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. 1
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