# Coupled Tanks Non Linear EOM Codes and Figures Mat Lab

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1.3 Lab 2: Tank 1 Control 1.3.1 Lab 2: Objective The objective of Lab 2 is to control the height of water in Tank 1 of the Coupled Tanks system. The Prelab focuses on linearizing the nonlinear dynamic system and representing the open-loop and closed-loop system dynamics as transfer functions for Proportional-Integral (PI) control design. The Lab Procedure designs PI control gains to meet desired control specifications and implements the PI controller in simulations of the experimental system. 1.3.2 Lab 2: Background Linearization In order to design and implement a linear controller for controlling the water level in Tank 1, we will use the Laplace transform to derive an open-loop transfer function of the system. However, by definition, a transfer function can only represent the linear dynamic relationship between the system's input and output. Therefore, the first step is to linearize the nonlinear EOM derived in the previous lab in the neighborhood of a desired operating condition. In the context of linearization, this operating condition is typically assumed to be a steady-state operating condition and is often referred to as the linearization point, nominal point, trim point, or quiescent point. Prior to discussing the linearization of dynamic Linear approximation y systems, first consider the linearization of a nonlin- of f(x) ear function, y = f(x), were y is the output of the y у function and x is the input (or argument) of the func- f(x) tion. Choosing the operating point (ī, ū) such that y = f(ī) defines where along the nonlinear function Operating f we would like to compute a linear approximation. point As shown in Fig. 1.3.1, the linear approximation will only be accurate within a neighborhood of the cho- sen operating point and the size of this neighborhood Figure 1.3.1: The linear approxima- depends on the severity of the nonlinearity and the tion of a nonlinear function f(x) at amount of linearization error considered acceptable the operating point (ī, ī). for the given application. Once the operating point is chosen, the linear model captures deviations from this oper- ating point. Therefore, it is useful to define incremental variables (also referred to as delta variables) such that y = ý + Ay and x = ī + Ax. Once these delta variables are defined, the linear approximation of f comes from applying the Taylor series expansion, truncated to include only the linear terms. Specifically, the output y is approximated as df y = f(ī) + (1.3.1) where f(ī) denotes the evaluation of function f at the operating point ī and pole denotes the derivative of f with respect to x evaluated at x = 7. If f is a function of two variables x x dă / 4x 11
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Lab-2: 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 non-linear 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. Open-loop dynamics: In order to obtain the open-loop 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 ̅̅̅

𝐶 𝐴 2𝑔
√ 1 𝑠 + 1) ( 𝑑 0 √ ) 𝑌(𝑠) = 𝑝 𝑈(𝑠)
⇒(
𝐶𝑑 𝐴0 2𝑔
2𝐴𝑐 ̅̅̅
𝐴𝑐
ℎ1
̅̅̅
𝐾𝑝 2𝐴𝑐
2𝐴𝑐 ̅̅̅

𝑌(𝑠)
ℎ1
√ 1 𝑠 + 1)
)√
⇒(
= ( )(
𝐶𝑑 𝐴0 2𝑔
𝑈(𝑠)
𝐴𝑐 𝐶𝑑 𝐴0 2𝑔
̅̅̅1
𝐾𝑝
2𝐴𝑐 ̅̅̅

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 closed-loop 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�...

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