MECH 6323 HW 3

MECH 6323 HW 3
yes
1. (a) For M ∈ C
n×m, show that for all x ∈ C
m,
kMxk2 ≤ kMk2→2 kxk2.
(b) Let {λi}
n
i=1 denote the eigenvalues of matrix A ∈ C
n×n
. Show that ρ(A) ≤ kAk2→2, where
ρ(A) is the spectral radius of matrix A, i.e., ρ(A) := max
i
|λi(A)|.
(c) Let A ∈ C
n×m and B ∈ C
m×k
. Prove the multiplicative property of the induced 2-norm:
kABk2→2 ≤ kAk2→2 kBk2→2.
(d) Let x ∈ C
m and y ∈ C
n
. Show that if kyk2 ≤ kxk2, then there exists a ∆ ∈ C
n×m such
that y = ∆x and ¯σ(∆) ≤ 1. The choice of ∆ should only be expressed in terms of x, y, and
their norms. Conversely, show that if kyk2 > kxk2, then there is no ∆ ∈ C
n×m such that
y = ∆x and ¯σ(∆) ≤ 1.
2. Consider the MIMO system P with state-space representation
x˙ =

−2 5
−5 −3

x +

−2 4
−2 −2

u
y =

1 2
−4 3 
x.
(a) Is P stable? Why?
(b) What is kPk∞? What is the frequency ωp that achieves the peak gain (¯σ(P(jωp)) = kPk∞)?
(c) What is the SVD of P(jωp)? Verify that the computed largest singular value satisfies
σ¯(P(jωp)) = kPk∞.
(d) Generate and submit the σ-plot of P. Verify that the peak of the singular value plot agrees
with the computed values for kPk∞ and ωp.
(e) Construct the vectors a, φ, b, and ψ ∈ R
2 with kak2 = 1 and kbk2 = kPk∞ such that the
input signal
u(t) =






a1 sin(ωpt + φ1)
a2 sin(ωpt + φ2)
.
.
.
am sin(ωpt + φm)






gives the steady-state output
y(t) =






b1 sin(ωpt + ψ1)
b2 sin(ωpt + ψ2)
.
.
.
bn sin(ωpt + ψn)






.
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Note that the amplitude vectors satisfy kbk2
kak2
= kPk∞, i.e., the peak gain is achieved by
this pair of (real) input/output signals.
(f) Simulate the linear system P with the input signal constructed in the previous part, e.g.,
using the lsim command in MATLAB. Plot the steady-state output signal predicted from
your construction in the previous part. Plot the simulated output on the same plot. Verify
that both results agree after the initial transient decays to zero.
3. Let S and T denote the sensitivity and complementary sensitivity closed-loop transfer functions.
Prove that
kSk∞ ≥ kTk∞ − 1.
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