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Equivalence Classes of the Partition Mathematics for Control Theory Introduction Logic and Sets Relations and Functions Mathematical Induction Hanz Richter Mechanical Engineering Department Cleveland State University Course rationale and aims The course is primarily directed at PhD students specializing in control and related areas (dynamic systems, mechatronics, optimization). The decision to offer the course was based on: 1. Even for students focusing more on control applications (as opposed to development of new algorithms and analysis methods), the controls literature stands out from other engineering fields for its widespread reliance on advanced mathematics (topics not normally covered in the typical graduate curriculum). PhD students in controls have much more pressure to catch up with the language and basic techniques used by leading researchers in the field. 2 / 40 Course rationale and aims... 2 The course prepares you to read: Frequently, the formality of the mathematical language used in the literature obstructs understanding of the key concepts presented in the paper. Familiarity with these concepts, even at a basic level, can make a big difference. 3 The course prepares you to write: Suppose you found an interesting pattern or result in your research, mostly by simulations. Formalizing the result in mathematical terms and offering a proof or a rational conjecture is key to publishing your work. The course will give you the opportunity to practice writing mathematical content for an engineering audience. 3 / 40 Course rationale and aims... 4 The topics themselves have been carefully chosen to bridge key topics in systems and control and to give you a good foundation for future independent study. 5 These slides are only the rythm of the piece. The accompanying book chapters assigned to you for reading are the rest of the music. If you only listen to the drum section, you will miss most of the piece. You must read along and do recommended exercises and homework. Maybe 1 hour a day of exclusive dedication to reading will be sufficient, plus time for homework. You will be asked to explain your solutions or concepts to the class as one way to earn points towards your grade. 4 / 40 Course organization The course is roughly divided in 3: 1. Foundations: We cover essentials such as set theory, functions, metric spaces, convergence. 2. Analysis: Includes: Normed spaces and inner product spaces, Lp spaces. 3. Geometric methods: Includes: Differential equations on manifolds, Lie groups, applications to nonlinear control and dynamics. Several important papers in controls and/or dynamics will be used as partial projects. A paper will be chosen by students for the final project, which could be a draft paper written by the student, with significant mathematical content (instructor will approve). Grading will be through progressive point accumulation, up to 80. The final project will be worth 20 points. 5 / 40 Reading materials We will use multiple book chapters, sometimes with overlapping material. Book chapters and papers will be made available at the beginning of each section or subsection. For this section (introduction, logic, sets, relations, functions, mathematical induction), we will use: ■ Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, ISBN 0-465-02656-7 (provided material from chapter 1). ■ Landin, Joseph, (1989) [1969], An Introduction to Algebraic Structures, Dover Publications, ISBN 0-486-65940-2 (provided material from chapter 1, pp. 1-28) ■ Anthony N. Michel and Charles J. Herget (2007) [1981], Algebra and Analysis for Engineers and Scientists, Birkhäuser, e-ISBN-13: 978-0-8176-4707-0 (chapter 1) ■ Kenneth R. Davidson and Allan P. Donsig [2002], Real Analysis with Real Applications, Prentice-Hall, ISBN 0-13-041647-9 (chapter 1, except 1.3 Linear Algebra and 1.4 Calculus). 6 / 40 A Recommended: use LTEX Incorporating mathematical notation into a document shouldn’t be a nightmare. The same goes for scientific documents containing numbered bibliographic references, figures, and tables. A 1. LTEXsources for sample documents will be made available. A 2. LTEX- oriented document sharing online tools are available for free. You may submit assignments simply by sharing them with the instructor using Overleaf. A 3. Online resources for learning LTEXand resolving issues are plentiful. A 4. Most students gain working knowledge of LTEXin a couple of weeks. A 5. LTEXis highly recommended, not mandatory. 7 / 40 Logic: the rules of the game in mathematics The ideas presented below are inspired by reading the book “Gödel, Escher, Bach: An Eternal Golden Braid”, by computer scientist Douglas Hofstadter. We will refer to this book as GEB. ■ Mathematics advances by generating new “truths” on the basis of previously known ones and following rules of deduction: logic. ■ This process is not unlike a game, where each new position of the pieces (a “truth”) is generated from others by followig the rules. ■ The process starts with some accepted truths (the initial layout of pieces is always the same and universally accepted). These are called axioms or postulates. One axiom of Euclidean geometry (one of Euclid’s five postulates) is “given a straight line and a point not belonging to it, there’s exactly one line passing through the point which does not intersect the first line”. ■ Axioms are not proven. Several axioms may be used to obtain new truths by a process of proof. Proof may be reduced to a series of symbol permutations per the rules of the game (see GEB). In other words, the generation of truth may be mechanized. 8 / 40 Hierarchy used in math writing These types of “truth” require a proof. Labeling results as proposition, lemma or theorem is a personal decision, but you will observe the following in the literature: ■ Truths are labeled “proposition”, “lemma”, “theorem” according to their generality and importance. ■ A proposition is a minor result, an accessory to build an upcoming more general result. ■ A lemma is more far-reaching and significant than a proposition, but still an accessory to build upcoming theorems. Sometimes a lemma ends up having the same reach as theorems: the Kalman-Yakubovich lemma of passivity theory in controls. The matrix inversion lemma. ■ A theorem is the type of truth at the top of the hierarchy in a theory. The Cayley-Hamilton Theorem has large importance in Linear Algebra. ■ A corollary is an immediate consequence of a theorem. It’s proven simply by invoking the theorem (applying it to a special situation, for example). 9 / 40 Relative truth, incomplete truth, vacuous truth Observe the following: 1. Axioms can be overriden, to create a new theory. A good example is hyperbolic geometry (due to Lobachevsky). Euclid’s parallel postulate is replaced with: “given a line and a point outside it, there are at least two distinct lines passing through the point which do not intersect the first line”. This results in a new theory, essential to Einstein’s relativity. 2. A consistent system with axioms and rules of deduction meant to derive truths about the natural numbers will never reach them all. Kurt Gödel proved this for arithmetic in 1931: there are truths that can never be reached by using the axioms and the rules, no matter the procedure. These are called undecidable propositions. Gödel’s result is known as Incompleteness Theorem. 3. Reading: Pages 33-41 in GEB, try the “MU puzzle” provided there. The following proposition is vacuously true: If x ∈ R is such that x2 = −1, then x =3.5. This kind of proposition establishes that the members of the empty set satisfy any arbitrary property. Vacuous propositions may arise in the proof process. 10 / 40 Logical Propositions A proposition in logics is a statement that can be evaluated as true or false: All cats have at most 4 legs There are exactly 24 primes between 1 and 100 Statements whose truth cannot be evaluated due to subjective factors or ambiguity are not propositions: Let’s eat snow This statement is false Conditional truth: Mathematical results (propositions, lemmas, theorems) are often implications, one-sided or two-sided: Suppose A. Then B. (A ⇒ B) Suppose A. Then B if and only if C. (A ⇒ (B ⇐⇒ C)) 11 / 40 Truth of Implication The validity of a statement like A ⇒ B is independent of whether A is true or not. What must be done is to establish the truth of the implication. Informally: If a tire becomes deflated, contact between it and the surface occurs at more than one point. The implication is true, we don’t have to think about the specific condition of the tire. In a statement like (A ⇒ (B ⇐⇒ C)), we must establish that if A is true, then B and C are equivalent (B will occur when and only when C occurs). Finally note that one-sided implications are not reversible. If contact with the surface occurs at more than one point, we can’t conclude that the tire is deflated. It could have been placed on a V-shaped surface. 12 / 40 Modus Ponens and Truth Tables Sometimes we do want to go farther than just the truth of the implication. We can establish the truth of the actual propositions. Modus ponens is one such scheme: A ⇒ B A is true Then B must be true. A truth table maps out all outcomes of assumed truths in a proposition. The truth table for A ⇒ B is: A B A ⇒ B T T T T F F F T T F F T Clearly, the implication is false (invalid) when B fails to occur when A has happened. Think about the interpretation of this table for the tire-surface example. 13 / 40 Logic Operators The following are used to construct statements: ■ AND, symbol ∧ ■ OR, symbol ∨ ■ NOT, symbol ∼ ■ XOR (exclusive OR), symbol ⊻ As an exercise, review the truth tables of all these.
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