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Lab2: Prelab
1. The given EOM is
𝐴𝑐 ℎ1̇ = 𝐾𝑝 (𝑣𝑝 − 𝑣𝑝𝑚𝑖𝑛 ) − 𝐶𝑑 𝐴0 √2𝑔ℎ1
𝐾𝑝
𝐶𝑑 𝐴0
⇒ ℎ1̇ =
(𝑣𝑝 − 𝑣𝑝𝑚𝑖𝑛 ) −
√2𝑔ℎ1
𝐴𝑐
𝐴𝑐
⇒ ℎ1̇ = 𝑓(ℎ1 , 𝑣𝑝 )
At steady state, the pump speed is ̅̅̅
𝑣𝑝 and water height ̅̅̅
ℎ1 .
̅̅̅
̅̅̅1 , ̅̅̅)
ℎ1̇ = 𝑓(ℎ
𝑣𝑝 = 0
Linearizing the above nonlinear EOM,
̅̅̅1 , ̅̅̅)
ℎ1̇ = ̅̅̅
ℎ1̇ + ∆ℎ1̇ = 𝑓(ℎ
𝑣𝑝 +
𝑑𝑓
𝑑𝑓

∆ℎ1̇ +

∆𝑣 ̇
𝑑ℎ1 (ℎ̅̅̅̅1,𝑣̅̅̅̅)
𝑑𝑣𝑝 (ℎ̅̅̅̅,𝑣̅̅̅̅) 𝑝
𝑝
1 𝑝
Obtaining the value of various term individually,
𝑑𝑓
𝐶𝑑 𝐴0 2𝑔

=−
√
𝑑ℎ1 (ℎ̅̅̅̅1 ,𝑣̅̅̅̅)
2𝐴𝑐 ̅̅̅
ℎ1
𝑝
𝐾𝑝
𝑑𝑓

=
𝑑𝑣𝑝 (ℎ̅̅̅̅,𝑣̅̅̅̅) 𝐴𝑐
1 𝑝
The linearized equation is:
∆ℎ1̇ =
𝑑𝑓
𝑑𝑓

∆ℎ1̇ +

∆𝑣 ̇
𝑑ℎ1 (ℎ̅̅̅̅,𝑣̅̅̅̅)
𝑑𝑣𝑝 (ℎ̅̅̅̅ ̅̅̅̅) 𝑝
1 𝑝
⇒ ∆ℎ1̇ = −
1 ,𝑣𝑝
𝐾𝑝
𝐶𝑑 𝐴0 2𝑔
∆𝑣
√ ∆ℎ1 +
2𝐴𝑐 ̅̅̅
𝐴𝑐 𝑝
ℎ1
2. Openloop dynamics: In order to obtain the openloop dynamics of the system under study,
the above obtained linearized time domain equation is transformed to frequency domain
through Laplace transform.
Replacing, ∆ℎ1 by 𝑦 and ∆𝑣𝑝 by 𝑢
∆ℎ1̇ = −
𝐾𝑝
𝐶𝑑 𝐴0 2𝑔
∆𝑣
√ ∆ℎ1 +
2𝐴𝑐 ̅̅̅
𝐴𝑐 𝑝
ℎ1
⇒ ∆𝑦̇ = −
𝐾𝑝
𝐶𝑑 𝐴0 2𝑔
𝑢
√ 𝑦+
2𝐴𝑐 ̅̅̅
𝐴𝑐
ℎ1
Laplace transform of the above equation is (after zeroing the initial condition)
𝑠𝑌(𝑠) = −
𝐾𝑝
𝐶𝑑 𝐴0 2𝑔
𝑈(𝑠)
√ 𝑌(𝑠) +
2𝐴𝑐 ̅̅̅
𝐴𝑐
ℎ1
⇒ 𝑠𝑌(𝑠) +
⇒ (𝑠 +
𝐾𝑝
𝐶𝑑 𝐴0 2𝑔
𝑈(𝑠)
√ 𝑌(𝑠) =
2𝐴𝑐 ̅̅̅
𝐴𝑐
ℎ1
𝐾𝑝
𝐶𝑑 𝐴0 2𝑔
𝑈(𝑠)
√ ) 𝑌(𝑠) =
2𝐴𝑐 ̅̅̅
𝐴𝑐
ℎ1
𝐾
2𝐴𝑐 ̅̅̅
ℎ
𝐶 𝐴 2𝑔
√ 1 𝑠 + 1) ( 𝑑 0 √ ) 𝑌(𝑠) = 𝑝 𝑈(𝑠)
⇒(
𝐶𝑑 𝐴0 2𝑔
2𝐴𝑐 ̅̅̅
𝐴𝑐
ℎ1
̅̅̅
𝐾𝑝 2𝐴𝑐
2𝐴𝑐 ̅̅̅
ℎ
𝑌(𝑠)
ℎ1
√ 1 𝑠 + 1)
)√
⇒(
= ( )(
𝐶𝑑 𝐴0 2𝑔
𝑈(𝑠)
𝐴𝑐 𝐶𝑑 𝐴0 2𝑔
̅̅̅1
𝐾𝑝
2𝐴𝑐 ̅̅̅
ℎ
𝑌(𝑠)
2ℎ
√ 1 𝑠 + 1)
)√
⇒(
=(
𝐶𝑑 𝐴0 2𝑔
𝑈(𝑠)
𝐶𝑑 𝐴0
𝑔
⇒
𝑌(𝑠)
=
𝑈(𝑠)
⇒
̅̅̅
𝐾𝑝
2ℎ
(
)√ 1
𝐶𝑑 𝐴0
𝑔
̅̅̅
𝐴
2ℎ
( 𝑐 √ 1 𝑠 + 1)
𝐶𝑑 𝐴0 𝑔
𝑌(𝑠)
𝐾𝐷𝐶
= 𝐺(𝑠) =
(𝜏𝑠 + 1)
𝑈(𝑠)
where,
̅̅̅1
𝐾𝑝
2ℎ
)√
𝐾𝐷𝐶 = (
𝐶𝑑 𝐴0
𝑔
and,
𝜏=
̅̅̅
𝐴𝑐
2ℎ
√ 1
𝐶𝑑 𝐴0 𝑔
3. The closedloop transfer function with PI control block is shown below:
R(s)
E(s)
C(s)
U(s)
G(s)
The closed loop transfer function is given by:
𝑌(𝑠)
𝐶(𝑠)𝐺(𝑠)
=
𝑅(𝑠) 1 + 𝐶(𝑠)𝐺(𝑠)
where,
Y(s)
𝐶(𝑠) = 𝑘𝑝 +
𝑘𝑖 𝑠𝑘𝑝 + 𝑘𝑖
=
𝑠
𝑠
and,
𝐾𝐷𝐶
(𝜏𝑠 + 1)
𝑌(𝑠)
𝐶(𝑠)𝐺(𝑠)
=
𝑅(𝑠) 1 + 𝐶(𝑠)𝐺(𝑠)
𝑠𝑘𝑝 + 𝑘𝑖
𝐾
(
) ( 𝐷𝐶 )
𝑌(𝑠)
𝑠
𝜏𝑠 + 1
⇒
=
𝑠𝑘𝑝 + 𝑘𝑖
𝑅(𝑠)
𝐾
) ( 𝐷𝐶 )
1+(
𝑠
𝜏𝑠 + 1
𝐾𝐷𝐶 (𝑠𝑘𝑝 + 𝑘𝑖 )
𝑌(𝑠)
⇒
= 2
𝑅(𝑠) 𝜏𝑠 + 𝑠 + 𝑠𝑘𝑝 𝐾𝐷𝐶 + 𝑘𝑖 𝐾𝐷𝐶
𝐾𝐷𝐶
𝑌(𝑠)
𝜏 (𝑠𝑘𝑝 + 𝑘𝑖 )
⇒
=
1 + 𝑘𝑝 𝐾𝐷𝐶
𝑅(𝑠)
𝑘𝐾
) 𝜏 + 𝑖 𝐷𝐶
𝑠2 + (
𝜏
𝜏
𝐺(𝑠) =
4. Obtaining the value of proportional and integral constant as function of 𝐾𝐷𝐶 , 𝜏, 𝜁 and 𝜔𝑛
A standard second order transfer function is given by,
𝑌(𝑠)
𝐾𝐷𝐶 𝜔𝑛2
=
𝑅(𝑠) 𝑠 2 + 2𝜁𝜔𝑛 𝑠 + 𝜔𝑛2
Comparing the closed loop transfer function for the system under study and the standard
form of the second order transfer function.
𝑘𝑖 𝐾𝐷𝐶
𝜔�...