## Transcribed Text

Problem 1
For the feedback system below, determine the largest gain K which preserves the stability
of the system. If K = 2K , compute the amplitude and frequency of the limit cycle
max
oscillations.
r
20
500K
S (s + 10) (s + 50)
y
20
Problem 2
In the feedback system below, the actuator is limited by the saturation, whose slope in the
linear regime is 1.
e
u
M
y
5
K
(10s+1)
3
-M
1/(sTi)
a) Adjust the parameters of the PI controller according to Ziegler-Nichols rules.
b) Determine the effect of the saturation limit M on the existence of self-oscillations
(limit cycles) in the closed loop system.
Problem 3
For the functions given below, determine whether they are continuous, continuously
differentiable, locally Lipschitz, globally Lipschitz. Justify your answer.
i(x) = x +/x f2(x) =sin(x)sgn(x)Problem 4
Let f : R²
R² be given by
f(x) = x 2
Is f(x) Lipschitz continuous? If yes, in what domain Dg R²: ? If Lipschitz continuous,
compute L>0 such that x,yeD.
Problem 5
x1=x2
Show that the origin of the system
is globally asymptotically stable. Hint: use a
=
quadratic Lyapunov function candidate.

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