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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 = bracketleftbigg 0 1 0 0 bracketrightbigg , b = bracketleftbigg 0 1 bracketrightbigg , k = bracketleftbig k 1 k 2 bracketrightbig , c = bracketleftbig 0 1 bracketrightbig 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
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Unformatted text preview: . (c) Compute a value for the initial state x ( ) : = ± x 1 ( ) x 2 ( ) x 3 ( ) ² ′ such that the solution to ˙ x = Ax , y = ± 1 1 1 ² 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 Φ ( 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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