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Problem 6.27PP
Consider the system given in Fig.
(a) Use Matlab to obtain Bode plots for K = 1, and use the plots to estimate the range of K for 
which the system will be stable.
(b) Verify the stable range of K by using margin to determine PM for selected values of K.
(c) Use hocus to determine the values of K at the stability boundaries.
(d) Sketch the Nyquist plot of the system, and use it to venfy the number of unstable roots for the 
unstable ranges of K.
(d) Sketch the Nyquist plot of the system, and use it to venfy the number of unstable roots for the 
unstable ranges of K.
(e) Using Routh’s criterion, determine the ranges of K for closed-loop stability of this system. 
Figure Control system
Step-by-step solution
step 1 of 15
Refer to Figure 6.93 in the textbook.
From the block diagram, the loop transfer function is.
L (» ) = A rC (*)W (j)
• ) + l
( i - l ) ( i ’ + 2 j + 2 )
Step 2 of 15
(a)
Substitute i for in loop transfer function.
i W = 7(j -1 ) ( s* + 2j + 2 )
( j - l ) ( s ’ + 2 i + 2 )
Write the MATLAB code to plot the bode diagram.
» num=[1 2];den=conv([1 -1],[1 2 2]);sys=tf{num,den)bode(sys)sys = s - 2 
s^2 - 2 Continuous-time transfer function.
Draw the bode diagram.
Bode Diigram
Freqaeaqr (rtd/s)
Figure 1
Step 3 of 15
From the bode plot, the closed loop system is unstable for AT s 1. But this can make the closed 
system stable with positive gain margin by increasing the gain K up to the crossover frequency 
reaches at ^ s 1.42rad/s. where phase plot crosses the —]80^ line.
Therefore, the range of x for the stable system is | | 0
K 0
K > \
Therefore, the range of x fo'’ fbe stable system is |} 0
K 0
^ > 1
Therefore, the range of x for the stable system is | |