Laplace Transforms 1. Standard notation in dynamics and control (shorthand notation) 2. Converts mathematics to algebraic operations 3. Advantageous for block diagram analysis Chapter 3

Laplace Transform Example 1: Usually define f(0) = 0 (e.g., the error) Chapter 3

Other Transforms etc. for Chapter 3 Note:

Chapter 3

Table 3.1 Laplace Transforms for Various Time-Domain Functions a f(t) F(s)

Chapter 3 Table 3.1 Laplace Transforms for Various Time-Domain Functions a f(t) F(s)

Table 3.1 Laplace Transforms for Various Time-Domain Functions a (continued) f(t) F(s)

Example 3.1 Solve the ODE, First, take L of both sides of (3-26), Rearrange, Take L -1, From Table 3.1 (line 11), Chapter 3

Example: system at rest (s.s.) Step 1 Take L.T. (note zero initial conditions) Chapter 3

Rearranging, Step 2a. Factor denominator of Y(s) Step 2b. Use partial fraction decomposition Multiply by s, set s = 0 Chapter 3

For  2, multiply by (s+1), set s=-1 (same procedure for  3,  4 ) Step 3. Take inverse of L.T. You can use this method on any order of ODE, limited only by factoring of denominator polynomial (characteristic equation) Must use modified procedure for repeated roots, imaginary roots Chapter 3 (check original ODE)

Laplace transforms can be used in process control for: 1. Solution of differential equations (linear) 2. Analysis of linear control systems (frequency response) 3. Prediction of transient response for different inputs Chapter 3

Factoring the denominator polynomial 1. Transforms to e -t/3, e -t Real roots = no oscillation

Chapter 3 2. Transforms to Complex roots = oscillation From Table 3.1, line 17 and 18

Chapter 3 Let h→0, f(t) = δ(t) (Dirac delta) L(δ) = 1 If h = 1, rectangular pulse input Use L’Hopital’s theorem (h→0)

Chapter 3 Difference of two step inputs S(t) – S(t-1) (S(t-1) is step starting at t = h = 1) By Laplace transform Can be generalized to steps of different magnitudes (a 1, a 2 ).

One other useful feature of the Laplace transform is that one can analyze the denominator of the transform to determine its dynamic behavior. For example, if the denominator can be factored into (s+2)(s+1). Using the partial fraction technique The step response of the process will have exponential terms e -2t and e -t, which indicates y(t) approaches zero. However, if We know that the system is unstable and has a transient response involving e 2t and e -t. e 2t is unbounded for large time. We shall use this concept later in the analysis of feedback system stability. Chapter 3

Other applications of L( ): A. Final value theorem “offset” Example 3: step response offset (steady state error) is a. B.Time-shift theorem y(t)=0 t < θ Chapter 3

C. Initial value theorem by initial value theorem by final value theorem Chapter 3

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