Question

Consider three (causal) LTI systems, corresponding to transfer functions described (except for gain K) by the following pole-zero plots. Im Im ++j10 X - -100 - 10 Re -10 : 5 * +-j10 Pole-Zero Plot for System A Pole-Zero Plot for System B Pole-Zero Plot for System C In each part below, determine which of the three systems can meet the given specifications, including any implications on what value(s) the transfer function's gain K must be. (a) (10 points) The zero-input response exhibits a time constant of decay of 1, while the steady-state response to an input 5u(t) is 3. System (circle one): A B C Value(s) of Gain K: (b) (10 points) The Bode magnitude plot has a straight-line asymptotic approximation sketched below. | HW dB 20+......... 1000 10 100 w (log scale) System (circle one): A B C Value(s) of Gain K:

Answer 1

→ consider system ® Transter function GCS) hos): (5+) (SHO) lime constant form of GCS) 10(1+5)(H ) -10 is an insignificant pole and it offers very less delay and it can be neglected and the equivalent system becomes first order 9 GCS) = k 10 (5+1) Standard first order system is k 1ST i to Time constant. Here GCS) 101 +500) Time constant = 1 → Given that input suct) should have steady state Value 3 input x(3) = ; C) = k 10(s+1) YCS) = X(5) G(5) 10 55+ LE SY09) sno Final value Lt 85k 1=3 540 10415H) 5K = 30 k=6] ay system

Consider system @ ks GCS): (5+10) (81100) -100 --to Ks 1000(1+3)(46) Poles giveas -20dB/dec slope Zero gives thodb/dec slope Bode plot a odbodec -zodid dec xer logue logue oas on 100 loool 2X+dades 1 > zero at origin = trodB dec slope -> pole at Toradisec ss -10dB/deck change at lorad/sec pole at 1008ad (sec >-20db/dec change at 100 rad/sec Given magnitude . at trad[sec is oda. ► 2ologik s 20logku = 0 dB Cosi ku. : о 61 KO) - 1 [K= 1 so ② + system ®