ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

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Presentation transcript:

ISAT 412 -Dynamic Control of Energy Systems (Fall 2005) The Laplace Transform ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

Second Order O.D.E. Solution General solution to the second order, linear, constant coefficient, homogeneous O.D.E. Assume

Characteristic Equation Assuming this form gives us the characteristic equation for the 2nd order equation Solve for the roots of the characteristic equation (factor or quadratic equation)

General 2nd Order, Homogeneous Solution The general homogeneous solution to the second order equation is given by We can express this result in three different ways depending upon the roots of the characteristic equation

Three cases Distinct real roots Repeated real roots Complex conjugate roots

Particular Solution The non-homogeneous solution is found in the same manner as for the 1st order equation Assume that the particular solution takes the form of the non-homogeneous term (forcing function)

Laplace Transform The Laplace transform solution method provides a systematic and general method for solving linear O.D.E.s

Laplace Transform The Laplace transform takes the function x from the time domain into the Laplace (or “s”) domain It transforms an O.D.E. in the time domain into an algebraic equation in the Laplace domain

Laplace Transform Notation Denote by using the capitalized letter corresponding to the lower-case symbol of the transformed function

Solving an O.D.E. using the Laplace Transform Transform the O.D.E. into the Laplace domain Simplify the resulting algebraic equation and solve for the transformed variable of interest Use the inverse Laplace transform to convert back to the time domain

Laplace Transform Pairs Relate common function in the time domain to their Laplace transform in the “s” domain Page 115 (and front cover) of your textbook

Properties of the Laplace Transform Linearity Note – this DOES NOT apply to products of functions!

Properties of the Laplace Transform Consult page 117 of the textbook for additional properties of the Laplace transform In particular, note the Laplace transform of first and second derivatives in the time domain Properties 2, 3, and 4