Shift-Invariant . . . Ideally, Membership . . . A Collection of Several . . . Definitions A Symmetry-Based Main Result What Is Chemical . . . Approach to Selecting Relation Between . . . Membership Functions and Connection Between . . . Discussion Its Relation to Chemical Home Page Kinetics Title Page ◭◭ ◮◮ Vladik Kreinovich, Olga Kosheleva Jorge Y. Cabrera, and Mario Gutierrez ◭ ◮ University of Texas at El Paso, El Paso, TX 79968, USA Page 1 of 14 olgak@utep.edu, vladik@utep.edu Go Back Thavatchai Ngamsantivong King Mongkut’s Univ. of Technology North Bangkok, Thailand Full Screen tvc@kmutnb.ac.th Close Quit
Shift-Invariant . . . Ideally, Membership . . . 1. Shift-Invariant Quantities: A Brief Reminder A Collection of Several . . . • In many physical theories, there is no fixed starting Definitions point for measuring the corr. physical quantities. Main Result What Is Chemical . . . • We can measure time based on the current calendar or Relation Between . . . starting with 1789 (the year of the French revolution). Connection Between . . . • If we select a new one which is q 0 units smaller, then Discussion the original numerical value q changes into q ′ = q + q 0 . Home Page • For such quantities, all the properties do not change if Title Page we change this starting point, i.e., if replace q by q + q 0 . ◭◭ ◮◮ • Strictly speaking, there is the absolute starting point ◭ ◮ for measuring time: the Big Bang. Page 2 of 14 • However, in most cases, the equations remains the same if we change a starting point for time. Go Back Full Screen • Similarly, in many practical applications, there is no absolute starting point for measuring potential energy. Close Quit
Shift-Invariant . . . Ideally, Membership . . . 2. Ideally, Membership Functions Should Reflect A Collection of Several . . . This Symmetry Definitions • Often, our knowledge is imprecise (“fuzzy”). Main Result What Is Chemical . . . • To describe and process such knowledge, L. Zadeh in- Relation Between . . . vented the ideas of fuzzy sets. Connection Between . . . • A fuzzy set on a universal set X is characterized by its Discussion membership function µ : X → [0 , 1]. Home Page • For shift-invariant quantities, our selection of µ ( x ) should Title Page reflect shift-invariance. ◭◭ ◮◮ • A seemingly natural idea is to require that µ ( x ) be ◭ ◮ shift-invariant, i.e., µ ( q ) = µ ( q + q 0 ) for all q and q 0 . Page 3 of 14 • Unfortunately, the only membership function µ ( q ) which Go Back satisfies this condition is the constant function. Full Screen • So, we cannot require that a single membership func- tion is shift-invariant. Close Quit
Shift-Invariant . . . Ideally, Membership . . . 3. A Collection of Several Membership Functions A Collection of Several . . . Should Be Shift-Invariant Definitions • A single membership function cannot be shift-invariant. Main Result What Is Chemical . . . • It is thus reasonable to require that a collection of sev- Relation Between . . . eral membership functions is shift-invariant. Connection Between . . . • In T-S fuzzy control, we often end up with a linear combination � c i · µ i ( q ) of these membership functions. Discussion Home Page • Thus, we consider sets of all such linear combinations Title Page – a linear space . ◭◭ ◮◮ • We are therefore looking for shift-invariant linear spaces. ◭ ◮ Page 4 of 14 Go Back Full Screen Close Quit
Shift-Invariant . . . Ideally, Membership . . . 4. Definitions A Collection of Several . . . • By a finite-dimensional linear space , we mean the class Definitions � n Main Result of all functions of the type c i · µ i ( q ), where: i =1 What Is Chemical . . . • n ≥ 1, Relation Between . . . • differentiable functions µ 1 ( q ), . . . , µ n ( x ) are fixed Connection Between . . . (and assumed to be linearly independent), and Discussion Home Page • the coefficients c 1 , . . . , c n can take any real values. Title Page • We say that a linear space L is shift-invariant if ◭◭ ◮◮ • for every function f ( q ) from the space L and ◭ ◮ • for every real number q 0 , Page 5 of 14 • the function f ( q + q 0 ) also belongs to the class L . Go Back • We say that a shift-invariant linear space L is basic if Full Screen L � = Lin( L 1 ∪ L 2 ) for shift-inv. linear spaces L 1 , L 2 . Close Quit
Shift-Invariant . . . Ideally, Membership . . . 5. Main Result A Collection of Several . . . • We say that a linear space of functions L is fuzzy-related Definitions if the following two conditions hold: Main Result What Is Chemical . . . • L is the set of all linear combinations of functions Relation Between . . . µ 1 ( q ), . . . , µ n ( q ) s.t. µ i ( q ) ∈ [0 , 1] for all q ≥ 0. Connection Between . . . • L does not include the constant functions f ( q ) ≡ 1. Discussion • Proposition. Each basic shift-invariant fuzzy-related Home Page linear space L is a linear combination of functions Title Page µ 1 ( q ) = exp( − λ · q ) , µ 2 ( q ) = q · exp( − λ · q ) , . . . , ◭◭ ◮◮ µ i ( q ) = q i − 1 · exp( − λ · q ) , . . . , µ n ( q ) = q n − 1 · exp( − λ · q ) , ◭ ◮ for some λ > 0 . Page 6 of 14 Go Back Full Screen Close Quit
Shift-Invariant . . . Ideally, Membership . . . 6. Proof of the Main Result: Ideas A Collection of Several . . . • Shift-invariance means that for every q 0 , there are some Definitions � n Main Result c ij ( q 0 ) for which µ i ( q + q 0 ) = c ij ( q 0 ) · µ j ( q ) . j =1 What Is Chemical . . . • Differentiating both sides w.r.t. q 0 and taking q 0 = 0, Relation Between . . . we get a system of linear differential equations Connection Between . . . n � Discussion µ ′ i ( q ) = C ij · µ j ( q ) . Home Page j =1 Title Page • Solutions to such systems are well-known: they are ◭◭ ◮◮ linear combinations of expressions x k · exp( − λ · q ). ◭ ◮ • Expressions corresponding to different λ form shift- Page 7 of 14 invariant spaces. Go Back • Thus, since L is basic, we can only have one value λ . Full Screen • The restrictions that µ i ( q ) ∈ [0 , 1] for all q ≥ 0 and µ ( q ) �≡ 1 imply that λ > 0. Q.E.D. Close Quit
Shift-Invariant . . . Ideally, Membership . . . 7. What Is Chemical Kinetics: Brief Reminder A Collection of Several . . . • Chemical kinetics describes the change in concentra- Definitions tion of chemical substances. Main Result What Is Chemical . . . • The reaction rate is proportional to the product of con- Relation Between . . . centrations of reagents. Connection Between . . . • For example, for a reaction A + B → C, the reaction Discussion rate is proportional to the product a · b . Home Page • Due to this reaction rate k · a · b : Title Page – the amounts a and b of substances A and B de- ◭◭ ◮◮ crease with this rate, while ◭ ◮ – the amount c of the substance C increases with this rate: Page 8 of 14 da db dc dt = − k · a · b ; dt = − k · a · b ; dt = k · a · b. Go Back Full Screen • If we have several reactions, then we add the rates cor- responding to different reactions. Close Quit
Shift-Invariant . . . Ideally, Membership . . . 8. Relation Between Membership Functions and A Collection of Several . . . Chemical Kinetics: An Intuitive Idea Definitions • Let us consider “small”, “medium”, and “large”. Main Result What Is Chemical . . . • The value q = 0 is absolutely small. As we increase q : Relation Between . . . • what was originally small starts slowly transform- Connection Between . . . ing into medium: s → m ; Discussion • then, what was originally medium starts slowly trans- Home Page forming into large: m → ℓ , etc. Title Page • It is reasonable to assume that both “chemical reac- ◭◭ ◮◮ tions” s → m and m → ℓ have the same rate k , then: ds dm dℓ ◭ ◮ dq = − k · s ; dq = k · s − k · m ; dq = k · m. Page 9 of 14 • It is natural to interpret the “concentrations” s ( q ), Go Back m ( q ), . . . , as degrees to which q is small, medium, . . . Full Screen • In other words, we take µ small ( q ) = s ( q ), µ medium ( q ) = Close s ( q ), . . . Quit
Shift-Invariant . . . Ideally, Membership . . . 9. Connection Between Chemical Kinetics and Mem- A Collection of Several . . . bership Functions: General Case Definitions • Let’s consider n ≥ 3 membership functions µ 1 ( q ), . . . , Main Result µ n ( q ), with reactions µ 1 → µ 2 , . . . , µ n − 1 → µ n . What Is Chemical . . . Relation Between . . . • The corresponding equations of chemical kinetics have the form: Connection Between . . . dµ 1 ( q ) = − λ · µ 1 ( q ) , . . . , dµ i ( q ) Discussion = λ · µ i − 1 ( q ) − λ · µ i ( q ) , . . . , Home Page dq dq Title Page dµ n ( q ) = λ · µ n − 1 ( q ) . dq ◭◭ ◮◮ • The initial values are µ 1 (0) = 1 and µ 2 (0) = . . . = ◭ ◮ µ n (0) = 0. Page 10 of 14 • Thus, this system allows us to uniquely determine the Go Back values µ i ( q ) for all q ≥ 0, as Full Screen λ i − 1 ( i − 1)! · q i − 1 · exp( − λ · q ) . µ i ( q ) = Close Quit
Recommend
More recommend
