Question

The open loop transfer function of a discrete-time system is given by k (z+0.9) G (2) = (z-1)(z-0.7) i) Draw the root locus for the system for variations in the value of K ii) Determine the marginal value of K for stability.

Answer 1

The open loop transfer function of the systeen given G (Z) 2 k(t+0.4) (2-1)(2+-0.7) Į Zen. So There are poles are I poles & Pial, P2 = 0.7 2,5 -0.9, Zero, on Z locus. domain, we So, By putting the zero & poles can get the moot for Root locus, There are two breaking points the break away & & break in ppoint. By from the transfer function 14 6(2) 20: KE_(2-1(2 -0.72 CS (2+54 ) Scanned with CamScanner

sor dk dz Z (27019) { •(2-1) + 83–(2+)(78 0-2)» 1 17+0.99 (2-1) (2 7019) + (709) (7-0-7)- (2-1). (7 +07 (7-8.7) Now by doing dk 22 28, we can get some values 1 of z. (2-1)(2+04)+(2+07)(2-10-4) 7-017) 0-(7-1( 7-6-7) zd Naz 41.82-2-23 20. -1.8 mg + 4x2.23 Ni 2 018435, -2.6435 - so there are two pound. b=2 to 0.8435 Break away point (As it is between poles E-216435. → Break in point CS C As it is rear zero). utwo & bez Scanned with CamScanner

no As, all the zero and poles are real, there are question of angle of departure and arrival. So the root locus is, Imaginary (2.8435 Koos k ka 0 Real. b2 ZiGorg) (-216495) Po Cort P, (1) k=0 Scanned with Root locus of given exuestion. Cams manner
The stability part can calculate from jury stability criterion.

The Characteristics equation of the systan is coi 20 (27)(z-0. Z can app It G (Z) 20 It k(z 40 19) 12-17 (7-0.0 k (Z to. 0.9) + (2 -7) { "+ (k-117) 7 +67+0.94) -0. for stability calculation, we apply Jury's stability criterion, ned to check for Fury stabilib from Plz) ů the P(2) 30 fw Z=1. Characteristics equation llila Herlin +0.4 +0.975) 1.9k. then the system in stable, we CS Janned With CamScanner

ir Again for Zaal, the equation should follow, (-1JN P(-1) ? (N=2) PE-1) yo to C---(k-1.7.) »f +601770196) 0 ar 3.4 -Dilk yo K< 34 Shaw So, for k <34 Again for absolute vaker of k, N. stability CS CamScanner

be for stability, another relationship should satiesfied, laol < and 10.7 +0.98 / 1 er 0:4k 50:3 k < 0.33 For stability of the system, it should satisfy. ork. 20133. so, 0 CS Scanned with CamScanner