ELEC 9732 Assignment 1 Instructions [see also course outline] 1 due in Moodle, Friday October 4, 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. 9 No use of Large Language Models Q1 (10) Piece-wise Linear System. Consider the piece-wise linear system x˙ = { −x if |x| ≤ 1 −2x if |x| > 1 with initial value x(0) = x0. 1a Solve the system to obtain x(t). 1b Hence or otherwise construct the phase plane plot. Mark time arrows on the plot. Q2 (20) Phase Plane Consider the following second order system with a nonlinear frictional component. x¨+ (1− |x|)x˙+ x = 0 2a (8) Construct a phase plane plot by the method of isoclines. 2b (4)♦ Construct a phase plane plot using matlab. Discuss the nature of the plot. 2c (8) Carry out a linearized stability analysis using the Lyapunov indirect method. Be sure to check validity of conditions. Q3 (20) Stability. Consider the following nonlinear system, x˙1 = x2 − x31 x˙2 = x1 − x32 3a (4) Establish existence by showing that a Lips- chitz condition holds in any bounded region, e.g. {(x1, x2) : |x1| ≤ R, |x2| ≤ R}. 3b (4) Find all equilibrium points. 3c (4) ♦Using matlab, construct a phase plane plot in the region {(x1, x2) : |x1| ≤ 4, |x2| ≤ 4} and discuss its features. 3d (8) Do a linearized stability analysis of one equi- librium point and compare the result with the phase plane plot. 1 51作业君版权所有