MEM255Su06.07_HW5.soln_studs

# MEM255Su06.07_HW5.soln_studs - MEM 255 Introduction to...

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MEM 255: Introduction to Controls Summer 06-07 Home work # 5 Due: 08/02/07 1. Consider a system with u as its input and y as its output, described by the following differential equation. (Items in ‘blue’ are matlab commands) 2 12 22 12 2 18 40 y y y y u u u + + + = + + a. Determine the transfer function Y(s)/U(s) . b. Determine the partial fraction expansion of Y(s)/U(s) . c. Obtain the state space representation (the one from class notes) of the system, and determine the {A,B,C,D} matrices. Let x denote the states. Verify that 1 ( ) / ( ) ( ) . Y s U s C sI A B D - = - + d. Generate a random non-singular (3 by 3) matrix T r [Tr = rand(3,3)] . Compute 1 1 { ; ; ; } r r r r r r r r A T AT B T B C CT D D - - = = = = . This amounts to rewriting the state space model in (c) above using the states 1 . r r x T x - = ; known as a similarity transformation .” Verify that 1 ( ) / ( ) ( ) . r r r r Y s U s C sI A B D - = - + Hence observe that transfer function is invariant under similarity transformations . e. Construct the matrix 2 r r r r r T B A B A B = , and determine 1 1 2 2 2 2 { ; ; ; } r r r r A T A T B T B C C T D D - - = = = = . This is another transformation x r to x 2 . T is called the controllability matrix . Verify that the transfer function remains the same. Can you observe any relationship between the states x 2 and x of (c)? f.

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MEM255Su06.07_HW5.soln_studs - MEM 255 Introduction to...

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