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Chapter 3 Laplace Transforms 1. Standard notation in dynamics and control (shorthand notation) 2. Converts mathematics to algebraic operations 3. Advantageous for block diagram analysis 1 Laplace Transforms Chapter 3 • Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. • Laplace transforms play a key role in important process control concepts and techniques. - Examples: • Transfer functions • Frequency response • Control system design • Stability analysis 2 Definition The Laplace transform of a function, f(t), is defined as Chapter 3 F ( s) L f (t ) f t e st dt 0 (3-1) where F(s) is the symbol for the Laplace transform, L is the Laplace transform operator, and f(t) is some function of time, t. Note: The L operator transforms a time domain function f(t) into an s domain function, F(s). s is a complex variable: s = a + bj, j 1 3 Inverse Laplace Transform, L-1: Chapter 3 By definition, the inverse Laplace transform operator, L-1, converts an s-domain function back to the corresponding time domain function: f t L1 F s Important Properties: Both L and L-1 are linear operators. Thus, L ax t by t aL x t bL y t aX s bY s (3-3) 4 where: - x(t) and y(t) are arbitrary functions Chapter 3 - a and b are constants - X s L x t and Y s L y t Similarly, L1 aX s bY s ax t b y t 5 Laplace Transforms of Common Functions Chapter 3 1. Constant Function Let f(t) = a (a constant). Then from the definition of the Laplace transform in (3-1), L a ae 0 st a st dt e s 0 a a 0 s s (3-4) 6 2. Step Function Chapter 3 The unit step function is widely used in the analysis of process control problems. It is defined as: S t 0 for t 0 1 for t 0 (3-5) Because the step function is a special case of a “constant”, it follows from (3-4) that 1 L S t s (3-6) 7 3. Derivatives Chapter 3 This is a very important transform because derivatives appear in the ODEs we wish to solve. In the text (p.53), it is shown that df L sF s f 0 dt (3-9) initial condition at t = 0 Similarly, for higher order derivatives: dn f L n dt n n 1 n 2 1 s F s s f 0 s f 0 n2 n 1 ... sf 0 f 0 (3-14) 8 where: - n is an arbitrary positive integer Chapter 3 - f k 0 dk f dt k t 0 Special Case: All Initial Conditions are Zero Suppose 1 n1 f 0 f 0 ... f 0 . Then dn f L n dt n s F s In process control problems, we usually assume zero initial conditions. Reason: This corresponds to the nominal steady state when “deviation variables” are used, as shown in Ch. 4. 9 4. Exponential Functions Consider f t ebt where b > 0. Then, Chapter 3 b s t L ebt ebt e st dt e dt 0 0 1 b s t 1 e 0 bs sb (3-16) 5. Rectangular Pulse Function It is defined by: 0 for t 0 f t h for 0 t t w 0 for t t w (3-20) 10 Chapter 3 6. Impulse Function (or Dirac Delta Function) The impulse function is obtained by taking the limit of the rectangular pulse as its width, tw, goes to zero but holding 1 the area under the pulse constant at one. (i.e., let h ) tw Let, t impulse function Then, L t 1 11 h Chapter 3 f t tw Time, t The Laplace transform of the rectangular pulse is given by F s h 1 e t w s s (3-22) 12 Other Transforms e jt cos t j sin t Note: e jwt cos t j sin t Chapter 3 j 1 e-jt e jt L(cosωt) = L 2 1 1 1 = 2 s jω s jω 1 s jω s jω = 2 2 s ω2 s2 ω2 s = 2 s ω2 e jt - e jt L(sin ωt) = L 2j ω = 2 s ω2 13 Difference of two step inputs S(t) – S(t-1) (S(t-1) is step starting at t = h = 1) Chapter 3 By Laplace transform 1 e s F ( s) s s Can be generalized to steps of different magnitudes (a1, a2). 14 Table 3.1. Laplace Transforms Chapter 3 See page 54 of the text. 15