LAPLACE TRANSFORMSChapter 3

Learning OutcomesUpon the completion of this chapter, students are able to:Convert Ordinary Differential Equation, ODE equations to algebraic equations using Laplace Transform

Standard notation in dynamics and controlConverts ordinary differential equation to algebraic operationsLaplace transforms play a key role in important process control concepts and techniques.Examples: Transfer functions Frequency responseControl system designStability analysis

Laplace Transforms (LT)

Definition

The Laplace transform of a function, f(t), is defined aswhere F(s) is the symbol for the Laplace transform, L is the Laplace transform operator, and f(t) is some function of time, t.

Definitions and Properties of LTLaplace Transform

Inverse Laplace Transform

Both L and L-1 are linear operators. Thus,

Similarly,

Usually define f(0) = 0 (e.g., the error)Exponential FunctionConstant FunctionDerivative FunctionLT of Common FunctionsHigher order Derivative Function3.43.163.9

Rectangular Pulse FunctionStep Function

Table 3.1 Laplace Transforms for Various Time-Domain Functionsa

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

Example 3.2Solve the ordinary differential equation With initial condition y(0)=y(0)=y(0)=0

SolutionTake LTs,term by term,using Table 3.1Rearranging and factoring out Y(s) ,we obtain

Partial Fraction Expansion (PFE)Basic idea: Expand a complex expression for Y(s) into simpler terms, each of which appears in the Laplace Transform table. Then you can take the L-1 of both sides of the equation to obtain y(t).Consider a general PFE (purposely for no complex and repeated factors appear)

Here D(s) is an n-th order polynomial with the roots all being real numbers which are distinct so there are no repeated roots.The PFE is:

Three methods to solve PFEMethod 2:Because the above eq must be valid for all value if s,we can specify two values of s and solve for the two constant Method 3:The fastest and most popular method is call HEAVISIDE EXPANSION.In this method multiply both side of the equation by one of the denominator term (s+bi) and then set s=-bi, which causes all terms except one to be multiplied by zero.

Method 2:Because the above eq must be valid for all value if s,we can specify two values of s and solve for the two constant

Method 3:The fastest and most popular method is call HEAVISIDE EXPANSION.In this method multiply both side of the equation by one of the denominator term (s+bi) and then set s=-bi, which causes all terms except one to be multiplied by zero.

Step 1 Take L.T. (note zero initial conditions)Example 3.3RearrangingStep 2a. Factor denominator of Y(s)

Step 2b. Use partial fraction decompositionMultiply by s, set s = 0For a2, multiply by (s+1), set s=-1 (same procedure for a3, a4)

Step 3. Take inverse of L.T.

Exercise Using partial fraction expansion where required, find x(t) for

Solutions

Repeated Factor

If a term s+b occurs r times in the denominators terms must be included in the expansion that incorporate the s+b factorExample for repeated factor problemEq A

Solution

A. Final value theoremExample: Time-shift theorem Important Properties of LT

C. Initial value theoremby initial value theoremby final value theoremExample:

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