ELEC 9732 Assignment 2
Instructions [see also course outline]
1 due in Moodle, Thursday October 7, 4pm
2 Signed School Cover Sheet attached
3 PDF only: typed - not handwritten.
4 Follow the Homework Rules.
5 Computeroutput : no discussion ⇒ no marks.
6 Analyticalresults : no working ⇒ no marks.
7 ♦ means you can use Matlab; else not.
8 No Copyingexcept from lectures ; No Discussion.
Q1 (16) Lyapunov Stability .
Consider the system
x˙1 = x2
x˙2 = −h(x1)− x2 − g(x3)
x˙3 = x2 − x3
where g(y), h(y) obey Lipschitz conditions and sat-
isfy,
g(0) = 0 = h(0), yh(y) > 0, yg(y) > 0, y 6= 0
Using the candidate Lyapunov function
V (x) =
∫ x1
0
h(y)dy +
∫ x3
0
g(y)dy +
1
2
x22
(i) Show that the system has a unique equilibrium
point.
(ii) Show V (x) is positive definite.
(iii) Show the equilibrium point. is asymptotically
stable.
(iv) Find conditions on h(y), g(y) that ensure the
equilibrium point is globally asymptotically sta-
ble.
Q2 (17) Input Output Stability .
Consider the system with input u, output y,
x˙1 = −x2
x˙2 = x1 − x2sat(x22 − x23) + x2u
x˙3 = x3sat(x
2
2 − x23)− x3u
y = x22 − x23
where the saturation function is,
sat(w) = w if |w| ≤ 1; = sign(w) if |w| > 1.
Find conditions under which this system is energy
input energy output stable. Also find an upper
bound on the system gain.
Q3 (17) Describing Functions .
Consider a unity feedback system with null refer-
ence signal and a forward loop consisting of a dead-
zone+saturation nonlinearity cascaded with a LTI
system with transfer function,
G(s) =
1− s
s(s + 1)
(i) Calculate the describing function of the dead-
zone+saturation nonlinearity.
(ii) With (a = 1, b = 32 , k = 1) use describing func-
tion stability analysis to see whether there is a
limit cycle and if so compute its approximate
amplitude and frequency.
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