Question

1. Consider the usual unity-feedback closed-loop control system with a proportional-gain controller: 19 r - PGS-Try P(s) Draw (by hand) and fully label a Nyquist plot with K = 1 for each of the plants listed below. Show all your work. Use the Nyquist plot to determine all values of K for which the closed-loop system is stable. Check your answers using the Routh-Hurwitz Stability Test. [15 marks] (a) P(s) = (b) P(s) = s(s+13 (6+2) (©) P(s) = 32(6+1)

Answer 1

open - Loop tocensfer function GHIS) = K PIS) PIS) = 3 1905 1 pujøl= Pujol = 3 I 5-2 och jwrz. i pljanlendi 2 Peja= = f 180 - tema tang ) 44 LP(jw) = -x + fasse Proj) = 32-1 , PC@j) = 0[=ß7 P (R) - L135° Svakti provenientes 6 -ftoj_ ye -135 Nyquist Criferian JA=p.2] . R-H side bate of 오더 AP=1 IP: RH side pale of OLTf 2 Rift side pale of CLTf po noo of encircement about it of S point in Acw din pal 1-1-21 thus S/H with stable if olak > 1 TK7213

Routh Horwitz Stability test KP(S) = 3% CIF = 1+ GIS) HIS) =0 a lt k PCS) = 0 It K3 Too =0 0 3 -2 = S-373k so S L $ = P-3R for stable SIM 2-3k co l 2 < 3.K. le pale should be in Left hand side of s-plare @ Pis) - 4 5 (5+1)() Rpcja)= 4K ses julltjau) (2+ja/ 2 P(jw) = -go-font -fonty (Plja E 4K WW 1644 [Proj) = 0 [gc) Ploj)= o LP70

one pale at osizin anay) 1(K) = @ - ancirclement +1+0) 2k lesest ceso formule GHIS)- K 5(549) (546) wpc = ab cupc = √3x =R lent I POVRj) = 4K U VB/ 3 x - 4K = 4€ - 2K x3x6 .pl p=p-z P=0 Naaz p=0-2 => HERZ for Stablo SIM Z=0 A=0 I should not encicle point (179) 24 cl k < 36 RH table H KPIS) =a 1+ 4K* "'20 5054178+2) S($+1)(572) TURCO, (5+35+2)+4K 50

5P+ 38°+75 +48=0 for Stacks 4K70 - 53 2 Se 3 . 4k si 6.4k nas ja G-4R 70 4K fer stable sin =) 67 4k < K [K<34 7 Sk . :$?(5+1) ·S'S+) KP(jw) 15KM - Calleja o [Pejarl = Ski- - wajitoz to Lpliw) =-{-tante? I!" ZPLW) = termiczny Plojit - 00/180 Jo rti pioj) = 0% 90' Ch'Plaj) - od Pecil Sk rger a paler filet Polls ten at wrot Ploj)=-ve 2 cf-vene '= -180

بود و لا دے کے wrot دوم + هجرا عددا - ۴۰) arost اساسا سے۔ < < کے ہ> سے ہے۔ >> noch to pose at onzin Piso = 5 this co-circle realius =2 {A) R-fffable [ PP(۲) = SR =0 s7st/)+SR =0 s?sti) $:+s't[0]s+sk s+2ء sts = 0 و ا اک | s2 » s'o. so sk. e (sin will be unstable)

tom Nyquirt plot of ① P= Paz (P=0) H = p-z = -2=0-2 ! Z=?| hell two pole on R-H side of s-planeasa I n this sit it unstable (for all value of k/ elka in this kool Ayse 1 @ for K 712 : SIM will stable | 6 © for to k <312 SIM will stable SIM will be unstable for all value of k ]

wpc calculation PIS) = 4K S(5+1) (542) pljupc) = 1 17-180" / pejupc) = -1809 Lpija) = -go-tan'w - tony Epljupc) = -180 = - 90 - Terri'w - teen 'cy. 1 - 180-go = don'w + torta, a go = formules tani'f wote 2 I T-w.cey Y I ton" atton =tonlath I ** 3i fonts nations go' fear's l-ab al fango = 3 1-a2 a possible if 1 = w ²2 w=8 [u pc = R rood ( Reel
one thumb's up PLEASE.

Thank you.