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practice-final1

# practice-final1 - × 2 matrix f = b f 1 f 2 B compute the...

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P RACTICE F INAL E XAM L INEAR S YSTEMS Jo˜ao P. Hespanha Please explain all you answers. 1. Consider the following nonlinear system ˙ x 1 = - x 1 + u 1 ˙ x 2 = - x 2 + u 2 ˙ x 3 = x 2 u 1 - x 1 u 2 y = x 2 1 + x 2 2 + x 2 3 (a) Linearize the system around the equilibrium point x 1 = x 2 = x 3 = 0. Is the linearized system controllable? (b) Linearize the system around the equilibrium point x 1 = x 2 = x 3 = 1. Is the linearized system controllable? 2. Consider the following system ˙ x = Ax + bu , y = cx + u , where c : = bracketleftbig 1 1 0 bracketrightbig , A : = - 2 0 0 0 1 0 0 0 - 1 , b : = 1 0 - 1 . (a) Compute the system’s transfer function. (b) Is the matrix A asymptotically stable, marginally stable, or unstable? (c) Is this system BIBO stable? (d) Is the system controllable and/or observable? 3. Find a realization for ˆ G ( s ) = bracketleftbigg - 6 s - 60 3 s + 33 s + 20 3 s + 33 bracketrightbigg . Make sure that your realization is both controllable and observable. 4. Consider the system ˙ x = Ax + bu with A : = bracketleftbigg 0 1 - 1 - 2 bracketrightbigg ,

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Unformatted text preview: × 2 matrix f : = b f 1 f 2 B compute the characteristic polynomial of A + b f . (b) Select f 1 and f 2 so that the eigenvalues of A + b f are both at zero. (c) For the matrix f computed above, is the closed-loop system ˙ x = ( A + b f ) x stable? 1 5. Consider the following LTI system ˙ x = Ax + Bu , y = Cx + Du , x ∈ R n , u ∈ R k , y ∈ R m (CLTI) and a state-feedback control u =-Kx + v , where v ∈ R k denotes a new input. (a) Compute the state-space model of the closed-loop and its transfer function from v to y . (b) Show that if the original system (CLTI) is controllable then the closed-loop system is also controllable. Hint: Use the eigenvector test. 2...
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practice-final1 - × 2 matrix f = b f 1 f 2 B compute the...

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