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Fundamental Theorems In your study of mathematics, you have already come across two “Fundamental Theorems” and another will be introduced in this course. The first was the Fundamental Theorem of Algebra, the second, the Fundamental Theorem of Calculus, and this course will introduce the Fundamental Theorem of Linear Systems. If you study a topic and it has a “fundamental theorem”, this is the single most important idea/concept/relationship in the subject area, and therefore should be among the ideas/concepts/relationships remembered forever. We will use the review of the first two Fundamental Theorems as an opportunity to review some other sub topics that will be of use in our study of control engineering. Fundamental Theorem of Algebra and other related topics. There are six operations associated with arithmetic and algebra: addition, subtraction, multiplication, division, raising a number to a power, and finding roots. Addition is the most fundamental of these operations because subtraction is the process of undoing addition (i.e. adding the additive inverse) multiplication is over and over addition o 3*4=4+4+4=3+3+3+3 division is the process of undoing multiplication (i.e. multiplication by the multiplicative inverse) o 12/3 answers the question: How many times can 3 be subtracted from 12? 12-3=9, 9-3=6, 6-3=3, 3-3=0, hence 12/3=4. raising a number to a power is defined using multiplication: For example o 30=1 o 31=3*30 o 32=3*31 o 3n=3*3n-1 Finding roots is the process of undoing the operation of raising a number to a power, that is y n x x y n More advanced topics in arithmetic include number theory, factoring techniques, prime numbers, etc. The arithmetic/algebraic operations satisfy certain rules. The most basic of these rules are the familiar “laws”. For numbers a, b, c Commutative law: a + b = b +a Associative law: a + (b + c) = (a + b) + c Distributive laws: a(b+c) = ab + ac (a+b)c = ac +bc Additive identity law: There is a special number, 0, such that for any and all a, a+0=0+a=a Additive inverse law: For any a, there exists (-a) such that a + (-a) = 0 Multiplicative identity law: There is a special number, 1, such that for any and all a, a1=1a=1 Multiplicative inverse law: For any and all non zero a, there exists (a-1) such that aa-1=a-1a =1. In addition, the operations satisfy the “order of operations” remembered mnemonically as “PEMDAS”. The letters mean that the algebraic operations are performed in the following order: o Parenthesis. Perform all operations within parenthesis and other grouping symbols first. This includes all other forms of “grouping symbols” such as the division bar, {}, [], etc. o Exponentials. Evaluate exponentials and any other functions next. o Multiplication/Division. Next perform multiplications and divisions as they occur from left to right. o Addition/Subtraction. Finally, perform additions and subtractions as they occur from left to right. From Arithmetic to Algebra. The transition from arithmetic to algebra involves replacing the numbers used in arithmetic by symbols (these symbols are called variables) that either represent numbers or represent more complicated combinations of variables. Hence, algebra is arithmetic applied to symbols rather than numbers. For example, an arithmetic expression might be 3+4 = 7 In algebra we encounter expressions more like 3+x = 7 where x might represent a number (i.e. 4) or a more complex expression like x=(y+5)2 which means that (y+5)2=4; y+5=(+/-)2; or y =-5+/-(2) and finally y =-3 or -7. Advanced topics in algebra include polynomials, techniques for factoring polynomials, prime polynomials, etc. One topic in algebra is solving equations. Remember that all equations contain an equals sign. The rules for solving equations are simple: Whatever you do to one side of the equals sign you must do to the other side Never divide by zero (or by a variable that is equal to zero, or by a combination of variables that is equal to zero). The Fundamental Theorem of Algebra concerns polynomials and their roots. The root of a polynomial (or of any function) is a value of the variable that caused the polynomial to evaluate (or become equal) to zero. A high school level web site that discusses the Fundamental Theorem of Algebra is: http://webpages.charter.net/thejacowskis/chapter6/section7.html A college level web site that discusses the Fundamental Theorem of Algebra is: http://ccrma.stanford.edu/~jos/mdft/Fundamental_Theorem_Algebra.html The Fundamental Theorem of Algebra. An nth order polynomial has n roots (some may be repeated). If, as in the important special case considered in control engineering, ALL of the coefficients of the polynomial are real then any complex root is accompanied by its complex conjugate. Connection between roots and factors. Suppose we have found the n roots of an nth order polynomial p(s), and these n roots are rk, k = 1, … n. Then p(s)=K(s-r1)(s-r2)...(s-rn). It is easy to see that the rk’s are roots of the factored form. Also, it is easy to see that if all of the complex rk’s are accompanied by their complex conjugates, then the coefficients of the expanded polynomial are all real. Exercises. 1. Construct the expanded polynomial from the given set of roots, assume K=1. An important partial answer is the polynomial in factored form. (Definition of expanded polynomial: In the final polynomial, each power of s must not occur more than once; no j’s can appear; the powers of s must decrease from left to right: the coefficients are as simple as possible.) a. {1, 1, 3, -3} b. {j, 1, -j, 5} c. {3} d. {1+j, 1-j, j, -j, 0} e. {a} f. j n n 1 2 , n jn 1 2 2. Given the following polynomials, find all of the roots (including multiplicity), and put the polynomial in factored form. a. p(s) = 4s+2 b. p(s) = 3s2+2s+3 c. p(s) = 5s4+3 d. p(s) = as+b e. p(s) = as2+bs+c 3. Commit to memory: a. Fundamental Theorem of Algebra i. Including the special case for real coefficients b. Definition of a root c. Relationship between roots of p(s) and factored form of p(s) 4. Using the information in (3 above) be able to a. Find the roots of a polynomial b. Put a polynomial in factored form c. Given roots, construct the factored form of the polynomial d. Beginning with the factored form of a polynomial, be able to produce the expanded form of the polynomial. Fundamental Theorem of Calculus and other related topics. As was pointed out earlier, the operations of arithmetic/algebra were addition, subtraction, multiplication, division, raising to powers, and taking roots. These operations come in pairs, an operation and one that undoes it. In arithmetic these operations are applied to numbers. In algebra they are applied to expressions that involve one or more variables, these expressions were called functions. Calculus introduces three more operations that are applied to functions: limits; differentiation; and integration. Differentiation and integration are defined in terms of limits, i.e. d f (t t ) f (t ) f (t ) lim dt t 0 t t n k f ( )d lim n f (ak k ) k , ak a l a k 0 k 0 l 1 There are two common forms of the Fundamental Theorem of Calculus. Each ties the operations of integration and differentiation together. d The Fundamental Theorem of Calculus (1). If F ( ) f ( ) (i.e. F is the d antiderivative of f) and some other technical assumptions that almost all functions of engineering interest satisfy, then b a f ( )d F (b) F (a) The Fundamental Theorem of Calculus (2). If some technical assumptions that almost t d dt a all functions of engineering interest satisfy, then f ( )d f (t ) Both versions show that integration and differentiation are (nearly) inverse operations, i.e. one undoes the other similar to subtraction undoing addition and division undoing multiplication and vice versa. The Fundamental Theorem of Calculus (1) is the basic technique for computing definite (as opposed to indefinite) integrals. It says that the integral (i.e. the area under the curve between the lower and upper limits of integration) is found by first finding the anti- derivative (i.e. a function whose derivative is the integrand) of the integrand, second evaluating this anti-derivative at the upper limit of integration, third evaluating it at the lower limit of integration and finally subtracting the second value from the first. The Fundamental Theorem of Calculus (2) says that if the integral is used to define a function of time by letting the upper limit be the time variable, then differentiating this function produces the integrand. You should note that care was taken to use as the dummy variable of integration. Further, all integrals have a dummy variable of integration and that this dummy variable NEVER shows up in the result of integration. I know you can find books, especially engineering books, that violate this. However, these books are WRONG and should not be followed in this practice. Limits. The idea of a limit is an investigation of what happens when some variable gets very small (or sometimes very large). We discuss the idea of a limit as a variable gets small. Consider the problem of determining how much the area of a square increases when the length of a side is increased by a small amount. The area of a square is A=L2. Now if the length is increased from L to L+dL, the area increases from A to A+dA = (L+dL)2 = L2+2LdL+dL2 = A + 2LdL +dL2. Hence dA = 2LdL + dL2. The derivative is dA/dL = limdL0 [2L + dL]. In the limit as dL approaches 0, we ignore all terms that go to 0, hence, dA = 2LdL approximately. See picture. Some derivative formulas. f (t ) df (t ) d , f (t ), f (t ) a, b, n constants (real or complex) dt dt tn nt n1 e at ae at d 1 ln(t ) dt t sin(t ) cos(t ) cos(t ) sin(t ) f ( g (t )) df dg dg dt f (t ) g (t ) f (t ) g (t ) g (t ) f (t ) f (t ) f (t ) g (t ) f (t ) g (t ) g (t ) g (t ) 2 af (t ) bg (t ) af (t ) bg (t ) Integration by parts. Integration by parts is an integration technique that is perhaps one of the most important. b b u ( )dv( ) u ( )v( ) a v( )du( ) b a a Antiderivative formulas. See table for derivative formulas. Fundamental Theorem of Linear Systems. Covered in this course. Consider a stable linear time-invariant system described by the transfer function G(s) where G( j) G( j) e jG( j ) . The steady-state output due to a sinusoidal input r (t ) R sin(t ) , is yss (t ) R G( j) sin(t G( j)) . Limits: Used in: Root locus; straight line Bode plots; error constants. Fundamental Theorem of Calculus: Used in: Laplace transforms; solving state equations; starting point for review of derivatives, integrals, limits, etc. Roots of polynomials: Used in: stability; root locus; Fundamental Theorem of Linear Systems: Used in: Foundation for Bode plots; foundation for phasors; foundation for experimental determination of transfer functions; Extra credit opportunity for students with a D or F: Turn in these exercises worked out and I will add up to 10 points to the total points earned on the tests, quizzes, final exam, etc. Due at time of first test. There is a very significant difference between a definition and a theorem. Recall the definition of the derivative d f (t t ) f (t ) f (t ) lim dt t 0 t We can use it to prove the “product rule of differentiation” f (t ) g (t ) f (t ) g (t ) f (t ) g (t ) d d d dt dt dt Proof: d f (t t ) g (t t ) f (t ) g (t ) f (t ) g (t ) t 0 lim dt t f (t t ) g (t t ) { f (t ) g (t t ) f (t ) g (t t )} f (t ) g (t ) lim t 0 t lim f (t t ) f (t ) g (t t ) f (t ) g (t t ) g (t ) t 0 t f (t t ) f (t ) g (t t ) g (t ) lim g (t t ) f (t ) lim t 0 t t 0 t d d f (t ) g (t ) f (t ) g (t ) dt dt This theorem, along with the Fundamental Theorem of Calculus can be used to develop the Integration-by-Parts theorem. udv uv vdu Proof: u (t )v(t ) u (t ) v(t ) u(t ) v(t ) d d d dt dt dt d d d dt u (t )v(t ) dt dt u (t ) v(t )dt u(t ) dt v(t )dt u (t )v(t ) v(t )du u (t )dv u (t )dv u (t )v(t ) v(t )du In short, definitions are arbitrary, whereas theorems, once the definitions are established are pre-determined and are logical consequences of the definitions and earlier postulates.