1. Introduction
Recent advancements in sensor technology have paved the way for a wide range of applications and services in areas encompassing safety and surveillance, climate prediction, and the environmental monitoring of pollution and temperature. These sensors are distributed across various locations and work together as a network. The primary task of these networks is to perform distributed data sampling, essentially sensing the environmental conditions to gather valuable information. Collecting data is an even more expensive and inefficient endeavor when such collection needs to be performed periodically. The challenge thus lies in optimizing sensor placement and movement to ensure accurate data collection with minimal resources. This involves using fewer sensors to cover large areas efficiently and relying on smart algorithms to determine the optimal positions. Placing sensors strategically allows for the creation of sensing patterns over space and time, ensuring that specific performance criteria such as energy efficiency, data quality, the accuracy of estimates, and the likelihood of detection are met [
1,
2,
3,
4,
5,
6,
7].
Complex controlled systems benefit from a systematic approach to selecting the most convenient outputs so that their states can be reconstructed/estimated reliably. For a system constituting a sensing network, these outputs can be defined in terms of the positioning of the network members in places that provide a better representation of spatial information of the phenomenon at hand. Some other applications of state reconstruction using sensor data include controlling fluid flows [
8], optimizing power grids [
9], and enhancing high-performance computing [
10]. Data-driven system approaches such as those based on Koopman theory also rely on sensor placement methods to enhance estimation and control [
11,
12]. In this context, the Kalman Filter (KF) framework allows us to assimilate the information over time to enhance the reconstruction of high-dimensional states using a minimal number of sensors [
13,
14]. Among the plethora of sensor placement methodologies, QR sensor placement has been applied across a range of measurement selection scenarios due to its efficiency and the fact that it maximizes the conditioning of reconstructing transformations [
15,
16,
17,
18,
19]. Thus, this technique is integral to the contribution of this paper.
A dynamic sensor network cannot disregard the dynamics of the platforms carrying the sensor of interest. This is even more important when such dynamics can change during monitoring missions in harsh environments. In this context, Model-Free Control (MFC) is an innovative non-linear control technique that adopts a data-driven approach to construct a linear surrogate model that replaces the complex dynamics of controlled systems, which allows control laws to change on the fly when the system unexpectedly changes [
20,
21,
22,
23,
24]. This surrogate model, commonly known as the
ultra-local model (ULM), encompasses two key parameters. The first parameter captures the overall time-varying dynamics of the system. The second parameter is considered as a tuning knob and is determined through expert knowledge of the system, thus employing a heuristic approach that is generally based on the system’s characteristics. Additionally, the system error dynamics is highly simplified, such that a control law can be designed for the purpose of stability and improving performance. The theoretical foundation of MFC was established in [
20], aiming to eliminate the requirement for an accurate model and precise knowledge of system parameters for effective control. In particular, the time-varying parameter in the ULM adaptively captures the overall system’s dynamics during a short time interval. Thus, this parameter is updated at every time step to make sure that the unmodeled dynamics of the operated system can be captured effectively. MFC demonstrates its versatility through a wide array of applications. For instance, MFC was applied to a gantry crane system whose outputs are unreliable but can recover performance under a Kalman Filter robustification approach [
21]. Furthermore, MFC has been successfully applied to control a quadcopter system but without the uncertainty quantification of the ULM performance or its position [
22,
25]. This uncertainty quantification is part of the work presented in this paper.
Various methodologies can be employed in the implementation of MFC when estimating the ULM. Traditionally, digital filters have been a common choice for this purpose [
20]. Another approach involves the use of a Luenberger observer, where the designer can tailor the convergence speed of the ultra-local model by adjusting the selected gains [
26]. Despite their success, these methodologies share a limitation: they do not provide an uncertainty quantification of the parameters or state of the system. Such enhancement can be carried out by incorporating a KF approach for estimating the ULM, as in [
21,
27]. This approach offers the advantage of quantifying the positional uncertainty generated from the MFC methodology in the form of the corresponding covariance matrix needed for the subsequent integration with spatial reconstruction methodologies.
Monitoring the spatial and temporal changes in a field with a minimal number of sensors mounted on systems controlled using MFC while achieving a specific performance constitutes a problem within the field of sensor networks [
28]. Environmental, geophysical, and biological processes, which are known for their complex variability in space and time, can be modeled based on their spatial and temporal correlations for stationary fields. For non-stationary fields, dynamic models help describe their evolution [
29]. When a field exhibits both stationary and non-stationary traits, a Kriged Kalman Filter (KKF) [
30], constituted by the merging of Kriging [
29,
31] with a KF, is known to effectively monitor these changes. Specifically, the KKF has been shown to be able to reconstruct spatial fields of information [
32]. Combining a minimum sensor-placement problem with a KKF can efficiently improve the reconstruction of a particular information field by decreasing the required points from which data need to be obtained. This becomes particularly advantageous in challenging environments like the Arctic, where data acquisition presents its own set of hurdles. The use of unmanned aerial vehicles (UAVs) gathering data from these critical points offers a viable solution to the problem. Moreover, MFC proves valuable in this context due to its ability to adapt to both external environmental and internal dynamic changes.
Methods such as those in [
2,
29,
31,
32] have been employed for the reconstruction of spatial fields, but these techniques do not take into account the positional uncertainty of the locations where data are taken. Additionally, the KKF does not provide an optimal procedure for deciding which locations for data acquisition are best given a dynamic field of information. While there is work on finding optimal sensor locations such as in [
15,
16,
17,
18,
19], their integration with the KKF is lacking. Lastly, MFC has been implemented for various applications [
20,
21,
27,
33,
34,
35,
36]; however, there is no application to our knowledge using the uncertainty obtained from MFC in spatial reconstruction methodologies. Therefore, the main contributions of this paper are as follows.
We provide an algorithm that determines the optimal number of points essential for reconstructing a given field. This procedure will be referred to as the optimal sensor placement procedure.
We integrate MFC on quadcopters to collect data from selected locations. MFC aims to enhance system robustness in the face of the challenging external conditions that may be encountered during terrain reconstruction in harsh environments.
We implement MFC employing a Kalman Filter (MFC-KF) to estimate the ULM for quadcopters. This method quantifies the position uncertainty of the sensing platform.
We augment the covariance obtained from the MFC-KF with that of the KKF to improve the reconstruction process. This integration takes into account the variance associated with the sensors used for data acquisition, further refining the accuracy of the reconstructed field.
The rest of the paper is organized as follows.
Section 2 provides the preliminaries of the techniques used in the proposed method (QR sensor placement, MFC, and a description of the KKF).
Section 3 presents the proposed methodology that integrates the concepts described in the preliminary section.
Section 4, provides numerical simulations that illustrate the proposed methodology with synthetic information. The final section gives the conclusions.
2. Preliminaries
In this section, we will first focus on compressed sensing in order to provide a foundation for how data reconstruction can be achieved by selecting key locations for measurements. This will be followed by sensor placement, which will focus on how to find the key data points in a set in order to reconstruct the data. Then, a brief summary of MFC will be provided that mainly focuses on how MFC based on the Kalman Filter (MFC-KF) works. This is followed by details on a robustification technique that can be used to make systems using MFC resilient to contaminated measurements. The robustification processes provides details on how to obtain reliable data and, more importantly, the quantification of the positional uncertainty. In this paper, the reconstruction of the data is carried out by quadcopters; therefore, a background on the modeling of a quadcopter is presented. Lastly, details on the KKF method are provided.
2.1. Compressed Sensing
The compressibility of most natural signals, like images and audio, implies that, in a suitable basis, only a limited number of modes are active. Consequently, compressed sensing leverages a signal’s sparsity on a universal basis to accomplish complete signal reconstruction from remarkably few measurements. Similarly, many high-dimensional physical systems can be effectively described by a low-dimensional attractor, enabling efficient prediction and control. States evolving through nonlinear dynamics often have compact representations in an appropriate transform basis. A compressible signal (i.e., a state)
can be represented as
where
is a sparse vector indicating the few active modes of
[
18,
37]. More specifically, the vector
is referred to as
K-sparse within the basis
when it contains exactly
K non-zero elements. In scenarios where the basis
is generic, such as the Fourier or wavelet bases, only a few active elements in
are essential for reconstructing the original signal
x. This sparsity allows for an efficient low-rank representation of the transform basis,
, for the sparse low-rank coefficients
, which can be written as
By collecting significantly fewer selected measurements, it becomes feasible to solve for the non-zero elements of
within the transformed coordinate system. One can then define the measurements
, where
, as
where
selects
p measurements of the signal
x. The selection of the measurement matrix
holds paramount significance in compressed sensing, which is one of the challenges in this study. Moreover, within compressed sensing, the objective is to identify the sparsest vector
that aligns with the provided measurements
. Combining (
1) and (
3), it follows that
The selection of aims to ensure that the operator is well conditioned for reconstructing x out of . In the case that some expert knowledge of the signal is known, custom sensors can be crafted specifically for those signals using the Proper Orthogonal Decomposition (POD). The POD identifies key features from data and represents high-dimensional signals using a limited set of orthonormal eigenmodes, forming a lower-dimensional embedding space, and allows efficient data handling. Thus, for instance, Singular Value Decomposition (SVD) can be used for reconstructing the complete signal from the low-dimensional representation while optimizing accuracy and minimizing errors.
2.2. Sensor Placement and the QR Factorization
The goal of sensor placement is to choose sensor positions that capture a low- dimensional subspace with significant characteristics to reconstruct high-dimensional signals efficiently. Such positions,
, are encoded in
, as mentioned before. In the often scenario where the vector
x is unknown, the basis coefficients
a in (
2) are approximated as
Note that (
5) is only valid when the number of measurements is equal to the number of significant modes in the basis
(
), for
, and we have
.
The most effective reconstruction of
x is achieved by pinpointing the rows of
that correspond to optimal sensor locations. Since
might not be square and depends on the choice of
, we denote
when
and
when
. The problem can thus be formulated by finding
such that the minimum singular value of
,
, is maximized. That is, we need to solve
The QR factorization is a fundamental numerical technique in linear algebra that decomposes a matrix into an orthogonal matrix
and an upper triangular matrix
[
38]. Equation (
6) is solved using QR factorization. QR factorization with column pivoting introduces a permutation matrix where the pivoting locations define
and, therefore, gives a
that enhances the numerical stability and accuracy of the reconstruction process. Thus, the QR factorization with column pivoting selects
p locations that most effectively sample the
r basis modes of
.
2.3. Model-Free Control
Since the objective is to reconstruct information with data acquired from a sensing platform operating in uncertain environments, the dynamics of the systems carrying the sensors must be accounted for. Consider an arbitrary system with
n dimensions, governed by the following differential equation:
where
is a function that encapsulates the relationships among the system’s inputs, outputs, and their respective derivatives (
derivatives of
and
derivatives of
u). A significant challenge arises in formulating
due to the inherent lack of complete knowledge about the system. This limitation often originates from unmodeled dynamics, uncertainties in modeled parameters, and unforeseen changes in the model dynamics due to operating in harsh environments.
To address these challenges, one effective approach is to employ a surrogate model. This model estimates the overall time-varying dynamics of the system using input–output data obtained from sensors in the system. One such surrogate model is the ULM. The ULM expresses the system dynamics for a short period of time as
where
v is the order of the ULM,
F represents the overall time-varying dynamics that need to be estimated,
is a tuning parameter, and
u is the input going into the system [
20,
21,
26]. Note that there does not exist a systematic methodology for selecting
, and expert knowledge of the application is often required to select or tune such a value. Nevertheless, some attempts have been proposed [
27,
39], where
is estimated using data-driven approaches. This work assumes that the selection of
is performed using expert knowledge. Therefore, the idea of MFC is to introduce a control law
u that cancels the time-varying dynamics and add an additional internal controller to produce a compensating control effort
for reducing tracking error. That is,
where
is the
v-th derivative of the desired reference trajectory. Traditionally, the control effort is implemented via a simple PID (Proportional, Integral, and Derivative) control strategy or one of its variants, PI, PD, and P. Due to the nature of MFC and its ability to adapt to time-varying dynamics, this implementation is often referred to as intelligent iPID, iPI, iPD, and iP [
20]. Substituting (
9) into (
8) leads to
Defining
, (
11) can be written as
, where the control effort
can be easily designed to minimize
. The choice of
v is based on the type of controller used. For example, if an iPID or iPD is used, then
. On the other hand, if iPI is required, then
. A basic implementation of MFC on a generic system is given in
Figure 1.
Traditionally, the estimation of the ULM parameters is performed using digital filter [
20,
22,
40]. Other methodologies include the use of Luenberger observers [
26], where the ULM given in (
8) is given the form of a linear state space representation. On the other hand, a KF approach can also be used to estimate the parameters of the ULM as shown in [
21]. First, (
8) is converted into a state space system. Assuming
, (
8) in continuous time is expressed as
A first-order discretization of (
12) gives
To estimate
F at time
k, a new state
is introduced, and (
13) is augmented as
which assumes that
does not change too fast. The steps for the KF using the ULM are then:
where
,
,
represents the standard deviation of the sensor noise,
is the standard deviation of the process noise,
K is the Kalman gain,
is the a priori covariance,
is the a posteriori covariance, and
is the output obtained from the system [
21].
2.4. Robustification of MFC
Since MFC is a data-driven approach, it is prone to measurement corruption or sensor failure. This is even more of a problem when MFC is used in harsh or unknown environments. Therefore, a robustification of MFC can be performed by integrating the
Robust Generalized Maximum-Likelihood Kalman Filter (RGMKF) [
21,
41,
42] into the MFC estimation methodology of the ULM. One of the key advantages of using RGMKF is its capability to handle process, sensor, and structural outliers while estimating a system’s states [
41]. In other words, the RGMKF offers a procedure for the rectification of states and their corresponding covariances by reducing the impact of outliers on the estimation process. The RGMKF operates through three main steps; the first consists of formulating the redundant observation vector. Here, the data coming from the system are stacked together. That is,
where
are the sensor measurements coming from the system and
are state predictions coming from (
15a). The vector
consists of multiple sensor values for each state and the observation matrix for
is
, where
has as many rows as the number of redundant sensors. The second step consists of detecting outliers in (
17), which is performed via Projection Statistics (PS) [
43]. Before obtaining the best estimate of the system state from (
17), one needs to pre-white the information to uncorrelate the data from the noise in the system. The data are pre-whitened as
where
is obtained via Cholesky decomposition of the noise covariance matrix associated with (
17). The next step estimates the system states via the Iterative Reweighted Least Squares method. That is,
where the recursion is performed over the weight matrix
. More detail on the RGMKF can be found in [
21,
41,
42]. Once the states have been rectified, the last step consists of correcting (
16d). That is,
where
is a diagonal matrix containing the PS weights is used on the quadcopters comprising the sensor network. As previously mentioned, robustification through the RGMKF is achieved by incorporating additional sensors, which can increase operational costs. A potential solution, as demonstrated in [
42], is to introduce redundancy through a swarm-based approach. However, this type of redundancy is not pursued here since it is outside the scope of the paper.
In the next subsection, a brief background on the modeling of a quadcopter system is provided.
2.5. Quadcopter System
The dynamic of a quadcopter is usually obtained based on force/moment dynamics and kinematics, with additional insights available in [
22,
25,
44]. That is,
where
,
, a
represent the position of the quadcopter in the Earth frame and
are the rotation angles in the body frame. Additionally,
represents the translation drag along each axis,
are the rotational drag coefficients,
is the difference between the rotor angular velocities, and
J is the rotor’s moment of inertia. A simple schematic of (
21) is shown in
Figure 2, where the lift force generated by each motor is denoted by
, and the angular velocity generated by each motor is denoted by
with
. The inputs to the system in terms of forces and angular velocities are
where
is a force velocity constant,
l is the length of the arm, and
d is the rotor’s reaction torque constant. When the model of the quadcopter is available, the traditional approach is to convert (
21) into control affine form [
45]. However, since MFC is being used, there is no need for this control-affine form in the formulation of appropriate quadcopter control laws. It is, however, important to emphasize that (
21) is presented here to show the intricacies of the acquisition network dynamics and that the model is solely employed for obtaining simulation results in subsequent sections while the inputs used to drive the quadcopter to the desired locations are provided by the robustified MFC-KF method. In the next section, a brief introduction to the KKF algorithm is provided since a modified KKF technique will subsequently be developed for reconstructing a field of information with the awareness of positional uncertainty originating from the acquisition network dynamics.
2.6. Kriged Kalman Filter
The Kriged Kalman Filter (KKF) is a technique designed for reconstructing spatially and temporally varying fields. This method integrates two well-established techniques: Kriging and the KF. Kriging is used for estimating unobserved data based on observed values from nearby locations, leveraging the spatial correlation among data points [
30,
32]. The KF estimates the states of a system by considering the statistical noise, observed values, and a model of the system [
46]. By combining these approaches, the KKF effectively manages the complexities of spatiotemporal data, providing accurate and robust estimates. The steps of KF have been covered in Equations (15) and (16), and similar steps are used in this section showing how to integrate the two techniques.
Consider field dynamics (e.g., produced by meteorological quantities such as wind or terrain variability such as snow with respect to historical bare land conditions). Thus, the spatially distributed information of interest at time
k is assumed to satisfy a dynamical equation of the form
where
denotes the predicted spatial information of the field parametrized by position
(i.e.,
and
),
is the prior best estimate of the field for time
,
is the redistribution of parameters from time
to time
k, and
is the Gaussian process noise associated with the spatial information with the covariance matrix
. The translation and diffusion of the spatial transition dynamics
can be modeled as
where
is a parameter aiming to keep (
22) stable and
and
are the translation and dilation parameters of an appropriate Gaussian kernel, respectively. The latter is related to a simple discretization of an advection and diffusion process [
1]. The sparse prediction of the measurements can be introduced in a formulation via the observation equation
where
,
is Gaussian noise associated with the measured values, and
is the associated covariance. Observe that the size of the matrix
varies with time, where
is the number of measurements taken over the sparse field. The matrix
is obtained by assuming the identity
, where
with 1 denoting the locations in
where measurements were taken and 0 otherwise. Notice that
is the same matrix used for sensor placement in
Section 2.1 but time-varying. Let
be the a priori covariance for the field dynamics prediction at time
k. Its propagation follows as
The measurements covariance,
, is given by
with
being the assumed noise caused by the process of sparsely measuring the field of information. The KKF Kalman gain can now be given as
and the field correction by
where
values are the measurements obtained (which can come from an arbitrary system capable of selecting data points within a field system such as a quadcopter),
is the spatial trend of the information field (for instance, obtained from ancillary information or historical data), and
is the prediction from information at time
in (
22). The covariance correction step follows naturally as
Next, the Kriging step for the stationary part of the field is
The final KKF step involves putting together the field update with the Kriging step of the stationary information. That is,
5. Conclusions
This paper presents a comprehensive approach to spatio-temporal field reconstruction using a combination of optimal sensor placement, MFC, and the KKF. The key contributions of this study addressed challenges in field estimation under dynamic and uncertain conditions.
First, an optimal sensor-placement procedure was provided to determine the minimal number of sensors required for effective field reconstruction. This method, based on QR decomposition, ensures efficient sensor usage while maintaining high accuracy. Second, MFC was implemented on quadcopters to collect data from the selected optimal points, enhancing the system’s robustness against challenging external conditions. Finally, the covariance obtained from the MFC-RGMKF was integrated with the KKF to improve the reconstruction process, accounting for sensor variance and refining accuracy.
The reconstructions shown in
Section 4 validated the effectiveness of these methodologies, demonstrating significant improvements in accuracy and efficiency. This study offers a robust framework for spatio-temporal field estimation, providing valuable insights for future research and applications in environmental monitoring, safety surveillance, and climate prediction.