Ali Hoang Hussain Dochain 2015

Computers and Chemical Engineering 76 (2015) 27–41
Contents lists available at ScienceDirect
Computers and Chemical Engineering
journal homepage: www.elsevier.com/locate/compchemeng
Review
Review and classification of recent observers applied in chemical
process systems
Jarinah Mohd Ali a , N. Ha Hoang b,∗∗ , M.A. Hussain a,∗ , Denis Dochain c
a
b
c
Department of Chemical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
Faculty of Chemical Engineering, University of Technology, VNU-HCM, 268 Ly Thuong Kiet Str., Dist. 10, HCM City, Viet Nam
CESAME, Université Catholique de Louvain, 4-6 Avenue G. Lemaître, B-1348 Louvain-la-Neuve, Belgium
a r t i c l e
i n f o
Article history:
Received 16 July 2013
Received in revised form 12 January 2015
Accepted 21 January 2015
Available online 11 February 2015
Keywords:
Review
Observer
State estimation
Chemical process
a b s t r a c t
Observers are computational algorithms designed to estimate unmeasured state variables due to the
lack of appropriate estimating devices or to replace high-priced sensors in a plant. It is always important to estimate those states prior to developing state feedback laws for control and to prevent process
disruptions, process shutdowns and even process failures. The diversity of state estimation techniques
resulting from intrinsic differences in chemical process systems makes it difficult to select the proper
technique from a theoretical or practical point of view for design and implementation in specific applications. Hence, in this paper, we review the applications of recent observers to chemical process systems
and classify them into six classes, which differentiate them with respect to their features and assists in
the design of observers. Furthermore, we provide guidelines in designing and choosing the observers for
particular applications, and we discuss the future directions for these observers.
© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Contents
1.
2.
3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classifications and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Methodology for observer design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.
Observability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.
Estimated variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.
Kinetics information of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.
Observer formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1.
Model of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2.
Observer equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3.
Observer gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4.
Error dynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.
Evaluating the observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current and future trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A. Observability matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B. Observability Gramians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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∗ Corresponding author. Tel.: +60 3 79675206; fax: +60 3 79675319.
∗∗ Corresponding author. Tel.: +84 8 38650484; fax: +84 8 38637504.
E-mail addresses: [email protected] (J. Mohd Ali), [email protected] (N. Ha Hoang), mohd [email protected] (M.A. Hussain), [email protected]
(D. Dochain).
http://dx.doi.org/10.1016/j.compchemeng.2015.01.019
0098-1354/© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
28
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
Nomenclature
A, B, C, E, F, G state space matrices
sign
component wise for vector argument of
z = col(z1 ,. . .,zn )
x1
measured component concentration
x̂
estimated state vector
ẋ, ż
dynamic of state vector
Z
process vector
measurement noise covariance vector
RV
u1 , u2
vector partition correspond to auxiliary variables
d
discrete
L, K
observer gain
.
estimated state vector with linear innovative term
of discontinuous function
external transfer vector
U1 , U2
O
observability matrix
Pk−1
covariance at time k − 1
Fk−1
nonlinear state transition function
auxiliary variable
D(s)
estimated disturbance
A1 , A2
unique solution of process vector
D1 (s), D2 (s) lumped disturbances in loop including external
disturbances
x̂
1. Introduction
The implementation of state feedback laws in a controlled plant
is often based on the assumptions that all states are available for
online measurement; however, in practice, some of them may not
be measurable due to a lack of appropriate estimating devices or
the high price of sensors (Dochain et al., 2009; Jana, 2010; Soroush,
1997; Wang et al., 1997). As a consequence, measuring the missing states or variables is expensive and time consuming due to the
significant technical standard requirements and the high cost of
installation of these devices (Gonzalez et al., 2001; Hulhoven et al.,
2006). For these reasons, devices called observers have been developed to reconstruct the state vector in order to estimate the missing
variables and, at the same time, to reduce the usage of high-priced
sensors (Dochain et al., 2009).
Luenberger (Luenberger, 1964, 1966, 1967, 1971) and Kalman
(Welch and Bishop, 1995) introduced the basic concepts of state
observers and Kalman Filter (KF)-based observers in the 1960s.
However, over the years, research in the design of observers has
become popular but challenging due to the requirements of high
accuracy, low cost and good prediction performances. In fact, many
observers today are simply modifications and extended versions
of the classical Luenberger observer and Kalman filter. In recent
years, various types of observers have been developed to accurately estimate state variables in linear and nonlinear chemical
processes (Aguirre and Pereira, 1998; Bastin and Dochain, 1990;
del-Muro-Cuellar et al., 2007; Gonzalez et al., 1998; Huang et al.,
2010; Lombardi et al., 1999; Pedret et al., 2009). They have been
widely used both theoretically and practically through simulations
and real plant testing (Bejarano and Fridman, 2010; Busawon and
Kabore, 2001; Farza et al., 2011; Lee, 2011; Lin et al., 2003; Oya and
Hagino, 2002).
Researchers have also developed observers for systems to tackle
problems such as disturbances, mismatches and faults. For this
purpose, different types of observers were developed with closely
similar formulations designed to overcome the drawbacks of each
other. For example, to estimate disturbances, the disturbance
observer (DOB) was introduced (Chen et al., 2009; Kim et al., 2011),
and later, the modified disturbance observer (MDOB) was developed to target large disturbances and mismatches (Yang et al.,
2011). After that, the fractional-order disturbance observer (FODOB) and Bode-ideal-cut-off observer (BICO-DOB) (Olivier et al.,
2012) were developed to include methodology for tuning and
optimizing the estimation performance. Another example is the
asymptotic observer, which was first developed based on available measurements of the temperature of a mixture and a subset of
the concentrations (Dochain et al., 1992) and later extended on the
basis of the energy balance almost similar to the thermodynamic
properties of the mixtures (Dochain et al., 2009; Hoang et al., 2012,
2013).
Due to the variety of methodologies in observer design for chemical process systems, combining and classifying them into several
different groups would be highly useful to serve as guidelines
to select and then design the appropriate observers for a specific chemical application. Previous surveys have only included the
study of one or two types of observers. For example, the reviews
by Spurgeon (2008) and Hidayat et al. (2011) focused respectively
on single observer types such as the sliding mode observers and
observers for linear distributed parameter systems. Another survey from Radke and Zhiqiang reviewed the design advantages of
a particular type of disturbance observers for practitioners (Radke
and Zhiqiang, 2006), whereas Ruhm solely explained the concepts
of open and closed loop observers (Ruhm, 2008). Dochain has
presented the available results of state and parameter estimation
approaches for chemical and biochemical processes, specifically the
extended Luenberger (ELO), Kalman (EKO), asymptotic and interval observers (Dochain, 2003). Kravaris and coworkers provided
an overview of recent developments regarding the design of nonlinear Luenberger observers, with special emphasis on the exact
error linearization techniques, and discussed general issues including observer discretization, sampled data observers and the use of
delayed measurements (Kravaris et al., 2012). In addition, Prakash
and coworkers reviewed recently developed Bayesian estimators
(Prakash et al., 2011), and Daum focused only on the extended nonlinear filters on the basis of the classical KF (Daum, 2005). Chen has
also reviewed Bayesian filtering from KF to particle filter, emphasizing the stochastic filtering theory based on Bayesian perspectives
(Chen, 2003). However, all of these reviews are specific in nature
and do not consider the whole spectrum of the different classes of
observers available.
Therefore, this review paper intends to provide a comprehensive survey considering the unique features of different types of
recent observers in chemical process systems (Dochain, 2003;
Kravaris et al., 2012) by categorizing them into different classes,
a level of organization not currently available in the literature. Six
classes are proposed, namely, Luenberger-based observers, finitedimensional system observers, Bayesian estimators, disturbance
and fault detection observers, artificial intelligence (AI)-based
observers and hybrid observers. In brief, the main contribution
of this review is to provide the list of recent observers that have
been applied in chemical process systems and to classify them
into six classes with emphasis on their positive highlights based
on their estimation performances in specific chemical process systems. Recent observers refer to the observers developed since the
year 2000, as most observers before that would be referred to as
the classical types (Elicabe et al., 1995; Gonzalez et al., 1998; Lee
and Ricker, 1994; Oliveira et al., 1996; Soroush, 1997; Wang et al.,
1997). All of these observers can be either linear or nonlinear and
have served as specific estimators to several unit operations. However, this review does not include some methodologies such as the
recursive error method (Lee et al., 2000) and the partial least square
method (Roffel et al., 2003), which have also been considered as
estimators.
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
This paper is organized as follows. The introduction is in Section
1, followed by the classification of observers and their applications
in chemical process systems in Section 2. Section 3 discusses the
observer design methodology. Section 4 focuses on the current and
future trends of observers, while Section 5 concludes the review.
2. Classifications and applications
The formulations of observer design methodologies in research
publications have normally been written and explained with
merely theoretical emphasis, which makes it difficult for practitioners and researchers, especially newcomers to this area, to
choose (potentially) appropriate observers for their systems. In
addition, selecting the most suitable observer for any specific system is an important but difficult task for installed systems due
to the diversity of the many available methods, observer types,
application range and nature of chemical process systems. So far,
this research area has been very active and attracts attention from
many researchers (Aguilar-Garnica et al., 2011; López-Negrete and
Biegler, 2012; Mesbah et al., 2011; Nagy Kiss et al., 2011; Olivier
et al., 2012). Based on our extensive review of the recent observers
applied to chemical process systems, we can clearly differentiate
them into six major classes. These classes are the Luenbergerbased observers, finite-dimensional system observers, Bayesian
estimators, disturbances and fault detection observers, artificial
intelligence-based observers and hybrid observers. The attributes,
advantages, limitations and guidelines for practitioners according
to each class are given in Table 1, while Table 2 sorts these recent
observers into their respective classes. In addition, the selection of
the recent observers according to those classes is depicted in Fig. 1
to guide and help researchers in their selection.
The category of Luenberger-based observers is the first class
that groups together all of the observers designed based on the
Luenberger observer methodology, or, in other words, it involves
the extended versions of the classical Luenberger observer itself
(Alonso et al., 2004; Dochain, 2003; Fissore et al., 2007; Tronci
et al., 2005; Vries et al., 2010). The extended Luenberger observer
(ELO), sliding mode observer (SMO), adaptive state observer (ASO),
generic and backstepping observers are examples of observers
falling into this class. This type of observer is suitable for less complex linear systems with relatively simpler computational methods
(Bejarano et al., 2007b).
The second category is the finite-dimensional system observers,
which include, among others, the reduced-order, low-order,
high-gain, asymptotic and exponential observers. These finitedimensional system observers are designed for chemical process
systems whose dynamics are described by ordinary differential
equations (ODEs) (Bitzer and Zeitz, 2002) and are quite straightforward to implement. They suit systems with less kinetic information,
but the accuracy of the convergence rate is uncertain. For example, for the case of asymptotic and exponential observers, the
convergence rate can only be shown if the process operating conditions are such that the dilution rate is bounded (Dochain et al.,
1992; Dochain, 2000; Hadj-Sadok and Gouze, 2001; Hoang et al.,
2013). It is worth noting that asymptotic/exponential and interval observers can also be extended to infinite dimensional systems
(i.e., distributed parameter systems) such as for tubular reactors
and plug flow reactors (Dochain, 2000; Aguilar-Garnica et al.,
2011).
Bayesian estimators, in the third category, provide an approach
based on the probability distribution estimation of state variables
by utilizing the available data of the system (Chen et al., 2004).
It assumes that all variables are stochastic in nature, and thus,
the distribution of state variables is achievable based on the measured variables. Examples of the Bayesian type of estimators are the
29
extended Kalman filter (EKF), particle filter (PF) and moving horizon estimator (MHE). These are based on probability distribution
and are therefore consistent and versatile estimators, which are
highly appropriate for fast estimation (Abdel-Jabbar et al., 2005;
Fan and Alpay, 2004; Patwardhan and Shah, 2005). However, the
computational complexity involved in using this approach makes
them infeasible for high-dimensional systems.
The fourth class is the disturbance and fault detection observers.
Although they can be of different classes, both are included in one
category because they are mostly applied to estimate irregularities in the system, either through disturbances or faults (Olivier
et al., 2012). Fault detection observers can also be applied to estimate parameters for fault diagnosis of chemical process systems.
Examples of disturbance and fault detection observers are the
disturbance observer (DOB), the modified disturbance observer
(MDOB), the unknown input observer (UIO) and the nonlinear
unknown input observer (NUIO). These are highly specific types of
observers and focus only on disturbances or fault detection related
variables during the estimation process (Chen et al., 2009; RochaCózatl and Wouwer, 2011; Sotomayor and Odloak, 2005; Yang et al.,
2011). They are mostly suitable for estimating disturbances and
faults, which provide early warning to operators prior to causing
disruption to the process units (Sotomayor and Odloak, 2005; Zarei
and Poshtan, 2010).
The fifth class is the artificial intelligence (AI)-based observers.
AI is the science of making the program perform intelligencebased tasks, which include methods such as fuzzy logic, artificial
neural networks (ANN), expert systems and genetic algorithms.
These types of observers have been widely utilized as estimators
in recent times. For example, the work by Hussain and coworkers utilized a hybrid neural network (HNN) to predict porosity in
a food drying process (Hussain et al., 2002), and the research by
Aziz and coworkers applied ANN to estimate the heat released
from a polymerization reactor (Aziz et al., 2000). Other applications
of AI-based observers can also be found in many papers (Barton
and Himmelblau, 1997; Islamoglu, 2003; Khazraee and Jahanmiri,
2010; Kordon et al., 1996; Kuroda and Kim, 2002; Liu, 2007; Ng and
Hussain, 2004; Turkdogan-Aydınol and Yetilmezsoy, 2010; Wang
et al., 2006, Wei et al., 2007). However, this review paper covers
only recent types of AI-based observers coupled with conventional (model-based) types, as purely AI observers are not recent
in nature. Examples of these recent types are the fuzzy Kalman filter (FKF) and the EKF-neural network observers (Porru et al., 2000;
Prakash and Senthil, 2008). These AI-based observers overcome the
limitations of single-based observers and are suitable for systems
with incomplete model structure and information. The formulation
of AI-based observer may be difficult and time consuming compared to the other hybrid observers in some systems (Senthil et al.,
2006). In addition, the AI elements must first be adapted for online
implementation (Himmelblau, 2008; Lashkarbolooki et al., 2012;
Rivera et al., 2010).
The sixth class is the hybrid observers, which are combinations
of more than one observer to obtain improved estimation in certain
systems. An example of this is the extended Luenberger observer
(ELO) combined with the asymptotic observer (AO) (Hulhoven et al.,
2006). ELO provides good convergence factors, while AO estimates
parameters without any kinetics data. Therefore, the combination
results in an improved hybrid observer that contains both features.
Hybrid observers are good at overcoming the limitations of the
single observer, but choosing the appropriate combination may
be tedious and time consuming (Aguilar-Lopez and Maya-Yescas,
2005; Bogaerts and Wouwer, 2004; Goffaux et al., 2009). Normally
this class of observer is suitable for conditions where the singlebased observer is not accurate enough for the process systems, for
instance, to compensate for offsets in estimation resulting from the
use of the single observer (Hulhoven et al., 2006).
30
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
Table 1
Observers’ overall evaluation according to classes.
No.
Class of
observers
1
Luenbergerbased
observers
2
3
Finitedimensional
system
observers
Bayesian
estimators
Example of observer equation
Attributes
Advantages
Limitations
Guidelines for practicing
engineers
For sliding mode observer:
Extension of classical
Luenberger observer
Simple computational
methods
Design is always based
on the perfect
knowledge of system
parameters
For exponential observer:
d
= F + Gx1 − LU1 + U2
dt
Knowledge of process
system kinetics is not
necessary
Easy implementation and
simple formulation
Convergence factor
depends strongly on
the operating condition
For less complex linear
systems, this type of
observer is sufficient for
crucial parameter
estimation
Suitable for systems with
less kinetics information
For extended Kalman filter:
T
+ Rv
Pk|k−1 = Fk−1 Pk−1|k−1 Fk−1
Based on probability
distribution and
mathematical
inference of the system
Fast estimation based on
prediction-correction
method and versatile
estimators
Good at estimating
disturbances and
predicting faults before
they can affect the unit
operations of the plant
Overcome limitations of
single observer and
suitable for systems with
incomplete model
structure
The complexity of their
computational method
is sometimes infeasible
for high dimensional
systems
May ignore other
uncertainties during
the estimation process
.
x̂ = Ax̂ + Bu + Lsign(y − C x̂)
4
Disturbance
and fault
detection
observers
For disturbance estimation:
D(s) = D̃1 (s) + D̃2 (s) + · · · + D̃n (s)
Focus on estimating
disturbances and
detecting faults within
the system
5
AI-based
observers
According to AI-elements, example
using fuzzy logic where the
IF-THEN rule is:
IF e is negative small AND e is zero
THEN x̂estimated = xactual
Combination of
observers with AI
elements
6
Hybrid
observers
For combination of extended
Luenberger and asymptotic
observer:
dZ(t)
= D(t)Z(t) + A1 u1 (t) + A2 u2 (t)
dt
Combination of two or
more observers
The detailed applications of the various observers under these
six classes are listed in Table 3. The table does not need any further
elucidation because it is comprehensive and self-explanatory in
nature, covering the objectives, the positive highlights, applications
in various unit operations and the relevant references involved for
each of the observer types.
3. Methodology for observer design
Observers were first designed based on linear formulation;
these original observers were known as linear observers and used
to estimate states and unknown variables in a linear process in
the eventual presence of disturbances or noise (Bara et al., 2001;
Overcome the limitations
of a single observer
May be difficult and
time consuming
For online
implementation, the AI
elements must first be
adapted to the system
Choosing appropriate
combination may be
tedious
For fast estimation results
based on probability theory,
Bayesian estimators may be
applied
If the objective is to
estimate disturbances and
parameters to predict faults,
then these type of observers
are the most appropriate
For highly nonlinear
systems with an incomplete
or unknown model
This is suitable for systems
where a single type of
observer is not accurate
enough
Bejarano and Fridman, 2010; Bejarano et al., 2007a,b; Bodizs et al.,
2011; Busawon and Kabore, 2001; El Assoudi et al., 2002; Fissore,
2008; Jafarov, 2011; Lee, 2011; Oya and Hagino, 2002; Vries et al.,
2010). Later, because most processes exhibit highly nonlinear
behavior, researchers formulated nonlinear observers (Besancon,
2007; Bitzer and Zeitz, 2002; Boulkroune et al., 2009; Busawon and
Leon-Morales, 2000; de Assis and Filho, 2000; Di Ciccio et al., 2011;
Dong and Yang, 2011; Farza et al., 1997, 2011; Floquet et al., 2004;
Hashimoto et al., 2000; Kalsi et al., 2009; Kazantzis and Kravaris,
2001; Kazantzis et al., 2000; Ko and Wang, 2007; Kravaris et al.,
2007; Maria et al., 2000; Schaum et al., 2008).
Most researchers developed observers based on the mathematical model of the systems and used the first principles model
prior to developing the observer’s equation (Dochain et al., 2009).
Table 2
Recent observers categorized under different classes.
Class
Luenberger-based
observers
Specific observer
1. Extended Luenberger
observer (ELO)
2. Sliding mode observer
(SMO)
3. Adaptive state
observer (ASO)
4. High-gain observer
5. Zeitz nonlinear
observer
6. Discrete-time
nonlinear recursive
observer (DNRO)
7. Geometric observer
8. Backstepping observer
Finite-dimensional
system observers
Bayesian estimators
Disturbance and fault
detection observers
Artificial
intelligence-based
observers
Hybrid observers
1. Reduced-order
observer
2. Low-order observer
3. High gain observer
4. Asymptotic observer
(AO)
5. Exponential observer
6. Integral observer
7. Interval observer
1. Particle filter (PF)
2. Extended Kalman filter
(EKF)
3. Unscented Kalman filter
(UKF)
4. Ensemble Kalman filter
(EnKF)
5. Steady state Kalman
filter (SSKF)
6. Adaptive fading Kalman
filtering (AFKF)
7. Moving horizon
estimator (MHE)
8. Generic observer
9. Specific observer
1. Disturbance observer
(DOB)
2. Modified disturbance
observer (MDOB)
3. Fractional- order
disturbance observer
4. Bode-ideal cut-off
observer
5. Unknown input observer
(UIO)
6. Nonlinear unknown
input observer
7. Extended unknown
input observer
8. Modified proportional
observer
1. Fuzzy Kalman filter
2. Augmented fuzzy
Kalman filter
3. Differential neural
network observer
4. EKF with neural network
model
1. Extended
Luenberger-asymptotic
observer
2. Proportional-integral
observer
3. Proportional-SMO
4. Continuous-discrete
observer
5. Continuous-discreteinterval observer
6. Continuous-discreteEKF
7. High-gaincontinuous-discrete
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
31
Fig. 1. General guideline for selecting recent observers according to their classification.
Therefore, most observer designs are model-based approaches
(Damour et al., 2010a, 2010b; Dochain et al., 2009; Salehi and
Shahrokhi, 2008) with the exception of the AI-based observers
(Bahar and Ozgen, 2010; Lahiri and Ghanta, 2008; Mohebbi et al.,
2011; Rezende et al., 2008; Wei et al., 2010). The gain of the
observer and its estimation error dynamics are also significant in
designing the model-based observers, as shown in several references (Aguilar-Lopez and Martinez-Guerra, 2007; De Battista et al.,
2011; Hulhoven et al., 2008; Porru et al., 2000). However, the success of observer designs is evaluated based on their capabilities in
estimating the difficult-to-measure states with satisfactory convergence rates (Bitzer and Zeitz, 2002; Ciccio et al., 2011; Hashimoto
et al., 2000; Kravaris et al., 2007; Lafont et al., 2011; Morel et al.,
2006) and with approximately zero estimation errors (Benaskeur
and Desbiens, 2002; Bogaerts and Wouwer, 2004; Kazantzis et al.,
2000; Liu, 2011).
The first important step before designing the observer is to
consider the detectability or observability condition of the system because observers have to be designed for a detectable or
observable system. Observability is the condition in which all initial states are observable and a system is said to be observable if,
for any initial condition vectors, its internals states can be inferred
by knowledge of its (external) outputs (Evangelisti, 2011; Moreno
and Dochain, 2008; Soroush, 1997). Detectability is a weaker condition than observability, where the non-observable states can
still decay to zero asymptotically (Evangelisti, 2011; Moreno and
Dochain, 2008). Both concepts will influence the feasibility conditions of the observers (Dochain et al., 1992; Hoang et al., 2013;
Moreno and Dochain, 2008). The concept of observability is central
in reconstructing unmeasurable state variables. This explains the
need of observers to estimate unknown states prior to developing
control laws and the fact that not all states are available directly
through on-line measurements (Ogata, 1995). Extensive discussions on observability and detectability can be found in various
references (Astolfi and Praly, 2006; Evangelisti, 2011; Hermann and
Krener, 1977; Moreno and Dochain, 2008; Soroush, 1997; Zuazua,
2007).
Once the system dynamics fulfill the observability or detectability conditions, observers can then be designed to estimate the state
variables. In this respect, the choice of a suitable observer according to the six classes provided in Section 2 is therefore of great
importance. Prior to that, the desired estimated states (i.e. the
exact values of the observed states) and initial conditions must
be defined clearly (Farza et al., 2011). After that, tests are run to
compare the estimates with the actual values to determine the performance of the proposed observer (Aamo et al., 2005; De Battista
et al., 2011; Hajatipour and Farrokhi, 2010; Jana et al., 2006; Nagy
Kiss et al., 2011; Salehi and Shahrokhi, 2008). The test not only is
important for the design of the single observer but also determines
whether a hybrid observer is further needed to estimate the parameter (Goffaux et al., 2009; Hulhoven et al., 2006; Sheibat-Othman
et al., 2008). If there are huge discrepancies between the actual
and estimated values, a hybrid observer should be developed to
improve the estimates. Furthermore, if systems are complex and
models are difficult to obtain from the first principles, hybrid AIbased observers would possibly be a suitable choice (Chairez et al.,
2007; Porru et al., 2000; Prakash and Senthil, 2008).
The design guideline for these observers based on the six classes
is depicted in Fig. 2 with detailed explanation in the following subsections.
32
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
Table 3
Application of recent observers in chemical process systems under different classes.
Observer
System
Positive highlight(s)
Ref.
Class 1: Luenberger-based observers
Crystal mass
ELO
Crystallization unit
Damour et al. (2010a,b)
ELO
ELO
Solutes concentration
Process kinetics, influent
concentrations
Substrate concentration,
specific growth rate
Specific growth rate
Substrate concentration
Biomass and substrate
concentration
Reactor parameters
Growth rate, kinetic coefficient
Liquid, vapor flow rate, reboiler
coefficient
Radical concentration
Fed-batch crystallizer
Fixed bed reactor
Good estimation without perfect
initial condition
Robust against model deviation
Easy to implement, simple structures
Fermentation process
Smooth estimates
Fed-batch bioreactor
Bioreactor
Bioreactor
Accurate and error free estimation
Proven stability factor
Proven stability factor
CSTR
Bioreactor
Debutanizer
Distribution coefficients
Compositions, partially known
parameters
Concentrate and tailing grade
Distillation column
Batch distillation column
Stable estimator
Guaranteed convergence factor
Precise estimates under mismatch
condition
Estimates without information of
initiator
Guaranteed convergence factor
Good convergence factor
Sheibat-Othman et al.
(2008)
Jana et al. (2006)
Murlidhar and Jana (2007)
Nitrogen oxide (NOx ) inlet
concentration, outlet reactant
conversion
Product compositions
Loop reactor
Guaranteed convergence, zero
estimation error
Fast, reliable estimates
Benaskeur and Desbiens
(2002)
Fissore et al. (2007)
Tronci et al. (2005)
Compositions, solid mass
fraction, production rate
Copolymerization reactor
Overcomes ill-conditioning of the
observability matrix
Accurate estimation
Class 2: Finite-dimensional system observers
Reduced-order
Down hole pressure
Reduced-order
Reactor concentration
Gas-lift well
CSTR
Stable estimates
Good concentration estimates
Reduced-order
Low-order
High-gain
High-gain
Bioreactor
30-tray distillation column
CSTR
CSTR
Robust estimation
Robust against noise
Robust against noise and disturbances
Precise estimates
Exponential
Substrate concentration
Steady state profiles
Reaction heat
Reactor concentration and
temperature
Reactor concentration
Tubular reactora
Exponential
Exponential
AO
Top tray compositions
Microorganisms concentration
Concentrations, enthalpy
Batch distillation column
Bioreactor
CSTR
AO
Reactor concentration
Tubular reactora
AO
Growth rate
Activated sludge process
Interval
Activated sludge process
Interval
Interval
Organic concentration, growth
rates
Reactant concentration
Residual parameters
Good estimation without process
kinetics
Good convergence properties
Guaranteed convergence
Good estimation, not sensitive to
noise
Good estimation without process
kinetics
Precise estimation without process
kinetics
Converge toward bounded interval
Aamo et al. (2005)
Salehi and Shahrokhi
(2008)
Kazantzis et al. (2005)
Singh and Hahn (2005c)
Aguilar et al. (2002)
Biagiola and Figueroa
(2004b)
Dochain (2000)
Integral
Heat of reaction
SMO
SMO
SMO
SMO
DNRO
ASO
ASO
ASO
ASO
ASO
Backstepping
Zeitz nonlinear
observer
Geometric
Geometric
Objective/estimate(s)
Polymerization process
Solid-solid separation unit
Distillation column
Mesbah et al. (2011)
Mendez-Acosta et al.
(2008)
Pico et al. (2009)
De Battista et al., 2011
Gonzalez et al. (2001)
Hajatipour and Farrokhi
(2010)
Huang et al. (2010)
Zhang and Guay (2002)
Jana et al. (2009)
López and Alvarez (2004)
Jana (2010)
El Assoudi et al. (2002)
Dochain et al. (2009)
Dochain (2000)
Plug flow reactora
Separator (grinding
process)
CSTR
Robust estimation
Good convergence factor
Hadj-Sadok and Gouze
(2001)
Hadj-Sadok and Gouze
(2001)
Aguilar-Garnica et al., 2011
Meseguer et al. (2010)
Robust estimation
Aguilar-Lopez (2003)
Class 3: Bayesian estimators
SSKF
Time-delay
Stirred tank heater
Consistent estimates even with noise
SSKF
Product compositions
Batch distillation column
Stable estimation
EKF
Interface temperature
Freeze-drying process
EKF
Component’s concentration
Batch distillation column
EKF
Product compositions
Batch distillation column
Good estimation without perfect
initial condition
Simple observer design yet accurate
estimation
Precise estimate even with noise
Patwardhan and Shah
(2005)
Venkateswarlu and
Avantika (2001)
Velardi et al. (2009)
EKF
EKF
Outlet reactor concentration
Liquid compositions
Accurate concentration estimation
Robust against modeling error
EKF
Top tray compositions and
flow rates
Solutes concentration
Solutes concentration
Particle size distribution
CSTR
Reactive distillation
column
Distillation column
Fed-batch crystallizer
Fed-batch crystallizer
Semi-batch reactor
Robust against model deviation
Robust against model deviation
Good estimation without accurate
model
EKF
UKF
UKF
Guaranteed convergence factor
Yildiz et al. (2005)
Venkateswarlu and
Avantika (2001)
Himmelblau (2008)
Olanrewaju and Al-Arfaj
(2006)
Jana et al. (2006)
Mesbah et al. (2011)
Mesbah et al. (2011)
Mangold et al. (2009)
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
33
Table 3 (Continued)
Observer
Objective/estimate(s)
System
Positive highlight(s)
Ref.
UKF
Biomass concentration
Fermentor
Wang et al. (2010)
UKF
EnKF
EnKF
AFKF
Uncertain parameters
Solute concentrations
Unmeasured disturbances
Product compositions
Hybrid tank system
Fed-batch crystallizer
Hybrid tank system
Batch distillation column
AFKF
Temperature
Heat exchanger
PF
Yield parameter
Fermentor
PF
Conditional density
CSTR
PF
Conditional density
Batch Reactor
MHE
MHE
Solutes concentration
Molecular weight distribution
Fed-batch crystallizer
Polymerization reactor
Effective estimation despite using the
simplified mechanistic model
Effective control and good estimation
Robust against model deviation
Effective control and good estimation
Precise estimate despite noisy
conditions
Good estimation without coefficient
adjustment
Good estimation based on
maximization algorithm theory
Few assumptions required for
estimation
Few assumptions required for
estimation
Robust against model deviation
Smooth estimates
MHE
Tray efficiencies
Binary distillation column
MHE
Generic observer
Biomass concentration
Carbon and nitrogen
concentrations
Carbon and nitrogen
concentrations
Prakash et al. (2010)
Mesbah et al. (2011)
Prakash et al. (2010)
Venkateswarlu and
Avantika (2001)
Bagui et al. (2004)
Chitralekha et al. (2010)
López-Negrete et al. (2011)
López-Negrete et al. (2011)
Animal cell cultures
Sequential batch reactor
Able to handle constraint during
estimation
Accurate estimates
Robust against modeling error
Mesbah et al. (2011)
López-Negrete and Biegler
(2012)
López-Negrete and Biegler
(2012)
Raïssi et al. (2005)
Boaventura et al. (2001)
Sequential batch reactor
Robust against modeling error
Boaventura et al. (2001)
Class 4: Disturbances and fault detection observers
Disturbances related to time
DOB
delay
FO-DOB
Disturbances due to mismatch
Conveyor (grinding
process)
Cyclone (grinding process)
Chen et al. (2009)
BICO-DOB
Disturbances due to mismatch
Cyclone (grinding process)
MDOB
Jacketed stirred tank heater
CSTR
Robust against uncertainties
Aguilar-Lopez and
Martinez-Guerra (2005)
UIO
Closed-loop system
disturbances
Uncertainties in reactive
concentration, reactor and
jacket temperature
Fault in actuator and sensor
Overcome the effect of internal
disturbances
Optimize the estimation even with
huge disturbances
Optimize the estimation even with
huge disturbances
Smooth disturbances estimate
Polymerization reactor
Accurate estimation
UIO
Fault in input sensor
CSTR
QUIO
Faults in concentration, flow
rates, light intensity
Fault in residuals
Fault in residuals
Bioreactor
Accurately estimating fault even in
the presence of disturbances
Satisfactory estimates
Sotomayor and Odloak
(2005)
Zarei and Poshtan (2010)
CSTR
CSTR
Acting as alternative fault alarm
Acting as alternative fault alarm
Rocha-Cózatl and Wouwer
(2011)
Zarei and Poshtan (2010)
Zarei and Poshtan (2010)
CSTR
Unbiased estimation
Prakash and Senthil (2008)
CSTR
Satisfactory unbiased estimates
Prakash and Senthil (2008)
Specific observer
Modified proportional
NUIO
EUIO
Class 5: AI-based observers
FKF
Olivier et al. (2012)
Olivier et al. (2012)
Yang et al. (2011)
Microreactor
Good agreement with the actual value
Poznyak et al. (2007)
Wastewater treatment
plant
Heterogeneous reactor
Guaranteed small estimation error
Chairez et al. (2007)
EKF-NN
Reactor temperature and
concentration
Reactor temperature and
concentration
Anthracene dynamics
decomposition and
contaminant concentration
Formic acid, fumaric acid,
maleic acid, oxalic acid
Outlet reactor concentration
Further reduction in estimation error
compared to EKF
Porru et al. (2000)
Class 6: Hybrid observers
ELO-AO
Continuous-discrete
Biomass concentration
Biomass concentration
Bioreactor
Batch reactor
Stable rate of convergence
Robust against modeling error
Process kinetics
Bioreactor
Biomass, substrate
concentration
Polymer molecular weight,
monomer concentration,
reactor temperature
Unknown inputs
Bioreactor
Avoids growth of interval sizes during
estimation
Accurate estimates, reduced error
Hulhoven et al. (2006)
Aguilar-Lopez and
Martinez-Guerra (2005)
Goffaux et al. (2009)
ASFKF
DNNO
DNNO
Continuous-discreteinterval
Continuous-discreteEKF
Proportional-SMO
Proportional-integral
High-gain-continuousdiscrete
a
Rate coefficient
Polymerization reactor
Robust against noise and uncertain
parameters
Bogaerts and Wouwer
(2004)
Aguilar-Lopez and
Maya-Yescas (2005)
Wastewater treatment
plant
Polymerization process
Stable estimation rate
Nagy Kiss et al. (2011)
Estimates without information of
initiator
Sheibat-Othman et al.
(2008)
Finite-dimensional system observers may be extended to the infinite-dimensional systems such as for tubular and plug flow reactor.
34
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
Fig. 2. General methodology for observer design.
3.1. Observability conditions
Two types of observability conditions typically applied for
observer designs are the observability matrix and the observability Gramian. The observability matrix appears with the alteration of
the state space models such as conversion to canonical forms, while
the observability Gramian arises when considering the operator
properties including system reduction and optimal linear quadratic
regulators (Curtain and Zwart, 1995; Singh and Hahn, 2005a). Both
the observability matrix and the observability Gramian provide
sufficient conditions for the observability of a system; however,
the observability matrix is related to the differential properties,
while the observability Gramian is based on the integral conditions (Tsakalis, 2013). Furthermore, the type of observability used
to detect the observable condition will depend on the formulation
of the systems. Brief formulations of both observability conditions
are given in Appendices A and B, respectively.
3.2. Estimated variables
The estimated variables are the difficult-to-measure parameters
intended to be estimated using observers. They are systemdependent and not specific to one parameter for a particular
process unit (Liu, 1999) such as reactor concentration in a CSTR
(Salehi and Shahrokhi, 2008), production rate and solid mass
fraction in a polymerization reactor (López and Alvarez, 2004)
or specific growth rate in a bioreactor (De Battista et al., 2011).
The estimated parameters are normally the crucial parameters
that may potentially lead to uncertainty in the process and can
affect product quality (Alanis et al., 2010; Fan and Alpay, 2004;
Mesbah et al., 2011; Olivier et al., 2012). The parameters should
also be updatable for online implementation and to eliminate bias
between simulation and the online estimation implementation
(Sandink et al., 2001). Examples of estimated parameters in various chemical process systems are given in Section 2 (we refer the
readers to the ‘Objective/Estimate(s)’ column in Table 3).
3.3. Kinetics information of the system
The kinetics information of the system determines the system’s
nonlinearities based on the mathematical model that represents
it (Biagiola and Figueroa, 2004b). This information is required to
aid in the selection of the appropriate observers. For a system
where this information is complete and system parameters are
well known, the Luenberger-based observer is appropriate, while
the Bayesian estimator is used for systems in which only certain
system parameters are known (Dochain, 2000, 2003). When less
kinetic information is available, researchers can apply exponential
or asymptotic observers (Dochain, 2000; El Assoudi et al., 2002;
Hadj-Sadok and Gouze, 2001; Hoang et al., 2013; Hulhoven et al.,
2008). AI-based observers may be suitable for systems with incomplete model information.
3.4. Observer formulation
Since most observers are model-based (Boaventura et al., 2001;
Chen et al., 2009; de Canete et al., 2012; Prakash et al., 2011; Vicente
et al., 2000; Yang et al., 2011), the design steps include modeling of
the systems prior to developing the observer equation, computing
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
the gain of the observer and deriving the estimation error model or
the error dynamic equation (D’Attellis et al., 1997; Di Ruscio, 2009;
Dochain, 2003; Gajic, 2003; Soroush, 1997). Classification of recent
observers into six classes as in Section 2 can be helpful in designing the observers, as we can now apply either method 1 or 2 in the
design (we refer the readers to Fig. 2). For example, modeling can
be simplified if the observers are from the finite dimensional and
AI-based classes because those two classes can be designed for systems with incomplete models or less kinetic information (Dochain
et al., 1992, 2009; Senthil et al., 2006). If the systems require specific parameter estimation such as disturbances, the focus can be
narrowed down to the disturbances and fault detection observer
types. Thus, we can directly apply the disturbances and faults
detection observers and obtain the best estimation performance.
Furthermore, if our systems are simple linear systems with easyto-formulate models, we can apply the Luenberger-based observers
instead of choosing from all other types of observers available
(Damour et al., 2010a, 2010b; Mesbah et al., 2011). In addition, we
can apply the hybrid observer to overcome the limitations of single observers and to improve the estimation performance (Goffaux
et al., 2009; Hulhoven and Bogaerts, 2002; Hulhoven et al., 2006).
3.4.1. Model of the system
The system for observer design is normally based on a mathematical model (Ahn et al., 1999; Bagui et al., 2004; Mohseni et al.,
2009), which is typically incorporated into the mass and energy balance of the systems (Bernard and Gouze, 2004; Dochain et al., 2009;
James et al., 2002; Salehi and Shahrokhi, 2008) and consequently it
may range from the finite-dimensional to the infinite-dimensional
case. It can also include several appropriate assumptions, for example, assuming perfect mixing and constant physical parameters to
simplify the modeling steps (Biagiola and Figueroa, 2004a).
3.4.2. Observer equation
The observer equation is developed to determine the observer
structure for a given system based on dynamic knowledge and
incorporated with the observer gain and the error dynamic equation (Bitzer and Zeitz, 2002; Cacace et al., 2010). For a linear
model-based observer, the state space representation is normally
used to describe the observer equation, and the measurement
equation is also involved in the formulation (Fuhrmann, 2008;
Patwardhan et al., 2006; Patwardhan and Shah, 2005; Senthil et al.,
2006). The number of measured variables will also affect the sensitivity of the estimation (Venkateswarlu and Avantika, 2001).
3.4.3. Observer gain
The design of the observer structure will require an appropriate
gain (Dochain, 2003), and it is chosen based on the stability of the
error dynamics of the system (Busawon and Kabore, 2001; Yang
et al., 2012). The observer gain can be solved using the Butterworth
polynomial and the Ackermann formula (Di Ruscio, 2009). Additionally, the Riccati equation is also applied to determine the gain
value by observing the error dynamic output (Farza et al., 2011).
3.4.4. Error dynamic equation
The error dynamic equation is needed to ensure the observer
structure is bounded to the modeling error (Wang et al., 1997) to
increase the robustness of the observer (Jung et al., 2008). It can be
represented in terms of the linear time-varying system (Mishkov,
2005; Röbenack and Lynch, 2004) and must be designed in such a
way that it is asymptotically or exponentially stable (Biagiola and
Figueroa, 2002; Härdin and van Schuppen, 2007; Iyer and Farell,
1996; Röbenack and Lynch, 2004; Zambare et al., 2003). In certain
systems, where the dynamic information is limited and it is difficult
to develop the error bounds due to large uncertainties (Dochain,
2000), the error dynamic equation can be ignored in the observer
35
design. This can be seen in the development of the asymptotic
observer (Dochain et al., 1992, 2009).
3.5. Evaluating the observer
The performance of an observer is usually tested in simulation followed by online implementation (BenAmor et al., 2004;
Bogaerts and Wouwer, 2004; Escobar et al., 2011). The test will
begin with a nominal or simulator model that is different from
the observer model and includes both the process and possibly
measurement noise (Di Ruscio, 2009). Normally, some reasonable model errors are introduced to the simulator model to test
its performance under plant model mismatches (Aguilar-Lopez
and Martinez-Guerra, 2005; Aguilar-Lopez and Maya-Yescas, 2005;
Jana et al., 2006; Lin et al., 2003). Poor noise assumptions can lead
to bias estimation and divergence and require more assumptions
for the design (Biagiola and Figueroa, 2004a).
4. Current and future trends
In previous years, single-based observers (i.e., Luenberger-based
and finite-dimensional observers) have commonly been used to
estimate the difficult-to-measure parameters in a chemical process
system prior to generating the control design and to replace highpriced sensors in the plant. These observers, however, started to
face challenges in handling uncertainties in both model and measurements (Dochain, 2003). In addition, many of these observers
require accurate knowledge of the process dynamics for the design,
which are difficult to obtain, especially in nonlinear chemical processes. AI-based observers have become popular for nonlinear
systems due to their reduced dependency on accurate models but
also are limited in terms of robustness. Thus, in order to overcome these limitations and challenges, hybrid observers have been
introduced (Hulhoven and Bogaerts, 2002; Hulhoven et al., 2006;
Poznyak et al., 2007). These hybrid observers also have the advantage of being easy to formulate and implement with the availability
of powerful modern computing resources.
Therefore, in general, the trend in the use of observers for chemical process has changed from single-based observer design (Bara
et al., 2001; Benaskeur and Desbiens, 2002; Hadj-Sadok and Gouze,
2001; Kazantzis et al., 2000) to hybrid observers design (Goffaux
et al., 2009; Hulhoven et al., 2006; Prakash and Senthil, 2008),
including AI-based designs, as seen in the trend pattern for the
usage of recent observers in chemical process systems shown in
Fig. 3. The statistics given in the figure show that although all
other classes of observers have always been in use, the trends
have been inconsistent and very much dependent on the particular system to be resolved. However, the hybrid observer, which
did not exist before 2000, has been consistently increasing in its
usage since then. This could be due to many factors but one main
reason is the availability of many types of observers (as seen in
Section 2) that could be easily combined in the modern software
available at present (de Assis and Filho, 2000; de Canete et al., 2012;
Escobar et al., 2011; Rivera et al., 2010). Hybrid observers has also
becoming popular due to their ability to obtain better estimation
performances, handle offsets efficiently and increase the rate of
convergence (Aguilar-Lopez and Martinez-Guerra, 2007; Hulhoven
and Bogaerts, 2002).
The sliding mode observer hybrid with an asymptotic/exponential observer is an example where a sliding mode
observer provides good convergence rate with guaranteed robustness and stability (Hajatipour and Farrokhi, 2010) while the
asymptotic/exponential observer supports a process system with
minimal kinetic data (Jana et al., 2006), which is beneficial in
situations where there is a lack of information or data regarding
36
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
Fig. 3. Current and future trend of recent observers in chemical process systems.
the process system. Another example involves the combination
of a reduced-order observer with both the sliding mode and
asymptotic/exponential observers where only certain parameter estimations are required (Zhu, 2012). The sliding mode
has also been merged with a low-order observer to cater for
high-dimensional processes and to overcome the uncertainties
in chemical process applications such as the distillation column
(Singh and Hahn, 2005c).
At the same time, many researchers have also shown their interest in applying pure AI techniques, such as the artificial neural
network (ANN), fuzzy logic and expert systems for predicting the
parameters in difficult-to-model systems (Bahar and Ozgen, 2010;
Beigzadeh and Rahimi, 2012; Brudzewski et al., 2006; de Canete
et al., 2012; Delrot et al., 2012; Hussain et al., 2002; Patnaik, 1997;
Porter Ii and Passino, 1995; Singh et al., 2005, 2007). AI algorithms
have also been merged with each other for estimation purposes
with different types of formulations (Arauzo-Bravo et al., 2004;
Chitanov et al., 2004; Chuk et al., 2005; Li et al., 2002; Ng and
Hussain, 2004; Sivan et al., 2007; Wei et al., 2007, 2010; Wilson and
Zorzetto, 1997; Yang and Yan, 2011; Yetilmezsoy et al., 2011). However, the problems with AI-based observers, especially in online
estimations, are issues related to robustness and stability, which
may be difficult to resolve if the data collected for the AI design do
not cover the whole range of the operating region for the observer
(Himmelblau, 2008; Wang et al., 2011; Wei et al., 2010). Furthermore some of the formulations of AI-based designs are black box in
nature and are difficult to envision with respect to real plant parameters. Hence, we foresee that the future direction of observer design
is toward the use of hybrid-based types including the AI-based
observers, although single-based observers (e.g., the Bayesian Estimators, the asymptotic and exponential observers) will still be
applicable in many applications involving chemical process systems.
5. Conclusions
For a chemical process plant, it is always vital to capture process states for monitoring purposes, to develop state feedback laws
for control, and to prevent process disruptions, plant shutdowns
and even failures. Unfortunately, not all of the important states
are measurable due to the lack of appropriate devices or the costs
and feasibility of installing sensors in the plant. For that purpose,
observers, which are computational algorithms designed to estimate unmeasured state variables, have been developed. However,
estimation techniques are available in various types and categories,
making it difficult to select one for any particular application. Thus,
we classified the observers into six classes with the aim to differentiate them with respect to their features and to assist in the design
of observers, emphasizing their positive highlights, advantages and
limitations based on their performance in chemical process systems. The classifications can also be used to assist researchers and
practitioners in selecting the appropriate observers for implementation or designing them for a specific application. In addition to
this, it was recognized that the trend of observer design would be
toward the use of hybrid observers due to the rapid development
in software that allows easy combination of observer algorithms.
Acknowledgments
The authors are grateful to the University of Malaya and
the Ministry of Higher Education in Malaysia for supporting
this collaborative work under the high impact research grant
UM.C/HIR/MOHE/ENG/25.
The Viet Nam National Foundation for Science and Technology
Development (NAFOSTED) is acknowledged for financial support
through project code 104.99-2014.74.
Appendix A. Observability matrix.
The main interest of the observability of a dynamic system is
that it allows a priori to come up with an observer which rebuilds
the system states with certain rate of convergence.
Consider a discrete-time system in the form of steady state:
x(k + 1) = Ad x(k)
(A.1)
With output measurement given by:
y(k) = Cd x(k)
(A.2)
If x(0) is known then the state variables at every instant of the
discrete-time system can also be determined. This is proven for
k = 0, 1,. . .,n − 1 as follows based on the substitution of k value into
Eqs. (A.1) and (A.2).
At k = 0:
x(1) = Ad x(0)
(A.3)
y(0) = Cd x(0)
(A.4)
At k = 1:
x(2) = Ad x(1)
(A.5)
y(1) = Cd x(1)
(A.6)
Substituting Eq. (A.3) into Eq. (A.6):
y(1) = Cd x(1) = Cd Ad x(0)
(A.7)
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
At k = 2:
x(3) = Ad x(2)
(A.8)
y(2) = Cd x(2)
(A.9)
By taking the derivative in the continuous-time measurements,
for first derivative:
ẏ(t0 ) = C ẋ(t0 )
(A.10)
As a summary, from k = 0: until n − 1:, assuming the ndimensional vector n(0) has n unknown components we obtain:
y(0) = Cd x(0)
(A.22)
Substituting Eq. (A.20) into Eq. (A.22):
Substituting Eqs. (A.3) and (A.5) into Eq. (A.9):
y(2) = Cd x(2) = Cd A2d x(0)
37
ẏ(t0 ) = C ẋ(t0 ) = CAx(t0 )
For second derivative:
ÿ(t0 ) = C ẍ(t0 ) = CA2 x(t0 )
(A.12)
y(2) = Cd x(2) = Cd A2d x(0)
(A.13)
(A.24)
For (n − 1)th derivative, similarly we have:
(A.11)
y(1) = Cd x(1) = Cd Ad x(0)
(A.23)
y
n−1
(t0 ) = Cxn−1 (t0 ) = CAn−1 x(t0 )
(A.25)
As a summary, we obtain
y(0) = Cx(t0 )
..
.
(A.26)
ẏ(t0 ) = C ẋ(t0 ) = CAx(t0 )
(A.27)
2
y(n − 1) = Cd x(n − 1) = Cd An−1
x(0)
d
(A.14)
each with dimenThe matrix blocks Cd , Cd Ad , Cd A2d , . . ., Cd An−1
d
sion p × n will stack on top of each other with overall dimension of
the matrix is np × n.
⎡
⎤(np)×1
y(0)
⎢ y(1) ⎥
⎢
⎥
⎢ y(2) ⎥
⎢
⎥
⎢
⎥
..
⎣
⎦
.
y(n − 1)
⎡
Cd
⎤(np)×n
⎢ C A ⎥
⎢ d d ⎥
⎢
⎥
2 ⎥
⎢
= ⎢ Cd Ad ⎥
⎢
⎥
⎢ .. ⎥
⎣ . ⎦
(A.15)
Cd An−1
d
(A.16)
⎡
C
(A.29)
⎤(np)×n
⎢ CA ⎥
⎢
⎥
⎢ CA2 ⎥
⎥
=⎢
⎢
⎥
⎢ .. ⎥
⎣ . ⎦
× x(t0 )
(A.30)
CAn−1
It will have unique solution provided the system matrix has rank
n(order of the system).
⎤
C
⎢ CA ⎥
⎢
⎥
⎢ CA2 ⎥
⎥=n
rank ⎢
⎢
⎥
⎢ .. ⎥
⎣ . ⎦
(A.31)
CAn−1
Cd An−1
d
Therefore, the observability matrix, denoted by O, must equal
to rank n (i.e. rank O = n) to determine the initial condition, x(0).
Cd
y(t0 )
⎤(np)×1
⎢ ẏ(t ) ⎥
⎢
⎥
0
⎢
⎥
⎢ ÿ(t0 ) ⎥
Thus ⎢
⎥
⎢
⎥
⎢
⎥
..
⎣
⎦
.
⎡
⎢ C A ⎥
⎢ d d ⎥
⎢
⎥
2 ⎥
⎢
rank ⎢ Cd Ad ⎥ = n
⎢
⎥
⎢ .. ⎥
⎣ . ⎦
⎡
yn−1 (t0 ) = Cxn−1 (t0 ) = CAn−1 x(t0 )
yn−1 (t0 )
⎤
Cd
(A.28)
..
.
⎡
It will have unique solution provided the system matrix has rank
n(order of the system).
⎡
ÿ(t0 ) = C ẍ(t0 ) = CA x(t0 )
⎤(np)×n
⎢ C A ⎥
⎢ d d ⎥
⎢
⎥
2 ⎥
⎢
O(Ad , Cd ) = ⎢ Cd Ad ⎥
⎢
⎥
⎢ .. ⎥
⎣ . ⎦
has rank n
(A.17)
Therefore, the observability matrix, denoted by O, must equal
to rank n (i.e. rank O = n) to determine the initial condition, x(t0 ).
⎡
⎤(np)×n
C
⎢ CA ⎥
⎢
⎥
⎢ CA2 ⎥
⎢
⎥
O(A, C) = ⎢
⎥
⎢ .. ⎥
⎣ . ⎦
has rank n
(A.32)
CAn−1
Cd An−1
d
Appendix B. Observability Gramians.
Now consider the linear continuous-time system:
ẋ(t) = Ax(t)
(A.18)
With output measurement given by:
y(t) = Cx(t)
(A.19)
The knowledge of x(t0 ) is sufficient to determine x(t) at any time
instant. At t = t0 :
ẋ(t0 ) = Ax(t0 )
(A.20)
y(t0 ) = Cx(t0 )
(A.21)
Besides that, another observability condition known as observability Gramians has also been discussed by several researchers
(Bru et al., 2004; Singh and Hahn, 2005b; Vaidya, 2007; Yasuda and
Skelton, 1991). Observability Gramians is based on the Gramians
matrices and must satisfy the Lyapunov equation.
For a discrete-time system, the k-step observability Gramian can
be defined as:
Qk = OTk Ok =
k−1
i=0
T
(Ai ) C T CAi
(B.1)
38
J. Mohd Ali et al. / Computers and Chemical Engineering 76 (2015) 27–41
The system is observable if and only if rank (Qk ) = n, thus the
observability Gramians is as follows:
Qk = limk→∞ Qk =
∞
i=1
T
(Ai ) C T CAi
(B.2)
Where Q (in Eq. (B.2)) satisfies a Lyapunov equation
AT QA − Q = − CT C.
Next, for a linear continuous-time system define as:
t
T
(eA ) C T CeA d
Qt =
(B.3)
0
The system is observable if and only if rank (Qk ) = n, thus the
observability Gramians is as follows:
∞
T
(eA ) C T CeA d
Q = limt→∞ Qt =
(B.4)
0
where similarly, Q (in Eq. (B.4)) satisfies a Lyapunov equation
AT QA − Q = − CT C.
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