A Flatness Based Active Disturbance Rejection Controller for the Four Tank
Benchmark Problem
Congzhi Huang, / H. Sira-Ram´ırez.
Abstract— A flatness based decentralized Active Disturbance
Rejection Control (ADRC) approach is here proposed as a
means to deal with the fully nonlinear, exogenously perturbed,
version, of the four-tank benchmark process problem under
non-minimum phase behavior for the traditionally chosen
system outputs. The liquid level control task, related to the
two lower tanks, is transformed into a corresponding, indi-
rect, multi-input multi-flat-output decoupling control problem
for the flat outputs. Decentralized robust active disturbance
rejection controllers are proposed for the flat outputs, thus
circumventing the well known non-minimum phase problems.
The effectiveness of the proposed approach is validated via
extensive realistic simulation results, wherein unknown chaotic
inputs are proposed as exogenous disturbances.
I. INTRODUCTION
The well-known four-tank benchmark problem, first pro-
posed by Johansson in [1], is a suitable test-bed for com-
paring the performance of different control strategies. It
can be regarded as a prototype for many process control
applications in practical engineering, such as chemical and
power plants. The problem is interesting and it has attracted
the interest of many researchers in the past decades as well
as in the teaching of multi-variable control [2]. It has been
even used as a benchmark problem in a European project [3].
The problem can be employed to ideally demonstrate some
crucial concepts in multivariable control, such as the relative
gain array (RGA) concept and it has the following promi-
nent features: a), the dynamics of the controlled process is
nonlinear; b), the subsystems among the process are highly
coupled and the degree of coupling can be easily adjusted
manually; c), all the variables, including the system inputs,
and states are available; d), the process can be operated very
safely.
ADRC was first initiated by Han [4], as a robust design
technique for disturbance rejection in nonlinear system with
parameter uncertainties and significant external disturbances.
The uncertainties are to be canceled by a nonlinear control
law [5]. ADRC has found remarkable applications in indus-
try, such as in power filter design [6,7], actuators [8], rotatory
speed regulation [9], power plant control [10], and so on.
Recently, Sira-Ram´ırez proposed an ADRC approach from
the perspective of differential flatness [11].
Gao [12] proposed a simplified linear version of the
original ADRC approach in 2003, where all the poles of the
Congzhi Huang is with School of Control and Computer Engineer-
ing, North China Electric Power University, Beijing, 102206, China
hcz190@ncepu.edu.cn
H. Sira-Ramirez is with the Department of Electrical Engineering,
Mechatronics Section, CINVESTAV-IPN, Av. IPN No. 2508, Col. San Pedro
Zacatenco 07360, D.F, Mexico hsira@cinvestav.mx
ESO and the closed-loop characteristic equation are placed
in the same location, and thus there is only one parameter
to be tuned.
The main objective of this paper is to propose an Active
Disturbance Rejection Control (Han [4]) approach to the
output regulation problem on a four-tank process and present
a comprehensive robust design procedure for the perturbed
nonlinear multi-input multi-output (MIMO) system from
the perspective of flatness. The prominent feature of the
proposed approach is that it is directly aimed at the nonlinear
system without any attempt of approximate linearization, and
with due consideration of complex (i.e., chaotic) exogenous
disturbances. Flatness has been shown to easily circumvent
non-minimum phase problems in output regulation schemes
for dynamical systems defined on a variety of dynamical
systems contexts (linear, non-linear, mono-variable, multi-
variable, continuous or discrete systems). The four tank
benchmark problem constitutes a typical nonlinear multi-
variable system, which may be subject to unknown distur-
bances, with an additional non-minimum phase behavior for
the set of naturally chosen outputs.
The paper is organized as follows. The four-tank bench-
mark problem is described in Section II, the flatness based
ADRC solution to the problem will be proposed in Section
III. Simulation results are given in Section IV to demonstrate
the effectiveness of the proposed approach in light of chaotic
disturbances. Section V presents the conclusions of the
article.
II. FOUR-TANK BENCHMARK PROBLEM FORMULATION
A. Process description
In this section, the four-tank benchmark problem will be
formulated first. The schematic diagram of the four-tank
benchmark process is shown in Fig. 1 (see [13]).
As can be seen from Fig.1, there are four tanks and two
pumps. Water discharged from pump a, coming from the
source tank, are delivered to Tank 1 and Tank 4. A valve is
used to split this water to feed Tank 1 with a fraction γ
a
and
Tank 4 with a fraction 1–γ
a
. Tank 2 and Tank 3 are fed by
the water extracted from pump b,with corresponding fraction
coefficients γ
b
, and 1–γ
b
, respectively. In addition, there is a
discharge hole at the bottom of each one of the tanks. The
water in the Tank 3 discharges into Tank 1 through the hole
at the bottom of Tank 3, while the water in Tank 4 flows into
Tank 2 thanks to the pressure and the force of gravity. There
are pressure transducers installed on all of the tanks and
thus the liquid level of each tank is available. The controlled
variables are represented by the levels of Tank 1 and Tank
2015 American Control Conference
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