MECH 6323 HW 4

MECH 6323 HW 4 
yes
Due Sunday 03/13/20 (10:00pm); Beware of daylight saving adjustments
1. Determine the H∞ and H2 norms of the following systems:
(a) H(s) = 1
s + a
, with a > 0. How do these norms compare to each other for different values
of a? What happens for a = 0?
(b)
x˙ 1 = x2
x˙ 2 = −x1 + u
y = x1.
2. Consider the system parameterized by k and R

x˙ 1
x˙ 2

=

−(1 + k
2
)/R 0
k −(2 + k
2
)/R   x1
x2

+

1
0

u
y = x2.
(a) For what values of k and R is this system stable?
(b) Derive the formula for the H2 norm of this system as a function of k and R. Using this
formula, plot the H2 norm as a function of k for R = 1 and R = 1000, and as a function of
R for k = 2.
(c) Find the solution of the unforced system (i.e., determine operator G(t) that maps the initial
conditions to the output y(t), y(t) = G(t)x0).
(d) Plot the maximal singular value of G(t) as a function of time (on the time interval t ∈
(0, 1000)) for two different cases: (i) R = 1000, k = 0; (ii) R = 1000, k = 2. How do these
two cases compare to each other? Explain the obtained results.
3. (a) Prove that σ = min
x6=0
kAxk2
kxk2
.
(b) Prove that ¯σ(A−1
) = 1
σ(A)
.
(c) Give and example of a 2 × 2 matrix A() that has stable eigenvalues that are constant and
independent of , but ¯σ(A()) → ∞ with  → ∞.
(d) Construct matrices A(), B, C, D (note that B, C, and D are constant matrices) such
that if G(s) = 
A() B
C D 
then kG(s)kH∞ → ∞ with  → ∞, but the poles of G are
independent of . Interpret this result.
4. Prove that if G1 and G2 have state-space realizations 
A1 B1
C1 D1

and 
A2 B2
C2 D2

, respectively,
then their serial and parallel interconnection yield
G1 G2 =


A1 B1C2 B1D2
0 A2 B2
C1 D1C2 D1D2

 =


A2 0 B2
B1C2 A1 B1D2
D1C2 C1 D1D2


1
and
G1 + G2 =


A1 0 B1
0 A2 B2
C1 C2 D1 + D2


respectively. Suppose G(s) = 
A B
C D 
is square and D is invertible then
G
−1 =

A − BD−1C BD−1
−D−1C D−1

.
5. Write T = GK(I + GK)
−1 as an LFT of K, i.e., find P such that T = F`(P, K).
6. Write K as an LFT of T = GK(I + GK)
−1
, i.e., find J such that K = F`(J, T).
7. For the state-space description represented by (A, B, C, D), find H such that
F`(H, 1/s) = C (sI − A)
−1 B + D.
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