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problemset7 - The University of Texas at Austin Department...

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The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Problem Set Seven C. Caramanis Due: Wednesday, November 4, 2009. This problem set is intended to get us started thinking about differential equations and their solution, as well as properties of the solution. 1. Computing Laplace Transforms: Compute the Laplace transform of each of the following: f ( t ) = t 0 cos( t - τ ) sin( τ ) . For f ( t ) as given in the first part of this problem, define the function h ( t ) = t 0 t 1 0 t 2 0 f ( τ ) dτ dt 1 dt 2 . Find L ( h ). f ( t ) = (sin t ) /t . 2. Inverse Laplace transform, and partial fraction expansions: If you have not seen these tech- niques in previous classes, you may want to look here: h ttp://en.wikipedia.org/wiki/Partial fraction. The relevant section of this article are: (1) the introduction, (2) Distinct first-degree factors in the denominator, (3) A repeated first-degree factor in the denominator, (4) Repeated fac- tors in the denominator generally. The remaining sections are interesting as well, but not necessary for solving the homework problems. H ( s ) = 2( s +2) ( s +1)( s 2 +4) ; H ( s ) = s - 2 ( s +3) 2 ( s +2) 2 ; (Note the repeated roots here). 3. All the transfer functions we will see in this class are rational (ratios of polynomials) where the degree of the numerator is at most the degree of the denominator. As we discussed in class, the stability is determined by the poles of the transfer function, that is, the zeros of the denominator. There is a classical technique known as the Routh-Hurwitz stability criterion, which allows us to determine whether the roots of a polynomial lie in the left half of the complex plane without explicitly computing the roots. 1 This is a simple test, and it goes as follows: Consider a polynomial p ( s ) = s n + a 1 s n - 1 + a 2 s n - 2 + · · · + a n - 1 s + a n . 1 For degree two equations, there is the well-known quadratic equation for solving for the roots. Similar expressions exist for third and fourth order polynomials. It was a long-standing open problem to find a similar formula for fifth
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