Question

Q.10- For the system shown in Figure 5 with K (s + 3)(s +5) Gs)s-2)s-4) Find the range of gain, K, which will cause the system to be stable. Cs) Q.11. Draw the Root Locus of the following systems. Find the points of intersection with the real and imaginary axis. 6(s)H(s)- s(s +2) K(s+5) of- Draw the Bode diagram of the following tmamsfer finction. His)- -100 s +12s +21s +10 213- Obtain the phase and gain margins of the system shown in following figure for the two cases where K-10 and K-100. C(s) Ris) sst 5)

Answer 1

813 RS) cls) G(s) S/ s8 s

K 7 K 7 Lt l S

2 0 k'

Command Window a-1,11 b-[1,2,01 >>z-tf (a,b) s + 1 s*2+2 Continuous-time transfer function >> rlocus (z) 2

Root Locus 0.4 0.986 0.965 0.998 0.996 0.993 :0.86 0.3 0.999 0.2 System z Gain: 0.019 Pole: -201 System z Gain: 0 Pole: 0 Tn 0.1 *Damping:-1 Damping t Overshoot(%): O Frequency (rad/s): 0 : Overshoot (%): 0 Frequency (rads) 201 System: z Gain: 131 Pole: -0.992 Damping: 1 Overshoot :0 Frequency (rad/s): 0.992 0.1 0.2 0.999 0.86 0.998- 0.993 0.986 0.996- 0.965 4 Real Axis (seconds

2 S-3 S 2S-3

Command Window > a 3] b-11,-2,-3 1 -3 >>z-cf (a,b) *2-25-3 Continuous-time transfer function >>rlocus (z)

Root Locus 2.5 0.5 1.5 System z Gain: 0 Pole: 3 Damping:-1 Overshoot (96): O Frequency (rad/s) 3 0.59.94.. 0985System: z Gain: 0.00573 Pole:-0.996 Damping: 1 Overshoot (%): 0 Frequency (rad/s): 0.996 0.50-94 0.5 , - 1.5 1.5 064 0.16 0.5 2.5 3.5 Real Axis (seconds)

2 S'İ3 K1o --lb

>a--10] -10 >b-1, 3,10] 3 0 >2-tf(a,b) -10 2+3 10 Continuous-time transfer function > rlocus (a,b) >>rlocus (a,b)

Root Locus 36024 .62 0.48 System: sys Gain: 0 Pole: -1.5 2.78 Damping: 0.474 Overshoot (%): 18.4 0.88 , Frequency (rad s) 316、… System: sys 0.88 Gain: 0.0204 Pole:-1.5-2.75i Damping: 0.479 Overshoot (%): 18 Frequency (rad/s): 3.13 0.62 0.12 0.48- 0.5 2.5 0.5 Real Axis (seconds 1)

SK-E2 s k-2 SK-3 25 /6 25+ /6 G@i H(S)-. S 25-3

Command Window > a [2,101 10 12-3 2-tf(a,b) 210 s 2-2 -3 Continuous-time transfer function

Root Locus 0974 0'4 0.9 0.82 0.945 System: Z Gain: 1.01 Pole:-0.00872+2.66i Damping: 0.00328 Overshoot (%): 99 Frequency (radis) 2.66 0-99 2ト.9.997… System: z Systemt z Gain: 0.00548 Pole: 2.98 Damping -1 Overshoot (%): 0 Frequency (rad/s): 2.98 Gain:0 Pole:-1 . … …-; . . . Damping: 1 . .,-s… , overshoot (%): 0 Frequency (rad/s):1 15 10 System z Pole: -5 Damping: Overshoot (%): 0 20-997..Gain: hf Frequency (rad/s): 5 System: z Gain: 1.02 Pole: 0.0176-2.68 Damping: 0.00657 Overshoot (%): 98 099...P Frequency (radis): 2.68 0.945 o974 0.82 15 10 Real Axis (seconds

s)冫 2 loo +125+215 +10

100o >>b-i,12,21,10] 1 12 21 10 -100 s*3 +123^2 + 21 , + 10 Continuous-time transfer function >>bode(z)

: Gain Margin (dB) -13.9 At frequency (rad/s):0.913 Bode Diagram System z Peak ain (d), 13.9 At frequency (rady's): 0.983 328 :Frequency (rad/s): 0.0981 Magnitude (dB); -0.251 20 Phase Margin (deg): 77.9 Delay Margin (sec): 13.5 Atfrequency (radis): 0.101 180 Closed loop stable? No System: z Frequency (rad/s): 0.906 Phase (deg) 180 Phase Margin (deg):-113 Delay Margin (sec); 0.555 At frequency (rad/s): 7.77 4- .----:-1: Closed loop stable? No : 102 Frequency (rad/s)

CF S)つ l o 2

10 >b-,6,5,0 >>z-tf (a,b) 10 Continuous-time transfer function >>bode(z)

Bode Diagram 60 40 20H At frequency (rad/s): 1e-20 Systemt z Peak gain (dB): 406 Gain Margin (dB): 9.54 At frequency (rad/s): 2.24 9 -20 -60 -80 100 90 135 System z Delay Margin (sec): 0.361 At frequency (rad/s) 1.23 Closed loop stabla? Yes -270 .2 10 102 10 10 Frequency (rad/s) 10

12-b' R (s) e S)

Command Window a 1001 100 b [1, 6,5,0] > z-tt(a, b) 100 93+6 s 2+5 Continuous-time transfer function bode (2)

Bode Diagram 80 System z Peak gain (dB) 426 System z 40F At frequency (radis): 1e-20 At frequency (rad/s):2.24 Closed loop stable? No 20- -80 -90 135 System z Phase Margin (deg):-23.7 Delay Margin (sec): 1.5 Closed loop stable? No 225 270 10 10 10 10 10 Frequency (rad/s)