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Unformatted text preview: EE 221 : Linear Systems HW #5 Solutions Sam Burden Oct 15, 2010 1 Note these solutions are overly terse and leave out some details; in your solutions, you should strive for greater clarity and rigor. Exercise 1. Time invariance is straightforward to verify. Let T s be the time- s shift operator and L an input/output map with ( Lu )( t ) = y ( t ). Then: T s Lu ( t ) = Z t + s- e- ( t + s- ) u ( ) d = Z t- e- ( t- ) u ( + s ) d = LT s u ( t ) with = + s. The issue of whether/how y may be seen as the readout or state transition map of a dynamical system was discussed in class. One way to think about it: consider the linear system x =- x + u . We know this is a dynamical system, and it yields the state transition map s ( t,,x ,u ) = e- ( t- ) x + Z t e- ( t- ) u ( ) d. Using the readout map ( t,,x ,u ) = s ( t,,x ,u ), we see that y ( t ) = lim - ( t,,x ,u )....
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
- Fall '10