EE222 Discussion 1/24
Tyler Westenbroek
January 24th 2020
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Problem 1
Consider the following second order model for a pendulum:
d
g
d2
θ(t) + β θ(t) + sine(θ(t)) = 0.
dt2
dt
l
(1)
Here, θ is the angle of the pendulum from the vertical, l > 0 is its length, g > 0
is the gravitational constant, and β ≥ 0 is a constant capturing the effects of
friction. Put the differential equation into state-space form, locate its equilibria
and linearize about these points. Next, classify each linearization for β > 0 and
β = 0. Do the linearizations agree with your intuition for how the nonlinear
system will behave?
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Problem 2
Consider the scalar system
ẋ = f (x)
(2)
where f : R → R is smooth. Suppose that we have Df (0) = −α for some α > 0.
Show that there exists δ > 0 such that for each initial condition x0 ∈ B(0, δ)
the solution for the system converges to the origin as t → ∞.
Now, suppose that we perturb the dynamics in the following way:
ẋ = f (x) + d(x).
(3)
Consider the following two conditions for the disturbance:
1. d(x) is such that kd(x)k ≤ c1 kxk2 for each x ∈ R
2. d(x) is such that kd(x)k ≤ c2 kxk for each x ∈ R
Characterize the behavior of the system near the origin in both cases for different
values of c1 , c2 > 0.
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Problem 3
Consider the system
ẋ = −y + x(1 − x2 − y 2 )
ẏ = x + y(1 − x2 − y 2 ).
Try sketching the phase portrait for the system. Show that the system has a
limit cycle. Can you prove which parts of the state-space have initial conditions
which converge to the limit cycle? Hint: Try working in polar coordinates!
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