2. For a system with governing equation of + 0.5y = f(t), find and plot response to a) a unit step f(t)=U(t); b) f(t)=U(t)-U(t-2); c) impulse f(t)=8(t). (a) Given equation is + 0.5 y = f(t), from the above equation t = 2, (y) ss = 2, y(0) = 0 Thus, yu(t)=2 (1-e),t0 1.5 0.5F 2 4 8 10 time [s] (b) By using superposition to sketch the response of f(t) = U(t) - U(t-2) with y(0) = 0,we have y(t)=2(1-e). U(t)-2-(1-e-(-2)/2). U(t2) ,0t2 For 0t2, the response is same as in Part(a) Fort 2, y(t) = 2(le). e(-2)/2 = 1.264.e-(1-2)/2 1.4 1.2 - 0.8 0.6- 0.4 0.2 X:2 Y: 1.264 (c) h(t): = dy dt U(t),t>0=e1/2,t>0 0.8 0.6- 0.4- 0.2 2 time [s] 4 6 8 10 time [s]