Question

Step Response Step Response 18 2 16 System 1 Peak amplitude: 2.19 Overshoot (9.00 Al time seconds):0.391 System: G2 Time seconds): 0.494 Amplitude 16 System: G2 Time seconds 31 Amplitude: 1.04 1.5 1.2 Amplitude Amplitude 1 08 06 0.5 0.2 O 0.1 0.2 0.6 0.3 04 0.5 Time (seconds) 07 5 Time (seconds) 1) For the step responses, obtained from some unknown systems, shown above, find dynamic system models using only the data points shown in, assume that all points are at the zero gradients (i.e., at that time instants t*, cos(wt - 6) = 1) a) Transfer Function (TF) b) ODE forms

Answer 1

First System (G.): a) Transfer function: Input step Signal = Qult) % overshoot 9.484 Peel time (Ap) c (fp) - (66) со &T 2. 19-2 » 0.095= erle ering こ 0.391 see % Mp X 600 4 Singer xo e c 2 In 0.096) >> ((0.085) * Pa* 2 مي c In 10.095) 0.5996 - 10.6 ** H(km 0.0963 Peal time. (p) = 1 = 0.391 see I Wa Wella 3 ñ WnF 5 0.391 0.397 Wha 6.39) -0.019 = 10.043 - o raslace on Transfer function, Tf a Windo 100 H $). sit a&Wastane s²+2 (0.68(6)st y (3) XS) gat last wo DODE form: y (5) (89+1287100) = loo x (S) whore, 46) + (a (6 + ) = 0 x loo woo X(t)= aut

En = 54 31.8 ralls Second System (Ca) Bord overshoot = 1.6 @ t= 0.494 see for 3rd overshoot, Wa 0.494 Waa 55 0.194 vadlo € 31.8 sample points are peal values only, We can use envelope, the Step response gad order system is, Wazon dire =&Wnt Sin (Watt & d = with Wn Tre Since the given of a standard C(t)= 1 - e 1-६ stanit - CE-13 е -&. Wat Sin (Watt 0) Sime points are given at reuo gradient. Bence envelop is -& wat y cha V- & @ Et Maga दर e & Walt, fa Wn) to е. - 0.6 @ t = F2 3 0.04

4 g (6) flon (ti-ko) 0.046 from een o= font & from @ e 0.6 15 0.0F 0.6 £Wni Co. hah - Bil] = ln (15) si & Wh ln (15) = 1.039 -11.04 3.1-0.LSL di = 0.494 da=3.) t (see) 0 +0 182 37,8 & 7.02 Tam o = 31.8 31.8 loh. LE W = 0.02966 0.03268 Wn 31.82 Transfer function Tag GB) 5% 28, Wn Stun? Gal) 8+21.02)5+ 31.8.239 1812.5124 G2(6) = * git + 2.085 st 1012.5724 BODE formas, (SP+ 2.085s + 1012.162) YB2 W19.512 X() ý (b) + 2.08 ġ(+612.5124 947 60/2,5124 *(t) 4 houboutin Fortunatella Tith 88,1268 Wn y 389