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Unformatted text preview: P RACTICE F INAL E XAM
L INEAR S YSTEMS
Jo˜ o P. Hespanha
Please explain all you answers.
1. Consider the matrix
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
∈ 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
˙ −1 10
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:
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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- Fall '09