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practice-final3

# practice-final3 - f computed before is the closed-loop...

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P RACTICE F INAL E XAM L INEAR S YSTEMS Jo˜ao P. Hespanha Please explain all you answers. 1. For which values of a , b , and c does the transfer function ( s - 1 )( s - 2 ) s 3 + as 2 + bs + c has a two-dimensional realization that is both controllable and observable. Explain. 2. Consider the n -dimensional system ˙ x = Ax , y = Cx and suppose that A has an eigenvector v for which Cv = 0. (a) Compute O v , where O denotes the observability matrix of the system. (b) Is the system observable? Explain. 3. Find a state-space realization for the following transfer matrix: ˆ G ( s ) = s s + 1 1 s + 2 4. Consider the system ˙ x = Ax + bu with A : = b 0 1 - 1 - 2 B , b : = b 0 1 B . (a) Is this system controllable? Explain. (b) Compute a 1 × 2 matrix f so that the eigenvalues of A + b f are both at zero. (c) For the matrix

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Unformatted text preview: f computed before, is the closed-loop system ˙ x = ( A + b f ) x stable? 5. Consider the system ˙ x = -2-1 1 1 x + 1 u , y = ± 1 1 ² x . (1) 1 (a) Compute the system’s transfer matrix T ( s ) . (b) Is (1) a minimal realization of T ( s ) ? If not compute a minimal realization. 6. Give examples of a Jordan block J with the following properties. Choose a 2 × 2 block whenever possible. (a) The system ˙ x = Jx is asymptotically stable. (b) The system ˙ x = Jx is unstable. (c) The system ˙ x = Jx is stable but not asymptotically stable. 2...
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practice-final3 - f computed before is the closed-loop...

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