Question

2. Given: [ 3x2(t) + 2x3t) x(t) - 4x1(t) + x2(t) ] Find, (a) use any method taught in class to determine the stability of the nonlinear system

Answer 1

- Solo x (t) = 3x3 (t) + 2zz(t) ] Lailt) – 4*,[+)+ x3 10 DALAM Determine the stability of non-linear system. - 312 + 2xă x2 = 2.-431 + x2 To find equilibrium point of the system, * die = 0 = 322e + 2a el Parts > Ke (2734 + 3) = 0 => Xze = 0 A-3 litro for * ine 0 = xie - 4x + Xge The 0. lie (xle - 4) = 0 => Xie = 0 of 4 it &e=-2 / xe - 4x16 - 3 these => 2 xic = + 164 +412X(3) 2x2 *** [i] the equilibrium states of the system.

TV CAR II To Lya punor's method. determine stability, we use . sufficient of for autonomous system, the condition stability is as follows: . Suppose that there exists a scalar function V(x) which, it for some real exo satisfies the following properties I for all x in this systent region llll«E. 11 Vex>0; x to Z G => vcx) is positive definite VC= 0 (ii) vex) has continuous partial denivatives wint all components of Xi L ovii să . Then the equilibrium point te=0 is asymptotically stable if y(x)20. Therefore, for Vie - [O], consider 1CX] > Piac? + P2xs? ; Pu>0,P270 which is opt o positive definite scalar quadratic function UCX) = 1 103 OS = 29,04 (32 + 2x2 + 2P2 % (at 4x + x)

= 69,12,22 + 4P, X1 222 + 29226² 22 - 8p226x2 + 202 203 22₂ 27 x₂ + 4 P 20, xz + (6p, -892) *, 2 + 21222 let, for simplicity, Pi=1, P2 =1, => ux) = 2xPaz + 4%, x2 - 2822 of 282 = 2x2 ( x + kamers 284 * 24a) . Since *(x) not always less than zero, system is aut asympto trcally stable writ Vie = [:].
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