢0 −1 0 ⎢ 1 −π/2 ⎢ x0 (3) = e ⎢ ⎢0 0 1 ⎣
⎤ ⎥
0⎥ ⎥ ⎥ ⎥ ⎦
0⎥
0 0 0 −1
⎛
⎜⎡ ⎤ ⎡ ⎜ ⎜ 1 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ ⎜⎢ 2.8 ⎥ ⎢1 ⎜ −⎢ ·⎜ ⎢
⎥ ⎢ ⎢ ⎜ ⎢ 0, −1.5(α − 1) ⎥ ⎥ ⎢0 ⎜⎣ ⎦ ⎣ ⎜ ⎜ 0 2.9 ⎜ ⎝
⎤ ⎥
0⎥ ⎥ ⎥ ⎥ ⎦
0⎥
0 0 0 −1
⎤
1 1 1 , = = 0.333, 3 2 3
1 0 0 0
[−1.292, −0.542 − 0.75α]
when α = 0, we get the biggest interval [−1.124, −0.27] and x02 = −0.9324 is located in this interval. We choose its membership grade in x01 (2) as
when α = 0, we get the biggest interval [−1.6, −0.434] and x01 = −1.142 is located in this interval. We choose its membership grade in x01 (1) as
⎤
0.5
(128)
(127)
μx01 (1) x01 = min
⎤⎡
⎢ ⎥ ⎢[−1.124, −0.27 − 0.854α]⎥ ⎢ ⎥ ⎥, =⎢ ⎢ ⎥ ⎢ ⎥ −1.046 ⎣ ⎦ −0.9532
⎟ ⎟ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ds⎟ ⎥ ⎟ ⎥ ⎟ ⎦ ⎟ ⎟ ⎟ ⎟ ⎠
⎤
0.85
1 0 0 0
1 0 0
⎤ ⎥
0 0 0⎥ ⎥ π/2 ⎥ ⎥ ⎦
0 0 1⎥
0
eπ/2−s
0 1 0
⎤ π π 0 0 ⎥ ⎢cos 2 − s −sin 2 − s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢sin π − s cos π − s 0 0 ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ·⎢ ⎥ ⎢ π π ⎢ 0 0 cos − s −sin − s ⎥ ⎢ 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ π π 0 0 sin − s cos − s
⎛
⎡
⎜⎡ ⎤ ⎡ ⎜ ⎜ 1 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ ⎜ ⎢[0.5α + 2.5, 3]⎥ ⎢1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢0 0.5 ⎜⎢ ⎦ ⎣ ⎜⎣ ⎜ 2.9 0 ⎜ ⎝
1 0 0
⎤ ⎥
0 0 0⎥ ⎥ π/2 ⎥ ⎥ ⎦
0 0 1⎥
0
eπ/2−s
0 1 0
2
2
⎤
⎞
0 0 ⎥ ⎢cos 2 − s −sin 2 − s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ π π ⎥ ⎢sin − s cos −s 0 0 ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ·⎢ ⎥ ⎢ π π ⎥ ⎢ 0 0 cos − s sin − s − ⎢ 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ π π 0 0 sin − s cos − s
⎟ ⎟ ⎟ ⎢ ⎥⎢ ⎥ ⎟ ⎟ ⎢0 1 0 0 ⎥ ⎢ ⎥ 0.85 ⎢ ⎥⎢ ⎥ ⎟ ⎟ s⎢ ⎥ ⎢ ⎥ ·e ⎢ ⎥ ds⎟ ⎥ ⎢
⎢0 0 1 0⎥ ⎢ 0, −0.5(α − 1) ⎥ ⎟ ⎣ ⎦⎣ ⎦ ⎟ ⎟ ⎟ 0 0 0 1 0.75 ⎟ ⎠
⎡
π
π
2
2
⎡
1 0 0 0
⎤⎡
0.5
⎤
M. S. N. Murty and G. Suresh Kumar ⎡
15 ⎤
−1.142
follows:
⎢ ⎥ ⎢ ⎥ −0.9324 ⎢ ⎥ ⎢ ⎥, =⎢ ⎥ ⎢[−1.25 + 0.5α, −0.438 − 0.312α]⎥ ⎣ ⎦ [−1.3532, −0.8532 − 0.5α]
⎤ −1.092 ⎢ ⎥ ⎢−0.664⎥ ⎢ ⎥ ⎥, X0 = ⎢ ⎢ ⎥ ⎢−0.584⎥ ⎣ ⎦ −0.812 ⎡ ⎤ [−1.6 + 0.8α, −0.488 − 0.312α] ⎢ ⎥ ⎢ [−1.164, −0.364 − 0.8α] ⎥ ⎢ ⎥ 2 ⎢ ⎥, x0 (1) = ⎢ ⎥ ⎢ ⎥ −0.584 ⎣ ⎦ −0.812 ⎡ ⎤ [−1.292, −0.492 − 0.8α] ⎢ ⎥ ⎢[−0.916, 0.092 − 1.008α]⎥ ⎢ ⎥ 2 ⎥, x0 (2) = ⎢ ⎢ ⎥ ⎢ ⎥ −0.584 ⎣ ⎦ −0.812 ⎡ ⎤ −1.092 ⎢ ⎥ ⎢ ⎥ −0.664 ⎢ ⎥ ⎥, x02 (3) = ⎢ ⎢ ⎥ ⎢[0.5α − 1.25, −0.23 − 0.52α]⎥ ⎣ ⎦ [−1.062, −0.562 − 0.5α] ⎡ ⎤ −1.092 ⎢ ⎥ ⎢ ⎥ −0.664 ⎢ ⎥ 2 ⎢ ⎥. x0 (4) = ⎢ ⎥ ⎢ [−0.834, −0.334 − 0.5α] ⎥ ⎣ ⎦ [−1.374, −0.458 − 0.916α]
(129) when α = 0, we get the biggest interval [−1.25, −0.438] and x03 = −1.046 is located in this interval. We choose its membership grade in x01 (3) as
μx01 (3) x03 = min 0.2, ⎡
1 ⎢ ⎢ 0 x01 (4) = e−π/2 ⎢ ⎢0 ⎣ 0 ⎛
0 −1 0 0
0 0 1 0
2 = 0.2, 3
⎤
0 ⎥ 0⎥ ⎥ 0⎥ ⎦ −1
⎜ ⎜⎡ ⎤ ⎡ ⎜ 1 ⎜ 0 ⎜⎢ ⎥ ⎢ ⎜⎢ ⎥ ⎢ 2.8 1 ⎜ ⎥−⎢ ·⎜ ⎢ ⎥ ⎢0 ⎜⎢ 0.5 ⎦ ⎣ ⎣ ⎜
⎜ 0.5(α − 1) + 3, 3 0 ⎜ ⎜ ⎝ ⎡
⎡
0
0
0
0
1 0 0
⎢ ⎢ s ⎢0 1 0 ·e ⎢ ⎣0 0 1
0 0 0
0 0 0 1
⎤
0 ⎥ π/2 0⎥ ⎥ eπ/2−s 1⎥ ⎦ 0 0 ⎤
π π − s −sin −s ⎢cos 2 2 ⎢ ⎢ ⎢ ⎢sin π − s cos π − s ⎢ 2 2 ⎢
·⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
1 0 0 0
0
0
⎥ ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ π π ⎥ cos − s −sin − s ⎥ 2 2 ⎥ ⎥ ⎥ ⎦ π π sin − s cos − s
2
⎞
2
⎟ ⎟ ⎟ 0.5 ⎟ 0 ⎥⎢ ⎥ ⎟ ⎢ ⎥ ⎟ 0.85 0⎥ ⎥⎢ ⎥ ds⎟ ⎢ ⎥ ⎟ ⎟ 0.4 0⎥ ⎦ ⎣
⎦ ⎟ ⎟ 1 0.5(α − 1) + 1, 1 ⎟ ⎟ ⎠ ⎤⎡
0 μx0 (1) x0
=1 x 2 0
=1 μx0 (1) x
−1.142 × 0.333 + (−1.092) × 0.333
0.333 + 0.333
= −1.117,
0 μx0 (2) x0
=1 x 2 x0 = 2 0
=1 μx0 (2) x 2
=
−0.9324 × 0.6 + (−0.664) × 0.75
1 1 (131) , 0.8 = = 0.5. 2 2 Similarly for rule base 2, by the use of formula (106), (107), we obtain the values of X0 , x02 (i), i = 1, 2, 3, 4 and given as
0.6 + 0.75
= −0.7833,
0 μx0 (3) x0
=1 x 3 x0 = 2 0
=1 μx0 (3) x 2
when α = 0, we get the biggest interval [−1.224, −0.62] and x04 = −0.9532 is located in this interval. We choose its membership grade in x01 (4) as
= =
(130)
Also the grades of the membership of x01 = −1.092, x02 = −0.664, x03 = −0.584, x04 = −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 X0 = (x01 , x02 , x03 , x04 ), where x01
⎤ −1.142 ⎢ ⎥ ⎢ ⎥ −0.9324 ⎥ =⎢ ⎢ [−1.296, −0.796 − 0.5α] ⎥ , ⎣ ⎦ [−1.224, −0.62 − 0.604α]
(132)
2
⎤
⎡
μx01 (4) x04 = min
⎡
=
−1.046 × 0.2 + (−0.584) × 0.2
0.2 + 0.2
= −0.815,
0 μx0 (4) x0
=1 x 4 x0 = 2 0
=1 μx0 (4) x 2
=
−0.9532 × 0.5 + (−0.812) × 0.25
0.5 + 0.25
= −0.9061.
(133)
16 6.
Advances in Fuzzy Systems 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. Example 1 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. ACKNOWLEDGMENTS The authors would like to thank Professor H. Ying (Associate-Editor) and the anonymous referees for their suggestions which helped to improve the quality of the presentation. REFERENCES [1] S. Barnett and R. G. Cameron, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, UK, 2nd edition, 1985. [2] M. S. N. Murty, B. V. Appa Rao, and G. S. Kumar, “Controllability, observability, and realizability of matrix Lyapunov systems,” Bulletin of the Korean Mathematical Society, vol. 43, no. 1, pp. 149–159, 2006. [3] M. Sugeno, “On stability of fuzzy systems expressed by fuzzy rules with singleton consequents,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 2, pp. 201–224, 1999. [4] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985. [5] A. Alwadie, H. Ying, and H. Shah, “A practical two-input twooutput Takagi-Sugeno fuzzy controllers,” International Journal of Fuzzy Systems, vol. 5, no. 2, pp. 123–130, 2003. [6] Y. S. Ding, H. Ying, and S. H. Shao, “Structure and stability analysis of a Takagi-Sugeno fuzzy PI controller with application 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-Sugeno PI and PD fuzzy controllers: analytical structures and stability analysis,” Information Sciences, vol. 151, pp. 245–262, 2003. [8] T. A. Johansen, R. Shorten, and R. Murray-Smith, “On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 3, pp. 297–313, 2000.
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