solution-manual-process-dynamics-and-control-donald-r-coughanowr

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SOLUTIONS MANUAL FOR SELECTEDSOLUTIONS MANUAL FOR SELECTEDSOLUTIONS MANUAL FOR SELECTEDSOLUTIONS MANUAL FOR SELECTED 
 
PROBLEMS INPROBLEMS INPROBLEMS INPROBLEMS IN 
 
 
 
PROCESS SYSTEMS ANALYSIS AND 
CONTROL 
 
DONALD R. COUGHANOWR 
 
 
 
 
 
 
 
COMPILED BY 
 
M.N. GOPINATH BTech.,(Chem)M.N. GOPINATH BTech.,(Chem)M.N. GOPINATH BTech.,(Chem)M.N. GOPINATH BTech.,(Chem) 
 
CATCH ME AT gopinathchemical@gmail.com 
 
 
 
 
 
 
 
 
 
Disclaimer: This work is just a compilation from various sources believed to be 
reliable and I am not responsible for any errors. 
 
 
 
 
 
CONTENTS 
 
 
 
 
 
 
PART 1: SOLUTIONS FOR SELECTED PROBLEMS 
 
 
 
 
 PART2: LIST OF USEFUL BOOKS 
 
 
 
PART3: USEFUL WEBSITES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
PART 1 
 
 
1.1 Draw a block diagram for the control system generated when a human 
being steers an automobile. 
 
 
 
 
1.2 From the given figure specify the devices 
 
 
 
Solution: 
 
 
 
 
 
 
Inversion by partial fractions: 
 
 
 
3.1(a) 0)0()0(1 '
2
2
===++ xxx
dt
dx
dt
dx
 
 
 
)0()0()( '2
2
2
xsxsXs
dt
dx
L −−=





 
 
)0()( xsXs
dt
dx
L −=



 
 
L(x) = X(s) 
L{1} = 1/s 
 
+−− )0()0()( '2 xsxsXs
s
sXxsXs
1
)()0()( =+− 
 
s
sXss
1
)()1( 2 =++= 
 
)1(
1
)(
2 ++
=
sss
sX 
 
Now, applying partial fractions splitting, we get 
 
)1(
11
)(
2 ++
+
−=
ss
s
s
sX 
 
2222
2
3
2
1
2
3
3
2
2
1
2
3
2
1
11
)(






+




 +











−






+




 +
+
−=
ss
s
s
sX 
 
 
tetCosesXL
tt
2
3
sin
3
1
2
3
1))(( 2
1
2
1
1
−−− −−= 
 






+





−=
−
tSintCosetX
t
2
3
3
1
2
3
1)( 2
1
 
 
b) 0)0()0(12 '
2
2
===++ xxx
dt
dx
dt
dx
 
 
 
 
 
when the initial conditions are zero, the transformed equation is 
 
 
s
sXss
1
)()1( 2 =++ 
 
 
 
)1(
1
)(
2 ++
=
sss
sX 
 
12)1(
1
22 ++
+
+=
++ ss
CBs
s
A
sss
 
 
CsBsssA ++++= 22 )12(1 
)(20
)(0 2
sofeffecientscotheequatingbyCA
sofeffecientcotheequatingbyBA
−+=
−+=
 
 
 
2,1,1
2
1
0
)(1
−=−==
−=
−=
=+
−=
CBA
AC
B
BA
constofeffecientscotheequatingbyA
 
 
 
 
12
21
)(
2 ++
+
−=
ss
s
s
sX 
( )
( ) 





+
++
−= −−
2
11
1
111
)}({
s
s
s
LsXL 
 
( ) 





+
+
+
−= −
2
1
1
1
1
1
1)}({
ss
LtX 
)1(1)}({ tetX t +−= − 
 
3.1 C 0)0()0(13 '
2
2
===++ xxx
dt
dx
dt
dx
 
 
 
by Applying laplace transforms, we get 
 
s
sXss
1
)()13( 2 =++= 
 
)13(
1
)(
2 ++
=
sss
sX 
 
13
)(
2 ++
+
+=
ss
CBs
s
A
sX 
 
 CsBsssA ++++= 22 )13(1 
)(30
)(0 2
sofeffecientscotheequatingbyCA
sofeffecientcotheequatingbyBA
−+=
−+=
 
 
 
 
 
3,1,1
33
1
0
)(1
−=−==
−=−=
−=
=+
−=
CBA
AC
B
BA
constofeffecientscotheequatingbyA
 
 
 






++
+
−= −−
13
31
)}({
2
11
ss
s
s
LsXL 
 




















−




 +
+
−= −−
22
11
2
5
2
3
31
)}({
s
s
s
LsXL 




















−




 +






−






−




 +
+
−= −−
2222
11
2
5
2
3
2
5
5
2
.
2
3
2
5
2
3
2
3
1
)}({
ss
s
s
LsXL 
 
t
t
CosetX
t
2
5
sinh
5
3
2
5
(1)( 2
3
+−=
−
 
 
 
3.2(a) 
1)0(
0)0()0()0(;
11
''''
3
3
4
4
=
====+
x
xxxtCos
dt
xd
dt
dx
 
 
 
Applying Laplace transforms, we get 
 
1
)0()0()0()()0()0()0()0()(
2
'''23'''''1234
+
=−−−+−−−−
s
s
xsxxssXsxsxxsxssXs 
 
 
1
)1()()(
2
34
+
=+−+
s
s
ssssX 
 
34
2
)1()1
1
()( sss
s
s
sX +





+++
+
= 
 = 
)1)(1(
12
)1)(1(
1
23
23
23
23
++
+++
=
++
++++
sss
sss
sss
ssss
 
 
 
 
 
11)1)(1(
12
23223
23
+
+
+
+
+++=
++
+++
s
FEs
s
D
s
C
s
B
s
A
sss
sss
 
 
 
)1()()1()1)(1()1)(1()1)(1(12 323222223 +++++++++++++=+++ ssFEssDssscssBsssAssss
 
 
A+B+E=0 equating the co-efficient of s5. 
A+B+E+F=0 equating the co-efficient of s
4
. 
A+B+C+D+F=0 equating the co-efficient of s
3
. 
A+B+C=0 equating the co-efficient of s
2
. 
B+C=2 equating the co-efficient of s. 
A+B+E=0 equating the co-efficient of s
2
. 
C=1equating the co-efficient constant. 
 
C=1 
-B=-C+2=1 
A=1-B-C=-1 
D+F=0 
E+F=0D+E=1 
D-E=0 
2D=1 
 
A=-1; B=1; C=1 
D=1/2; E=1/2; F =-1/2 
 
 
 
{ }






+
−
+
+
+++
−
= −−
1
)1(2/1
1
2/1111
)(
232
11
s
s
ssss
LsL 
 
 
{ }






+
−
+
+
+++
−
= −−
1
)1(2/1
1
2/1111
)(
232
11
s
s
ssss
LsXL 
{ } int
2
1
2
1
2
1
2
1)(
2
StCose
t
ttX t −++++−= − 
 
 
2)0(;4)0(2 12
2
2
−==+=+ qqtt
dt
dq
dt
qd
 
 
applying laplace transforms,we get 
 
23
'2 22)0()(()0()0()(
ss
qssQqsqsQs +=−+−− 
 





 +=−+−+ 1
12
424))((
2
2
ss
ssssQ 
 
 
)(
)24(
)1(2
)(
2
3
ss
s
s
s
sQ
+
++
+
= 
 
 
= 
)1(
2422
4
34
+
+++
ss
sss
 
 
 
)1(
3*2
)1(
2
1
1
4)(
4 +
+
+
+





+
=
sssss
sQ 
 
 
31
3
1
)1(24)())(( teetqsQL tt +−+== −−− 
 
 
therefore te
t
tq −++= 2
3
2)(
3
 
 
 
3.3 a) 



+
−
+
=
++ 4
1
1
1
3
3
)4)(1(
3
2222 ss
s
ss
s
 
 
 




+
−
+
=
2222 2
1
1
1
ss
 
 
tCosCost
ss
L 2
2
1
1
1
2222
1 −=



+
−
+
− 
 
b) [ ] 522)1(
1
)52(
1
2222 +−
+
+=
+−
=
+− ss
CB
s
A
sssss
 
 
 
 
A+B=0 
-2A+C=0 
5A=1 
A=1/5 ;B=-1/5;C=2/5 
 
 
We get 



+−
−
+=
52
21
5
1
)(
2 ss
s
s
sX 
 
 
Inverting,we get 
 
 
= 


 −+ tCosetSine tt 22
2
1
1
5
1
 
= 










 −+ tCostSinet 22
2
1
1
5
1
 
 
 
 
c) 2222
22
)1(1)1(
233
−
+
−
++=
−
+−−
s
D
s
C
s
B
s
A
ss
sss
 
 
 
 
233)1()1()1( 232222 +−−=+−+−+− sssDssCssBsAs 
 
 
233)()12()2( 2322323 +−−=+−++−++− sssDsssCssBsssA 
 
 
A+C=3 
-2A+B-C+D=-1 
A-2B=-3 
B=2; 
A=2(2)-3=1 
C=3-1=2 
D=2(1)-2+2-1=1 
 
We get 
22 )1(
1
1
221
)(
−
+
+
++=
ssss
sX 
By inverse L.T 
 
[ ] tt teettXL +++=− 221)(1 
[ ] )2(21)(1 tettXL t +++=− 
 
 
3.4 Expand the following function by partial fraction expansion. Do not evaluate 
co-efficient or invert expressions 
 
)3()1)(1(
2
)(
22 +++
=
sss
sX 
 
3)1(11
)(
222 +
+
+
+
+
+
+
+
+
=
s
F
s
EDs
s
CBs
s
A
sX 
 
 
 
22222 )1)(1()3)(1)(()1)(3)(1)(()3()1( ++++++++++++++= ssFssEDssssCBsssA
 
 
 
)14)(1()34)(()1)(34)(()3)(12( 222224 ++++++++++++++++= sssFssEDssssCBssssA
 
 
 
2333)4
34()346()342()43()( 2345
=++++++
+++++++++++++++++=
FEACAFE
BCAsCBCAsBCBAsFBCAsFBAs
A+B+F=0 
-3A+C+4B+F=0 
2A+B+4C+3B=0 
6A+C+4B+3C=0 
A+4C+3B+3D+4E+F=0 
3A+3C+3E+F=2 
 
by solving above 6 equations, we can get the values of A,B,C,D,E and 
33 )3()1)(1(
1
)(
+++
=
ssss
sX . 
 
3232 )3()3(321
)(
+
+
+
+
+
+
+
+
+
+++=
s
H
s
G
s
F
s
E
s
D
s
C
s
B
s
A
sX 
 
by comparing powers of s we can evaluate A,B,C,D,E,F,G and H. 
c)
)4()3)(2(
1
)(
+++
=
ssss
sX 
 
 
4321
)(
+
+
+
+
+
+
+
=
s
D
s
C
s
B
s
A
sX 
by comparing powers of s we can evaluate A,B,C,D 
 
3.5 a) 
)15.0)(1(
1
)(
++
=
sss
sX 
 
 
)15.0(1)15.0)(1(
1
+
+
+
+=
++ s
C
s
B
s
A
sss
Let 
 
1)(
2
1
2
3
2
2
22
=++





++





++= ssCs
s
B
ss
A 
A=1 
 
2
1
2
0
22
−=+==++ C
B
C
BA
 
 
 
2
3
0
2
3
−=+==++ CBCB
A
 
 
 
 
B/2=1/2 *-3/2=-1; 
 
B=-2; 
C= -3/2+2=1/2 
 
 
 






+
+
+
−=
15.0
1
2
1
1
21
)(
sss
sX 
 
 
tt eetxsXL 21 21)())(( −−− +−=== 
 
 
b) 0)0(;22 ==+ xx
dt
dx
 
 
Applying laplace trafsorms 
 
 
ssXxssX /2)(2)0()( =+− 
 
 
)2(
2
))((1
+
=−
ss
sXL 
 





+
= −−
)2(
2
2))(( 11
ss
LsXL 
 
= 



+
−= −−
2
2/12/1
2))(( 11
ss
LsXL 
 
=
te 21 −− 
 
3.6 a) 
52
1
)(
2 ++
+
=
ss
s
sY 
 
= 
52
1
)(
2 ++
+
=
ss
s
sY 
 
4)1(
1
2 ++
+
=
s
s
 
 
= 





++
+
= −−
4)1(
1
))((
2
11
s
s
LsYL 
 
using the table,we get 
 
tCosetY t 2)( −= 
 
 
b) 
4
2 2
)(
s
ss
sY
+
= 
 
 
32
21
)(
ss
sY += 
 
Y(t)= 21 ))(( ttsYL +=− 
 
 
c) 
3)1(
2
)(
−
=
s
s
sY 
 
= 
3)1(
222
−
+−
s
s
 
 
 
32 )1(
2
)1(
2
−
+
−
=
ss
 
 
 






−
+





−
= −−
3
1
2
1
)1(
2
)1(
2
)(
s
L
s
LtY 
 
 
= )2(
2
(2 2
2
ttee
t
te ttt +=+ 
3.7a) 
)1()1()1(
1
)(
222 +
+
+
+
+
=
+
=
s
DCs
s
BAs
s
sY 
 
 
1)1)(()( 2 =++++ sDCsBAsthus 
 
= 1)()(23 =+++++ DBsCADsCs 
 
 
C=0,D=0 
Also A=0;B=1 
222222 )()()()()()(
1
)1(
1
)(
is
D
is
C
is
B
is
A
isiss
sY
−
+
−
+
+
+
+
=
−+
=
+
= 
 
 
1)())(()())(( 2222 =+++−+−+−+ isDisisCisBisisA 
 
 
1)()22()()( 23 =−+−−+++−++++−++ DCiBAiDiCBiAsDCiBAisCA 
 
 
Thus,A+C=0 
-Ai+B+Ci+2Di=0 ; B=D 
A-2Bi+C+2Di=0 
-Ai-B+Ci-D=1 Also D=-Ci;B=-Ci, 
 A=-C,C=-i/4 
 
 
 
A=i/4 ; B=-1/4; D=-1/4 
 
 
22 )(
4/1
)(
4/
)(
4/1
)(
4/
)(
isis
i
isis
i
sY
−
−
+
−
−
+
+
−
+
+
= 
 
 
 
22 )(
4/1
)(
4/
)(
4/1
)(
4/
)(
isis
i
isis
i
tY
−
−
+
−
−
+
+
−
+
+
= 
 
 
22 )(
4/1
)(
4/
)(
4/1
)(
4/
)(
isis
i
isis
i
tY
−
−
+
−
−
+
+
−
+
+
= 
 
itititit teeeeitY 4/14/14/14/)( −−−= −− 
 
)(4/1)( itititit teieteietY −−−= −− 
)()()()((4/1)( tSinitCosttiSintCositSiniCostttiSinCostitY +−+−−−−= ) 
 
)22(4/1)( tCosttSintY −= 
 
)(2/1)( tCosttSintY −= 
 
 
 
3.8 
)1(
1
)(
2 +
=
ss
sf 
 
= 
1
)(
2 +
++=
s
C
s
B
s
A
sf 
 
1)1()1( 2 =++++= CssBssA 
 
Let s=0 ; A=1 
 s=1; 2A+B+C=1 
s=-1: C=1 
B=-1 
1
111
)(
2 +
++=
sss
sf 
 
tettf −+−= )1()( 
 
 
 
PROPERTIES OF TRANSFORMS 
 
 
 
 
4.1 If a forcing function f(t) has the laplace transforms 
 
s
e
s
ee
s
sf
sss 3
2
21
)(
−−−
−
−
+= 
 
2
231
s
ee
s
e sss −−− −
+
−
= 
 
)]2()2()1()1[()]3()([)}({)( 1 −−−−−+−−== − tuttuttutusfLtf 
 
 )3()2()2()1()1()( −−−−−−−+= tututtuttu 
 
graph the function f(t) 
 
 
 
 
 
 
4.2 Solve the following equation for y(t): 
 
1)0(
)(
)(
0
==∫ ydt
tdy
dy
t
ττ 
 
Taking Laplace transforms on both sides 
 





=∫ dt
tdy
LdtyL
t
)(
})({
0
τ
 
)0()(.)(.
1
ysyssy
s
−= 
 
 
1)(.)(.
1
−= syssy
s
 
 
1
)(
2 −
=
s
s
sy 
 
)cosh(
1
)}({)(
2
11 t
s
s
LsyLty






−
== −− 
 
4.3 Express the function given in figure given below the t – domain and the 
s- 
 
domain 
 
 
This graph can be expressed as 
 
)}6()6()5()5()5({)}3()3()2()2({)}5()1({ −−+−−−−+−−−−−+−−−= tuttuttututtuttutu
 
)6()6()5()5()3()2()2()2()1()( −−+−−−−−−−−+−= tuttuttuttuttutf 
 
2
6
2
53
2
3
2
2
)}({)(
s
e
s
e
s
e
s
e
s
e
s
e
tfLsf
ssssss −−−−−−
+−−−+== 
 
 
2
53623
s
eeee
s
ee ssssss −−−−−− −−+
+
−
= 
 
 
 
4.4 Sketch the following functions: 
 
 
)3()1(2)()( −+−−= tutututf
 
 
 
 
)2()1(3)(3)( −+−−= tututtutf
 
 
 
 
 
 
4.5 The function f(t) has the Laplace transform 
 
 22 /)21()( seeSf ss −− +−= 
 
 obtain the function f(t) and graph f(t) 
 
 
2
221
)(
s
ee
sf
ss −− +−
= 
 
 
 
2
2
2
1
s
ee
s
e sss −−− −
−
−
= 
 
 )]2()2()1()1{()()1()1()}({)( 1 −−−−−−+−−−== − tuttutttututsfLtf 
 
 
 )2()2()1()1(2)( −−+−−−= tuttutttu 
 
4.6 Determine f(t) at t = 1.5 and at t = 3 for following function: 
 
 )2()3()1(5.0)(5.0)( −−+−−= tuttututf 
 
 
 At t = 1.5 
 
 )2()3()1(5.0)(5.0)( −−+−−= tuttututf 
 
 )1(5.0)(5.0)5.1( −−= tutuf 
 
 05.05.0)5.1( =−=f 
 
 At t = 3 
 
 0)33(5.05.0)3( =−+−=f 
 
 
 
 
RESPONSE OF A FIRST ORDER SYSTEMS 
 
 
5.1 A thermometer having a time constant of 0.2 min is placed in a 
temperature bath and after the thermometer comes to equilibrium with 
the bath, the temperature of the bath is increased linearly with time at the 
rate of I deg C / min what is the difference between the indicated 
temperature and bath temperature 
 (a) 0.1 min 
(b) 10. min 
after the change in temperature begins. 
 
© what is the maximum deviation between the indicated temperaturew 
and bath temperature and when does it occurs. 
 
(d) plot the forcing function and the response on the same graph. After the 
long enough time buy how many minutes does the response lag the input. 
 
 
 Consider thermometer to be in equilibrium with temperature bath at 
temperature Xs 
0,)/1()( >°+= ttmXtX S 
 as it is given that the temperture varies linearly 
 
X(t)-Xs = t 
 
Let X(t) = X(t) - Xs = t 
 
Y(s) = G(s).X(s) 
 
22 1
1
1
1
)(
s
C
s
B
s
A
ss
sY ++
+
=
+
=
ττ
 
 
 
 A = 12 =−= CB ττ 
 
 
 
2
2 1
1
)(
sss
sY +−
+
=
τ
τ
τ
 
 
 tetY t +−= − ττ τ/)( 
 
(a) the difference between the indicated temperature and bath temperature 
 at t = 0.1 min = X(0.1)_ Y(0.1) 
 
 = 0.1 - (0.2e-0.1/0.2 - 0.2+0.1) since T = 0.2 given 
 = 0.0787 deg C 
 
(b) t = 1.0 min 
 
X(1) - Y(1) = 1- (0.2e
-1/0.
2 - 0.2 +1) = 0.1986 
 
(c) Deviation D = -Y(t) +X(t) 
 
 = -τe-t/T+T =τ (-e-t/T+1) 
 
For maximum value dD/dT = τ (-e-t/T+(_-1/T) = 0 
 
-e
-t/ 
= 0 
 as t tend to infinitive 
D = τ (-e-t/T+(_-1/T) = τ =0.2 deg C 
 
 
 
5.2 A mercury thermometer bulb in ½ in . long by 1/8 in diameter. The 
glass envelope is very thin. Calculate the time constant in water flowing 
at 10 ft / sec at a temperature of 100 deg F. In your solution , give a 
summary which includes 
 
(a) Assumptions used. 
(b) Source of data 
(c) Results 
 
 T = mCp/hA = 
)(
)(
DLAh
CAL p
π
ρ
+
 
 
Calculation of 
 
 
 nmed CR
K
hD
NU (Pr)== 
 
4.9677
10
10)3048.0*10)(10*54.2*8/1(
Re
3
32
=== −
−
µ
ρDv
d 
 
KgKKJ
K
C p
/2.4Pr ==
µ
 
 
 Source data: Recently, Z hukauskas has given c,m ,ξ,n values. 
 
 For Re = 967704 
 
C = 0.26 & m = 0.6 
 
 NuD = hD/K = 0.193 (9677.4)*(6.774X10
-3) = 130 
 
 .h = 25380 
 
 
 
 
5.3 Given a system with the transfer function Y(s)/X(s) = (T1s+1)/(T2s+1). 
Find Y(t) if X(t) is a unit step function. If T1/T2 = s. Sktech Y(t) Versus 
t/T2. Show the numerical values of minimum, maximum and ultimate values 
that may occur during the transient. Check these using the initial value 
and final value theorems of chapter 4. 
 
 
 
1
1
)(
2
1
+
+
=
sT
sT
sY 
 
 X(s) =unit step function = 1 X(s) = 1/s 
 
 
 
 
 
 
 
 
siT
B
s
A
sTs
sT
sY
22
1
)1(
1
)( +=
+
+
= 
 
 A = 1 B = T1 - T2 
 
 
sT
TT
s
sY
2
21
1
1
)(
+
−
+= 
 
2/
2
211)( Tte
T
TT
tY −
−
+= 
 
 If T1/T2 = s then 
 
2/41)( TtetY −+= 
 
 Let t/T2 = x then 
x
etY
−+= 41)( 
 
 Using the initial value theorem and final value theorem 
 
 )()(
0
ssYLimTYLim
ST ∞→→
= 
 
 
 = 5
1
1
1
1
2
1
2
1
2
1 ==
+
+
=+
+
∞→∞→ T
T
s
T
s
T
Lim
sT
sT
Lim
SS
 
 
)()(
0
ssYLimTYLim
ST ∞→→
= = 1
1
1
2
1
0
=
+
+
→ sT
sT
Lim
S
 
 
 
 
 
 
Figure: 
 
 
 
 
5.4 A thermometer having first order dynamics with a time constant of 1 
min is placed in a temperature bath at 100 deg F. After the thermometer 
reaches steady state, it is suddenly placed in bath at 100 deg F at t = 0 and 
left there for 1 min after which it is immediately returned to the bath at 
100 deg F. 
(a) draw a sketch showing the variation of the thermometer reading with 
time. 
(b) calculate the thermometer reading at t = 0.5 min and at t = 2.0 min 
 
min)1(
1
1
)(
)(
=
+
= τ
ssX
sY
 
 






−=
−
s
e
s
s
s1
10)( 
 




 −
=
−
s
e
sY
s1
10)( 






+
−
+
=
−
)1()1(
1
10)(
ss
e
ss
sY
s
 
1)1(10)( <−= − tetY t 
 
( ) 1)1()1(10)( )1( ≥−−−= −−− teetY tt 
 
At t = 0.5 T = 103.93 
 
 At = 2 T =102.325 
 
 
 
5.5 Repeat problem 5.4 if the thermometer is in 110 deg F for only 10 sec. 
 
 If thermometer is in 110 deg F bath for only 10 sec 
 
 60/10110 teT −−= 
 sec60&sec100 =<< Tt 
 535.101sec)10( ==tT 
 sec10535.1100 60/)10( >+= −− teT t 
 T(t=30sec) = 101.099 deg F 
 
 T(t=120sec) = 100.245 deg F 
 
 
 
5.6 A mercury thermometer which has been on a table for some time,is 
registering the room temperature ,758 deg F. Suddenly, it is placed in a 400 
deg F oil bath. The following data are obtained for response of the 
thermometer 
 
Time (sec) Temperature, Deg F 
0 75 
1 107 
2.5 140 
5 205 
8 244 
10 282 
15 328 
30 385 
Give two independent estimates of the thermometer time constant. 
 






−
=
T
t
400
325
ln
τ 
 
From the data , average of 9.647,11.2,9.788,10.9,9.87,9.95, and 9.75 is 10.16 sec. 
 
 
 
5.7 Rewrite the sinusoidal response of first order system (eq 5.24) in 
terms of a cosine wave. Re express the forcing function equation (eq 5.19) 
as a cosine wave and compute the phase difference between input and 
output cosine waves. 
 
 
τ
τ
ω
ω
τ 1
1
)(
1
1
)(
22
+






+
=
+
=
s
s
A
s
s
sY 
 
 splitting into partial fractions then converting to laplace transforms 
 
 
)sin(
11
)(
22
/
22
φω
ωτωτ
αωτ τ +
+
+
+
= − t
A
e
A
tY t 
 
where φ = tan-1 (ωτ) 
 
As t →∝ 
 
)




 −−
+
=+
+
= φ
π
ω
ωτ
φω
ωτ 2
cos(
1
)sin(
1
)(
2222
t
A
t
A
stY 
 
 





 −=+= tAtAtY ω
π
φω
2
cos)sin()( 





 −=
2
cos)(
π
ωtAtY 
 
The phase difference = φ
ππ
φ =




−−−
22
 
 
 
5.8 The mercury thermometer of problem 5.6 is allowed to come to 
equilibrium in the room temp at 75 deg F.Then it is immersed in a oil 
bath for a length of time less than 1 sec and quickly removed from the 
bath and re exposed to 75 deg F ambient condition. It may be estimated 
that the heat transfer coefficient to the thermometer in air is 1/ 5
th
 that in 
oil bath.If 10 sec after the thermometer is removed from the bath it reads 
98 Deg F. Estimate the length of time that the thermometer was in the bath. 
 
 t < 1 sec τ/
1
1325400 teT −−= 
 
 Next it is removed and kept in 75 Deg F atmosphere 
 
 Heat transfer co-efficient in air = 1/5 heat transfer co-efficient in oil 
 
 
 hair = 1/5 hoil 
 
 
hA
mC
=τ sec10=oilτ 
 sec50=airτ 
 
 50/1
2)75(75 tF eTT
−−+= 
 
 CTempFinalTF deg98== 
 
 50/10
10/
)325325(7598 1 −
−−+= ee t 
 
91356.010/ =−te 
 
t 1 = 0.904 sec. 
 
 
 
5.9 A thermometer having a time constant of 1 min is initially at 50 deg C. 
it is immersed in a bath maintained at 100 deg C at t = 0 . Determine the 
temperature reading at 1.2 min. 
 
 τ = 1 min for a thermometer initially at 50 deg C. 
 
 Next it is immersed in bath maintained at 100 deg C at t = 0 
 
 At t = 1.2 
 
 
)1()( /τteAtY −−= 
 
50)1(50)2.1( 1/2.1 +−= −eY 
 
Y(1.2) = 84.9 deg C 
 
 
5.10 In Problem No 5.9 if at, t = 1.5 min thermometer having a time 
constant of 1 minute is initially at 50 deg C.It is immersed in a bath 
maintained at 100 deg C at t = 0.Determine the temperature reading at t = 
1.2 min. 
 
At t = 1.5 
 
CY °= 843.88)5.1( 
 
Max temperature indicated = 88.843 deg C 
 
AT t = 20 min 
 
)1(843.13843.88 1/8.18−−−= eT
 
 
T = 75 Deg C. 
 
 
5.11 A process of unknown transfer function is subjected to a unit impulse 
input. The output of the process is measured accurately and is found to be 
represented by the function Y(t) = t e
-t
. Determine the unit step response in 
this process. 
 
 X(s) = 1 Y(t) = te
-t 
 
 
 
2)1(
1
)(
+
=
s
sY 
 
2)1(
1
)(
)(
)(
+
==
ssX
sY
sG 
 
For determining unit step response 
 
 
2)1(
1
)(
+
=
s
sY 
 
 
 
22 )1(1)1(
1
)(
+
+
+
+=
+
=
s
C
s
B
s
A
s
sY 
 
 A = 1 B = -1 C = -1 
 
 
2)1(
1
1
11
)(
+
−
+
−=
sss
sY 
 
 tt teetY −− −−= 1)( 
 
 
 
 
 
 
 
 
 
 
 
Response of first order system in series 
 
 
 
 
 
 
7.1 Determine the transfer function H(s)/Q(s) for the liquid level shown in 
figure P7-7. Resistance R1 and R2 are linear. The flow rate from tank 3 is 
maintained constant at b by means of a pump ; the flow rate from tank 
3 is independent of head h. The tanks are non interacting. 
 Solution : 
 
 
 
 
 A balance on tank 1 gives 
 
 
dt
dh
Aqq 111 =− 
 
 where h1 = height of the liquid level in tank 1 
 
similarly balance on the tank 2 gives 
 
dt
dh
Aqq 2221 =− 
 
and balance on tank 3 gives 
 
 
dt
dh
Aqq 302 =− 
 
here 
1
1
1
R
h
q =
2
2
2
R
h
q = bq =0 
So we get 
 
 
dt
dh
A
R
h
q 11
1
1 =− 
 
 
 
dt
dh
A
R
h
R
h 2
2
2
2
1
1 =− 
 
 
dt
dh
Ab
R
h
3
2
2 =− 
 
writing the steady state equation 
 
011
1
1 ==−
dt
dh
A
R
h
q SsS 
dt
dh
A
R
h
R
h SSS 2
2
2
2
1
1 =− 
0
2
2 =−b
R
h S 
Subtracting and writing in terms of deviation 
 
dt
dH
A
R
H
Q 11
1
=− 
 
dt
dH
A
R
H
R
H 2
1
2
2
1
1 =− 
 
dt
dH
A
R
H
3
2
2 = 
 
where Q = q –qS 
 H1= h1-h1S 
 
 H1= h2-h2S 
 
 H = h - hS 
 
 Taking Laplace transforms 
 
 
)(
)(
)( 11
1
1 sHsA
R
sH
sQ =− ---------(1) 
 )(
)()(
22
2
2
1
1 sHsA
R
sH
R
sH
=− --------(2) 
 
)(
)(
3
2
2 sHsA
R
sH
= ----------(3) 
We have three equations and 4 unknowns(Q(s),H(s),H1(s) and H2(s). So we 
can express one in terms of other. 
 
From (3) 
 
sAR
sH
sH
31
2
2
)(
)( = -------------(4) 
 
)1(
)(
)(
21
12
2 +
=
sR
sHR
sH
τ
 where 222 AR=τ ------------(5) 
 
From (1) 
 
 
)1(
)(
)(
1
1
1 +
=
s
sQR
sH
τ
, 111 AR=τ ---------(6) 
 
Combining equation 4,5,6 
 
)1)(1)((
)(
)(
213 ++
=
sssA
sQ
sH
ττ
 
 
 
 
)1)(1)((
1
)(
)(
213 ++
=
sssAsQ
sH
ττAbove equation can be written as 
 
i.e, if non interacting first order system are there in series then there overall 
transfer function is equal to the product of the individual transfer function in 
series. 
 
 
7.2 The mercury thermometer in chapter 5 was considered to have all its 
resistance in the convective film surrounding the bulb and all its 
capacitance in the mercury. A more detailed analysis would consider both 
the convective resistance surrounding the bulb and that between the bulb 
and mercury. In addition , the capacitance of the glass bulb would be 
included. 
 
Let 
Ai = inside area of bulb for heat transfer to mercury. 
Ao = outside area of bulb, for heat transfer from surrounding fluid. 
.m = mass of the mercury in bulb. 
mb = mass of glass bulb. 
C = heat capacitance of mercury. 
Cb = heat capacity of glass bulb. 
.hi = convective co-efficient between the bulb and the surrounding fluid. 
.ho = convective co-efficient between bulb and surrounding fluid. 
T = temperature of mercury. 
Tb = temperature of glass bulb. 
Tf = temperature of surrounding fluid. 
 
Determine the transfer function between Tf and T. what is the effect of bulb 
resistance and capacitance on the thermometer response? Note that the 
inclusion of the bulb results in a pair of interacting systems, which give an 
overall transfer function different from that of Eq (7.24) 
 
Writing the energy balance for change in term of a bulb and mercury 
respectively 
 
 
 
 
Input - output = accumulation 
 
dt
dT
CmTTAhTTAh bbbbiibf =−−− )()(00 
 
dt
dT
CmTTAh bii =−− 0)( 
 
Writing the steady state equation 
 
0)()(00 ==−−−
dt
dT
CmTTAhTTAh bsbbsbsiibsfs 
 
0)( =− sbsii TTAh 
 
 
Where subscript s denoted values at steady subtracting and writing these 
equations in terms of deviation variables. 
 
dt
dT
CmTTAhTTAh bbbmbiibf =−−− )()(00 
 
 
dt
dT
CmTTAh mmbii =−− 0)( 
 
 
Here TF = Tf - TfS 
 TB = Tb - TbS 
 Tm = T - TS 
 
 Taking laplace transforms 
 
 
)()())()((00 sTCmTTAhsTsTAh BbbmBiiBF =−−− ----(1) 
 
 
And )())()(( ssTmCsTsTAh BmBii =− ------(2) 
 
 
= )()())()((00 ssTCmsmCSTsTsTAh BbbmBF =−− 
 
From (2) we get 
 
)1()(1)()( +=





+= ssTs
Ah
mC
sTsT im
ii
mB τ 
 
 
Where 
ii
i
Ah
mC
=τ 
 
 Putting it into (1) 
 
 
 0)1))(1()()(
00
0 =





+++− s
Ah
mC
sssTsT imF ττ 
 
= 





+++= s
Ah
mC
sssTsT imF
00
0 )1))(1()()( ττ 
 
= 
1)(
1
)(
)(
00
0
2
0 ++++
=
s
Ah
mC
s
sT
sT
ii
F
m
ττττ
 
 
= 
1)(
1
)(
)(
00
0
2
0 ++++
=
s
Ah
mC
s
sT
sT
ii
F
m
ττττ
 
 
Or we can write 
 
1)(
1
)(
)(
00
0
2
0 ++++
=
s
Ah
mC
s
sT
sT
ii
f ττττ
 
ii
i
Ah
mC
=τ and 
00
0
Ah
Cm bb=τ 
 
 
We see that a loading term mC/ hoAo is appearing in the transfer function. 
The bulb resistance and capacitance is appear in 
0τ and it increases the 
delay i.e Transfer lag and response is slow down. 
 
 
 
 
 
 
 
 
 
 
7.3 There are N storage tank of volume V Arranged so that when 
water is fed into the first tank into the second tank and so on. Each tank 
initially contains component A at some concentration Co and is equipped 
with a perfect stirrer. A time zero, a stream of zero concentration is 
fed into the first tank at volumetric rate q. Find the resulting 
concentration in each tank as a function of time. 
 
Solution: 
 
 
. i
th
 tank balance 
 
dt
dC
VqCqC iii =−−1 
 
0)1( =−− issi qCqC 
 
 






=
=−−
q
V
dt
dC
q
V
CC iii
τ
)1(
 
 
 
Taking lapalce transformation 
 
 
)()()()1( ssCisCsC ii τ=−− 
 
)()1()()1( sCissC i τ+=− 
 
 
ssC
sC
i
i
τ+
=
− 1
1
)(
)(
1
 
 
Similarly 
 
issC
sC
sC
sC
sC
sC
sC
sC
sCo
sC
i
i
i
ii
)1(
1
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
2
1
1
2
0
1
τ+
=×−−−−−−−−−××=
−
− 
 
 
Or 
 
N
N
ssCo
sC
)1(
1
)(
)(
τ+
= 
 
NN ss
C
sC
)1(
)( 0
τ+
−
= 
 
 






+
−−−−−−−
+
−
+
−−=
− ssss
CsC
NNN τ
τ
τ
τ
τ
τ
1)1()1(
1
)(
10
 
 
 
 
 










−−−−−
−
−
−
−−=
−−
−
−
−
−
−
τ
ττ
ττ
tN
N
t
N
N
t
N e
N
te
N
te
CtC
)!2(
.
)!1(
.1)(
2
2
1
10
 
 
 














+−−−−−
−






+
−









−−=
−−
−
1
)!2(
.
)!1(
.1)(
21
0
N
t
N
t
eCtC
NN
t
N
τττ 
 
 
7.4 (a) Find the transfer functions H2/Q and H3/Q for the three tank system 
shown in Fig P7-4 where H1,H3 and Q are deviation variables. Tank 1 and 
Tank 2 are interacting. 
 
7.4(b) For a unit step change in q (i.e Q = 1/s); determine H3(0) , H3(∞) 
and sketch H3(t) vs t. 
 
Solution : 
 
 Writing heat balance equation for tank 1 and tank 2 
 
 
 
dt
dh
Aqq 111 =− 
 
 
 
dt
dh
Aqq 2221 =− 
 
 
1
21
1
R
hh
q
−
=
2
2
2
R
h
q = 
 
 
Writing the steady state equation 
 
01 =− ss qq 
 
 
021 =− ss qq 
 
Writing the equations in terms of deviation variables 
 
dt
dH
AQQ 111 =− 
 
dt
dH
AQQ 2221 =− 
 
1
21
1
R
HH
Q
−
=
2
2
2
R
H
Q = 
 
Taking laplace transforms 
 
)()()( 111 ssHAsQsQ =− 
 
)()()( 2121 ssHAsQsQ =− 
 
)()()( 2111 sHsHsQR −= 
 
)()( 222 sHsQR = 
 
Solving the above equations we get 
 
 
( )[ ]1)(
)(
2121
2
21
22
++++
=
sRAs
R
sQ
sH
ττττ
 
 
Here 111 AR=τ 
 
 222 AR=τ 
 
Now writing the balance for third tank 
 
 
dt
dh
Aqq 3332 =− 
 
Steady state equation 
 
 
032 =− SS qq 
 
 
 
3
3
3
R
h
q = 
 
 
dt
dh
A
R
H
Q 33
3
3
2 =− 
 
Taking laplace transforms 
 
)(
)(
)( 3
3
3
2 ssHA
R
sH
sQ =− 
 
( )1)()( 3
3
3
2 += s
R
sH
sQ τ where 333 AR=τ 
 
 
From equation 1,2,3,4 and 5 we got 
 
 
 
[ ]1)(
1
)(
)(
2121
2
21 ++++
=
sRAssQ
sQs
ττττ
 
 
 
Putting it in equation 6 
 
 
 
[ ]( )11)()(
)(
32121
2
21
33
+++++
=
ssRAs
R
sQ
sH
τττττ
 
 
 
Putting the numerical values of R1,R2 and R3 and A1,A2,A3 
 
 
[ ]( )12164
4
)(
)(
2
3
+++
=
ssssQ
sH
 
 
[ ]164
2
)(
)(
2
2
++
=
sssQ
sH
 
 
 
Solution (b) 
 
 
s
sQ
1
)( = 
 
[ ]( )12164
41
)(
23 +++
=
ssss
sH 
 
 
From initial value theorem 
 
 
 
)()0( 33 ssHLimH
S ∞→
= 
 
 
 = 
)164)(12(
4
2 +++∞→ sss
Lim
S
 
 
 
 = 
)1
6
4()12(
4
2
3
ss
ss
Lim
S
+++
∞→
 
 
 
H3 (0) = 0 
 
 
 
 
From final value theorem 
 
)()( 3
0
3 ssHLimH
S→
=∞ 
 = 
)164)(12(
4
20 +++→ sss
Lim
S
 
 
H3 (∞) = 4 
 
 
 
 
7.5 Three identical tanks are operated in series in a non-interacting fashion 
as shown in fig P7.5 . For each tank R=1, ττττ = 1. If the deviation in 
flow rate to the first tank in an impulse function of magnitude 2, 
determine 
(a) an expression for H(s) where H is the deviation in level in the third 
tank. 
(b) sketch the response H(t) 
(c) obtain an expression for H(t) 
 
solution : 
 
 
 
 writing energy balance equation for all tanks 
 
 
dt
dh
Aqq 11 =− 
 
 
dt
dh
Aqq 221 =− 
 
 
 
dt
dh
Aqq =− 32 
 
 
R
h
q 11 =
R
h
q 22 = 
R
h
q =3 
 
 
So we get 
 
 01 =− SSqq 
021 =− SS qq 
 032 =− SS qq 
 
writing in terms of deviation variables and taking laplace transforms 
 )(
)(
)( 1
1 sHA
R
sH
sQ S=− 
 
 )(
)()(
2
21 sHA
R
sH
R
sQ
S=− 
)(
)()(2 sHA
R
sH
R
sH
S=− 
 
solving we get 
 
33 )1(
1
)1()(
)(
+
=
+
=
ss
R
sQ
sH
τ
 
 
33 )1(
2
)1(
)(
)(
+
=
+
=
ss
sQ
sH
τ
 
{ } tetsHLtH −− ==
2
2)()(
2
1 
 
tettH −= 2)( 
02
)(
=−= −− tt tete
dt
tdH
 
 
22 tt == 
 
at t = 2 max will occur. 
 
 
7.6 In the two- tank mixing process shown in fig P7.6 , x varies from 0 lb 
salt/ft3 to 1 lb salt/ft
3
 according to step function. At what time does the 
salt concentration in tank 2 reach 0.6 lb/ ft
3
 ? The hold up volume of each 
tank is 6 
ft
3
. 
 
Solution 
 
Writing heat balance equation for tank 1 and tank 2 
 
 
dt
dy
Vqq yx =− 
dt
dl
Vqq cy =− 
steady state equation 
 
0=− ysxs qq 
0=− csys qq 
 
 
writing in terms of deviation variables and taking laplace transforms 
 
)()()( sYs
q
V
sYsX =− 
 
q
V
s
s
q
VsX
sY
=
+
=






+
= τ
τ
;
1
1
1
1
)(
)(
 
 
2)1(
)(
)1(
)(
)(
+
=
+
=
s
sX
s
sY
sC
ττ
 
 
2)1(
1
)(
)(
+
=
ssX
sC
τ
 
 
s
sX
1
)( = 
2
3
6
===
q
V
τ 
 
2)12(
)(
)(
+
=
ss
sX
sC 
2)
2
1
(
)4/1(
)(
+
=
ss
sC 



+







=
2)
2
1
(
11
4
1
)(
s
s
sC 





 +
−



 +






−=
2
1
1
2
1
1
2
1
1
)(
2
s
s
sC 
 
 
22
2
1
1)(
tt
etetC
−−
−−= 
 
3/61.0)( ftsaltlbtC = 
 
t = 4.04 min 
 
 
 
 
7.7 Starting from first principles, derive the transfer functions H1(s)/Q(s) 
and H2(s)/Q(s) for the liquid level system shown in figure P7.7. The 
resistance are linear and R1= R2 = 1. Note that two streams are flowing 
from tank 1, one of which flows into tank 2. You are expected to give 
numerical values of the parameters and in the transfer functions and to 
show clearly how you derived the transfer functions. 
 
Writing heat balance equation for tank 1 
 
dt
dh
Aqqq a
1
11 =−− 
 
1
1
1
R
h
q =
a
a
R
h
q 1= 
 
dt
dh
A
R
h
R
h
q
a
1
1
1
11 =−−= 
 
writing the balance equation for tank 2 
 
dt
dh
Aqq 2221 =− 
 
dt
dh
A
R
h
R
h 2
2
2
2
1
1 =− 
 
writing steady state equations 
 
0
1
1 =−−
R
sh
R
h
q
a
s
s
 
0
2
2
1
1 =−
R
sh
R
sh
 
writing the equation in terms of deviation variables 
 
dt
dH
A
RR
HQ
a
1
1
1
1
11
=





+− 
 
dt
dH
A
R
H
R
H 2
2
2
2
1
1 =− 
 
taking laplace transforms 
 
sHA
RR
RR
sHsQ S
a
11
1
21
1 )()( =




 +
− -----------(1) 
 
and )(
)()(
22
2
2
1
1 sHsA
R
sH
R
sH
=− -----------(2) 
 
from (1) we get 





 +
+
=
a
a
RR
RR
sA
sQ
sH
1
1
1
1 1
)(
)(
 
 
 






+
+






+
=
1
)(
)(
1
11
1
1
1
s
RR
ARR
RR
RR
sQ
sH
a
a
a
a
 
 
[ ]1)(
)(
1
1
1
1
+






+
=
s
RR
RR
sQ
sH a
a
τ
 ; 
a
a
RR
ARR
+
=
1
11
1τ 
 
and from (2 ) we get 
[ ]( )11)(
)(
21
1
2
1
2
1
++












+
=
ss
R
R
RR
RR
sQ
sH a
a
ττ 222
AR=τ 
 
putting the numerical values of parameters 
 
 





 +






=
1
3
4
3
2
)(
)(1
s
sQ
sH
 
 
( )11
3
4
3
2
)(
)(2
+




 +






=
ss
sQ
sH
 
 
 
 
8.1 A step change of magnitude 4 is introduced into a system having the 
transfer 
 
46.1
10
)(
)(
2 ++
=
sssX
sY
 
Determine (a) % overshoot 
 (b)Rise time 
 (c)Max value of Y(t) 
 (d)Ultimate value of Y(t) 
 (e) Period of Oscillation. 
 
 
 
 Given 
s
sX
4
)( = 
)46.1(
40
)(
2 ++
=
sss
sY 
 
 
 The transfer function is 
)1)
4
6.1
()(2.0
25.010
)(
)(
2 ++
×
=
ss
sX
sY
 = 
)14.025.0
5.2
2 ++ ss
 
 
 
5.0;25.02 == ττ 
 
4.02 =τξand )1(4.0
)5.0(2
4.0
dunderdampeissystem=<==ξ 
 
we find ultimate value of Y(t) 
 
10
4
40
)46.1(
40
)()(
2
00
==
++
==
→→∞→ ss
s
LtssYLttYLt
SSt
 
 
thus B= 10 
 
now, from laplace transform tables 
 








+
−
−=
−
)sin(
1
1
110)(
2
φα
ξ
τ
ξt
etY 
 
where 
ξ
ξ
φ
τ
ξ
α
22 1
tan,
1 −
=
−
= − 
 
 
 
 
 
 
(a) Over shoot = 




 ×−
=








−
−
=
84.0
4.0
exp
1
exp
2
π
ξ
πξ
B
A
 = 0.254 
 
thus % overshoot = 25.4 
 
c)thus, max value of Y(t) = A+B = B(0.254)+B 
 = 2.54+10 = 12.54 
 
e) Period of oscillation = 
21
2
ξ
πτ
−
= 3.427 
 
b) For rise time, we need to solve 
 
r
t
ttforte ==








+
−
−
−
10)sin(
1
1
110
2
φα
ξ
τ
ξ
 
 
 
= )sin( φατ
ξτ
+
−
rte
r
= 0 
= 0)1589.1833.1sin(5.0
4.0
=+
−
rte
rτ
 
 
solving we get tr = 1.082 
 
thus 
 
SOLUTION: % Overshoot = 25.4 
 Rise time = 1.0842 
 Max Y(t) = 12.54 
 U(t) Y(t) = 10 
 Period of oscillation = 3.427 
 
Comment : we see that the Oscillation period is small and the decay ratio 
also small = system is efficiently under damped. 
 
 
 
8.2 The tank system operates at steady state. At t = 0, 10 ft
3
 of wateris 
added to tank 1. Determine the maximum deviation in level in both tanks 
from the ultimate steady state values, and the time at which each 
maximum occurs. 
 
A1 = A2 = 10 ft
3
 
 R1 = 0.1ft/cfm R2 = 0.35ft/cfm. 
 
 
As the tanks are non interacting the transfer functions are 
 
 
)1(
1.0
1)(
)(
1
1
+
=
+
=
ss
K
sQ
sH
τ
 
 
)15.3)(1(
35.0
)1)(1()(
)(
21
22
++
=
++
=
ssss
R
sQ
sH
ττ
 
 
Now, an impulse of providedisftt 310)( =∂ 
 
 
te
s
sHsQ −=
+
===
1
1
)(10)( 1 
 
and 
15.45.3
5.3
)15.3)(1(
5.3
)(
22 ++
=
++
=
ssss
sH 
 
Now 871.15.32 === ττ 
 
 202.1
2
5.4
5.42 ====
τ
ξξτ 
 
thus, this is an ovedamped system 
 
Using fig8.5, for 2.1=ξ , we see that maximum is attained at 
min776.1,95.0 == t
t
τ
 
 
And the maximum value is around 325.02 =τ Y2 (t) = 0.174 
 
= H2(t) = 0.174x3.5 = 0.16ft 
 
thus max deviation is H1 will be at t = 0 = H1 = 1 ft 
 
 max deviation is H2 will be at t = 1.776 min = H2Max = 0.61 ft. 
 
comment : the first tank gets the impulse and hence it max deviation turns out 
to be higher than the deviations for the second tank. The second tank exhibits an 
increase response ie the deviation increases, reaches the H2Max falls off to zero. 
 
 
 
 
8.3 The tank liquid level shown operates at steady state when a step 
change is made in the flow to tank 1.the transaient response in critically 
damped, and it takes 1 min for level in second tank to reach 50 % of total 
change. If A1/A2 = 2 ,find R1/R2 . calculate ττττ for each tank. How long does 
it take for level in first tank to reach 90% of total change? 
 
 
For the first tank, transfer function 
1
11
1)(
)(
+
=
s
R
sQ
sH
τ 
 
For the second tank 
)1)(1()(
)(
21
2
++
=
ss
R
sQ
sH
ττ
 
 
 
= 
1)()(
)(
21
2
21
22
+++
=
ss
R
sQ
sH
ττττ
 
 
 
1)(
1
)(;
1
)(
21
2
21
2
2 +++
==
ss
R
s
sH
s
sQ
ττττ
 
 
21)( τττ =parameterFor 21)( τττ =parameter 
 
 
 
















+−==
−
21
21
22 11)(,1
ττ
ττ
ξ
t
e
t
RtHfor 
 
 
given, t = 1 for 21)( τττ =parameter 
 
( ) 222 )0(1)( RRtH =−=∞→ 
 
 
I
R
eR −=
















+−=
−
2
1
11 2
1
21
2
21ττ
ττ
 
 
 
also 212 ττξτ += 
 
 
5.0
2
1
1
2
2
1
2211
21 ======
+
==
A
A
R
R
RARAτ
ττ
ξ 
 
 
from I 
 
 
τ
τ
1
1
15.01
−





 +=− e 
 
 
 
min372.1
min596.0
5.0
min372.1;1.0
)1(9.0
)1()(94.0
)1()(;
)1(
)()3.8
%90
21
2
1
596.0
596.0
11
1
11
1
1
1
1
1
=
==
=
==
−=
−=∞→
−=
+
=
−
−
−
−
t
R
R
thus
te
eRR
eRt
eRtH
ss
R
sH
t
t
t
t
ττ
τ
τ
τ
 
 
 
 
 
 
Comment : 
.tan
tansec,., 2121
kfirstthanchangestoslowlymoreresponds
ondtheRRasAlsoquicklystatesteadytheregainssystemtheindicateofvaluesSmall >ττ
 
 
 
8.4 Assuming the flow in the manometer to be laminar function between 
applied pressure P1 and the manometer reading h. Calculate a) steady sate 
gain ,b) τ ,c) ξ . Comment on the parameters and their relation to the 
physical nature of this problem. 
 
 
Assumptions: 
 
Cross-sectional area =a 
Length of mercury in column = L 
Friction factor = 16/Re (laminar flow) 
Mass of mercury = mrg 
 
Writing a force balance on the mercury 
 
 
Mass X acceleration = pressure force - drag force - gravitational force 
 
)(
2
)(
2
12
2
ghA
u
AfAp
dt
hd
AL ρ
ρ
ρ −−=
 
 
 
g
p
h
dt
dh
gDdt
hd
g
L
ρρ
µ 1
2
2 8
=++
 
 
At Steady state, g
p
h ss ρ
1=
 
 
= 
g
p
H
dt
dH
gDdt
Hd
g
L
ρρ
µ 1
2
2 8
=++ 
 
= [ ]
g
sp
sHssH
gD
sHs
g
L
ρρ
µ )(
)()(
8
)( 12 =++ 
 
 
= [ ] )()(1 13221 spksHsksk =++ 
 
= 
)1()(
)(
2
2
1
3
1
1
++
=
sksk
k
sp
sH
 
 
 
Where ;1
g
L
k = ;
8
2
gD
k
ρ
µ
= ;
1
3
g
k
ρ
= 
 
Thus 
)12()(
)(
22
1
1
++
=
ss
R
sp
sH
ξττ
 
 
 
Where ;
1
g
R
ρ
= ;2
g
L
=τ ;
8
2
gDρ
µ
ξτ = 
 
 
Now ;)
g
L
b =τ 
 
1
4
2
1
.
8
)
−














==
g
L
gDgD
c
ρ
µ
τρ
µ
ξ 
 
Steady state gain 
 
;
1
)(
0 g
RsGLt
S ρ
==
→
 
 
Comment : a) τ is the time period of a simple pendulum of Length L. 
 b) ξ is inversely proportional to τ , smaller the τ ,the system will 
tend to move from under damped to over damped characteristics. 
 
 
 
 
8.5 Design a mercury manometer that will measure pressure of upto 2 
atm, and give responses that are slightly under damped with ξ = 0.7 
 
Parameter to be decide upon : 
 
 
.a) Length of column of mercury 
.b) diameter of tube. 
 
 
Considering hmax to be the maximum height difference to be used 
 
 
;
13600*81.9
10*01325.1*2 5
maxmax1 === hghp ρ 
;51.1max mh = 
 
Assuming the separation between the tubes to be 30 cm, 
 
We get an additional length of 0.47 m; 
 
Which gives us the total length L= 1.5176.47 
 
L = 2 M 
 
Now, ξ = 0.7 = 7.0
4
=













L
g
gDρ
µ
 
 
00015.0
10*5.1
81.9*13600*74.0
2
81.9
*10*6.1*4
7.0
4
7
3
=
=== −
−
g
L
g
D
ρ
µ
 
 
 
As can be seen, the values yielded are not proper, with too small a diameter 
and too large a length. A smaller ξ value and lower measuring range of 
pressure might be better. 
 
 
8.6 verify that for a second order system subjected to a step response, 
 
[ ]
ξ
ξ
τ
ξ
ξ
τ
ξ 2
12
2
1
tan1sin
1
1
1)(
−
+−
−
−= −
− t
etY
t
 
 
 
With ξ <1 
 
)12(
11
)(
22 ++
=
sss
sY
ξττ
 
 
 
baswhere
ssssss
+−=
−
+
−
=
−−=++
τ
ξ
τ
ξ
ξττ
1
))((12
2
1
21
22
 
 
bas −−=
−
+
−
=
τ
ξ
τ
ξ 12
2 
 
 
))((
1
)(
21
2
sssss
sY
−−






=
τ
 






−
+
−
+=
)()(
1
)(
21
2 ss
C
ss
B
s
A
sY
τ
 
 
 
 






−
+
−
+=
)()(
1
)(
21
2 ss
C
ss
B
s
A
sY
τ
 
 
 
0
1)()())(( 12211
2
=++
=−+−+++−
CBA
ssCsssBsssssssA s
 
 
1)( 121 =−−+− CsBsssA s 
 
2121
1
11
;1
ss
CB
ss
AsAs s −=+=== 
 
 
 
1
22
1
s
CsBs −=+= 
 
2121
21
21
11
ssss
ss
CsCs =−
+
=+= 
)(
11
)(
1
)(
1
12121122122 ssssssss
B
sss
C
−
−=−
−
−==
−
== 
 
 
( ) ( ) ( )




−−
+
−=
−





=
2122112121
2
1
.
)(
11
.
11
.
11
)(
sssssssssssss
sY
τ
 
8.6 





−
+
−
−= tstS e
sss
e
sssss
tY 21
)(
1
)(
111
)(
12212121
2τ
 
 
 
21
2
1
ssτ
= 1 
 
 






−
−
−= tStS e
s
e
sss
tY 21
2112
2
11
)(
1
1)(
τ
 
 
 [ ]tStS eses
ss
tY 21 12
12 )(
1
1)( −
−
−= 
 
 
 [ ]tStS esestY 21 12
2 12
1)( −
−
+=
ξ
τ
 
 
 
 
 
 
 
[ ]btjbtjbabtjbtjbajetY
t
sin)(cos()sin)(cos(
12
1)(
2
−+−−+−−
−
−=
−
ξ
τ τ
ξ
 
 
 
 
 [ ])sincos(2
12
1)(
2
btabtbjb
je
tY
t
+−
−
−=
−
ξ
τ τ
ξ
 
 
 [ ])sin()cos(1
1
1)( 2
2
tt
e
tY
t
αξαξ
ξ
τ
ξ
+−
−
−=
−
 
 
[ ]
ξ
ξ
α
21−
= 
[ ]







 −
= −
ξ
ξ
φ
2
1 1tan 
verified 
 
 
8.14 From the figure in your text Y(4) for the system response is 
expressed 
 
 
 
 
 
 
 
 
 
 
 
 
 
b) verify that for ,1=ξ and a step input 
 
 
τ
τ
t
e
t
tY
−





 +−= 11)( 
1
11
)(
22 ++
=
sss
sY
ττ
 
 
 
22 )1()1(
1
)(
+
+
+=
+ ττ
CBs
B
A
ss
sY 
 
1)12(
222 =+++ CsBsssA ττ 
 
 
02 =+ BAτ 
 
 
02 =+CAτ 
 
 
A=1; 2τ=B ; τ2=C 
 
 
( )21
)1(1
)(
+
++
−=
s
s
s
sY
τ
τττ
 
 
 
( ) 2)1(1
1
)(
+
−
+
−=
sss
sY
τ
τ
τ
τ
 
 
 
ττ
τ
tt
teetY
−−
−−=
1
1)( 
 
τ
τ
t
te
t
tY
−
+−= )1(1)( 
 
proved 
 
 
 
c) for ,1>ξ prove that the step response is 
 
 
[ ])sinh()cosh(1)( ttetY
t
αβατ
ξ
+−=
−
 
 
 
1
1
2
2
−
=
−
=
ξ
ξ
β
τ
ξ
α 
 
 
 
Now 
))((
/1
)(
21
2
ssBss
sY
−−
=
τ
 
 
 
 
 
Where 
τ
ξ
τ
ξ 12
1
−
+−=s 
 
τ
ξ
τ
ξ 12
2
−
−−=s 
 
from 8.6(a) 
 
 
 






−−
+
−−
−=
)(
1
)(
1
)(
1
)(
1111
)(
2122112121
2 sssssssssssss
tY
τ
 
 
[ ]tStS eses
ss
tY 21 12
12 )(
1
1)( −
−
−= 
 



















 −+−
−







 −−−
−
+=
−
−−








 −
−
teeeetY
ttt
τ
ξ
τ
ξτ
ξ
τ
ξ
τ
ξξ
τ
ξξ
ξ
τ 12
1
1
2
2
2
2
11
12
1)( 
 












−−−−+−
−
+=
−
−
−
−−








 −
−
t
ttt
t
eeeee
e
tY τ
ξ
τ
ξ
τ
ξτ
ξ
τ
ξ
ξξξξ
ξ
1
2
1
2
1
2
22
2
11
12
1)( 
 
 













 +
−




 −
−
−+=
−−−
221
1)(
2
ttttt
eeee
etY
αααα
τ
ξ
ξ
ξ 
[ ])sinh()cosh(1)( ttetY
t
αβατ
ξ
+−=
−
 
 
8.7 Verify that for a unit step-input 
 
(1) overshoot = 








−
−
21
exp
ξ
πξ
 
(2) Decay ratio = 








−
−
21
2
exp
ξ
πξ
 
 
For a unit step input the response (ξξξξ<1): 
 















 −
+−
−
−= −





−
ξ
ξ
τ
ξ
ξ
τ
ξ
2
12
2
1
tan1
1
1)(
t
Sin
e
tY
t
 
(1) we have to find time t where the maxima occurs 
 
= dY/dt = 0 
 















 −
+−
−
= −





−
ξ
ξ
τ
ξ
ξτ
ξ τ
ξ
2
12
2
1
tan1
1
) t
Sin
e
dt
dY
t
0
1
tan1
2
12 =















 −
+−− −





−
ξ
ξ
τ
ξ
τ
τ
ξ
t
Cos
e
t
 
= 















 −
+− −
ξ
ξ
τ
ξ
2
12 1tan1tan
t
=
ξ
ξ 21−
 
 
π
ξ
ξ
n
t
=
− 21
 
for maxima 
 
= π
ξ
ξ
n
t
2
1 2
=
−
 
 
= 
21 ξ
π
−
=
t
t 
 
 
 
 
 
 
8.8 Verify that for X(t) =A sin ωωωωt, for a second order system, 
 
( ) ( )
)sin(
2)(1
)(
222
φω
ξτω
+
+−
= t
t
A
tY2
1
)(1
2
tan
ωτ
ξωτ
φ
−
−= − 
 
)12(
1
)(
)(
2222 +++
=
sss
A
sY
ξττω
 
 
 






−
+
−
+
+
+
−
=
)()()(
)(
2
1
1
111
2 ss
D
ss
C
js
B
js
AA
sY
ωωτ
ω
 
 
Now as tBtAtYt ωω sincos)(, 1111 +=∞→ 
 
 
Where 1111 BAA += 
 
)( 1111 BAjB −= 
 
to determine ordertheinjjsputBA ωω −= ,, 11 
 
 
))((2 21
1
sjsj
j
A
−−
−
=
ωωω ))((2 21
1
sjsj
j
B
++
=
ωωω
 
 
 
 






−−
−
++
=
))((
1
))((
1
2 2121
11
sjsjjsjs
j
A
ωωωωω
 
 
 
 






++
+++−−+−−−
=
))((
)()(
2 22
2
22
1
2121
2
2121
2
11
ωω
ωωωωωω
ω ss
ssjsjsssjsjsj
A 
 






++
+
=
))((
)(
22
2
22
1
2111
ωω ss
ss
A similarly 





++
−
=
))((
)(
22
2
22
1
2
2111
ωωω
ω
ss
ss
B 
 
 
using 
22121
12
ττ
ξ
=
−
=+ ssss 
 
 
=
2
2
22
2
2
2
2
1
)12(224
τ
ξ
ττ
ξ −
=−=+ ss 
 












+−+
−
=
42
2
2
4
2
11
)12(
21
2
ωξ
τ
ω
τ
τ
ξ
τ
ωA
A 
 
 
22
2
2
3
21
2





+




 −
−
=
τ
ξω
τ
ω
τ
ωξA
 = 
222 )2())(1(
2
ξωτωτ
ωξτ
+−
− A
 
 
 
and 



























+




 −





 −
=
22
2
2
2
2
11
21
1
τ
ξω
ω
τ
ω
ω
τ
τ
ϖA
B 
 
222
2
)2())(1(
))(1(
ξωτωτ
ωτ
+−
−
=
A
 
 
 
 
Thus 
211
11
)(1
2
tan
ωτ
ωτξ
φ
−
−
==
B
A
 
 
And,
222 )2())(1(( ξωυωτ +−
=
A
ANew (using NewABA =+
22
1111 
 
 
Thus, )(
)2())(1((
)(
222
φω
ξωυωτ
+
+−
= tSin
A
tY 
 
proved 
 
 
8.9) If a second- order system is over damped, it is more difficult to 
determine the parameters τξ & experimentally. One method for determining 
the parameters from a step response has been suggested by R.c Olderboung 
and H.Sartarius (The dynamics of Automatic controls,ASME,P7.8,1948),as 
described below. 
 
(a) Show that the unit step response for the over damped case may 
be written in the form. 
 
21
21
12
1)(
rr
erer
ts
trtr −
−= 
 
Where r1 and r2 are the roots of 
 
01222 =++ ss ξττ 
(b) Show that s(t) has an inflection point at 
 
)(
)/ln(
12
12
rr
rr
ti −
= 
 
© Show that the slope of the step response at the inflection point 
 
)(
)( 1
itt
ts
dt
sd
i
=
−
 
 
 
Where, i
trtr
i ererts
21
21
1 )( −=−= 
 
)(
1
2
1
211 rrr
r
r
r
−






−= 
(d) Show that the value of step response at the inflection point is 
)(1)( 1
21
211
ii ts
rr
rr
ts += and that hence 
21
1
11
)(
)(
1
rrts
ts
i
i −−=− 
 
 
(e) on a typical sketch of a unit step response show distances equal to 
 
)(
1
&
)(
)(
1
11
ii
i
tsts
ts
− 
(f) Relate 
21 && rrtoτξ 
 
(a) 
))((
1
12
1
)12
1
)(
21
2
2
2
2
22 rsrs
ss
ss
sG
−−
=






+




+
=
++
= τ
ττ
ξ
τ
ξττ
 
 
= 
))((
1
)(
21
2
rsrss
sY
−−
= τ 
 
 
2121 ))((
1
rs
C
rs
B
s
A
rsrss −
+
−
+=
−−
 
)()())((1 1221 rscsrsBsrsrsA −+−+−−= 
 
Put s= 0 = Ar1r2 =1 ; 
2τ=A 
 
Put 
)(
1
;1)(
211
2111
rrr
BrrBrrs
−
==−== 
)(
1
;1)(
122
1222
rrr
CrrCrrs
−
==−== 
 
 






−−
+
−−
+=
))((
1
))((
11
)(
21221211
2
2 rsrrrrsrrrs
sY
τ
τ
 
 
 
 






−
+
−
+=
)()(
1
)(
122211
2
2
21
rrr
e
rrr
e
tY
trtr
τ
τ
 
 
[ ]





−
−
−= trtr erer
rr
tY 12 21
21
1
1)( 
 
)(
1)(
21
21
12
rr
erer
tY
trtr
−
−
−=
φ
 
 
(b) For inflection point , 0&0
2
3
2
2
==
dt
sd
dt
sd
 
 
21
21 )(
22
rr
eerr
dt
ds
trtr
−
−
−= 
 
 
0
)(
21
1221
2
2 22
=
−
−
−=
rr
ererrr
dt
sd
trtr
 
 
 
 
itrrtrtr e
r
r
ereu
)(
1
2
12
2112 −==== 
 
 
 
21
1
2ln
rr
r
r
ti −






== 
 
 
(c ) )(
)( '
itt
ts
dt
tds
i
=
=
] 
 
 
















−





−
−=
−− 12
1
21
1
2
1
2
21
21
rr
r
rr
r
r
r
r
r
r
rr
rr
 
















−





−
−=
−
−
− 12
1
21
1
2
21
2
1
1
2
21
21
rr
r
rr
r
r
r
r
rr
r
r
r
rr
rr
 
 
= 
















−
−
−
− 12
1
1
2
21
211 )(
rr
r
r
r
r
rr
rrr
 
 
 
















−=
−
=
12
1
1
2
1
)( rr
r
tt
r
i r
r
r
dt
tds
 
 
 
Also 
)(
()(
21
)
21
12
rr
eerr
dt
tds
trtr
tt i −
−
−=
= 
 
 
 






−
−
−
−=
=
1
)(
)(
2
1
21
21
1
r
r
rr
err
dt
tds
tr
tt i
 
 
 
 
trtr
tt
erer
dt
tds
i
21
21
)(
−=−=
=
 
 
 
(d) 
21
1
2
2
11
21
21
)(
11)(
12
rr
r
r
r
r
ts
rr
erer
ts
irttr
i
ii
−






−
+=
−
−
−= 
 
 
= 
21
1
2
2
11 )(
1)(
rr
r
r
r
r
ts
ts
i
i −






−
+== 
 
 
Now 






+==−
21
1 11)(1)(
rr
tsts ii 
 






−−==
−
21
'
11
)(
)(1
rrts
ts i 
 
21
21221
1
;
11
rr
rrrr ====+ τ
ττ
 
 
;
2
21 =
−
=+
τ
ξ
rr ξ2121 2 rrrr −=+ 
 
 








+−=
1
2
2
1
2
1
r
r
r
r
ξ 
 
proved. 
 
 
8.10 Y(0),Y(0.6),Y(∞ ) if 
 
 
)12(
)1(251
)(
2 ++
+
=
ss
s
s
sY 
 
 
 
)1
25
2
25
(
11
1)(
2
++





 +=
sss
sY 
 
 
Y(s) impulse response + step response of G(s) 
 
 
Where 
)1
25
2
25
(
1
)(
2
++
=
ss
sG 
 
 
ξ
ξ
τ
ξ
τξ
τ
ξ 2
12
2
1
tan1sin
1
1
)(
−
+−
−
= −
− t
etY
t
 
 
 
 
Y(t) = 1+5.0.3e
-t
 sin (4.899t)-1.02e
-t
 sin(4.899t+1.369) 
 
 
Y(0)= 1-1=0 
Y(0.6) = 1+0.561+0.515 
Y(∞ ) =1 
 
Comment : as we can see ,the system exhibits an inverse response by 
increasing from zero to more than 1 and as t tend to ∞ ,will reach the 
steady state value of 1. 
 
 
8.11 In the system shown the dev in flow to tank 1 is an impulse of 
magnitude 5 . A1 = 1 ft
2
, A2 = A3 = 2 ft
2
 , R1 = 1 ft/cfm R2 = 1.5 ft/cfm . 
 
(a) Determine H1(s), H2(s), 
H3(s) 
 
 
Transfer function for tank 1 
)1(
1
)(
)(
1
1
+
=
ssQ
sH
τ
 
 
 
 
 
)1(
5
)(1 +
=
s
sH 
 
 
from tank 2, 
)13)(1(
5.1
)1)(1()(
)(
21
22
++
=
++
=
ssss
R
sQ
sH
ττ
 
 
 
for tank 3, 
dt
dh
Aqq c
3
32 =− 
 
dt
dh
AQconstqq c
3
323 )( === 
 
 
dt
dh
A
R
H 3
3
2
2 = 
 
 
thus, 
ssH
sH
R
sH
sSHA
3
1
)(
)()(
)(
2
3
2
2
33 === 
 
 
ssH
sH
R
sH
sSHA
3
1
)(
)()(
)(
2
3
2
2
33 === 
 
)13)(1(
5.0
)(
)(3
++
=
ssssQ
sH
 
 
 
8.11© 
)1(
5
)(1 +
=
s
sH 
 
 
 tetH −= 5)(1 
 
 
 AH 155.0)46.3(1 = 
 
143
5.1
)(
)(
2
2
++
=
sssQ
sH
 
 
143
5.7
)(5)1(
22 ++
===
ss
sHQ 
 
 
3=τ 
 
 
42 =ξτ 
 
155.1
32
4
2
4
===
τ
ξ 
 
 
from fig 8.5 
τ
ξ
t
and155.1= = 2
3
46.3
===
τ
ξ
t
 
 
 
 
 
5.7265.0)(2 XtH =τ 
 
 
147.1
5.7265.0
)(2 == τ
X
tH 
 
 
 
 
 
)143(
5.0
)(
)(
2
3
++
=
ssssQ
sH
 
 
 
 
)143(
5.2
)(5)(
23 ++
===
sss
sHsQ 
3=τ
3
2
=ξ 
 
from fig 8.2 at 155.1,2 == ξ
τ
t
 
 
 
Y(t) =0.54 
 
H3(t) =0.54*2.5 = 1.35 
 
 
 
8.12 sketch the response Y(t) if 
)12.1(
)(
2
2
++
=
−
ss
e
sY
S
 
 
 
Determine Y(t) for t = 0,1,5,∞ 
 
22
2
22
2
2
2
)8.0()6.0(
)8.0(
8.0
1
)8.0()6.0()12.1(
)(
++
=
++
=
++
=
−−−
s
e
s
e
ss
e
sY
SSS
 
0)(,
14.0)5(,5
0)1(,1
0)0(0
2))2(8.0sin(25.1)( )2(6.
=∞∞=
==
==
==
≥−= −−
Yt
Yt
Yt
Ytfor
ttetY t
 
 
 
 
 
Problem 8.13 The system shown is at steady state at t = 0, with q = 10 
cfm 
A1 = 1ft
2
,A2=1.25ft
2
, R1= 1 ft/cfm, R2= 0.8 ft/cfm. 
 
a) If flow changes fro 10 to 11 cfm, find H2(s). 
b) Determine H2(1),H2(4),H2(∞ ) 
c) Determine theinitial levels h1(0),h2(0) in the tanks. 
d) obtain an expression for H1(s) for unit step change. 
 
 
 
Writing mass balances, 
 
( )
)1tan(11
1
21 kfor
dt
dh
A
R
hh
q =
−
− 
 
At steady state 
( )
SS
ss hh
R
hh
q 21
1
21
2 −=
−
− 
 
Also for tank 2 
( )
dt
dh
A
R
h
R
hh 2
2
2
2
1
21 =−
−
 
 
At steady state 
( )
810*8.0
8.01
2
221 ====
−
S
Sss h
hhh
 
 
181 =Sh 
 
C) 181 =Sh ft fth 8)0(2 = 
 
 
 
 
The equations in terms of deviation variables 
 
 
dt
dH
AQQ 111 =− where 
1
21
1
R
HH
Q
−
= 
 
dt
dH
AQQ 2221 =−
2
2
2
R
H
Q = 
18.2
8.0
1)()(
)(
2
2121
2
21
22
++
=
++++
=
sssRAs
R
sQ
sH
ττττ
 
 
))(31.8(
)18.2(
8.0
)(
22
aAns
sss
sH
++
= 
 
 
 
Step response of a second order system 
 
4.1
2
8.2
;8.22
112
===
===
ξξτ
ττ
 
 
 
 
 
)(176.0)22.0(8.0)(;11) 2 figfromfttH
t
ta =====
τ
)(624.0)78.0(8.0)(;44) 2 figfromfttH
t
tb =====
τ
 
fttHtc 8.0)() 2 =∞→=∞→ 
 
 
Thus 
ftH 176.0)1(2 = 
ftH 624.0)4(2 = 
ftH 8.0)(2 =∞ 
 
 
8.13(d) we have 
 
)()()()( 111 sHsAsQsQ = 
 
)()()()( 2221 sHsAsQsQ =− 
 
 
)()()()()()( 22112 sHsAsHsAsQsQ +=− 
 
 






+=− )()(
1
)()()( 22
2
11 sHsA
R
sHsAsQ 













 +
= )(
1
2
2
2 sH
R
sτ
 
 
 





 −
=
sH
ssHAsQR
sH
2
112
2
)()((
)(
τ
 
 
 
 
We have 
Deg
R
sRAs
R
sQ
sH 2
2121
2
21
22
1)()(
)(
=





++++
=
ττττ
 
 
 






+
−
=
)1(
)()(()(
2
11222
s
ssHAsQR
Deg
sHR
τ
 
 






−−=
+
)(
)(
1
1 1
1
2
sQ
sH
sA
Deg
sτ
 
 
 
 





 −−
=





−=
Deg
sDeg
sADeg
sH
sAsQ
sH
2
1
2
1
11)(
1
1
)(
)( ττ
 
 
 





 −−++++
=
Deg
ssRAs
sAsQ
sH
22121
2
21
1
11)(1
)(
)( τττττ
 
 
 
 
=




 ++
=
Deg
sRAs
sAsQ
sH )1
)(
)(
21121
1
τττ
 
 
 






++++
++
=
1)(
)1(
)(
)(
2121
2
21
212
sRAs
sRR
sQ
sH
ττττ
τ
 
 
 
 
8.14 
 
 
( )
)18.04(
422
)(
2 ++
+
=
ss
s
s
sY 
 
 
 
( )
)18.04(
24
)(
2 ++
+
=
ss
s
s
sY 
 
 
)18.04(
12
14)(
2 ++





 +=
sss
sY 
 
 
)18.04(
8
)(
2 ++
=
sss
sY +
)18.04(
4
2 ++ ss
 
 
 
 = (step response) + (impulse response) 
 
 
24, ==τNow ; 8.02 =ξτ 
 
 
 2.0=ξ 
 
 
also, 2
2
4
==
τ
t
 
 
 
impulse response τY(t) = 4*0.63 = 2.52 (from figure) 
 
step response = 8*1.15 = 9.2 (from figure) 
 
Y(4) = 1.26+9.2 
 
Y(4) =10.46 
 
 
 
 
 Q 9.1. Two tank heating process shown in fig. consist of two identical, 
well stirred tank in series. A flow of heat can enter tank2. At t = 0 , the flow rate of heat 
to tank2 suddenly increased according to a step function to 1000 Btu/min. and the temp 
of inlet Ti drops from 60oF to 52oF according to a step function. These changes in heat 
flow and inlet temp occurs simultaneously. 
(a) Develop a block diagram that relates the outlet temp of tank2 to inlet 
temp of tank1 and flow rate to tank2. 
(b) Obtain an expression for T2’(s) 
(c) Determine T2(2) and T2(∞) 
(d) Sketch the response T2’(t) Vs t. 
 
Initially Ti = T1 = T2 = 60
o
F and q=0 
W = 250 lb/min 
Hold up volume of each tank = 5 ft3 
Density of the fluid = 50 lb/ft
3 
Heat Capacity = 1 Btu/lb (oF) 
 
 
Solution: 
(a) For tank 1 
w 
Ti 
T1 
T2 
w 
q 
 
Input – output = accumulation 
WC(Ti – To) - WC(T1 – To) = ρ C V 
dt
dT1 -------------------------- (1) 
At steady state 
WC(Tis – To) - WC(T1s – To) = 0 ------------------------------------(2) 
 
(1) – (2) gives 
WC(Ti – Tis) - WC(T1 – T1s) = ρ C V 
dt
dT 1
'
 
WTi
’
 - WT1
’
 = ρ V 
dt
dT 1
'
 
Taking Laplace transform 
 
WTi(s) = WT1(s) + ρ V s T1(s) 
 
ssTi
sT
τ+
=
1
1
)(
)(1 , where τ = ρ V / W. 
 
From tank 2 
 
q + WC(T1 – To) - WC(T2 – To) = ρ C V 
dt
dT2 -------------------------- (3) 
At steady state 
 qs + WC(T1s – To) - WC(T2s – To) = 0 ------------------------------------(4) 
 
(3) – (4) gives 
Q ‘ + WC(T1 – T1s) - WC(T2 – T2s) = ρ C V 
dt
dT 2
'
 
Q ‘ + WCT1
’ - WCT2
’ = ρ C V 
dt
dT 2
'
 
Taking Laplace transform 
 
Q (s) + WC(T1(s) - T2(s)) = ρ C V s T2(s) 
 



 +
+
= )(
)(
1
1
)( 12 sT
WC
sQ
s
sT
τ
 , where τ = ρ V / W. 
 
(b) τ = 50*5/250 = 1 min 
 WC = 250*1 = 250 
 Ti(s) = -8/s and Q(s) = 1000/s 
Now by using above two equations we relate T2 and Ti as below and after taking laplace 
transform we will get T2(t) 
( )
( )
( )
4)84()(
1
1
1
11
8
)1(
11
4)(
1
8
)1(
4
)(
)(
1
1
250
)(
1
1
)(
2
22
22
22
−+=






+
−
+
−−





+
−=
+
−
+
=
+
+
+
=
−t
i
ettT
sssss
sT
ss
sT
sT
s
sQ
s
sT
ττ
 
 
(c) T2’(2) = -1.29 
 T2(2) = T2’(2) + T2s = 60 – 1.29 = 58.71 
o
F 
T2’(∞) = -4 
 T2(∞) = T2’(∞) + T2s = 60 – 4 = 56 
o
F 
 
 
 
Q – 9.2. The two tank heating process shown in fig. consist of two identical , well stirred 
tanks in series. At steady state Ta = Tb = 60
o
F. At t = 0 , temp of each stream changes 
according to a step function 
 Ta’(t) = 10 u(t) Tb’(t) = 20 u(t) 
(a) Develop a block diagram that relates T2’ , the deviation in the temp of tank2, 
to Ta’ and Tb’. 
(b) Obtain an expression for T2’(s) 
(c) Determine T2(2) 
 
W1 = W2 = 250 lb/min 
V1 = V2 = 10 ft
3 
ρ1 = ρ2 = 50 lb/ft
3 
C = 1 Btu/lb (oF) 
0.5 
0.85 
-4 
0 
T2’(t) 
t 
 
 
Solution: 
(a) For tank1 
 
ssTa
sT
1
1
1
1
)(
)(
τ+
= , where τ1= ρ V / W1. 
 
 For tank2 
W1C(T1 – To) +W2C(Tb – To) – (W1+W2)C(T2 – To)= ρ C V 
dt
dT2 ------ (1) 
At steady state 
W1C(T1s – To) +W2C(Tbs – To) – (W1+W2)C(T2s – To)= 0 -----------------(2) 
 
(1) – (2) 
W1T1
’
 + W2Tb
’
- W3T2
’
 = ρ V 
dt
dT 2
'
 
Taking L.T 
W1T1(s) + W2Tb(s)- W3T2(s) = ρVs T2(s) 
W1 
Ta 
T1 
T2 
W3=W1+W2 
Tb 
W2 
 
W1 
[ ])(
3
2)(
3
1
1
1
)( 12 STW
WST
W
W
s
sT b++
=
τ
 where τ= ρ V / W3. 
 
 
 
 
(b) τ1 = 50*10/250 = 2 min 
 τ = 50*5/250 = 1 min 
 W1/W3 = 1/2 = W2/W3 
 Ta(s) = 10/s and Tb(s) = 0/s 
Now by using above two equations we relate T1 and Ta as below and after taking laplace 
transform we will get T2(t) 
( )
( )
( )
( )
2
2
2
2
2
2
1
2
10515)(
21
20
1
515
)(
)21)(1(
2015
)(
1
10
)21)(1(
5
)(
1
)(
2
1
)21)(1(
)(
2
1
)(
1
)(
2
1
)1(
)(
2
1
)(
t
t
ba
b
eetT
sss
sT
sss
s
sT
sssss
sT
s
sT
ss
sT
sT
s
sT
s
sT
sT
−− −−=






+
−
+
−=
++
+
=
+
−
++
=
+
−
++
=
+
−
+
=
 
 
(c) T2’(2) = 10.64 
o
F 
 T2(2) = T2’(2) + T2s = 60 + 10.64 = 70.64 
o
F 
 
 
Q – 9.3. Heat transfer equipment shown in fig. consist of tow tanks, one nested inside the 
other. Heat is transferred by convection through the wall of inner tank. 
1. Hold up volume of each tank is 1 ft
3
 
2. The cross sectional area for heat transfer is 1 ft
2
 
3. The over all heat transfer coefficient for the flow of heat between the tanks is 10 
Btu/(hr)(ft
2
)(
o
F) 
4. Heat capacity of fluid in each tank is 2 Btu/(lb)(
o
F) 
5. Density of each fluid is 50 lb/ft
3
 
Initially the temp of feed stream to the outer tank and the contents of the outer tank are 
equal to 100 oF. Contents of inner tank are initially at 100 oF. the flow of heat to the inner 
tank (Q) changed according to a step change from 0 to 500 Btu/hr. 
(a) Obtain an expression for the laplace transform of the temperature of inner 
tank T(s). 
(b) Invert T(s) and obtain T for t= 0,5,10, ∞ 
 
Solution: 
 (a) For outer tank 
WC(Ti – To) + hA (T1 – T2)- WC(T2 – To) = ρ C V2 
dt
dT2 -------------------------- (1) 
At steady state 
WC(Tis – To) + hA (T1s – T2s)- WC(T2s – To) = 0 ------------------------------------ (2) 
 
(1) – (2) gives 
 
WCTi’ + hA (T1’ – T2’)- WCT2’ = ρ C V2 
dt
dT '2 
Substituting numerical values 
10 Ti’ + 10 (T1’ – T2’) – 10 T2’ = 50
dt
dT '2 
Taking L.T. 
Ti(s) + T1(s) – 2T2(s) = 5 s T2(s) 
Now Ti(s) = 0, since there is no change in temp of feed stream to outer tank. Which gives 
ssT
sT
52
1
)(
)(
1
2
+
= 
Q 
10 lb/hr 
T1 
T2 
 For inner tank 
Q - hA (T1 – T2) = ρ C V1 
dt
dT1 --------------------- (3) 
Qs - hA (T1s – T2s) = 0 ------------------------------- (4) 
(3) – (4) gives 
Q’ - hA (T1’– T2’ ) = ρ C V1 
dt
dT '1 
Taking L.T and putting numerical values 
Q(s) – 10 T1(s) + 10 T2(s) = 50 s T1(s) 
Q(s) = 500/s and T2(s) = T1(s) / (2+ 5s) 
)(50
52
)(10
)(10
500
1
1
1 ssT
s
sT
sT
s
=
+
+− 



 +
+
−= 1
52
1
5)(
50
1
s
ssT
s
 
)11525(
)52(50
)(
21 ++
+
=
sss
s
sT 
 
( )( )
50
18.26
50
82.3
)52(2
)(1
++
+
=
sss
s
sT 
( ) ( )
50
18.26
29.5
50
82.3
71.94100
)(1
+
−
+
−=
sss
sT 
50
18.26
50
82.3'
1 e 5.29 -e 94.71 - 100 (t)T
tt −−
= 
and 
50
18.26
50
82.3
1 e 5.29 -e 94.71 - 200 (t)T
tt −−
= 
For t=0,5,10 and ∞ 
T(0) = 100 
o
F 
T(5) = 134.975
 o
F 
T(10) = 155.856 oF 
T(∞) = 200
 o
F 
 
 Q – 10.1. A pneumatic PI controller has an output pressure of 10 psi, 
when the set point and pen point are together. The set point and pen point are suddenly 
changed by 0.5 in (i.e. a step change in error is introduced) and the following data are 
obtained. 
 
 
 
 
 
 
 
 
Determine the actual gain (psig per inch displacement) and the integral time. 
Soln: 
e(s) = -0.5/s 
for a PI controller 
Y(s)/e(s) = Kc ( 1 + τI
-1
/s) 
Y(s) = -0.5Kc ( 1/s + τI
-1
/s
2
) 
Y(t) = -0.5Kc ( 1 + τI
-1 t ) 
At t = 0
+
 y(t) = 8 � Y(t) = 8 – 10 = -2 
2=0.5Kc 
Kc = 4 psig/in 
At t=20 y(t) = 7� Y(t) = 7-10 = -3 
3 = 2 ( 1 + τI
-1 20 ) 
τI = 40 sec 
 
Q-10.2. a unit-step change in error is introduced into a PID controller. If KC = 10 , τI = 1 
and τD = 0.5. plot the response of the controller P(t) 
Soln: 
P(s)/e(s) = KC ( 1 + τD s+ 1/ τIs) 
For a step change in error 
Time,sec Psig 
0
-
 10 
0
+
 8 
20 7 
60 5 
90 3.5 
P(s) = (10/s)(1 + 0.5 s + 1/s ) 
P(s) = 10/s + 5 + 10/s
2
 
P(t) = 10 + 5 δ(t) + 10 t 
 
 
Q – 10.3. An ideal PD controller has the transfer function 
 P/e = KC ( τD s + 1) 
 An actual PD controller has the transfer function 
 P/e = KC ( τD s + 1) / (( τD/β) s + 1) 
 Where β is a large constant in an industrial controller 
 If a unit-step change in error is introduced into a controller having the second 
transfer function, show that 
 P(t) = KC ( 1 + A exp(-βt/ τD)) 
 Where A is a function of β which you are to determine. For β = 5 and KC =0.5, 
plot P(t) Vs t/ τD. As show that β � ∞, show that the unit step response approaches that 
for the ideal controller. 
Soln: 
P/e = KC ( τD s + 1) / (( τD/β) s + 1) 
For a step change, e(s) = 1/s 
P(s) = KC s( τD s + 1) / (( τD/β) s + 1) 
10 
15 
10(1+t) 
P(t) 
t 
 = 












+




 −
+
β
τ
βτ
ss
K
D
D
C
1
11
1
 
P(t) = 
















 −
+
−
D
t
D
D
C eK
τ
β
β
τ
βτ
11
1 
 = 





−+
−
D
t
C eK
τ
β
β )1(1 
So, A = β – 1 
P(t) = 0.5 ( 1 + 4 exp(-5t/ τD)) 
 
As β � ∞ then τD/β � 0 and 
P/e = KC ( τD s + 1) / (( τD/β) s + 1) becomes 
P/e = KC ( τD s + 1) that of ideal PD controller 
 
Q – 10.4. a PID controller is at steady state with an output pressure of a psig. The set 
point and pen point are initially together. At time t=0, the set point is moved away from 
2.5 
0.5 
P(t) 
t/τD 
the pen point at a rate of 0.5 in/min. the motion of the set point is in the direction of lower 
readings. If the knob settings are 
 KC = 2 psig/in of pen travel 
 τI = 1.25 min 
 τD = 0.4 min 
plot output pressure Vs time 
Soln: 
Given de/dt = -0.5 in/min 
s e(s) = -0.5 
Y(s)/e(s) = KC ( 1 + τD s+ 1/ τIs) 
Y(s) = -( 1/s + 1/ τIs
2
 + τD ) 
Y(t) = -( 1 + t/1.25 + 0.4 δ(t) ) 
Y(t) = y(t) – 9 = - ( 1 + t/1.25 + 0.4 δ(t) ) 
y(t) = 8 – 0.8 t – 0.4 δ(t) 
 
 
Q – 10.5. The input (e) to a PI controller is shown in the fig. Plot the output of the 
controller if KC = 2 and τI = 0.5 min 
9 
8 
7.6 
10 
y(t) 
t 
 
 
e(t) = 0.5 ( u(t) - u(t-1) - u(t-2) + u(t-3) ) 
e(s) = (0.5/s) ( 1 – e
-s 
 - e
-2s
 + e
-3s
 ) 
P(s)/e(s) = KC ( 1 + (1 / τI s) ) = 2 ( 1+ 2/s ) 
P(s) = ( 1/s + 2/s2 ) (1 – e-s - e
-2s + e-3s ) 
P(t) = 1 + 2t 0 ≤ t < 1 
 = 2 1 ≤ t < 2 
 = 5 – 2t 2 ≤ t < 3 
 = 0 3 ≤ t < ∞ 
 
 
Q – 12.1. Determine the transfer function Y(s)/X(s) for the block diagrams shown. 
Wxpress the results in terms of Ga, Gb and Gc 
 
0 1 
2 3 
4 
t, min 
0.5 
e 
-0.5 
Soln. 
(a) Balances at each node 
(1) = GaX 
(2) = (1) – Y = GaX – Y 
(3) = Gb(2) = Gb(GaX – Y) 
(4) = (3) + X = Gb(GaX – Y) + X 
Y = Gc(4) = Gc (Gb(GaX – Y) + X) 
 = GaGbGcX – GbGcY + GcX 
GbGc
GaGbGc
X
Y
+
+
=
1
)1(
 
(b) Balances at each node 
 (1) = X – (4) 
 (2) = Gb(1) = Gb( X – (4)) 
 (5) = GcX/Ga 
 (3) = Gc(2) = GbGc( X – (4)) 
 (4) = (3) + (5) --------------------------- 5 
 = GbGc( X – (4)) + GcX/Ga 
 Y = Ga(4) 
From the fifth equation 
 (4) = GbGcX – GbGc(4) + GcX/Ga ----------- 6 
 
GaGbGc
XGcGaGbGc
)1(
)(
)4(
+
+
= 
From the sixth equation 
 
)1(
)1(
GbGc
GcGaGb
X
Y
+
+
= 
Q – 12.2 
Find the transfer function y(s)/X(s) of the system shown 
 
 
Soln: 
Balance at each node 
(1) = X – Y ---------(a) 
(2) = (1) + (3) ----------(b) 
(3) = G1(2) where G1 = 1/(τ1s + 1) ----------(c) 
(4) Y = G2(3) where G2 = 0.5/(τ1s/2 + 1) ----------(d) 
From (d) and (c) 
 Y = (2)G1G2 
 = G1G2 (X – Y + (3) ) ----------(e) 
Also from (b) and (c) 
 (3) = G1((1) + (3)) 
 (3)(1 – 1/(τ1s + 1)) = 1/(τ1s + 1) 
 (3) τ1s = 1 
 (3) = 1/(τ1s ) = (X – Y) / (τ1s) 
Substitute this in (e) 
 ( )YX
sss
Y −





+
++
=
11
1
1
1
)1
2
)(1(
5.0
τττ
 
 
12
1
1
22
1 ++
=
ssX
Y
ττ
 
 
Q – 12.3. For the control system shown determine the transfer function C(s)/R(s) 
 
 
Soln. 
Balances at each node 
(1) = R – C ------------------(a) 
(2) = 2 (1) = 2(R – C) ------------------(b) 
(3) = (2) – (4) = 2(R – C) – (4) -------------------(c) 
(4) = (3)/s = (2(R – C) – (4))/s -------------------(d) 
(5) = (4) – C -------------------(e) 
C = 2(5) -------------------(f) 
Solving for (4) using (d) 
 s (4) = 2(R – C) – (4) 
 (4) = 2(R – C) / (s +1) 
Using (e) 
(6) = 2(R – C) / (s +1) – C 
( ) 


 −−
+
= CCR
s
C
1
2
2 
( )
73
4
)1(24)1(4
+
=
++++=
sR
C
ssCR
 
 
Q – 12.4. Derive the transfer function Y/X for the control system shown 
 
 
Soln. 
Balance at each node 
(1) = (5) + X -----------------(a) 
(2) = (1) – (4) -----------------(b) 
(3) = (2)/s ------------------(c) 
Y = (3)/s ------------------(d) 
(5) = 2 (3) ------------------(e) 
(4) = 25Y ------------------(f) 
From (b) 
(4) = (1) – (2) 
= (1) – s (3) from (c) 
= (1) – s
2
 Y from (d) 
= (5) + X - s2 Y from (a) 
= 2 (3) + X - s
2
 Y from (e) 
= 2 s Y + X - s2 Y 
From (f) 
Y = (2 s Y + X - s
2
 Y)/25 
X = Y( 25 – 2s + s
2
 ) 
252
1
2 +−
=
ssX
Y
 
 
 
 
13.1 The set point of the control system in fig P13.1 given a step change 
of 0.1 unit. Determine 
(a) The maximum value of C and the time at which it occurs. 
(b) the offset 
(c) the period of oscillation. 
Draw a sketch of C(t) as a function of time. 
 
 
)12)(1(
5
1
)12)(1(
5
++
+
++
=
ss
K
ss
R
C
 
 
932
8
2 ++
=
ssR
C
 
 
s
R
1.0
= 
b) 0889.0
9
8.0
932
8.0
)(
20
==
++
=∞
→ ss
LtC
S
 
 
offset = 0.0111 
c) 
22
1
3
1
2;
3
2
;
9
8.0
=⇒=== ξξττK 
 
overshoot = 305.0
1
exp
2
=








−
−
ξ
πξ
 
= Maximum vslue of C = 1.0305*0.0889=0.116 
Maximum value of C = 0.116 























 −
+−
−
−= −
−
ξ
ξ
τ
ξ
ξ
τ
ξ 2
122
1
tan1sin
1
1
1
9
8.0
116.0
t
e
t
 
 
6.1
1
tan
1
2
1
2
=
−
−
= −
ξ
ξ
ξ
τ
t 
 
Time at which Cmax occurs = 1.6 
(c ) Period of ociullation is 166.3
1
2
2
=
−
=
ξ
πτ
T 
T =3.166 
 
Decay ratio = (overshoot)
2
 = 0.093 
 
 
 
 
 
13.2 The control system shown in fig P 13.2 contains three-mode controller. 
(a) For the closed loop, develop formulas for the natural period of oscillation 
τ and the damping factor ξ in terms of the parameters K, Dτ , Iτ and 1τ . 
(b) Calculate ξ when K is 0.5 and when K is 2. 
(c ) Do ξ & τ approach limiting values as K increases, and if so, what are 
these values? 
(d ) Determine the offset for a unit step change in load if K is 2. 
(e ) Sktech the response curve (C vs t) for a unit-step change IN LOAD 
WHEN k is 0.5 and when K is 2. 
(f) In both cases of part (e) determine the max value of C and the time at 
which it occurs. 
 






++
+
+






++
+
=
s
s
s
k
s
sk
s
R
C
a
I
D
I
D
τ
τ
τ
τ
τ
τ
1
1
1
1
1
1
1
1
)
1
1
 
 
( )














++
+
+
+
=
s
sk
s
s
U
C
I
D τ
τ
τ
τ
1
1
1
1
1
1
1
1
1 
 
 
= 
1
12 +




 ++





+ s
k
k
s
k
k
s
I
II
DI
I
τ
ττ
ττ
τ
 
 






++++






++
=
s
sks
s
sk
R
C
I
D
I
D
τ
ττ
τ
τ
1
11
1
1
1
 
 
 
( )
ksksk
ssk
R
C
IIDI
IID
++++
++
=
τττττ
τττ
)1()(
1
2
1
2
 
 
k
k
k
k IIDI ττξ
τττ
τ
)1(
2;
)(2 +=
+
= 
 
k
k
k
k IIDI τξ
τττ )1()(
2
+
=
+
×= 
 
 
)(2
)1(
1ττ
τ
ξ
+
+
==
D
I
kk
k
 
 
)(2
)1()(4
)(
2
1
2
1
2
1
1
2
ττ
τττ
τττ
π
ξ
τπ
+
+−+
+
×
=
−
==
D
DD
DI
kk
kkk
k
k
T 
 
 
21
1
)1(4
)(4
+−













+





+
=
kkk
k
T
II
D
D
τ
τ
τ
τ
ττπ
 
 
 
B) Dτ = Iτ =1; 1τ .=2 
 
For k = 0.5 ; ξ =0.75 671.0
)5.2(5.0
1
= 
For k = 2 ; ξ =1.5 530.0
32
1
=
×
 
C) 






+





 +
=
+
+
==
k
k
kk
k
I
D
I
D
I
τ
τ
τ
ττ
τ
ξ
2
1
2
1
1
2
1
)(
)1(
2
1
 
 
 
As 3535.0
2
1
,
1
==∞→
τ
τ
ξ Ik 
 
21
1
)1(4
)(4
+−













+





+
=
kkk
k
T
II
D
D
τ
τ
τ
τ
ττπ
 
 
 
k
k IDI τττττ 1
+
= 
 
 
k
I
DI
1τττττ += 
 
DIK τττ =∞→ =1 
 
2552.7
14
4
, =
−
=∞→
I
D
DTkAs
τ
τ
πτ
 
 
(d) 






++
+
+
+
=
s
s
s
k
s
U
C
I
D τ
τ
τ
τ
1
1
1
1
1
1
1
1 
 
ksksk
s
U
C
IDI
I
++++
=
2
1 )()1( ττττ
τ
 
 
s
U
1
= 
 
ksksk
C
IDI
I
++++
=
2
1 )()1( ττττ
τ
 
 
0
0
)(
0
==
→ k
ssfLt
S
 
 
 
K=2 
 
For a unit step change in U 
 
 
0)( =∞C 
 
Offset = 0 
 
(e) k = 0.5, ξ =0.671 & τ =2.236 
 
95.18
1
2
2
=
−
=
ξ
πτ
T 
 
 
If k = 0.5 
 
135
2
2 ++
=
ss
s
U
C
; 
135 2 ++
=
ss
s
C 
 
 
If k = 2 
15.12
5.0
2 ++
=
ss
s
U
C
;
15.12
5.0
2 ++
=
ss
C 
 
 
In general 
τ
ξ
ξτ
τ
ξ
t
etC
t
2
2
1sin
1
11
)( −
−
=
−
 
 
The maximum occurs at 
ξ
ξ
ξ
τ 21
2
1
tan
1
−
−
= −t 
 
If k = 0.5 tmax = 2.52 Cmax=0.42 
If k = 2 tmax = 1.69 Cmax=0.19 
 
 
13.3 The location of a load change in a control loop may affect the system 
response. In the block diagram shown in fig P 13.3, a unit step chsange 
in load enters at either location 1 and location 2. 
(a) What is the frequency of the transient response when the load enters at 
location Z? 
(b) What is the offset when the load enters at 1 & when it enters at 2? 
(c) Sketch the transient response to a step change in U1 and to a step 
change in U2. 
 
0;
1
21 == U
s
U 
C
ss
UCRS =





+






+
+−
12
1
12
2
))(( 1 
R = 0 
 
1144
2
1)12(
2
5
)12(
2
2
2
2
1 ++
=
++
×
+
=
ss
s
s
U
C
 
 
1144
2
2
1 ++
=
ssU
C
 
 
 
 
11
1
11
4
2;
11
2
;
11
2
=⇒=== ξξττK 
 
2516.0
2
10
2
11
2
11
2
==
−
===
πτ
ξ
πT
fFrequency 
 
C(∞) = 2/11 
Offset = 2/11 =0.182 
 
U1=0;U2=1/s 
 
C
s
UCR
s
=





+





 +−
+
×⇒
12
1
)(
12
2
5 2 
 
 
R=0 
 
1
12
1
12
10
12
1
2 +
+
×
+
+=⇒
ss
s
U
C
 
 
1144
12
2
2 ++
+
=
ss
s
U
C
 
 
C(∞) = 1/11 
 
Offset = 1/11=0.091 
 
a) if ;
1
1
s
U = frequency = 0.2516 
if ;
1
2
s
U = frequency = 0.2516 
b)if ;
1
1
s
U = frequency = 0.182 
if ;
1
2
s
U = frequency = 0.091 
 
 
13.5A PD controller is used in a control system having first order process and a 
measurement lag as shown in Fig P13.5. 
(a) Find the expressions for ξ and τ for the closed –loop response. 
(b) If τ1 = 1 min, τm = 10 sec, find KC so that ξ = 0.7 for the two cases: (1) 
τD =0,(2) τD =3 sec, 
(c) Compare the offset and period realized for both cases, and comment on the 
advantage of adding derivative mode. 
 
 
)1)(1(
)1(
1
)1(
)1(
)
1
1
++
+
+
+
+
=
ss
sK
s
sK
R
C
a
m
DC
DC
ττ
τ
τ
τ
 
 
)1()(
)1)(1(
1
2
1 +++++
++
=
CDCmm
mDC
KsKs
ssK
R
C
τττττ
ττ
 
 
= 
1
12
+
=
C
m
K
ττ
τ 
1
1
+
=
C
m
K
ττ
τ 
 
)1(2
1
1
1
+
++
=
Cm
DCm
k
k
ττ
τττ
ξ 
 
b) 7.0;10min;11 === ξττ sm 
 
 
1)
)1(600
70
2
1
7.0
0
+
×=⇒
=
C
D
k
τ
 
 
kc=3.167 
2) 
255.5
)1(600
370
2
1
7.0
3
=
+
+
×=⇒
=
C
C
C
D
k
k
k
sτ
 
 
c)for 
1
)(;
1
+
=∞=
C
C
k
k
c
s
R 
 
periodsperiodsfor
periodperiodfor
offsetCsfor
offsetCFor
D
D
D
D
==
−
==
==
−
×
==
==∞=
==∞=
17.86
)7.0(1
255.6
600
2
;3
57.105
1
167.4
600
2
;0
16.0;84.0)(;3
24.0;76.0)(;0
2
2
π
τ
ξ
π
τ
τ
τ
 
 
Comments: 
 
Advantage of adding derivative mode is lesser offset lesser period 
 
13.6The thermal system shown in fig P 13.6 is controlled by PD controller. 
Data ; w = 250 lb/min; ρ = 62.5 lb/ft3; 
 V1 = 4 ft
3,V2=5 ft
3; V3=6ft
3; 
 C = 1 Btu/(lb)(°F) 
 
Change of 1 psi from the controller changes the flow rate of heat of by 500 
Btu/min. the temperature of the inlet stream may vary. There is no lag in the 
measuring element. 
(a) Draw a block diagram of the control system with the appropriate transfer 
function in each block.Each transfer function should contain a numerical values of 
the parameters. 
(b) From the block diagram, determine the overall transfer function relating the 
temperature in tank 3 to a change in set point. 
(c ) Find the offset for a unit steo change in inlet temperature if the controller 
gain KC is 3psi/°F of temperature error and the derivative time is 0.5 min. 
 
 
CWTTTCVqCWT 10110 )( +−=+ ρ 
 
CWTTTCVCWT 21221 )( +−= ρ 
 
CWTTTCVCWT 32332 )( +−= ρ 
 
)()( 1110 CVCWTqCVWCT ρρ +=++ 
 
1
01
CVWC
q
TT
ρ+
+= 
 
T1= T2 = T3 
 
⇒
+
+=
1
03
CVWC
q
TT
ρ ( )sCVWC
sq
sT
1
3
)(
)(
ρ+
= 
13.6 (b) 
 
)15.1)(125.1)(1(
2
)1(1
)15.1)(125.1)(1(
2
)1(
)(
)('3
+++
++
+++
+
=
sss
sk
sss
sk
sR
sT
DC
DC
τ
τ
 
 
 
= 
)1(2)175.2875.1)(1(
)1(2
)(
)(
2
'
3
sksss
sk
sR
sT
DC
DC
τ
τ
+++++
+
= 
 
12)275.3(625.4875.1
)1(2
)(
)(
23
'
3
+++++
+
=
cDC
DC
kskss
sk
sR
sT
τ
τ
 
 
 
c) kC=3;
s
soffsetD
1
)(?,,5.0 '0 === ττ 
 
 
 
 
12)275.3(625.4875.1
1
)(
)(
230'
'
3
+++++→ CDCsi kskss
Lt
sT
sT
τ
 
 
 
= 143.0
7
1
12
1
==
+Ck
 
 
Offset =0.143 
 
13.7 (a) For the control system shown in fig P 13.7, obtain the closed loop 
transfer function C/U. 
 (b) Find the value of KC for whgich thre closed loop response has a ξ of 
2.3.(c) find the offset for a unit-step change in U if KC = 4. 
 
 
C
s
UCR
s
s
KC =




 +−
+
+
×
1
)(
125.0
1
 
 
 
= 
125.0
1
.1
1
+
+
+
=
s
s
s
K
s
U
C
C
 
 
 
 
)1(25.0
125.0
2 +++
+
=
sKss
s
U
C
C
 
 
 
CC KsKs
s
U
C
4)1(4
4
2 +++
+
= 
 
b) ξ=2.3 
 
C
C
C
K
K
K
1
2;
4
1 +
== ξττ 
 
= 
C
C
C
K
K
K
1
3.22
4
1 +
=× 
 
3.2
1
=
+
=
C
C
K
K
 
KC=2.952 
 
C) KC=4,U = 1/s 
 
= 
1620
41
2 ++
+
×=
ss
s
s
C 
 
4
1
)( =∞C 
offset = 0.25. 
 
13.8 For control system shown in Fig 13.8 
(a) C(s)/R(s) 
(b) C(∞∞∞∞) 
(c) Offset 
(d) C(0.5) 
(e) Whether the closed loop response is oscillatory. 
 
 
 
(a) 
)1(
4
1
)1(
4
+
+
+
=
ss
ss
R
C
 
 
4
4
2 ++
=
ssR
C
 
 
b) C(∞) =2*1=2 
 
C(∞) =2 
 
C) offset = 0 
 
 
d) 
4
1
4
1
2;
2
1
=⇒== ξξττ 
 






+





−
−= −
−
15tan
4
15
sin
4
1
1
1
1
2
)( 12
2 τ
t
e
tC
t
 
 
= 














+−= −
−
15tan
4
15
sin
15
4
12)5.0( 14
1
eC 
C(0.5)=0.786 
 
.e) ξ<1, the response is oscillatory. 
 
13.9 For the control system shown in fig P13.9,determine an expression for 
C(t) 
if a unit step change occurs in R. Sketch the response C(t) and compute 
C(2). 
 
 
 
1̀2
1
1
11
1
1
+
+
=





 ++
+
=
s
s
R
C
s
s
R
C
 
 
2
2
1
1)(
12
11
)12(
1
1
t
etC
ssss
s
C
s
R
−
−=
+
−
+=
+
+
=
=
 
 
C(2) = 0.816 
 
 
 
 
13.10 Compare the responses to a unit-step change in a set point for the 
system shown in fig P13.10 for both negative feedback and positive feedback.Do 
this for KC of 0.5 and 1.0. compare the responses by sketching C(t). 
 
 
-ve feed back : 
 
 
 
 
))1(( ++
=
C
C
Kss
K
C 
 
 
 
+ve feed back 
 
1
1
1
1
1
+
−
+
×
=
s
K
s
K
R
C
C
C
 
 
))1(( C
C
Kss
K
C
−+
= 
 
For KC = 0.5 , response of -ve feed back is 
 
 
32
3
2
3
1
)32(
1
+
−
+=
+
=
ssss
C 
 
 
 
)1(
3
1
3
1
3
1
)( 2
3
2
3 tt
eetC
−−
−=−= 
 
response of +v feed back is 
 
 
21)(
12
21
)12(
1
t
etC
ssss
C
−
−=
+
−
+=
+
=
 
 
For KC = 1, response of -ve feed back is 
 
tetC
ssss
C
2
2
1
2
1
)(
2
2
1
2
1
)2(
1
−−=
+
−
+=
+
=
 
 
 
response of +ve feed back is 
 
 
ttC
s
C
=
=
)(
1
2
 
 
 
 
14.1 Write the characteristics equation and construct Routh array for 
the control system shown . it is stable for (1) Kc= 9.5,(ii) KC =11; (iii) Kc= 
12 
 
 
 
Characteristics equation 
 
0)66(116
0)66(116(
06)3)(2)(1(
0
)3)(2)(1(
6
1
23
2
=++++
=++++
=++++
=
+++
+
Kcsss
Kcsss
Kcsssor
sss
Kc
 
 
 
Routh array 
 
)1(6
)66(6
111
2
3
Kcs
Kcs
s
+
+ 
 
 
 
 
For Kc=9.5 
 
= 10-(Kc)= 10-9.5=0.5>0 therefore stable. 
 
For Kc=11 
 
= 10-(Kc)= 10-11=-1<0 therefore unstable 
For Kc=121 
 
= 10-(Kc)= 10-12=--2<0 therefore unstable 
 
 
 
 
14.2 By means of the routh test, determine the stability of the system shown 
when KC = 2. 
 
 
Characteristic equation 
 
 
0
1042
10
2
3
121
2
=





++





 ++
sss
 
 
 
 
060252
01205042
0120401042
010.2).3(2)1042(
23
23
23
2
=+++
=+++
=++++
=++++
sss
sss
ssss
ssss
 
 
 
Routh Array 
 
1 25 
2 60 
-10/2 
 
The system is unstable at Kc = 2. 
14.4 Prove that if one or more of the co-efficient (a0,a1,….an) of the 
characteristic equation are negative or zero, then there is necessarily an 
unstable root 
 
Characteristic equation : 
0..................................110 =+++
−
n
nn axaxa 
 
0)/......................./( 0
1
010 =++
−
aaxaaxa n
nn 
 
0.............................,
0).(..........).........)((
21
210
<
=−−−
n
n
haveWe
xxxa
ααα
ααα
 
 
 
 
As we know the second co-efficient a1/a0 is sum of all the roots 
2/)1(
11
2
0
1






−= ∑∑
==
n
j
ji
n
ia
a
αα 
 
 
 
Therefore sum of all possible products of two roots will happen twice as 
21αα dividing the total by 2. 
 
And 
00/
)00(0
202 >⇒>∴
<<>
aaa
jiji αααα
 
 
Similarly 
 
),.......1(0
0/
00)1()1(
0/01)1(
)()1(
0
0
0
0
njforaso
aacasebothin
againis
a
a
soissumtheandisoddjif
aasoissumtheandisevenjif
rootsjofproductspossibleaollofsum
a
a
j
j
jj
j
j
jj
=>
>
><−−=
>>−=
−=
 
 
 
 
14.5 Prove that the converse statement of the problem 14.4 that an 
unstable root implies that one or more co-efficient will be negative or 
zero is untrue for all co-efficient ,n>2. 
 
Let the converse be true, always .Never if we give a counter example we 
can contradict. 
Routh array 
 
0
2
3
23
01
31
21
32
s
s
s
s
sss
−
+++
 
 
System is unstable even when all the coefficient are greater than 0; hence a 
contradiction, 
14.6 Deduce an expression for Routh criterion that will detect the 
Presence of roots with real parts greater than σ for any rectified σ >0 
 
Characteristic equation 
 
0.................................110 =+++
−
n
nn axaxa 
 
 
Routh criteria determines if for any root, real part > 0. 
 
Now if we replace x by X such that 
 
.x + σ =X 
 
Characteristic equation becomes 
 
0.................................)()( 110 =++−+−
−
n
nn aXaXa σσ 
 
 
Hence if we apply Routh criteria, 
 
We will actually be looking for roots with real part >σ rather than >0 
 
0.................................22
1
10 =+++
−−
n
nnn axaxaxa 
 
 
Routh criterion detects if any root jα is greater than zero. 
 
Is there any )1(0.,...............,,........., 21 −−−−−>= njx αααα 
Now we want to detect any root 
 
)1(
0
From
j
j
>
−>
α
σα
 
 
σα
σα
σ
σασ
σασ
σασ
σασ
σ
α
α
α
α
αααα
−>
>+
+=
>+=+
>+=+
>+=+
>+=+
>=
>=
>=
>=
>=
j
j
n
j
n
j
nj
or
anyecttocriteriaRouthapplyand
xXLetso
x
x
x
x
anythereis
sidesbothonadd
x
x
x
x
anythereisimplies
x
0det
,
0
.
.
.
0.
.
.
.
0
0
0
.
.
.
0
.
.
.
0
0
0,.................,............................,.........,
2
1
2
1
21
 
 
14.7 Show that any complex no S1 satisfying ,1<S yields a value of 
 
s
s
Z
−
+
=
1
1
that satisfies Re(Z)>0, 
Let S=x+iy, 122 <+ yx 
 
s
s
Z
−
+
=
1
1
 
 
 
 
( )






+−
+=
>
<−++<
==
=−=
−++
<+>+−
<<−
<<−
<+
<+
>++−
>+−>
++−
+−
=
++−
++−
=
+−+
−−+=+−
+−
+−
−−
++
−
)5.05.0(
1
)5.05.0(
:
0)Re(
4)2)(1(0
00&1
40&1
2)(1
10)(1
int11
11
1
1
0)(21
0)(10)Re(
)(21
)(1
)Re(
)(21
2)(1
21
))11(1(
)1(
)1(
)1(
1
1
22
22
2222
22
22
22
22
22
22
22
22
22
22
is
L
partrealthetodueunstableissystemtheisif
example
z
xyx
isitthenyxif
isitthenyxif
xyx
Now
yxthereforetrueisyx
circleunittheinsPoy
xareRanges
yx
yxhavewe
yxx
andyxthenzif
yxx
yx
Z
yxx
iyyx
yxx
yiyxxx
iyx
iyx
iyx
iyx
 
 
)5.05.0(
)5.05.0(
1 5.01 tSintCose
is
L t +=





+−
− 
14.8 For the output C to be stable, we analyze the characteristic 
equation of the system 
 
 
0)1(
)1)(1(
11
1 3
21
=+×
++
+ s
sssI
τ
τττ
 
 
01)()(
01)1(
3
2
21
3
21
3212
2
1
=+++++
=+++++
sss
sssss
III
I
ττττττττ
ττττττ
 
 
 
Routh Array 
 
1
0
1)(
0
21
2
321
3
s
s
s
s
I
II
α
τττ
τττττ
+
+
 
 
 
 
)(
))((
21
21321
τττ
ττττττττ
α
+
−++
=
I
III 
 
 
 
Now 
 
0)1( 12 >τττI 
 
Since 21 &ττ are process time constant they are definitely +ve 
0;0 21 >> ττ 
 
(2) 0)( 21 >+τττ I 
(3) 
21321 ))((0 ττττττττα III >++⇒> 
 
 
0
)()(
0
3
21
21
2132121
21322311
>
−
+
>
+−>+
>−+++
I
I
I
II
also τ
τ
ττ
ττ
τ
ττττττττ
ττττττττττ
 
 
 14.9 In the control system shown in fig find the value of Kc for which 
the system is on the verge of the instability. The controller is replaced by a 
PD controller, for which the transfer function is Kc(1+s). if Kc = 10, 
determine the range for which the system is stable. 
 
 
Characteristics equation 
 
 
0)66(116
06)3)(23(
06)3)(2)(1(
0
)3)(2)(1(
6
1
23
2
=++++
=++++
=++++
=
+++
+
Kcsss
Kcsss
Kcsssor
sss
Kc
 
 
 
Routh array 
 





 +−
+
)
3
1
3
13
31
2
3
Kc
s
Kcs
s
 
 
For verge of instability 
8
)
3
1
3
=





 +=
Kc
Kc
 
 
 
 
Characteristics equation 
 
0
3)1(
)1(10
1 =
+
+
+
s
kcs
 
 
011)103(3 23 =++++ Kcssss 
 
Routh Array 
 
30/2
230
11)103(3
113
1031
2
3
>
>
>+
+
D
S
s
D
vegefor
s
s
s
τ
τ
τ
τ
 
 
 
14.10 (a) Write the characteristics equation for the central system shown 
(b) Use the routh criteria to determine if the system is stable for Kc=4 
© Determine the ultimate value of Kc for which the system is unstable 
 
(a) characteristics equation 
02)1(32
0)2()12)((
0
)1(
1
12
1
3
2
1
23
2
=++++
=++++
=
+






=





 ++
kcskcss
skcsss
ss
s
kc
 
 
0)1(33 23 =++++ kcsss 
 
Kc=4Routh array 
 
3;03
0
3
4)1(3
3/1
83
52
2
3
==−
=
−+
−
KcKc
kckc
stablenot
s
s
s
 
 
 
For verge of instability 
 
14.11 for the control shown, the characteristics equation is 
0)1(464 234 =+++++ kssss 
(a) determine value of k above which the system is unstable. 
(b) Determine the value of k for which the two of the roots are on 
the imaginary axis, and determine the values of these imaginary roots and 
remaining roots are real. 
 
 0)1(464 234 =+++++ kssss 
 
 
 
k
ks
ks
s
ks
+
+−
+
+
11
)1(
5
4
4
15
44
)1(61
2
3
4
 
 
 
 
For the system to be unstable 
 
1
1
01
4
5
1
1
0
5
1
14
−>
−<
<+
>
+
<
<










 +−
k
k
k
k
k
k
 
 
 
The system is stable at -1<k<4 
 
(b) For two imaginary roots 
 
4);1(
5
4
4 =+= kk 
 
 
Value of complex roots 
is
s
±=
=+ 055 2
 
 
 
 
 
5454641 22342 +++++++ sssssss 
 
 24 0 ss ++ 
 
 
0
55
55
404
54
2
2
3
23
+
+
++
+
s
s
ss
ss
 
SOLUTION: 
 
i
i
s ±−=
±−
=
−±−
= 2
2
24
2
20164
 
 
 
 
 
 
PART 2 
 
LIST OF USEFUL BOOKS FOR PROCESS 
CONTROL 
 
 
 
 
1. PROCESS CONTROL BY R.P VYAS, CENTRAL TECHNO 
PUBLICATIONS, INDIA ( WIDE VARIETY OF SOLVED PROBLEMS ARE 
AVAILABLE IN THIS BOOK) 
 
2. ADVANCED CONTROL ENGINEERING BY RONALD.S .BURNS , 
BUTTERWORTH AND HIENEMANN. 
 
3. PROCESS MODELLING SIMULATION AND CONTROL FOR 
CHEMICAL ENGINEERS, WILLIAM.L.LUYBEN, MCGRAW HILL. 
 
4. A MATHEMATICAL INTRODUCTION TO CONTROL THEORY BY 
SCHOLOMO ENGELBERG, IMPERIAL COLLEGE PRESS 
 
 
 
 
LIST OF USEFUL WEBSITES 
 
www.msubbu.com FOR BLOCK DIAGRAM REDUCTION AND 
OTHER CHEMICAL ENGG. LEARNING RESOURCES 
 
 
Readings,Recitations,Assignments,Exams,StudyMateri
als,Discussion Group,Video Lectures 
now study whatever u want with respect to chemical 
engg. 
 
http://ocw.mit.edu/OcwWeb/index.htm 
 
 
 
 
 
 
***********************