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.