Using Partial Fraction Expansion Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Using Partial Fraction Expansion

Outline of Today’s Lecture Review Laplace Transform Inverse Laplace Transform Properties of the Laplace Transform Final Value Theorem

Laplace Transform Traditionally, Feedback Control Theory was initiated by using the Laplace Transform of the differential equations to develop the Transfer Function The was one caveat: the initial conditions were assumed to be zero. For most systems a simple coordinate change could effect this If not, then a more complicated form using the derivative property of Laplace transforms had to be used which could lead to intractable forms While we derived the transfer function, G(s), using the convolution equation and the state space relationships, the transfer function so derived is a Laplace Transform under zero initial conditions

Laplace Transform CAUTION: Some Mathematics is necessary! The Laplace transform is defined as Fortunately, we rarely have to use these integrals as there are other methods

Some Common Laplace Transforms The Laplace Transform of the Impulse Function The Laplace Transform of the Step Function The Laplace Transform of a Unit Ramp: The Laplace Transform of the 2nd power of t: The Place Transform of the nth power of t:

Some Common Laplace Transforms Laplace Trans Form of the exponentials: Laplace Transforms of trigonometric functions:

Important Inverse Transforms

Properties of the Laplace Transform Laplace Transforms have several very import properties which are useful in Controls Now, you should see the advantage of having zero initial conditions

Final Value Theorem If f(t) and its derivative satisfy the conditions for Laplace Transforms, then This theorem is very useful in determining the steady state gain of a stable system transfer function Do not apply this to an unstable system as the wrong conclusions will be reached!

The Transfer Function

Partial Fraction Expansion When using Partial Fraction Expansion, our objective is to turn the Transfer Function into a sum of fractions where the denominators are the factors of the denominator of the Transfer Function: Then we use the linear property of Laplace Transforms and the relatively easy form to make the Inverse Transform.

Partial Fraction Expansion There are three cases that we need to consider in expanding the transfer function: Case 1: All of the roots are real and distinct Case 2: Complex Conjugate Roots Case 3: Repeated roots

Case 1: Real and Distinct Roots

Case 1: Real and Distinct Roots Example

Case 1: Real and Distinct Roots An Alternative Method (usually difficult)

Case 1: Real and Distinct Roots Example

Case 2: Complex Conjugate Roots

Case 2: Complex Conjugate Roots Example

Case 3: Repeated Roots

Case 3: Repeated Roots Example

Heaviside Expansion

State Space Formulation Example: The Nose Wheel

Example Now, we have a good view of the system structure so that we can choose and adjust values, if needed

Example

Summary Next Class: Nyquist Stability Criteria