Question

a) True or false

If a second order system has a settling time of 7 seconds, and a peak time of 3 seconds we may say that the system is stable.

b)

c)

Answer 1

solution: settling time Isa 7 see peak time, To: 3 see we know that, settling time, Isa gon & a damping natural frequency where ratio wn a 7 4 وخا qwn: Jlitt 4 Fly we know thalt! peak time Tha ہم لها where Wat ? damped natural frequency 7/ Wa = wn Sie A 32 ولا : و له kilm → an 1- 72 z D 국 Kit { put value of wn from 7% ega + % 2 7q 4

both sider: Squaring (7-2) ? Ily) 2 (4-223.3589² 1 = 4.4583? 3 ??? House 4.458 Ş: 0.47 A is tot than to Hence Puf the value of & in en we get l- 4 wna 2x0.474 Wn a 1.206 Hence stond and send onder system transfer, funtion Gofa wn? CIS) res سام و حماچ 2 مي characteris hy equation: S²12? wm se wn? 27 27 2x0.474 x 1.2065+(1.206) 20 => $24111435+ 1.454 - 0

By Routh hurmity criteria. column column 1 2 S² 143 s' 1.454 so 1.4554X143-140 0 Housy 5 1/43 Hence there 1. no is sign change in column the given system is stable. > true 162 TIS) z 10 (517) 15+10) ($130) 10/512) szos tros stemalard compare the above bransfer function se and order transfer function ShpqZq was op wat 20 C. 3 fwr14,14 2? Wn? 30 30 2x14:19 - Z-1.06 > 1 the damping sahio, & is greater than L. Hence the giver system overdampeel. is

le Percentage over sherrt. %OS: 10% 2 ܬܨ 0 Peak time passee we know that, (3* f-32) ·l eter /T-3")) Inoin - za SI-92 & squeaning both sides get 2 2 let 5.301 ??* 9.869 92 (1-22) 1-3 $1201 - 5.3013: 9,86992 > 1517 [?: 5.302 {?> S-304 15-12 lą 0.59 <1 L. Honce As the uphom damping rehio & is less than the given system is underdamped. Geo M میرا wniz? म I 0.807 wn w I-0.392 & Wn .78

Standard beend onder system transfer function.. ces res) wn? S21622, wys own Characteristics egna. s?+2ą, woms amo s², 2x0.59x0.785 p0.782 20 2) S2, 0.925 0.61 20 by routh hertuito criteria. $2. + L O. 92 S' fo.64 sº 0.67% 0.93180) +0.92 0.51 Hence the → There is given system sign change in Column a i's stable. the giren bystem is Hable and underdampeda