ELEC 9732 Assignment 2
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ELEC 9732 Assignment 2
Q1 (16) Lyapunov Stability Consider the system
x 1 = -X2X3 + 1
X 2 = X1X3 — X2
x 3 = x3 (1 — X3 )
(i) Show that the system has a unique equilibrium point.
(ii) Using linearization, show that the equilibrium point is asymptotically stable.
(iii) Determine whether the equilibrium point is globally asymptotically stable.
Q2 (17) Input Output Stability .
Consider the system with input u, output y
X1 = X2
X 2 = Xi — ksat (2xi + X2)+ u
y = Xi where k is a constant and
[ y if |y|< 1
\ sign(y) if |y| > 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 reference signal and a forward loop consisting of a 'contact' nonlinearity (a = 1,k = 2) cascaded with a LTI system with transfer function,
(i) Calculate the describing function of the contact nonlinearity.
(ii) Hence use describing function stability analysis to see whether there is a limit cycle and if so compute its approximate amplitude and frequency.
Gain
2022-10-19