Báo cáo nghiên cứu khoa học: "THE COMPARISON OF SHEAF- SOLUTIONS IN FUZZY CONTROL PROBLEM" pdf

In [5], we have offered the neccesary conditions of Sheaf-Optimal Control Problem in Fyzzy type SOFCP, that means the controls u t U E∈ ⊂ p not belong to R.. p This paper shows some co

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 2-2006 THE COMPARISON OF SHEAF- SOLUTIONS

IN FUZZY CONTROL PROBLEM Nguyen Dinh Phu , Tran Thanh Tung

Faculty of Mathematics and Computer Sciences, University of Natural Science,

VNU-HCM

( Manuscript Received on December 14 th , 2005, Manuscript Revised March 8 th , 2006)

ABSTRACT: In [2 ] the author considered the Sheaf-Optimal Control Problem

(SOCP) by differential equations: dx(t) f (t, x(t), u(t))

wherex0∈ ⊂Q R , u U R , t [0, T] Rn ∈ ⊂ p ∈ ⊂ +, and sheaf of solutions:

Ht,u ={x(t) x(t, x u(t)) | x= 0, 0∈H0 ⊆Q, t I [0, T] R , u(t) U∈ = ⊂ + ∈ }

with the goal functionI(u)→min

In [5], we have offered the neccesary conditions of Sheaf-Optimal Control Problem in

Fyzzy type (SOFCP), that means the controls u (t) U E∈ ⊂ p not belong to R p

This paper shows some comparison of sheaf-solutions Ht,uand Ht,u for many kinds of

fuzzy controls u (t), u(t) U E∈ ⊂ p in Sheaf Fuzzy Control Problem(SFCP)

Keywords: Fuzzy Theory, Optimal Control Theorey, Differential Equations

1 INTRODUCTION :

For Sheaf-Optimal Control Problem(SOCP) many controls u(t) and u(t) u(t)= + Δu

are considered with Δ =u u(t) u(t)− ≤δ, where u (t), u(t) U R∈ ⊂ p [2] For Sheaf-Optimal

Control Problemin Fuzzy Type(SOFCP) we have fuzzy controls u (t) and u(t) U E∈ ⊂ p with

u(t) u(t)− ≤T p [5]

For the Sheaf Fuzzy Control Problem (SFCP) we have the same fuzzy controls

u (t)and u(t) U E∈ ⊂ p, that was defined by definition 5 in [5] The paper is organized as

follows:

In the second section, offering the Sheaf Fuzzy Control Problem (SFCP) we get

estimations of the norms • Cand • Lof

Δ =x x(t, x , u(t)) x(t, x , u(t))0 − 0 and

Δ =f f (t, x(t, x , u(t)), u(t) ) f (t, x(t, x , u(t)), u(t))0 − 0

In section 3, we study some comparisons of sheaf solutions Ht u, in many kinds of fuzzy

controls u (t) , u(t) U E∈ ⊂ p, that means we have to compare the measure μ(HT,u)− μ(H )T,u

2 THE SHEAF FUZZY CONTROL PROBLEM (SFCP)

As we know, the solutions of differential equations depend locally on initial, right hand

side and parameters Now, we consider a control system of differential equations

dx(t) f (t, x(t), u(t))

where x(0) x= 0∈H0 ⊂ ⊂Q R ,x(t) Q R , u(t) U E , t I [0,T] Rn ∈ ⊂ n ∈ ⊂ p ∈ = ⊂ +

and f : I R× n×Ep →Rn

Science & Technology Development, Vol 9, No.2 - 2006

Definition 1. The sheaf - solution ( or sheaf-trajectory) lx t x u( , , )0 q which gives at the

time t a set

Ht,u={x(t) x(t, x , u)| x= 0 0 ∈H0⊂Q, x(t) solution of (1)− }, (2)

where x0∈H0 ⊂ ⊂Q R , u(t) U E , t In ∈ ⊂ p ∈

In the case, when a control u(t) is fuzzy, we have Sheaf Fuzzy Control Problem (SFCP)

Suppose at time t=0 0, ( )u =0 and x( )0 =x0 ∈H0 For two admissible controls

p

u(t) and u (t) ∈ ⊂U E , we have two sets of sheaf-solutions

Ht,u= {x(t) x(t, x , u)| x= 0 0 ∈H0 ⊂Q, x(t) a solution of (1) by control u(t)− }

Ht,u ={x(t) x(t, x , u (t))|x= 0 0 ∈H0 ⊂Q, x(t) a solution of (1) by control u(t)− },

where t I∈ (See fig.1)

Fig 1.The sheaf-solutions of Sheaf Fuzzy Control Problem (SFCP)

If μ(H )t,u is a measure of the set Ht,u then μ(H )t,u is called a cross-area of sheaf

trajectory at (t,u), in particular it is a square of set Ht,u.That is

t ,u

H

t ,u

H

μ = ∫ is a square of Ht,u.

Assumption 1. Suppose that the vector function f (t,x(t),u(t)) satisfies

i) ∂ Δ + ∂ Δ ≤ Δ + Δ

ii) +∞

iii) f (t, x(t, x , u(t)), u(t)0

x

∂

=

for all x(t) Q R , u(t), u(t) U E , t I∈ ⊂ n ∈ ⊂ p ∈ , where M, m, L are real positive constants and

spA is trace of matrix A

Lemma 1 For the fuzzy controls u(t) and u(t) U E∈ ⊂ p, the norm of

u u(t) u(t)

Δ = − is estimated as follows:

a) || u ||Δ C≤ p (6)

b)

T L

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 2 -2006

Proof of Lemma 1: Let u (t) , u(t) U E∈ ⊂ p are fuzzy controls In [5], we defined a fuzzy

function u : I→ ⊂U Ep = × × ×E E E, that means u(t) (u (t), u (t), , u (t))= 1 2 p Because

every u (t)k satisfies u (t) 1k ≤ ( k=1,2, p) then a norm of

a) Δu C =max u(t) u(t) : t I{ − ∈ }

p i i 2

i 1

max u (t) u (t) : t I p

=

where u (t) , u(t) U E∈ ⊂ p

b)

L

u || u(t) || dt p dt T p

Δ =∫ Δ ≤ ∫ ≤ (■)

Theorem 1 Suppose that u (t) , u(t) U E∈ ⊂ p are fuzzy controls If the function

f(t,x(t), u(t)) satisfies (3) and (4) then the norm of Δ =x x(t, x , u(t)) x(t, x , u(t))0 − 0

is estimated as follows:

a) ΔxC≤( T m M p ) exp(MT)+ (8)

Proof of Theorem 1: Let u (t) , u(t) U E∈ ⊂ p are fuzzy controls with Δ =u u(t) u(t)−

satisfies(6) or (7)

a) The solutions of (1) are equivalent the following integrals:

t 0 0

x(t) x= +∫f (s, x(s), u(s)) ds and

t 0 0

x(t) x= +∫f (s, x(s), u(s)) ds

Estimating Δx(t) as follows

t 0

x(t) f (s, x(s), u(s) f (s, x(s), u(s)) ds

t

k

k 2 0

f (s, x(s), u(s))dx f (s, x(s), u(s))du d f (s, x(s), u(s)) ds

∫

M dx du ds M x(s) ds M u(s) ds mT

t

0

M x(s) ds MT p mT

By Gronwall-Bellmann’s Lemma, it implies that

∈

t [0,T]

x max x(t) T(m M p ) exp(MT)

b) Δx(t) t t

M x(s) ds M u(s) ds mT

Δx(t)

t 0

M x(s) ds MT p mT

≤T(m M p ) exp(MT)+

For

T

2 L

0

x x(t) dt T (M p m) exp(MT)

Δ = Δ∫ ≤ + we have (9) (■)

Science & Technology Development, Vol 9, No.2 - 2006

Theorem 2 Suppose that u (t) , u(t) U E∈ ⊂ p are fuzzy controls, if the function

f(t,x(t), u(t)) satisfies (3) and (4) then the norm of

f f (t, x(t, x , u(t)), u(t) ) f (t, x(t, x , u(t)), u(t))

is estimated as follows:

a) Δf C≤MT[(M p +m) exp(MT)+ p] m+ (10)

b) Δf L≤ T M T( m M p ) exp(MT){ ⎡⎣ + + p⎤⎦+m} (11)

Proof of Theorem 2:

a) For max df 1 d f2 1 d f : t I3

k

k 2

1

k!

+∞

=

k

k 2

+∞

=

≤M ( xΔ C + Δu ) mC +

≤M[T(M p +m) exp(MT) T p] m+ +

≤MT[(M p +m) exp(MT)+ p] m+

0 f (s, x(s, x , u(s)), u(s)) f (s, x(s, x , u(s)), u(s)) ds

M ( x(t) dt u(t) dt) m dt

≤M ( xΔ L + Δu )L +mT

≤ ⎡ + + ⎤+

≤T M T( m M p ) exp(MT){ ⎡⎣ + + p⎤⎦+m} (■)

3 THE COMPARISON OF SHEAF SOLUTIONS IN THE SFCP

Lemma 2. For A, B 0≥ there exists a real number K such that eA −eB≤K eA B−

Proof of Lemma 2: We have eA −eB = e (eB A B− − ≤1) K eA B− , K e> B (■)

Now, suppose that μ(H )0 is given. There are many following results of comparison of

sheaf- solutions :

Theorem 3. Suppose that u (t) , u(t) U E∈ ⊂ p are fuzzy controls If the function

f(t,x(t), u(t)) satisfies (3) ,(4) and (5) then we have the following estimation:

| (H )μ T,u − μ(H ) |T,u ≤ μ(H ) exp(LT p)0 (12)

Proof of Theorem 3: We have

t,u

H

0

0 0

x(t,x , u)

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 2 -2006

∫

T

0 0

x(t,x , u) f( ,x( ,x , u),u( )

that means

C

f max f (t, x(t, x , u(t)), u(t)) f (t, x(t, x , u(t)), u(t)) : t I

t,u

(H )

0

0 0

x(t,x ,u)

x

∂

0

T

0

0

f( ,x( ,x ,u), u( ))

x

T

0

(H ) (H ) exp(L u(t) dt)

It is analogous of proof a) above, we have

T

0

(H ) (H ) exp(L u(t) dt)

Estimating| (H )μ T,u − μ(H ) |T,u we have

| (H ) (H ) | (H ) exp(L u(t) dt) exp(L u(t) dt)⎡ ⎤

T 0

0

(H ) K exp[ L ( u(t) u(t) )dt]

0 T

0

(H ) K exp[ L u(t) dt]

≤ μ ∫ Δ ≤ μ(H ) K exp[ LT p]0

Corollary 1 Suppose that u (t) , u(t) U E∈ ⊂ p are fuzzy controls If the function

f(t,x(t), u(t)) satisfies (3) and (4), then for (1) when n=1 we have the following estimation:

| (H )μ T,u − μ(H ) | (bT,u ≤ 0−a ) exp(2LT p)0 , (13)

where K exp(LT p) =

Proof of Corollary: When n 1= we have μ(H ) b0 = 0−a0 , finally we get (13) (see

fig.2)

Fig 2.The sheaf-solutions of Sheaf Fuzzy Control Problem (SFCP), when n = 1 (■)

4 CONCLUSION

In the Sheaf Fuzzy Control Problem (SFCP) for many different fuzzy controls

p

u (t) , u(t) U E∈ ⊂ we have the comparison (7)-(13).There are differences between the Sheaf

Fuzzy Control Problem (SFCP) and the Sheaf Optimal Control Problem in Fuzzy Type

(SOFCP) what was offered in [5]

5 ACKNOWLEDGMENT

Science & Technology Development, Vol 9, No.2 - 2006

This work is a part of research supported by The Research Fund of Ministry of Educations of Vietnam, under contract NCCB 2003/18 The Authors gratefully acknowledge the financial support of this Research Fund The Authors would like to thank the referee for his (her)

careful reading and valuale remarks which improve the presentation of the paper

SO SÁNH BÓ NGHIỆM TRONG BÀI TOÁN ĐIỀU KHIỂN MỜ

Nguyễn Đình Phư , Trần Thanh Tùng

Khoa Toán – Tin học, Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

TÓM TẮT: Trong [2] tác giả đã xét bài toán điều khiển tối ưu bó (SOCP) cho bởi hệ phương trình vi phân:

dx(t) f (t, x(t), u(t))

ở đây x0∈ ⊂Q R , u U R , t [0, T] Rn ∈ ⊂ p ∈ ⊂ +, và bó nghiệm:

Ht,u ={x(t) x(t, x u(t)) | x= 0, 0∈H0 ⊆Q, t I [0, T], u(t) U∈ = ∈ }

với hàm mục tiêu I(u)→ min.

Trong [5] lại trình bày các điều kiện cần của bài toán điều khiển tối ưu bó dạng mờ

(SOFCP), với các điều khiển mờ u (t) U E∈ ⊂ p thay vì thuộc Rp

Bài báo này đưa ra các so sánh các bó nghiệm Ht,u và Ht,u ứng với các điều khiển mờ khác nhau u (t), u(t) U E∈ ⊂ p của bài toán điều khiển bó dạng mờ (SFCP).

Từ khóa: Lý thuyết mờ, Lý thuyết điều khiển tối ưu, Phương trình Vi phân

REFERENCES

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sets and systems 119 (2001), pp.367-373

[4] Phu N D.,General views in theory of Systems, VNU – Publishing House, HCM City,

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[5] Phu N D , Tung T T , Sheaf-Optimal Control Problems in Fuzzy Type, J Science and Technology Devolopment Vol 8, No 12 , 2005, pp.5-11

[6] Phu H X., Some necessary conditions for optimaty for a class of optimal control problems which are linear in the control variable, J Systems and Control Letters.Vol

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