# A sensitivity-based approach to optimal sensor selection for process
networks

Siyu Liua,b, Xunyuan Yinc, Zhichao Pana, Jinfeng Liub,

a School of Internet of Things Engineering, Jiangnan University, Wuxi 214122,
China

b Department of Chemical & Materials Engineering, University of Alberta,

Edmonton, AB, Canada, T6G 1H9

c School of Chemical and Biomedical Engineering, Nanyang Technological
University,

62 Nanyang Drive, Singapore, 637459
Corresponding author: J. Liu. Tel: +1-780-492-1317. Fax: +1-780-492-2881.
Email<EMAIL_ADDRESS>
###### Abstract

Sensor selection is critical for state estimation, control and monitoring of
nonlinear processes. However, evaluating the performance of each possible
combination of $m$ out of $n$ sensors is impractical unless $m$ and $n$ are
small. In this paper, we propose a sensitivity-based approach to determine the
minimum number of sensors and their optimal locations for state estimation.
The local sensitivity matrix of the measured outputs to initial states is used
as a measure of the observability. The minimum number of sensors is determined
in a way such that the local sensitivity matrix is full column rank. The
subset of sensors that satisfies the full-rank condition and provides the
maximum degree of observability is considered as the optimal sensor placement.
Successive orthogonalization of the sensitivity matrix is conducted in the
proposed approach to significantly reduce the computational complexity in
selecting the sensors. To validate the effectiveness of the proposed method,
it is applied to two processes including a chemical process consisting of four
continuous stirred-tank reactors and a wastewater treatment plant. In both
cases, the proposed approach can obtain the optimal sensor subsets.

Keywords: Degree of observability; sensitivity analysis; sensor selection;
orthogonalization; nonlinear process.

##  1 Introduction

The sensor selection problem arises in various applications, including target
tracking [1, 2], agro-hydrological systems [3, 4], power systems [5, 6] and
wireless networks [7, 8]. Some useful methods on sensor selection have been
developed. One straightforward solution to sensor selection is to perform an
exhaustive search, which evaluates the performance given by each possible
combination of $m$ out of $n$ candidate sensors [9]. The number of
combinations for an exhaustive search is $\frac{n!}{m!(n-m)!}$. Therefore,
this type of solution is not suitable for large-scale systems. In [10], sensor
selection is performed for the reduced-order WWTP model with small $m$ and
$n$, in which case the exhaustive search is feasible. If sensors need to be
selected from a large number of candidate sensors, then it may be difficult or
even impossible to find the optimal solution.

Graph-based methods are widely used for the sensor node selection in networked
systems based on the structural observability theory [11]. By examining the
strongly connected components (SCCs) in a graph describing node connections,
graph-based methods can determine the number and locations of sensor nodes for
ensuring structural observability [12, 13]. The minimal set includes one
sensor in each root SCC (an SCC with no incoming edges) of the observability
inference diagram. Graph-based methods has become an influential method and
has been used to provide insights into the relation between network topology
and observability. However, since these approaches do not explicitly take into
account model parameters, in some cases, the estimator designed based on the
sensors selected by the graph-based approaches may lead to less accurate state
estimates [14, 15]. A workaround is to simultaneously deal with the sensor
selection and state estimation, which has the potential to improve the overall
estimation performance [16, 17].

Another line of research on sensor selection is to rely on suitable
observability criteria that are based on the properties of the empirical
observability Gramian, e.g., the trace [18], the determinant [5], and the
smallest eigenvalue [19]. The trace of an empirical observability Gramian
matrix can be used as a measure of observability. However, a system can be
unobservable if it has an eigenvalue of zero, despite a potentially large
trace. The direction that has the lowest degree of observability can be
determined by the smallest eigenvalue and the corresponding eigenvector. When
the determinant of a Gramian matrix is used as a measure of observability, the
smallest eigenvalue can be used as an additional measure in order to avoid the
situation that the eigenvalue of Gramian matrix is almost zero and the system
is not fully observable. A main limitation of the above methods is that the
computation will become very expensive when applied to nonlinear systems,
because these methods would require simulating the system many times for each
perturbed initial condition in order to compute the ensemble average.

The observability of a nonlinear system can be determined by using the Lie
derivative [20]. However, this type of method only determines whether a system
is observable but not the degree of observability. Also, due to the heavy
computational burden, the Lie derivative is only suitable for small-scale
systems. Sensitivity analysis can used as a tool for assessing the degree of
observability, which has been often applied to quantitatively investigate
parameter effects [21, 22, 23] and simultaneous state and parameter estimation
[24, 25] but yet to be used for sensor selection.

In this work, we propose a sensitivity-based approach to select a minimum
sensor set in a computationally efficient manner. A local sensitivity matrix
describing the sensitivities between states and output measurements is
constructed, which reflects the degree of observability of the different
states. Once the minimum number of sensors to make the local sensitivity
matrix full rank is determined, we perform successive orthogonalization of
each column of the sensitivity matrix to rank sensors, and propose a measure
of the degree of observability to determine the optimal sensor subset that
satisfies the full-rank condition and maximizes the degree of observability.
The proposed sensitivity-based approach avoids solving mixed-integer
programming and exhaustive search, making it highly computationally efficient.
Compared the aforementioned existing methods, the proposed approach maximizes
the system observability and better correlates with the sensitivities that
guide the development of the degree of observability.

The remainder of this paper is organized as follows. Section 2 describes the
sensor selection problem and introduces two types of the existing methods. The
construction of sensitivity matrix is shown in Section 3. Section 4 defines
the criteria for degree of observability and performs optimal sensor
selection. In Section 5, we show the effectiveness of the proposed approach
based on a chemical process consisting of four continuous stirred-tank
reactors, and a wastewater treatment plant. The conclusions are drawn in
Section 6.

##  2 Preliminaries

###  2.1 System description and problem formulation

Consider the following nonlinear systems

$\bm{x}(k+1)=f(\bm{x}(k),\bm{u}(k))+\bm{w}(k),$ (1)

where $\bm{x}(k)\in{\mathbb{R}}^{n}$ denotes the state vector,
$\bm{u}(k)\in{\mathbb{R}}^{r}$ is the input vector,
$\bm{w}(k)\in{\mathbb{R}}^{n}$ denotes the system disturbance, and $f$
represents the state transition function.

The first objective of sensor selection is to determine the minimum number of
sensors such that the system is observable; the second objective is to
determine the optimal locations of the sensors. Consider a set of sensors
$\mathcal{S}^{(m)}$ consisting of $m$ sensors such that the output
$y_{i}(k)\in{\mathbb{R}}$ of sensor $i\in\mathcal{S}^{(m)}$ is given by

$y_{i}(k)=h_{i}(\bm{x}(k))+v_{i}(k),$ (2)

where $h_{i}(\bm{x}(k))\in{\mathbb{R}}$ is the sensor output function
corresponding to sensor $i$, $v_{i}(k)\in{\mathbb{R}}$ denotes measurement
noise. Define the output vector
$\bm{y}(k):=[y_{1}(k),y_{2}(k),\ldots,y_{m}(k)]^{\tiny\mbox{\rm
T}}\in{\mathbb{R}}^{m}$ and noise vector
$\bm{v}(k):=[v_{1}(k),v_{2}(k),\ldots,v_{m}(k)]^{\tiny\mbox{\rm
T}}\in{\mathbb{R}}^{m}$. Then, we formulate the following measurement equation
containing the measurements of the $m$ sensors

$\bm{y}(k)=h(\bm{x}(k))+\bm{v}(k).$ (3)

In the following, we seek to select an optimal sensor subset
$\mathcal{S}^{(o)}\subseteq\mathcal{S}^{(m)}$, $o\leq m$ from the potentially
available measured outputs in Equation (3). For solving the sensor selection
problem, some existing methods are summarized in the next subsection.

###  2.2 Existing approaches

Sensor selection problems on linear systems have attracted extensive efforts
over the years. More recently, methods to address the nonlinear systems have
also emerged. We will first briefly review the works on linear systems,
followed by a more thorough review of the efforts on nonlinear systems.

For linear systems, efforts have been made to address the problem of sensor
selection by minimizing the error in estimating its states. Such selection
problems are in general NP-hard. For example, Zhang et al., minimized the
trace of a priori and a posteriori error covariance matrices produced by a
Kalman filter, and they showed that the optimization problem is NP-hard [26].
Another widely used method is greedy algorithms, a class of approximation
algorithms. Such methods are guaranteed to have the desirable performance if
the cost function is submodular [27].

For nonlinear networks, the sensor placement is typically formulated as a
mixed-integer programming problem, for example, to maximize the determinant of
the empirical observability Gramian as follows [5]:

$\displaystyle\max_{\bm{\theta}}\det\bm{W}(\bm{\theta}),$ (4a)
$\displaystyle{\rm s.t.}\sum_{i=1}^{m}\theta_{i}=\bar{m},$ (4b)
$\displaystyle\theta_{i}\in\\{0,1\\},\quad i=1,2,\ldots,m,$ (4c)

where $\bm{W}(\bm{\theta})$ denotes the empirical observability Gramian which
is dependent on the value of the parameter $\bm{\theta}$,
$\bm{\theta}\in\\{0,1\\}^{m}$ is a binary decision vector to denote which
sensors are selected, i.e., $\theta_{i}=1$ if $i\in\mathcal{S}^{(m)}$ is
selected and $\theta_{i}=0$ otherwise, $\bar{m}$ is the number of sensors to
be placed. The empirical observability Gramian $\bm{W}(\bm{\theta})$ is
defined using the original system model, which reflects the observability of
the system state in a given domain. However, the cost of computing the
empirical observability Gramian is high because it requires simulating the
process networks for different perturbations of the initial condition.

In (4a), the determinant of the observability Gramian is considered as a
measure of the degree of observability. In addition to the determinant, other
measures based on the empirical observability Gramian are also used including
the trace [18], the smallest eigenvalue [19], and the condition number [9] of
the Gramian matrix. A comparison of these degree of observability measures is
reported in [28].

There are some other variants for the optimization problem in (4a), for
example, using the Jacobian matrix to replace the empirical observability
Gramian matrix. Nonetheless, these variations are still mixed-integer
programming problem [15]. mixed-integer nonlinear programming problems are in
general challenging to solve especially when the number of potential sensors
is large. The solution proposed in this paper is operative and less
computationally demanding. In the following section, we will elaborate on the
method for selecting the optimal sensor set.

##  3 Output to initial state sensitivity and observability test

In this work, we propose to use the sensitivity matrix of the output $\bm{y}$
to the initial state $\bm{x}(0)$ as a measure of the observability and
introduce a computationally efficient optimal sensor selection algorithm based
on the sensitivity matrix. In this section, we describe how the sensitivity
matrix can be constructed and the corresponding observability test.

###  3.1 Construction of the sensitivity matrix

In sensor selection, we need to determine a sensor set such that the state of
the system is observable (or can be constructed) based on the output
information provided by the selected sensors. For a nonlinear system, the
local observability of the state (or the initial state $\bm{x}(0)$) can be
checked using the sensitivity matrix of the outputs to the initial state [29].
To construct the sensitivity matrix, we need the sensitivity of each output to
each of the state element of $\bm{x}(0)$ as follows:

$\displaystyle s_{ij}(k)=\frac{\partial y_{i}(k)}{\partial
x_{j}(0)}=\frac{\partial h_{i}}{\partial x_{j}}(k)\bm{S}_{x,x(0)}(k),$ (5)

where $\bm{S}_{x,x(0)}(k):=\frac{\partial\bm{x}(k)}{\partial\bm{x}(0)}$ is the
sensitivity of the state $\bm{x}(k)$ with respect to the initial condition
$\bm{x}(0)$. The sensitivity coefficient describes how sensitive the system
output is to changes in a single state. At sampling time $k$, a time dependent
matrix $\bm{S}_{y,x(0)}(k):=\frac{\partial\bm{y}(k)}{\partial\bm{x}(0)}$ is
constituted where each column denotes the sensitivity of different outputs to
each state element:

$\displaystyle\bm{S}_{y,x(0)}(k)=$
$\displaystyle\left[\begin{array}[]{ccc}s_{11}(k)&\cdots&s_{1n}(k)\\\
\vdots&\ddots&\vdots\\\
s_{m1}(k)&\cdots&s_{mn}(k)\end{array}\right]\in{\mathbb{R}}^{m\times n}.$ (9)

The above sensitivity $\bm{S}_{y,x(0)}(k)$ provides the sensitivity
information based on the measurements obtained at time $k$. By collecting the
sensitivity of all the past measurements from sampling time $0$ to sampling
time $k$, the following sensitivity matrix can be constructed:

$\displaystyle\bm{S}(k,0)=\left[\begin{array}[]{c}\bm{S}_{y,x(0)}(0)\\\
\bm{S}_{y,x(0)}(1)\\\ \vdots\\\
\bm{S}_{y,x(0)}(k)\end{array}\right]\in{\mathbb{R}}^{(k+1)m\times n},$ (14)

The sensitivity matrix should be normalized before using it to check the
observability so that the rank calculation will not be affected by the
parameter values. Each column of the normalized sensitivity $\bm{S}(k,0)$ is a
list of output sensitivities over time to the same element in the state
vector, indicating the direction of sensitivity for that state element. If the
norm of a column is larger than other columns, the output $\bm{y}$ is most
sensitive to changes of the state element corresponding to the column.

###  3.2 Observability test

This local sensitivity matrix is similar to the observability matrix of a
linear system and can be used to check the local observability of a nonlinear
system [24]. Therefore, the local observability test can be formulated based
on the rank test of the sensitivity matrix $\bm{S}(k,0)$. A full-rank
sensitivity matrix indicates local observability. Singular value decomposition
(SVD) is used for evaluating the rank of a given matrix. Specifically, SVD is
performed for $\bm{S}(k,0)$ as follows:

$\displaystyle\bm{S}(k,0)=\bm{U}\bm{\varSigma}\bm{V}^{\tiny\mbox{\rm T}},$
(15)

where $\bm{U}\in{\mathbb{R}}^{(km+m)\times(km+m)}$ and
$\bm{V}\in{\mathbb{R}}^{n\times n}$ are orthonormal matrices,
$\bm{\varSigma}\in{\mathbb{R}}^{(km+m)\times n}$ contains the $n$ singular
values $\\{\sigma_{r},r=1,2,\dots,n\\}$ of $\bm{S}(k,0)$ in decreasing order
on the diagonal. If $\sigma_{n}=0$, or the smallest singular value is smaller
than a pre-determined threshold, $\bm{S}(k,0)$ is rank-deficient.

In the proposed sensor selection algorithm, observability test is conducted to
find the sensor combinations that can make the system state observable. In
addition to observability, we also resort to the degree of observability to
determine the optimal sensor selection in the next section.

##  4 Proposed optimal sensor selection algorithm

Based on the local sensitivity matrix, we propose a measure to quantify the
degree of observability of different combinations of sensors based on
successive orthogonalization. Then, sensors are ranked based on their
contributions to the overall degree of observability. The minimum sensor set
and optimal locations of sensors are determined through maximizing the degree
of observability of the system while ensuring the local sensitivity matrix is
full column rank.

###  4.1 A measure for degree of observability

Since each column of the sensitivity matrix contains the sensitivities of the
output $\bm{y}$ to a state element, it is proposed to use the norm of each
column for the initial screening of the degree of observability of each of the
state elements. The output is most sensitive to the state element that
corresponds to the column with the maximum norm. This also implies that the
output measurements carry the most information of this state element and this
state element is most observable based on the output measurements.

While the norm of the columns is useful in identifying the most observable
state element, it should not be used directly to examine the other columns due
to the potential linear dependence of the columns. The highly linear
correlation between any two columns suggests that changes in the corresponding
state elements would have similar effects on the outputs. Thus, the two state
elements are indistinguishable based on the measurements. It is proposed to
use the orthogonalization method to check the dependence of the columns of the
sensitivity matrix. The idea of this method is to calculate the perpendicular
distance of one column to the vector space spanned by the other columns and
then uses the distance as a measure of the linear dependency. By successive
orthogonalization, it is possible to rank the columns in terms of their
information contained in the measurements. Also, the proposed measure for
degree of observability is based on total information contained in the
measurements regarding the state elements.

Figure 1: Illustration of the orthogonolization

The way to calculate the proposed measure is illustrated in the following
example with three elements in the state vector. Consider that the sensitivity
matrix $\bm{S}(k,0):=[\bm{a},\bm{b},\bm{c}]$ where $\bm{a}$, $\bm{b}$,
$\bm{c}$ are the columns. In the first step, we calculate the norm of each of
the three column vectors. Without loss of generality, we assume the vector
$\bm{a}$ has the largest norm denoted as $F_{1}$. $F_{1}$ is an indicator of
the amount of information contained in the measurements regarding the state
element corresponding to $\bm{a}$. Due to potential linear dependence between
the columns, the other two columns $\bm{b}$, $\bm{c}$ may also contain
information that is already expressed by column $\bm{a}$. To proceed, we
define $\bm{X}_{1}:=\bm{a}$. The direction of $\bm{X}_{1}$,
$\bm{q}_{1}=\bm{X}_{1}/\|\bm{X}_{1}\|$ is used as the first basis for the
orthogonalization.

In the second step, the other two vectors are projected onto $\bm{q}_{1}$. The
information in $\bm{S}(k,0)$ that cannot be represented by $\bm{X}_{1}$ can be
calculated by

$\displaystyle\tilde{\bm{b}}=\bm{b}-(\bm{q}_{1}^{\tiny\mbox{\rm
T}}\bm{b})\bm{q}_{1},$ (16)
$\displaystyle\tilde{\bm{c}}=\bm{c}-(\bm{q}_{1}^{\tiny\mbox{\rm
T}}\bm{c})\bm{q}_{1}.$ (17)

$\tilde{\bm{b}}$ and $\tilde{\bm{c}}$ are now orthogonal to $\bm{q}_{1}$. Let
us define the residual matrix
$\bm{R}_{1}:=[\tilde{\bm{b}},\tilde{\bm{c}}]=\bm{S}(k,0)$
$-\bm{X}_{1}(\bm{X}_{1}^{\tiny\mbox{\rm
T}}\bm{X}_{1})^{-1}\bm{X}_{1}^{\tiny\mbox{\rm T}}\bm{S}(k,0)$, which
represents the information that cannot be captured by the first basis in the
sensitivity matrix.

In the third step, we evaluate the norm of each column of $\bm{R}_{1}$.
Suppose that $\tilde{\bm{c}}$ has a larger norm $F_{2}$. $F_{2}$ indicates the
extra information contained in the measurements regarding the state element
corresponding to $\bm{b}$ that has not been covered by column $\bm{a}$. Next,
we add $\bm{c}$ from $\bm{S}(k,0)$ to $\bm{X}_{1}$ to form a new matrix with
two columns $\bm{X}_{2}=[\bm{a},\bm{c}]$. The second basis
$\bm{q}_{2}=\tilde{\bm{c}}/\|\tilde{\bm{c}}\|$ is formed and the information
that cannot be expressed by $\bm{X}_{2}$ is evaluated by projecting
$\tilde{\bm{b}}$ to the line/plane (denoted $P_{1}$) that is perpendicular to
$\bm{X}_{2}$. The new vector

$\displaystyle\bar{\bm{b}}=\tilde{\bm{b}}-(\bm{q}_{2}^{\tiny\mbox{\rm
T}}\tilde{\bm{b}})\bm{q}_{2}$ (18)

with the minimum norm $F_{3}$, which represents the extra information
contained in the measurements regarding the state element corresponding to
$\bm{b}$ that is not captured by $\bm{a}$ or $\bm{c}$. The corresponding
residual matrix is $\bm{R}_{2}:=[\bar{\bm{b}}]=\bm{S}(k,0)-$
$\bm{X}_{2}(\bm{X}_{2}^{\tiny\mbox{\rm
T}}\bm{X}_{2})^{-1}\bm{X}_{2}^{\tiny\mbox{\rm T}}\bm{S}(k,0)$. By appending
$\bm{b}$ to $\bm{X}_{2}$, we can form a new matrix
$\bm{X}_{3}=[\bm{a},\bm{c},\bm{b}]$. In $\bm{X}_{3}$, the columns are ordered
according to their contributions to the information contained in the
measurements regarding the state elements. The above procedure is also
illustrated in Figure 1.

For a sensor set $\mathcal{S}^{(o)}$, we assume that
$\mathcal{S}^{(o)}=\mathcal{S}^{(m)}=\\{1,2,\dots,m\\}$, and we can construct
the sensitivity matrix $\bm{S}(k,0)\in{\mathbb{R}}^{(km+m)\times n}$ according
to the sensor measurements $\bm{y}$ provided by $\mathcal{S}^{(o)}$. The
projection is repeated for $m-1$ steps, and we can obtain multiple norms
denoted as $F_{i}$, $i=1,2,\dots,m$. These $F_{i}$ are indicators of the
amount of information contained in the measurements regarding the state
elements. Let us define

$\displaystyle F=\sum_{j=1}^{m}F_{j}.$ (19)

In (19), $F$ represents the amount of information contained in the
measurements provided by the current sensor combination $\mathcal{S}$
regarding the state. We propose to use $F$ as the measure of the degree of
observability of the state. However, at this time, whether the system is
observable is not considered. According to the SVD presented in Subsection
3.2, we calculate the rank of the current sensitivity matrix $\bm{S}(k,0)$.
The proposed measure for degree of observability is as follows:

$\displaystyle D(\mathcal{S}^{(o)})=\alpha F,$ (20)

where

$\alpha=\left\\{\begin{aligned} 1&,&{\rm rank}(\bm{S}(k,0))=n,\\\ 0&,&{\rm
rank}(\bm{S}(k,0))<n.\end{aligned}\right.$ (21)

If rank($\bm{S}(k,0))=n$, then $\alpha=1$. The degree of observability
$D(\mathcal{S}^{(o)})$ is effective; If rank($\bm{S}(k,0))\textless n$,
$\alpha$ is set to zero, then $D(\mathcal{S}^{(o)})$ is also equal to zero.
This implies that we should only evaluate the degree of observability of
observable systems.

The procedure of calculating the degree of observability using the
orthogonalization method is summarized in Algorithm 1:

Input: Sensor set $\mathcal{S}^{(o)}$, sensitivity matrix $\bm{S}(k,0)$ based
on $\mathcal{S}^{(o)}$

Output: Degree of observability $D(\mathcal{S}^{(o)})$

1

2Calculate the rank of normalized $\bm{S}(k,0)$.

3 if  _rank $(\bm{S}(k,0))<n$_ then

4 $\alpha=0$

5else

6 $\alpha=1$

7Calculate the norm of each column of the normalized $\bm{S}(k,0)$.

8 Select the column with the largest norm $F_{1}$, and denote it as
$\bm{X}_{1}$.

9 for  _$l=1:m-1$_ do

10 Calculate the residual matrix:
$\bm{R}_{l}=\bm{S}(k,0)-\bm{X}_{l}(\bm{X}_{l}^{\tiny\mbox{\rm
T}}\bm{X}_{l})^{-1}\bm{X}_{l}^{\tiny\mbox{\rm T}}\bm{S}(k,0)$.

11 Select the column from $\bm{S}(k,0)$ that corresponds to the column with
the maximum norm $F_{l+1}$ in $\bm{R}_{l}$, and add it to $\bm{X}_{l}$ as a
new column to form $\bm{X}_{l+1}$.

12Calculate $F$ by (19).

13 Calculate $D(\mathcal{S}^{(o)})=\alpha F$.

return $D(\mathcal{S}^{(o)})$

Algorithm 1 Calculating the degree of observability

###  4.2 Selection of the minimum sensor set

In this section, we discuss the proposed computationally efficient approach
for determining the minimum number of sensors to ensure the observability of
the state vector while maximizing the degree of observability given the number
of the sensors.

In the proposed approach, we select the minimum sensor set by removing sensors
from the available sensor set one by one based on the degree of observability.
Initially, all the sensors ($\mathcal{S}^{(m)}$) are selected and the system
is assumed to be observable. Then, the sensor that contributes the least to
the degree of observability is determined and removed from the selected sensor
set. To determine this specific sensor, we remove one sensor from the selected
sensor set. There are $m$ possible options. For each of these $m$ options, we
calculate the corresponding degree of observability. Among the $m$ degrees of
observability, we find the maximum one. The sensor that is not included in the
set corresponding to the maximum degree of observability is the sensor that
contributes the least to the degree of observability and should be removed at
this step. Following this way, we remove the sensors one by one until the
system is not observability. At certain step, if removing a sensor makes the
system unobservable, the algorithm stops. The last observable sensor set is
the selected optimal minimum sensor set.

Below is a strategy for removing a single sensor at a time. The optimal sensor
$i^{*}$ to be removed is selected by

$i^{*}=arg\max_{i\in\mathcal{S}^{(o)}}D(\mathcal{S}^{(o)}\backslash\\{i\\}).$
(22)

The above strategy is to remove the sensor $i\in\mathcal{S}^{(o)}$ and
calculate the degree of observability for each possible option. From $m$
options, the sensor to be removed is the solution of maximizing the degree of
observability $D(\mathcal{S}^{(o)}\backslash\\{i\\})$. When the sensor $i^{*}$
is removed, the current sensor set
$\mathcal{S}^{(o)}=\mathcal{S}^{(o)}\backslash\\{i^{*}\\}$. To proceed, we
will continue screening sensors from the current sensor $\mathcal{S}^{(o)}$
with $m-1$ sensors. At this time, the dimension of the sensitivity matrix will
change with the updated sensor set $\mathcal{S}^{(o)}$. In the next step,
there are $m-1$ possible options. For each of these $m-1$ options, we
construct the sensitivity matrix $\bm{S}(k,0)\in{\mathbb{R}}^{(m-2)(k+1)\times
n}$, and calculate the corresponding degree of observability. The sensor that
is not included in the set corresponding to the maximum degree of
observability is selected to remove from $m-1$ sensors at this step.

The procedure of the proposed method is summarized in Algorithm 2 for solving
two optimization objectives: 1) determining the minimum sensor set while
ensuring the system observable; and 2) obtaining the maximum degree of
observability of the minimum set.

1

2The set containing all sensors is $\mathcal{S}^{(m)}=\\{1,2,\ldots,m\\}$. The
initial set is: $\mathcal{S}^{(o)}=\mathcal{S}^{(m)}$.

3

4Set $o=m$. Perform the following steps:

5

6for  _$i\in\mathcal{S}^{(o)}$_ do

7 Remove the $i$th sensor from $\mathcal{S}^{(o)}$.

8 Perform Algorithm 1 and obtain $i$ degrees of sensitivity
$D(\mathcal{S}^{(o)})$.

9if  _all $D(\mathcal{S}^{(o)})=0$_ then

10 Terminate the procedure.

11else

12 Choose $i^{*}$ by (22) and remove the corresponding sensor from
$\mathcal{S}^{(o)}$.

13 $\mathcal{S}^{(o)}\leftarrow\mathcal{S}^{(o)}\backslash\\{i^{*}\\}$,
$o\leftarrow o-1$, go to Step 3.

14

return $\mathcal{S}^{(o)}$

Algorithm 2 Determine the best minimum sensor set

In Algorithm 2, according to the criterion for quantifying the degree of
observability of different sensor combinations, we determine to remove the
sensor that contributes less to the observability of the system based on the
the degrees of observability. Therefore, the optimal sensor set is selected so
that the degree of observability is maximized. The minimum number of sensors
are determined for the observability of the system and the optimal sensor
subset is determined on maximizing the degree of observability of the system.

######  Remark 1.

The sensor selection algorithm proposed in this paper is a good option from an
economic perspective since it requires the minimum number of sensors. However,
the minimum sensor set may not be able to provide sufficiently large degree of
observability. In order to have robust state estimation performance in the
presence of significant process disturbances and measurement noise, additional
sensors can be added to the minimum set according to the ordering of sensors
given by the algorithm. A detailed illustration is given through the second
application in Section 5.

######  Remark 2.

The cost of different sensors is an important factor that may affect the
result of sensor selection. Taking into account the cost will make the
proposed approach more applicable, and it will be investigated in the future
work.

###  4.3 Complexity of the algorithm

It is well known that the main challenge associated with the sensor selection
is the computational complexity. A comparison on the problem of computational
complexity of some classical algorithms for the sensor selection, like global
searching method, solving the mixed-integer programming, and greedy algorithm
will be discussed briefly in the following. First of all, the procedure
proposed in this work is computationally efficient by removing the sensors one
by one from the sensor set based on the degree of observability. The number of
all possible sensor combinations that needs to be considered is
$m+(m-1)+\cdots+(o+1)+o=\frac{1}{2}(m-o+1)(m+o)$ (when selecting $o$ sensors
from a set of $m$ sensors).

  * •

As the global searching method in [10], when selecting minimum number sensors
from a set that can make the system observable, the number of the sensors
should be increased form $1$ to $m$ until the minimum number $o$ is found.
Therefore, a total of $C_{m}^{1}+C_{m}^{2}+\cdots+C_{m}^{o}$ sensor
combinations need to be checked.

  * •

The sensor selection is typically formulated as a mixed-integer programming
problem, as shown in Equation (4). Solving the mixed-integer programming is
challenging especially for nonlinear cases. Even for the mixed-integer linear
programming, it is still NP-hard. In particular, the special case of $0-1$
integer linear programming, in which unknowns are binary, the number of all
possible solutions is $2^{m}$.

  * •

The greedy method is another widely used algorithm for the sensor selection
problem. For example, Zhang et al., applied the greedy methods to select
sensors that yield the largest incremental reduction in steady-state (a priori
or a posteriori) error covariance [26]. Similarly, an alternative strategy is
to add the sensors one by one using the greedy method to maximize the degree
of observability. This approach would have
$m+(m-1)+\cdots+(m-o+1)=\frac{o}{2}(m-o+1)$ possible sensor combinations that
needs to be considered.

Compared with above methods, when $m=2o-1$, the number of sensor combinations
of the adding method is the same as the removing strategy, in which the
computational complexity is acceptable. However, the adding strategy is
severely flawed by the lack of meaning of observability in the early stages
where there are too few sensors. That is, it is likely that the earlier sensor
selection is not optimal and that a larger number of sensors must be selected.
In contrast, the system in removal strategy is always observable by virtue of
the method of eliminating the least contributing sensors. Thus the minimum
number of sensors is guaranteed. For this consideration, the proposed removal
strategy seems to be a superior method than the adding one.

##  5 Simulation results

In this section, we apply the proposed procedure and algorithms to determine
the minimum number of sensors and the optimal sensor placement for two
nonlinear process networks.

###  5.1 Application to a four-CSTR process

First, we consider a chemical process consisting of four continuous stirred-
tank reactors (CSTRs), of which a schematic is shown in Figure 2. Pure
reactant $A$ is fed into the $i$th reactor for reactions at temperature
$T_{0i}$, flow rate $F_{0i}$, and molar concentration $C_{0i}$, $i=1,2,3,4$.
During the reaction process, the outlet flow of each of the first three
reactors is supplied to its adjacent downstream tank. A portion of the outlet
stream of the second reactor is recycled back to the first reactor at flow
rate $F_{r1}$, temperature $T_{2}$ and molar concentration $C_{A2}$. The
effluent of the fourth reactor is constituted by a recycle flow back to the
first reactor and a discharging flow for further processing. Three parallel
irreversible exothermal reactions $A\rightarrow B$, $A\rightarrow C$, and
$A\rightarrow D$ take place in the reactors. The heat $Q_{i}$ is removed
from/provided to the $i$th reactor by using the equipped jacket. Based on mass
and energy balances, a first-principles process model has been established,
which can be found in [30].

Figure 2: A schematic of the four-CSTR process

The molar concentration $C_{Ai}$ and temperature $T_{i}$ in the $i$-th reactor
are system states, $i=1,2,3,4$. Table 1 shows the values of model parameters,
where $V_{i}$, $E_{i}$, $\Delta H_{1}$, $k_{i}$ are the volume of reactors,
the activation energies, enthalpies, and pre-exponential constants of the
reactions, $\rho$ and $c_{p}$ denote the density and heat capacity of the
mixture in the reactors.

Table 1: Parameter values of the four-CSTR process $T_{01}=300$ K |  $C_{01}=4.0$ kmol/$\rm m^{3}$ |  $k_{1}=3.0\times 10^{6}$ $\rm h^{-1}$ |  $\Delta H_{1}=-5.0\times 10^{4}$ kJ/kmol
---|---|---|---
$T_{03}=300$ K |  $C_{02}=2.0$ kmol/$\rm m^{3}$ |  $k_{2}=3.0\times 10^{5}$ $\rm h^{-1}$ |  $\Delta H_{2}=-5.2\times 10^{4}$ kJ/kmol
$T_{02}=300$ K |  $C_{03}=3.0$ kmol/$\rm m^{3}$ |  $k_{3}=3.0\times 10^{5}$ $\rm h^{-1}$ |  $\Delta H_{3}=-5.0\times 10^{4}$ kJ/kmol
$T_{04}=300$ K |  $C_{04}=3.5$ kmol/$\rm m^{3}$ |  $F_{1}=35$ $\rm m^{3}$/h |  $E_{1}=5.0\times 10^{4}$ kJ/kmol
$F_{01}=5$ $\rm m^{3}$/h |  $V_{1}=1$ $\rm m^{3}$ |  $F_{2}=45$ $\rm m^{3}$/h |  $E_{2}=7.5\times 10^{4}$ kJ/kmol
$F_{02}=10$ $\rm m^{3}$/h |  $V_{2}=3$ $\rm m^{3}$ |  $F_{3}=33$ $\rm m^{3}$/h |  $E_{3}=7.53\times 10^{4}$ kJ/kmol
$F_{03}=8$ $\rm m^{3}$/h |  $V_{3}=4$ $\rm m^{3}$ |  $F_{r1}=20$ $\rm m^{3}$/h |  $R=8.314$ kJ/(kmol$\cdot$K)
$F_{04}=12$ $\rm m^{3}$/h |  $V_{4}=6$ $\rm m^{3}$ |  $F_{r2}=10$ $\rm m^{3}$/h |  $c_{p}=0.231$ kJ/(kg$\cdot$K)
$\rho=1000$ kg/m3 |  |  |

In the simulations, heating inputs to the four reactors are chosen to be
$Q_{1}=1.0\times 10^{4}$kJ/h, $Q_{2}=2.0\times 10^{4}$kJ/h, $Q_{3}=2.5\times
10^{4}$kJ/h, and $Q_{4}=1.0\times 10^{4}$kJ/h. The continuous model is
discretized using the fourth-order Runge-Kutta method with a sample time
$\Delta t=\frac{1}{120}$h. We assume that sensors are available to measure all
the states ($m=N=8$). The objective is to find the optimal minimum sensor set.
At each sampling point, the sensitivity matrix $\bm{S}(k,0)$ is calculated.
Initially, the sensor set $\mathcal{S}^{(o)}=\mathcal{S}^{(m)}$ where
$\mathcal{S}^{(m)}=\\{C_{A1},T_{1},C_{A2},T_{2},C_{A3},T_{3},C_{A4},T_{4}\\}$
denoting as $\\{1,2,3,4,5,6,7,8\\}$ for simplicity, and the system is fully
observable. Furthermore, we consider that the four-CSTR model is affected by
the process noise with zero mean and standard deviation $0.1\bm{x}_{s}$
($\bm{x}_{s}$ is the steady-state point of $\bm{x}$ corresponding to the
constant values of the inputs picked earlier), and measurement noise with zero
mean and standard deviation $0.1\bm{y}_{s}$ where $\bm{y}_{s}$ is the value of
$\bm{y}$ at steady-state.

Figure 3: Degrees of observability by removing different sensor from remaining sensor set. Table 2: The process of sensor selection for four-CSTRs. $m$ | $\mathcal{U}$ | $\mathcal{S}^{(o)}$ |  rank($\bm{S}(k,0)$) |  degreemax | sensor to remove
---|---|---|---|---|---
8 | $\\{\\}$ | $\\{1,2,3,4,5,6,7,8\\}$ | 8 | 261.47 | 5
7 | $\\{5\\}$ | $\\{1,2,3,4,6,7,8\\}$ | 8 | 259.61 | 3
6 | $\\{5,3\\}$ | $\\{1,2,4,6,7,8\\}$ | 8 | 255.97 | 6
5 | $\\{5,3,6\\}$ | $\\{1,2,4,7,8\\}$ | 8 | 251.65 | 4
4 | $\\{5,3,6,4\\}$ | $\\{1,2,7,8\\}$ | 8 | 241.89 | 2
3 | $\\{5,3,6,4,2\\}$ | $\\{1,7,8\\}$ | 8 | 223.15 | 7
2 | $\\{5,3,6,4,2,7\\}$ | $\\{1,8\\}$ | 8 | 164.74 | 1
1 | $\\{5,3,6,4,2,7,1\\}$ | $\\{8\\}$ | 5 | 67.7 |

To find the optimal minimum sensor set, the sensors will be removed one by one
from $\mathcal{S}^{(o)}$ by performing the proposed sensor selection
algorithm. The corresponding degree of observability of remaining sensor
combinations is shown in Figure 3 and the steps are recorded in Table 2
($\mathcal{U}$ is the set of removed sensors, $m$ is the number of sensors in
$\mathcal{S}^{(o)}$, the maximum degree of observability is denoted as
degreemax). The initial value of the degree of observability is 261.47 when
all the sensors are selected. For $S^{(8)}$, as shown in Figure 3, when the
fifth sensor is removed, the corresponding degree of observability of
remaining seven sensors is the largest, which is 259.61 as shown in Table 2.
This means that removing the fifth sensor has the smallest impact on the
system observability. Therefore, the sensor for state 5 ($C_{A3}$) is removed
in this step. In the next step, another sensor is to be removed and we aim to
find the optimal set containing six sensors. It is found that if the third one
is removed from $S^{(7)}$, the degree of observability is the largest (255.97)
among all possible combinations of six sensors. Therefore, the third sensor is
removed at this step. The remaining steps for removing sensors are similar
until the 7th sensor is removed from $S^{(3)}$. From Table 2, it can be seen
that degreemax value becomes smaller with the decrease of the number of
sensors. In the case of ensuring the observability of the system, the decrease
in degreemax value is not very significant (from $m=8$ to $m=2$). However, if
we further remove any of the remaining two sensors, the rank is deficient
(rank($\bm{S}(k,0)$)=5), and the degreemax value decreases dramatically to
only 67.7. Therefore, the sensor selection process is terminated at $m=2$, and
sensors $5,3,6,4,2,7$ is removed in sequence. It can be seen from Figure 3 and
Table 2 that the selected sensor set is $\\{1,8\\}$, i.e.,
$\\{C_{A1},T_{4}\\}$, with the maximum degree of obserbability for the minimum
number of sensors.

Figure 4: The trajectories of the actual states ($C_{A1}$, $T_{1}$, $C_{A2}$,
$T_{2}$) and their state estimates based on the sensor set {$C_{A1}$,
$T_{4}$}.

Now, we have determined the minimum number of sensors and the optimal
placement of sensors. Next, we consider state estimation using the moving
horizon estimation (MHE) to show that the selected sensor set
$\\{C_{A1},T_{4}\\}$ provides good state estimation performance. The process
and measurement noises are the same as that of the sensor selection algorithm.
The input variables and sampling time are also the same as before. The output
vector here is $[C_{A1},T_{4}]^{\tiny\mbox{\rm T}}$ corresponding to the
selected set. The steady-state value
$\bm{x}_{s}:=[2.78,363,2.58,356,2.6,355,2.6,392]^{\tiny\mbox{\rm T}}$, and the
initial value guess is set to be $1.2\bm{x}_{s}$. The moving horizon length is
40. The covariance matrices of process noise and measurement noise in the MHE
are $\bm{Q}_{w}={\rm diag}((0.1\bm{x}_{s})^{2})$, $\bm{R}_{v}={\rm
diag}((0.1\bm{y}_{s})^{2})$. The actual states and the state estimates are
shown in Figures 4 and 5. It can be observed that the state estimates are
close to their actual states.

Figure 5: The trajectories of the actual states ($C_{A3}$, $T_{3}$, $C_{A4}$,
$T_{4}$) and their state estimates based on the sensor set {$C_{A1}$,
$T_{4}$}.

The root-mean-square error (RMSE) is used to evaluate the performance of state
estimation. In order to reduce the impact of noise randomness on the
estimation results, we can perform the state estimation several times and
average it as an evaluation of performance. For further showing the validity
of selected sensor set, all possible two sensor combinations are performed as
a comparison. There are a total of $C_{8}^{2}=28$ sets, of which the mean and
standard deviation of the RMSEs (based on 10 different simulation runs) for 15
sensor combinations are shown in Figure 6, and the corresponding degrees of
the observability are shown in Table 3. The other 13 cases cannot make the
system observable. In Figure 6, it can be found that the selected sensor set
$\\{1,8\\}$ gives the smallest estimation errors compared to other sensor sets
with the same number of sensors, and the degree of observability is largest in
Table 3. For the sets $\\{1,3\\}$, $\\{3,6\\}$ and $\\{5,6\\}$, their mean
values of RMSE in Figure 6 are large, and the corresponding degrees of
observability in Table 3 are relatively small.

Figure 6: Mean and standard deviation of RMSEs with 10 runs for 15 sensor combinations with 1% measurement and process noises  Table 3: Degree of observability | {1,2} | {1,3} | {1,4} | {1,6} | {1,7} | {1,8} | {2,3} | {2,5}
---|---|---|---|---|---|---|---|---
degree | 90.1 | 48.1 | 63.7 | 67.0 | 65.7 | 141.4 | 91.1 | 73.4
| {3,4} | {3,6} | {3,7} | {3,8} | {4,5} | {5,6} | {5,8} |
degree | 77.2 | 51.8 | 72.1 | 99.0 | 71.4 | 45.0 | 109.6 |

###  5.2 Application to a wastewater treatment plant

In this subsection, we apply the proposed approach to a wastewater treatment
plant (WWTP), which is a large-scale process network [31]. A schematic diagram
of the WWTP is presented in Figure 7, which consists of a five-compartment
activated sludge reactor and a spiltter. The biological reactor has two
sections: the non-aerated section containing the first two anoxic chambers and
the aerated section consisting of the remaining three chambers.

Wastewater is fed into the first anoxic reactor at concentration $Z_{0}$ and
flow rate $Q_{0}$. A portion of the effluent with $Z_{a}$ and $Q_{a}$ of the
last aerated reactor is recycled back to the first anoxic reactor and the rest
of the effluent is fed into the splitter at concentration $Z_{f}$ and flow
rate $Q_{f}$. The processed water leaves from the top of the splitter at
concentration $Z_{e}$ and flow rate $Q_{e}$. The generated sludge is withdrawn
from the bottom of the splitter at $Z_{e}$ and $Q_{w}$. A second recycle
stream from the splitter is fed back to the first anoxic reactor at $Z_{r}$
and $Q_{r}$. This model describes 8 basic biological reaction processes and
considers 13 major compounds. The concentrations of these 13 compounds in the
5 reactors and 1 splitter are the state variables of the bioreactor model. The
state set for the spiltter and each reactor is {$S_{\rm I}$, $X_{\rm I}$,
$S_{\rm S}$, $X_{\rm S}$, $X_{\rm B,H}$, $X_{\rm B,A}$, $X_{\rm P}$, $X_{\rm
ND}$, $S_{\rm NO}$, $S_{\rm NH}$, $S_{\rm ND}$, $S_{\rm O}$, $S_{\rm ALK}$}.
The definition of these state variables can be found in [32]. There are a
total of 78 states that describe the dynamics of the entire process. For the
WWTP, up to 48 measurements are available for state estimation. In each
reactor, eight states/state-related variables can be measured and the
definitions and expressions of these measurements are shown in Table 4. In
this simulation, we detail how sensitivity analysis can be used to determine
the minimum number and optimal placement of sensors, and how degree of
observability can improve state estimation.

Figure 7: A schematic of the wastewater treatment plant  Table 4: List of measurements. Measurement | Expression in the form of process states
---|---
Oxygen | $S_{\rm O}$
Alkalinity | $S_{\rm ALK}$
NH$4^{+}$ and NH3 nitrogen | $S_{\rm NH}$
Nitrate and nitrite nitrogen | $S_{\rm NO}$
Chemical oxygen demand (COD) |  $S_{\rm I}$+$S_{\rm S}$+$X_{\rm I}$+$X_{\rm S}$+$X_{\rm B,H}$+$S_{\rm B,A}$
Filtered chemical oxygen demand (CODf) |  $S_{\rm I}$+$S_{\rm S}$
Biological oxygen demand (BOD) |  $S_{\rm S}$+$X_{\rm S}$
Total suspended solids (TSS) |  $X_{\rm I}$+$X_{\rm S}$+$X_{\rm B,H}$+$S_{\rm B,A}$+$X_{\rm P}$+$X_{\rm ND}$

Based on the results of [32], the initial sensor set is chosen as follows.

  * •

Reactor 2: $COD_{2}$, $COD_{\rm f,2}$, $TSS_{2}$, $S_{\rm O,2}$, $S_{\rm
ALK,2}$.

  * •

Reactor 3: $COD_{3}$, $COD_{\rm f,3}$, $TSS_{3}$, $S_{\rm O,3}$, $S_{\rm
ALK,3}$.

  * •

Reactor 4: $S_{\rm O,4}$.

  * •

Reactor 5: $S_{\rm O,5}$.

  * •

Ideal splitter: $COD_{6}$, $S_{\rm NH,6}$, $S_{\rm NO,6}$, $S_{\rm ALK,6}$.

The proposed algorithm is applied to select the minimum number of sensors from
the given set and to provide the optimal locations of the sensors for the
minimum set. In the simulations, the process measurements are sampled every
$\Delta=15$min. The data file of dry weather can be found on the International
Water Association website. Initially, the sensor set
$\mathcal{S}^{(16)}=\\{S_{\rm O,2}$, $S_{\rm O,3}$, $S_{\rm O,4}$, $S_{\rm
O,5}$, $COD_{2}$, $COD_{3}$, $COD_{6}$, $COD_{\rm f,2}$, $COD_{\rm f,3}$,
$S_{\rm ALK,2}$, $S_{\rm ALK,3}$, $S_{\rm ALK,6}$, $TSS_{2}$, $TSS_{3}$,
$S_{\rm NO,6}$, $S_{\rm NH,6}\\}$ denoting as $\\{1,2,...,16\\}$ for
simplicity, and the system is fully observable. Figure 8 shows the process of
removing the sensors one by one, and the corresponding degree of observability
of remaining sensor combinations. It is observed that for $S^{(16)}$, when the
first sensor is removed, the corresponding degree of observability of the
remaining 15 sensors is the largest. This means that removing the first sensor
has the smallest impact on the system observability. Therefore, sensor 1 is
removed first. In the next step, another sensor is to be removed and we aim to
find the optimal set containing 15 sensors. It can be seen that the tenth one
is removed from the 15 sensors, and the degree of observability is the largest
among all possible combinations of 14 sensors. The remaining steps to remove
sensors are performed similarly. It can be found that the 16 measurements used
in [32] can be reduced to 10. If we continue to remove any of the 10 sensors,
the system becomes unobservable, so the selection process is terminated at
$m=10$. The selected 10 sensors are shown in Table 5 (Case 1).

Figure 8: Degree of observability by removing different measurements from.

Once the set of sensors is determined, we consider the state estimation using
the extended Kalman filtering (EKF), the same as used in [32], to verify the
effectiveness of the selected sensors. The initial guess is set to be
$1.2\bm{x}_{s}$. The tuning parameters used in the EKF are $\bm{Q}_{w}={\rm
diag}((0.1\bm{x}_{s})^{2})$, $\bm{R}_{v}={\rm diag}((0.1\bm{y}_{s})^{2})$ and
$\bm{P}(0)=\bm{Q}_{w}={\rm diag}((0.1\bm{x}_{s})^{2})$. The actual states of
reactor 2 and their state estimates are shown in Figure 9. It is obvious that
the state estimates can track the actual state trajectory well.

Figure 9: Trajectories of the actual process states (solid lines) and the
estimates (dashed lines) with 10 measurements of reactor 2.

In order to further show the validity of the selected minimum number of sensor
combinations, three randomly chosen ten-sensor combinations are constructed
(Cases 2-4 in Table 5). The mean values of the normalized estimation error can
be defined as

${\rm Mean}|e|=\frac{1}{N}\sum_{t=1}^{N}e(k),$ (23)

where

$e(k)=\sqrt{\sum_{i=1}^{78}\left(\frac{\hat{x}_{i}(k)-x_{i}(k)}{\max(\hat{x}_{i}-x_{i})}\right)^{2}}$
(24)

is the normalized error at sampling time $k$ based on its maximum estimation
error. Mean$|e|$ and RMSE are listed in Table 5, which are used to evaluate
the performance of state estimation for each sensor combination. It can be
found that the Mean$|e|$ and RMSE obtained by Case 1 are the smallest among
the four cases. The trajectories of the actual states and the state estimates
of reactor 6 for four cases are presented in Figure 10. From Table 5 and
Figure 10, it can be found that the selected sensor set
$\\{2,5,6,7,8,9,13,14,15,16\\}$ in Case 1 gives the best estimation results
compared to other sensor combinations with the same number of sensors.

If we want to have more robust state estimation performance in the presence of
significant process disturbances and measurement noise, additional sensors can
be added to the minimum set of 10 sensors. In this case, it is necessary to
select the sensor that contributes most to the degree of observability, and
then include it in sensor set. From Figure 8, we can see the order in which
the sensors are removed. Therefore, the order in which the sensors are added
can be based on the reverse order of removal. When we only want to add one
sensor, we should select the sensor 11. If there is more budget or need to add
another, then we should further select sensor 12.

Figure 10: Trajectories of the actual process states (solid lines) and the state estimates based on 10 measurements (Case 1: blue dashed lines; Case 2: red dash-dotted lines; Case 3: dotted lines; Case 4: green dash-dotted lines) of reactor 6. Table 5: Performance comparison for four measurement configurations. | RMSE |  Mean$|e|$ | Sensor selection
---|---|---|---
Case 1 | 0.1322 | 2.1618 |  {$S_{\rm O,3}$, $S_{\rm NH,6}$, $S_{\rm NO,6}$, $COD_{2}$, $COD_{3}$, $COD_{6}$, $COD_{\rm f,2}$, $COD_{\rm f,3}$, $TSS_{2}$, $TSS_{3}$}
Case 2 | 0.2066 | 3.2992 |  {$S_{\rm O,2}$, $S_{\rm O,5}$, $S_{\rm ALK,2}$, $S_{\rm ALK,5}$, $S_{\rm NH,3}$, $S_{\rm NO,5}$, $COD_{1}$, $COD_{\rm f,4}$, $BOD_{2}$, $BOD_{5}$}
Case 3 | 0.2252 | 3.5492 |  {$S_{\rm ALK,2}$, $S_{\rm ALK,3}$, $S_{\rm ALK,6}$, $S_{\rm NO,1}$, $S_{\rm NO,4}$, $COD_{2}$, $COD_{\rm f,1}$, $COD_{\rm f,5}$, $BOD_{1}$, $BOD_{4}$}
Case 4 | 0.1873 | 2.9660 |  {$S_{\rm O,1}$, $S_{\rm O,4}$, $S_{\rm ALK,3}$, $S_{\rm NH,1}$, $S_{\rm NH,3}$, $S_{\rm NO,2}$, $COD_{4}$, $COD_{\rm f,4}$, $BOD_{3}$, $TSS_{6}$}

##  6 Conclusions

In this work, a sensitivity-based method is proposed to quantify the degree of
observability of nonlinear process networks, and a computationally efficient
optimal sensor selection method for state estimation is developed to determine
the minimum number of sensors and their optimal locations based on the degree
of observability. The proposed approach is applied to a four-CSTRs process and
a large-scale wastewater treatment process. The results show that the proposed
algorithm is very effective in determining the minimum number of measurements
and their optimal sensor locations. Compared with the existing approaches, the
proposed approach does not need to solve mixed-integer programming and is very
computatinally efficient. For the future work, we will consider the cost of
sensors in selecting the sensors and apply the proposed method to other
fields.

##  7 Acknowledgement

The first author, S.Y. Liu, is a visiting Ph.D. student in the Department of
Chemical and Materials Engineering at the University of Alberta from March
2021 to February 2023. She acknowledges the financial support from the China
Scholarship Council (CSC) during this period.

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