Question

Consider the following closed-loop system, in which the plant model is P(s) = elave R()2-CO POTY() a) Assume C(s) = K. Determine the range of K for which the closed-loop system is stable via: (1.) the routh-hurwitz stability criteria, (ii.) the margin() command in Matlab, and (lii.) the rlocus command in Matlab. b) Assume a proportional controller of C(s) = K = 40, and a time delay T, located between the controller and plant. Determine the maximum T, value that can be tolerated before the closed-loop system becomes unstable. Use MATLAB to generate two plots, closed- loop unit step responses when 7, is at 95% and 105% of its maximum allowable val (1.e., T, = 0.95 Tmax and T, = 1.05 Tmax). Provide your code with your answer. (Hint: you can use MATLAB margin() function to determine the relevant margin and crossover frequency. Also, you only need to show the first 40 seconds of the unit step response.) c) Assume the plant has a time delay of 1 second. Determine the maximum gain K of the proportional controller C(S) = K for which the closed-loop system will remain stable Use MATLAB to plot the closed-loop unit step response when K is at 95% and 105% of its maximum allowable value (i.e., K = 0.95 Kmax and K = 1.05 Kmax). Provide your code with your answer. (Hint: You can again use MATLAB margin() function to determine the relevant margin and crossover frequency. Also, it is enough to show only the first 40 seconds of the unit step response.) Note that the Routh-Hurwitz and root locus techniques are not suited to analysis of systems with time delay, while bode plots are.

Answer 1

| Sali Giss = ((s) x pls) = 8k s(5+852 The closed loop Chalectastic squetion is It Go so i 17 20; 8K__ s(5 +64 +165) st 165 + 645+k820 Routh my da $ 116 4 8K sl 16x64-8k 16 tu stablilit S 16x64-8k zoi KE 128 b) for from - mat lab Gis) = 4088 the in plant phare magin is S(57812 tem est Contabilion 36.9 at way. The delay iť phare is -WT i -wr— qo sfon w = -180° to w=4 T= o. 1608 manc inum oo The allouable delay is 0.1008 seconds.

a TS fran matles with plant transfer function. Gs) a ēls x 8 have gain mag in s(s+8) og 20.3dB. The maximum allsueble gain is- - 20 los kz 2013 dB (203/20) ka 10 h 10-3514 o 10

matlab part: a

clc;

clear all;

close all;

s=tf('s');

g=8/(s*(s+8)^2);% plant

margin(g);grid

legend('bode of plant')

figure

rlocus(g);

legend('root locus of plant')

File Edit View Insert Tools Desktop Window Help 08H3 hayo. I E DO Bode Diagram Gm = 42.1 dB (at 8 rad's), Pm = 88.2 deg (at 0.125 rad's) Magnitude (dB) -100 -1504 -90 E bode of plant -135 Phase (deg) -180 -225 -270 10" 100 102 103 Frequency (rad/s)

File Edit View Insert Tools Desktop Window Help 08H3 hayo. I E DO Root Locus root locus of plant System: root locus of plant Gain: 125 Pole: -0.0422 + 7.921 Damping: 0.00533 Overshoot (%): 98.3 Frequency (rad/s): 7.92 Imaginary Axis (seconds') -15 -10 Real Axis (seconds)

matlab part b)

clc;

clear all;

close all;

s=tf('s');

k=40;

t95=0.1528;% 95% of delay time

t105=0.1688;% 105% of delaytime

gc1=exp(-t95*s);

gc2=exp(-t105*s);

g=8/(s*(s+8)^2);% plant

step(feedback(k*g*gc1,1));grid

legend('step response with 95% td')

figure

step(feedback(k*g*gc2,1));grid

legend('step response with 105% td')

File Edit View Insert Tools Desktop Window Help 08H3 hayo. I E DO Step Response Step response with 95% td 1.6 HD 12 HHH 1.1 Amplitude .1 MAAAAAAAMAAAAAAAAAAA mumenti HHHH 0.6 HHHHHH 0.4 IN HILL 80 120 Time (seconds)

File Edit View Insert Tools Desktop Window Help 08H3 Way. I E = 0 Step Response step response with 105% td Amplitude wwwwww 150 200 250 Time (seconds)
matlab part c)

clc;

clear all;

close all;

s=tf('s');

k95=9.8338;

k105=10.869;

gc=exp(-s);

g=8/(s*(s+8)^2);% plant

step(feedback(k95*g*gc,1));grid

legend('step response with 95% k')

figure

step(feedback(k105*g*gc,1));grid

legend('step response with 105% k')

File Edit View Insert Tools Desktop Window Help 08H3 Way. I E = 0 Step Response step response with 95% 12 HHH Amplitude |-----|....................... ................... ............................................. . T HII 150 200 250 Time (seconds)

File Edit View Insert Tools Desktop Window Help 08H3 Way. I E = 0 Step Response A s tep response with 105% Amplitude wmANAAAAAAAAA 100 150 300 350 400 450 500 250 Time (seconds)