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

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

Chapter 3 Let h→0, f(t) = δ(t) (Dirac delta)

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

Please see Table 3.1 in Text Chapter 3

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

Example 2: system at rest (s.s.) Step 1 Take L.T. (note zero initial conditions) Chapter 3 To find transient response for u(t) = unit step at t > 0 1. Take Laplace Transform (L.T.) 2. Factor, use partial fraction decomposition 3. Take inverse L.T.

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

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

Properties of Laplace Transform

Linearity

Sifting Theorems

Final Value Theorem “offset” Example 3: step response offset (steady state error) is a. Chapter 3

Initial Value Theorem by initial value theorem by final value theorem Chapter 3

Transform of Time Integration

Differentiation of F(s)

Integration of F(s)