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Problem 3.60PP
Find constraints on the two gains K^ and K2 that guarantee a stable closedloop system, and plot 
the allowable region(s) in the (/Cl, K2) plane.You may wish to use the computer to help solve this 
problem.
Suppose the characteristic polynomial of a given closed-loop system is computed to be
s4 + (11 +K2)s3 + (121 +K■\)s2 + {K^ + K1K2 + 110K2 + 210)s + 11/C1 +100 = 0.
Step-by-step solution
step 1 of 3
Step 1 of 3
The characteristic polynomial of the closed loop system is. 
j^ + ( i i+ A : 2 ) j ’ + ( i2 i+ ic , y + ( x , + A : ,A :2 + i io ^ : 2 + 2 io ) i+ i iK |+ 10 0=0 
Write the Routh array for the polynomial. 
s*-. I I2 I+ £ , HAT,+100
i i + a:2 ac, + a:|A:2+iiojc2+21o o
ŝ : a IIA:,+100
s : b
iia:, + ioo
Where.
^ ( ii+ A r2 )( i2 i+ A :i) - (A : i+ A :|A :2 + iiO A :2 + 2 io )( i)
l I + ATj
133i + i i a :,+ 1 2 ia :2 + a:,a: 2 - a:i -AT,AC;-1 1 0 ^ 2 -2 1 0 
11+AC2
10AC,+MAC;+ 1121
II + AC2
And,
^ a(AC| + AC|AC2 + 110AC2+2l0)-(llAC, + 100)(ll+A:2) 
a
+AC,AC2 + iioa:2 + 210)-(1 lAC,+100)(11+AC2 ) 
V. l l + AT; )___________________________________________
io / : i+ n A :a + i i2 i
11+^2
(ioa: , -1-11^ 2 + ^ 1^ 2 -*-iioa:2-(-21o) - ( i i + a:2)(iia : ,+ ioo)( i i -i-a:2) 
ioa: , + i i / :2 + i i 21
step 2 of 3
For a stable system, first column elements of Routh array must be greater than zero. 
Therefore,
i u a :2 > o
And,
ii/:,-i-ioo>o
\ \ K t > - m
-100
K t > -
11
Therefore, the two gain of the system are AC, and AC2 > -11 
' 11 ^
Step 3 of 3
Draw the allowable region in the ( ac„ a:2) plane.
i 1C2
3
3
•lO O /ld
-150 -120 -90 -AO -30 ^
100
A llo w ab le re g io n 
50
0 30 60 90 120
................................. ...................
1 -50
i
1
-100
1
1 -150
1
1
•200
1
1
-250
Hence, the plot is drawn.