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Fecha de aceptación: 18/ 01/2023
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Optimal tracking of the water level for a coupled tank system using Linear
Quadratic Regulator
Pedro Teppa-Garran
pteppa@unimet.edu.ve
Universidad Metropolitana, Caracas, Venezuela
Seguimiento óptimo del nivel de agua de un sistema de tanques acoplados
empleando el regulador lineal cuadrático
Abstract
It is proposed an optimal controller that will lead to zero asymptotic steady-state tracking error. The reference
inputs can include steps, ramps, and other persistent signals. For a step signal, it is well known that zero steady-
state tracking error can be achieved with a type-one open loop system. This idea is formalized in this work, by
augmenting the coupled tank system model with an internal model of the reference input. Then, through the
Linear Quadratic Regulator (LQR) method, the desired performance objectives are addressed by minimizing a
quadratic function of the state and control input. Experimental results on the coupled tank system have been
provided to illustrate the effectiveness of the method.
Keywords
:
Reference tracking, Optimal control, LQR, Internal model, Coupled tank system.
Resumen
Se propone un controlador óptimo que resultará en un error de seguimiento asintótico en estado estacionario
nulo. Las entradas de referencia pueden incluir escalones, rampas y otras señales persistentes. Para una señal tipo
escalón, es bien conocido que se puede lograr un error de seguimiento de estado estacionario cero con un sistema
de lazo abierto de tipo uno. Esta idea se formaliza en este trabajo, aumentando el modelo del sistema de tanques
acoplados con un modelo interno de la entrada de referencia. Luego, a través del regulador lineal cuadrático
(LQR, por sus s
iglas en inglés), los requerimientos de diseño sobre la respuesta temporal son incorporados
mediante la minimización de una función cuadrática dependiente de las variables de estado y de la señal de
control. Se proporcionan resultados experimentales en el sistema de tanques acoplados para ilustrar la efectividad
del método.
Palabras clave: Seguimiento de señales de referencia, Control óptimo, LQR, Modelo interno, Sistema de
tanques acoplados.
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Suivi optimal du niveau d'eau pour un système de réservoirs couplés
utilisant le régulateur quadratique linéaire
Rastreamento ideal do nível de água para um sistema de tanque acoplado
usando regulador quadrático linear
Resumo
É proposto um controlador ótimo que levará a zero erro de rastreamento assintótico em estado estacionário. As
entradas de referência podem incluir degraus, rampas e outros sinais persistentes. Para um sinal de passo, é bem
conhecido que o erro de rastreamento em estado estacionário zero pode ser alcançado com um sistema de malha
aberta tipo um. Esta ideia é formalizada neste trabalho, aumentando o modelo do sistema de tanque acoplado
com um modelo interno da entrada de referência. Em seguida, por meio do método do Regulador Quadrático
Linear (LQR, por sua sigla em inglês), os objetivos de desempenho desejados são abordados minimizando uma
função quadrática do estado e da entrada de controle. Resultados experimentais no sistema de tanque acoplado
foram fornecidos para ilustrar a eficácia do método.
Palavras-chave:
Rastreamento de referência, controle ideal, LQR, modelo interno, sistema de tanque acoplado
Resumé
Il est proposé un contrôleur optimal qui produira une erreur de poursuite nulle en régime permanent
asymptotique. Les entrées de consigne peuvent inclure des échelons, des rampes et d'autres signaux persistants.
Pour un signal type échelon, il est bien connu qu'une erreur de poursuite en régime permanent nulle peut être
obtenue avec un système à boucle ouverte de type un. Cette idée est formalisée dans ce travail, en augmentant le
modèle du système de réservoirs couplés avec un modèle interne de l'entrée de référence. Ensuite, grâce au
régulateur quadratique linéaire (LQR, pour son sigle en anglais), les objectifs de performance souhaités sont
atteints en minimisant une fonction quadratique de l'état et de l'entrée de commande. Des résultats
expérimentaux sur le système de réservoirs couplés ont été fournis pour illustrer l'efficacité de la méthode.
Mots clés:
Suivi de référence, Commande optimale, LQR, Modèle interne, Système de réservoirs couplés.
Revista TEKHNÉ Nº 25.3
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
i. INTRODUCCIÓN
Technologies such as chemical reactors,
fermentation vessels, and steam and surge drums
benefit from accurate level instrumentation and
control [1, 2, 3, 4]. Tank level control systems are
used frequently in different processes. For
example: pharmaceutical industries, petrochemical
plants, food/beverages industries and nuclear
plants; depend upon tank level control systems.
Often the tanks are coupled, so there is interaction
between their levels, resulting in a nonlinear
behavior and may also exhibit non minimum phase
characteristics [5, 6].
There are numerous control policies developed for
coupled tank systems. Among them, we can
mention: Proportional-Integral-Derivative (PID) type
controllers [7, 8, 9], Fuzzy control [10, 11], Model
Predictive Control [12, 13], Backstepping Control
[14, 15], Sliding-Mode Control [16, 17], Fractional
PID type controllers [18, 19], Robust control [20]
and Active Disturbance Rejection Control [21].
In this work, it is used Linear Quadratic Regulator
Optimal Control (LQR) [22, 23]. This method is well
known in modern optimal control theory and has
been widely used in many applications [22]. LQR
design has already been employed to control the
liquid level of the tank system [24, 25, 26]. The
relevance of this work aims to focus on determining
an optimal solution by using the LQR method to the
tank system tracking problem. Specially, it is
desired to find asymptotic optimal tracking with zero
steady-state error. This is formalized by introducing
an internal model of the reference input [27, 28] in
the state feedback control law previously obtained
from a LQR problem.
The article is organized as follows. Section II
describes the coupled tank system. In section III,
the internal model principle and optimal LQR design
are used to formulate a method that guarantees
zero steady-state tracking error for the water level.
Finally, in section IV, it is shown the effectiveness
of the optimal design through its application to the
coupled tank system and Conclusions.
ii.
COUPLED TANK SYSTEM
The coupled tank system is given in Fig. 1. The
setup experiment also includes a personal
computer and the Matlab/Simulink software
interface. The apparatus is used in the control
laboratory at the Simón Bolívar University in
Venezuela. It consists of a single pump with two
tanks. Each tank is instrumented with a pressure
sensor to measure the water level. The pump
drives the water from the bottom basin up to the top
of the system. Depending on how the outflow
valves are configured, the water then flows to the
top tank, bottom tank or both. One configuration is
shown in Fig. 2, where the output of the pump is
connected to the first tank.
The nonlinear state space model [6, 29] is shown in
(1) where the state vector is equal to the tanks
levels, the control signal corresponds to the input
voltage applied to the pump and the output is
chosen as the second tank level.
(
)
=
() 0
()
()
(
)
+
0
(
)
(
)
=
[
0 1
]
()
(1)
With
=
2
,
=
2
,
and
denote
the cross-sectional area of the tanks 1 and 2,
respectively.
,
give the cross-sectional areas
of the corresponding orifices, is the gravitational
constant on Earth and
is the pump flow constant.
The nonlinear model is linearize in the operating
point
[
,
]
resulting in the equations
(
)
=
0
(
)
+
0
(
)
(
)
=
[
0 1
]
()
(2)
The description and numerical values of the
physical parameters for the tank system are given
in Table 1.
By employing the numerical values from Table 1. It
is possible to perform the following calculations
=
=
(
4.445 2
)
= 15.53
=
(
0.635 2
)
= 0.317
=
(
0.476 2
)
= 0.178
=
= 15
Replacing the previous values in (2) allows to
obtain the linear model of the coupled tank system
as
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
(
)
=
(
)
+
(
)
(
)
= ()
(3)
where
=
0
, =
0
, =
[
0 1
]
With = 0.1168, = 0.0656 and = 0.2577.
Table I. Physical parameters of the coupled tank system.
Description
Value
Unit
Pump flow constant
4
//
Out 1 Orifice Diameter
0.635
Out 2 Orifice Diameter
0.476
Tanks Diameter
4.445
Tanks Height
30
Gravitational constant on Earth
981
Maximum flow
100
Pump peak voltage
22
Figure 1. Coupled tank system.
Figure 2. Standard configuration of the coupled tank
system.
iii. OPTIMAL TRACKING CONTROL SYSTEM
DESIGN
We begin by considering the design problem to
enable the optimal tracking of a step reference
input
() = with zero steady-state error ()
defined as
(
)
=
() ()
(4)
Taking the time-derivative of the auxiliary signal
(
)
= () and using the output equation in (3)
yields
(
)
=
(
)
=
(
)
0 = ()
Applying now, time-derivative to the state equation
in (3) and defining the intermediate variables
(
)
=
() and
(
)
= () leads to the step
tracking system
()
()
=
0
0
()
()
+
0
()
(5)
Or equivalently, in compact form
(
)
=
(
)
+
()
(6)
for the vector
(
)
=
[() ()]
and augmented
matrices
=
0
0
,
=
0
In order to have a LQR formulation of the tracking
problem, the following quadratic cost is considered
=
[
()
(
)
+
()
]
(7)
where is a positive semi-definite matrix which has
an impact on the closed-loop transient response
and parameter > 0 can be used to tune the
amplitude of the control signal. It is well known [22,
23] that the state feedback control
(
)
= ()
(8)
minimizes Eq. (7) and stabilizes system (6) with the
vector equal to
=
(9)
Here is the symmetric positive definite solution of
the Continous Algebraic Riccati Equation given by
+
+ =
(10)
Since, it can be easily verified that the system (5)
with matrices , and from (3) is controllable
[30]. We can find constants
and
by
expressing gain vector
()
as =
in (8). The resulting equation is
(
)
=
[
]
()
()
(11)
After performing time-integration in (11), the control
signal () in (3) applied to the coupled tank system
is
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
(
)
=
()
()
(12)
Equation (12) guarantees system (6) is stable and
the quadratic cost (7) is minimized. The above
results are summarized in Theorem 1 and the
corresponding control system block diagram in Fig.
3
Theorem 1: If the state feedback control law (12) is
applied to the coupled tank system. The error signal
(4) for a step reference input
(
)
= will tend
asymptotically to zero minimizing the quadratic cost
(7) for a given positive semi-definite matrix and
parameter > 0.
Figure 3. Optimal LQR internal model design for the coupled
tank system for a step reference input.
It is straightforward to extend the method for a ramp
reference input
(
)
= , 0. Taking twice the
time-derivative of
(
)
= () and using (3) yields
(
)
=
(
)
() =
(
)
0 =
()
Defining the intermediate variables
(
)
=
() and
(
)
= () gives
()
()
()
=
0 1 0
0 0
0 0
()
(
)
()
+
0
0
()
(13)
Again, the previous equation can be expressed in
the compact form given by (6) if the following
augmented matrices are defined
=
0 1 0
0 0
0 0
,
=
0
0
Since, system (13) with matrices , and from
(3) is controllable [30]. We can compute constants
,
and
in (8) to form
(
)
=
(
)
()
()
(14)
The control signal () applied to the coupled tanks
system is found by integrating (14) twice, this signal
will cause the error signal (4), for a ramp reference
input, tend asymptotically to zero minimizing the
quadratic cost (7) for a given positive semi-definite
matrix and parameter > 0. Fig. 4 shows the
block diagram of the control system.
Figure 4. Optimal LQR internal model design for the coupled
tank system for a ramp reference input.
iv. DESIGN TESTS ON A LABORATORY SETUP
The design technique is demonstrated through the
implementation on the coupled tank system in real
time and the corresponding results are presented in
this section.
A. Step reference input
The reference input applied to the coupled tank
system is shown in Fig. (5). It consists of two
changes in the set point, the first at 60 and the
second at 160 , after the start of the experiment.
Figure 5. Input reference signal to validate the optimal controller
design with step-like internal model.
Table 2 collects several experiments carried out to
study the impact of the selection of matrix and
the constant in (7) on the closed-loop
performance of the system. The performance
measures considered are: rise time
(
)
, maximum
percentage overshoot %
, integral square error
(
)
and root mean square of the control signal
(
)
[31].
Gain in (8) is computed using the lqr function of
Matlab. The syntax of the command is =
(
,
, ,
)
. Experiments 1 to 4 show the
influence of the constant in (7) on the amplitude
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
of the control signal applied to the coupled tank
system. This is confirmed by the decrease in the
value of
as increases. Although the energy
of the control effort decreases, on the other hand,
the transient response tends to deteriorate slightly.
The remaining experiments evaluate the
contribution of the matrix in the transient
response of the liquid level of the tank. The
increase in the values of the diagonal of the matrix
generates a faster system response at the expense
of a greater overshoot. The time responses of the
seven experiments of Table 2, from a practical point
of view, are satisfactory. In order not to overload
the article with an excessive amount of graphics,
we select the first experiment to display some
results. Figure 6 confirms the good tracking of the
liquid level in the tank and Fig. 7 shows the control
signal applied to the pump of the tank always within
the voltage technological limits of the pump.
B. Ramp reference input
The reference input applied to the coupled tank
system is shown in Fig. 8.The seven experiments
carried out to evaluate the ramp internal model and
LQR design are summarized in Table 3. Each
controller designed exhibits excellent results. In this
case, we choose the response of the seventh
experiment to display the performance of the
control system, through figures 9 and 10.
Table II. Different experiments implemented on the coupled tank
system to validate the optimal design of the controller for a step-
type internal model in the reference input
.
#
,
%
1
= , = 0.5
23.70
4.53
1411
6.59
2
= , = 1
28.44
5.13
1541
5.64
3
= , = 2
30.81
5.53
1709
5.54
4
= , = 4
38.71
5.80
1912
5.50
5
= 5, = 1
17.38
8.13
1308
6.59
6
= 10, = 1
11.85
14.00
1291
6.62
7
= 20, = 1
11.06
21.53
1354
6.76
Figure 6. Closed-loop liquid level response for experiment 1 of
Table: 2.
Figure 7. Control signal for experiment 1 of Table 2.
C. Disturbance rejection
The disturbance rejection ability of the design from
the experiment 1 of the step reference case is now
studied through the following experiment. The
system starts in the configuration of Fig. 2, but at
= 100 switches to that of Fig. 11, where the
pump output feeds both tank 1 and tank 2 and at
= 200 , it returns to the initial interconnection.
This experiment allows to model a trapezoidal
perturbation that operates in the interval
[100; 200] , causing a decrease in the inlet flow to
tank 1 and the appearance of a direct flow in tank 2.
Figure 12 shows the very satisfactory property of
disturbance rejection achieved with the design
method.
Figure 8. Input reference signal to validate the optimal controller
design with ramp-like internal model.
Table III. Different experiments implemented on the coupled
tank system to validate the optimal design of the controller for a
ramp-type internal model in the reference input.
#
,
%
1
= , = 0.5
4.00
15.5
4.74
2
= , = 1
4.32
19.7
4.74
3
= , = 2
4.68
25.1
4.73
4
= , = 4
5.10
32.6
4.72
5
= 5, = 1
3.60
11.5
4.74
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
6
= 10, = 1
3.32
9.5
4.74
7
= 20, = 1
3.08
7.8
4.74
Figure 9. Closed-loop liquid level response for experiment 7 of
Table 3.
v. CONCLUSIONS
It is proposed a method to design a controller that
guarantees the optimal tracking of the water level in
a coupled tank system. The major advantage of the
proposed design is the framework on which it is
based, the combination of the internal model
principle and the LQR optimal method, provides
simplicity and flexibility in tuning the small number
of controller gain parameters that are necessary for
the implementation of the method. The experiments
are performed on the coupled tank system and the
results illustrate the effectiveness of the method.
Currently, research is being carried out on how to
transfer specifications of the desired closed-loop
transient response, for example, over shoot and
settling time, in elements of the matrix of the
quadratic cost of the LQR technique.
Figure 10. Control signal for experiment 7 of Table 3.
Figure 11. Connection scheme of the coupled tanks system to
create a disturbance that decrease the inlet to tank 1.
Figure 12. Closed-loop liquid level response for disturbance
rejection.
ACKNOWLEDGMENTS
The author is grateful for the support provided by the
Research Program of the Metropolitan University in
Caracas, Venezuela through project number PG-A-14-
21-22.
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
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Optimal tracking of the water level for a coupled tank system using Linear Quadratic Regulator
PEDRO TEPPA-GARRAN
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Revista TEKHNÉ Nº 25.3
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ISSN electrónico: 2790-5195
ISSN: 1316-3930
83
Esta obra está bajo una licencia de Creative Commons CC BY-NC-SA 3.0 y pueden ser reproducidos para cualquier uso no-
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