Laplace Transforms1. Standard notation in dynamics and control (shorthand notation)

2. Converts mathematics to algebraic operations

3. Advantageous for block diagram analysisChapter 3

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

Other Transformsetc. forChapter 3Note:

Chapter 3

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

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

Table 3.1 Laplace Transforms for Various Time-DomainFunctionsa (continued) f(t) F(s)

Example 3.1Solve 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 decompositionMultiply by s, set s = 0Chapter 3

For a2, multiply by (s+1), set s=-1 (same procedurefor a3, a4)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 rootsChapter 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 inputsChapter 3

Chapter 3Factoring the denominator polynomial1.Transforms to e-t/3, e-t Real roots = no oscillation

Chapter 32.Transforms to Complex roots = oscillationFrom Table 3.1, line 17 and 18

Chapter 3Let h0, f(t) = (t) (Dirac delta) L() = 1If h = 1, rectangular pulse inputUse LHopitals theorem(h0)

Chapter 3Difference of two step inputs S(t) S(t-1)

(S(t-1) is step starting at t = h = 1)

By Laplace transformCan be generalized to steps of different magnitudes(a1, a2).

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, ifthe denominator can be factored into (s+2)(s+1).Using the partial fraction techniqueThe step response of the process will have exponential terms e-2t and e-t, which indicates y(t) approaches zero. However, ifWe know that the system is unstable and has a transient response involving e2t and e-t. e2t 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 theoremoffsetExample 3: step responseoffset (steady state error) is a.Time-shift theorem y(t)=0 t < Chapter 3

C. Initial value theoremby initial value theoremby final value theoremChapter 3

Chapter 3Previous chapterNext chapter