PLEASE HELP!!! 1. (30 points) Consider the following closed-loop system r(k) rk)

Question

PLEASE HELP!!!

1. (30 points) Consider the following closed-loop system r(k) rk) (A) Find Gzo You can leave the sampling period as a constant T (B) Draw the root locus for the discrete-time closed-loop system with K-0 to oo for sample period T- 1 second. Make sure you calculate the appropriate break-in and break-away points and the value of K when the closed-loop system is at the stability boundary. (C) Use Matlab draw the root locus for the discrete-time closed-loop system with K-0 to for sample period T 4 second. Use either the Jury's test or bilinear transformation Routh Hurwitz test to find the value K at the stability boundary. Confirm it on your root locus plot and comment on the difference in the limits on K with different sample rate

Explanation / Answer

enter the following program in matlab to get the tone mary had a little lamb

fsamp=44100;

% 44100 samples per sec

% 261.63 Hz => Nper = 44100/261.63

song = zeros(1,fsamp/4);

A = [1.0000 -2.4584 1.8539 -0.2387 -0.0466 -0.1614 -0.1004 0.1980 -0.0420];

fc=261.63;

fd=293.66;

ff=349.23;

fg=392.00;

Nfc=ceil(fc);

Nfd=ceil(fd);

Nff=ceil(ff);

Nfg=ceil(fg);

Npc = ceil(44100/fc);% period of note c

Npd = ceil(44100/fd);% period of note d

Npf = ceil(44100/ff);% period of note f

Npg = ceil(44100/fg);% period of note g

b=0.1;

Nper=Npc;

M = Nfc/2;

xin = zeros(1,M*Nper);

for ii=1:M

xin(1+(Nper-1)*ii) = 1;

end

yout=filter(b,A,xin);

Ny=size(yout,2);

u=[0:Ny-1];

wt=sin(pi*u/Ny);

wt=wt.*wt;

yout1=yout.*wt;

song=[song yout1 yout1];

Nper=Npd;

M = Nfd/2;

xin = zeros(1,M*Nper);

for ii=1:M

xin(1+(Nper-1)*ii) = 1;

end

yout=filter(b,A,xin);

Ny=size(yout,2);

u=[0:Ny-1];

wt=sin(pi*u/Ny);

wt=wt.*wt;

yout1=yout.*wt;

song=[song yout1];

Nper=Npc;

M = Nfc/2;

xin = zeros(1,M*Nper);

for ii=1:M

xin(1+(Nper-1)*ii) = 1;

end

yout=filter(b,A,xin);

Ny=size(yout,2);

u=[0:Ny-1];

wt=sin(pi*u/Ny);

wt=wt.*wt;

yout1=yout.*wt;

song=[song yout1];

Nper=Npg;

M = Nfg/2;

xin = zeros(1,M*Nper);

for ii=1:M

xin(1+(Nper-1)*ii) = 1;

end

yout=filter(b,A,xin);

Ny=size(yout,2);

u=[0:Ny-1];

wt=sin(pi*u/Ny);

wt=wt.*wt;

yout1=yout.*wt;

song=[song yout1];

Nper=Npg;

M = Nfg/2;

xin = zeros(1,M*Nper);

for ii=1:M

xin(1+(Nper-1)*ii) = 1;

end

yout=filter(b,A,xin);

Ny=size(yout,2);

u=[0:Ny-1];

wt=sin(pi*u/Ny);

wt=wt.*wt;

yout1=yout.*wt;

song=[song yout1];

Nper=Npf;

M = Nff/2;

xin = zeros(1,M*Nper);

for ii=1:M

xin(1+(Nper-1)*ii) = 1;

end

yout=filter(b,A,xin);

Ny=size(yout,2);

u=[0:Ny-1];

wt=sin(pi*u/Ny);

wt=wt.*wt;

yout1=yout.*wt;

song=[song yout1];

plot(song)

Fs=44100;

bits=16;

soundsc(song,fsamp);

audiowrite('hb.wav', song, Fs, 'BitsPerSample', bits);

amp=10;

audioplayer(amp*song,fsamp)

the tone is as follows

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