practice-final2

practice-final2 - P RACTICE F INAL E XAM L INEAR S YSTEMS...

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Unformatted text preview: P RACTICE F INAL E XAM L INEAR S YSTEMS Jo˜ o P. Hespanha a Please explain all you answers. 1. Consider the matrix A= 0 1 −2 . 3 Compute A100 and eAt . Hint: Diagonalize A. 2. (a) Show that if J is a stable Jordan block then for every t ≥ 0 all the eigenvalues of eJt have magnitude smaller or equal to 1. (b) Show that if A is a stable matrix then for every t ≥ 0 all the eigenvalues of eAt have magnitude smaller or equal to 1. 3. Prove that if the single-input/single-output system x = Ax, ˙ x ∈ Rn , y ∈ R y = cx, is observable then the null space of the matrix A−λI ∈ R( n + 1 ) × n c (1) only contains the zero vector, for every λ ∈ C. Hint: Prove the statement by contradiction assuming that the observability matrix is nonsingular and yet the null space of the matrix (1) contains a nonzero vector for some λ ∈ C. 4. Consider the system x= ˙ −1 10 1 x+ u, 0 1 0 y= 1 (a) Is this system controllable? observable? (b) Compute the system’s transfer matrix. (c) Is this system BIBO stable? (d) Is this system stable in the sense of Lyapunov? 5. Find a minimal state-space realization for the following transfer matrix: ˆ G(s) = 1 2 s+ 1 s+ 1 s+ 2 s+ 1 1 x. 6. Suppose we want ﬁnd the control input u to the system x ∈ Rn , u ∈ Rm x = Ax + Bu, ˙ that minimizes the quadratic cost ∞ J= x(t ) 2 + u(t ) 2dt . 0 As you know, the optimal control is of the form u = −B′ Px where P is a positive deﬁnite solution to the equation PA + A′P + I − PBB′ P = 0. Show that the resulting closed-loop system is asymptotically stable. Hint: Try to ﬁnd in (2) a Lyapunov equation for the closed-loop system. 2 (2) ...
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practice-final2 - P RACTICE F INAL E XAM L INEAR S YSTEMS...

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