practice-midterm2

practice-midterm2 - . (c) Compute a value for the initial...

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P RACTICE M ID - TERM E XAM L INEAR S YSTEMS Jo˜ao P. Hespanha Please explain all you answers. 1. Consider the nonlinear systems ¨ y + ˙ y + y = u 2 1 . (a) Compute a state-space representation for the system with input u and output y (b) Linearize the system around the solution y ( t ) = 0, u ( t ) = 1, t 0. 2. Consider the system ˙ x = ( A bk ) x + bu , y = cx , where A = b 0 1 0 0 B , b = b 0 1 B , k = ± k 1 k 2 ² , c = ± 0 1 ² where k 1 and k 2 are constant scalars. (a) Compute the system’s transfer function. (b) Determine values for k 1 and k 2 such that the transfer function is equal to s s 2 + s + 1 Hint: Your answer to (a) should appear as a function of the constants k 1 and k 2 . 3. Consider the matrix A = 2 1 0 0 2 0 0 0 3 (a) Compute the characteristic polynomial of A . Is A diagonalizable? (b) Compute e At
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Unformatted text preview: . (c) Compute a value for the initial state x ( ) : = x 1 ( ) x 2 ( ) x 3 ( ) such that the solution to x = Ax , y = 1 1 1 x is equal to y ( t ) = te 2 t , t . 1 Hint: To solve (c), start by writing y ( t ) as a function of x 1 ( ) , x 2 ( ) , x 3 ( ) and then determine values for these constants to get the desired output. 4. Compute a matrix A ( t ) such that ( t , t ) = 1 e t 2 t 2 2 1 e t 2 t 2 2 is the state transition matrix of the homogeneous ODE x = A ( t ) x 2...
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This note was uploaded on 12/29/2011 for the course ME 243a taught by Professor Abamieh during the Fall '09 term at UCSB.

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practice-midterm2 - . (c) Compute a value for the initial...

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