A vector-measurement-sensor-selection problem in the undersampled and oversampled cases is considered by extending the previous novel approaches: a greedy method based on D-optimality and a noise-robust greedy method in this paper. Extensions of the vector-measurement-sensor selection of the greedy algorithms are proposed and applied to randomly generated systems and practical datasets of flowfields around the airfoil and global climates to reconstruct the full state given by the vector-sensor measurement.

Optimal sensor placement is an important challenge in the design, prediction, estimation, and control of high-dimensional systems. High-dimensional states can often leverage a latent low-dimensional representation, and this inherent compressibility enables sparse sensing. For example, in the applications of aerospace engineering, such as launch vehicles and satellites, optimal sensor placement is an important subject in performance prediction, control of the system, fault diagnostics and prognostics, etc. This is because there are limitations of installation, cost, and downlink capacity for transferring measurement data. Reduced-order modeling has been gathering a lot of attention in various fields. A proper orthogonal decomposition (POD) [

Here,

Although Joshi et al. proposed a convex approximation method for this objective function [

The previous studies introduced so far consider the scalar-sensor measurement that the selected sensors obtain a single component of data at the sensor location. There are several applications of vector-sensor measurement, such as two components of velocity, or simultaneous velocity, pressure, and temperature measurements used in weather forecasting. For example, the real-time particle-image-velocimetry (PIV) measurement of the flowfield is required for the feedback control of a high-speed flowfield in laboratory experiments. The velocity field is calculated from the cross-correlation coefficient for each interrogation window of the particle images in the PIV measurement, but the number of windows that can be processed in a short time is limited because of the high computational costs of the calculation of cross-correlation coefficients. We have been developing a sparse processing particle-image-velocimetry (SPPIV)-measurement system [

Here, the difference between scalar and vector-sensor measurements is the number of components of data: scalar and vector-sensor measurements obtain a single and multiple components of data at the sensor location, respectively. The extension of the vector-measurement-sensor selection of the convex approximation method has already been addressed in Section C, Chapter V of the original paper [

Scalar | Vector | |
---|---|---|

D-optimality-based Convex (DC) | Joshi et al. [ |
Joshi et al. [ |

D-optimality-based Greedy (DG) | Saito et al. [ |
Present study |

Bayesian D-optimality-based Greedy (BDG) | Yamada et al. [ |
Present study |

The main novelties and contributions of this paper are as follows:

The present study extends the DG and BDG methods proposed for scalar-sensor measurements to vector-sensor measurements in the undersampled and oversampled cases.

The extensions of the vector-sensor measurement of the DG and BDG methods for the undersampled and oversampled cases have significant novelties for the sparse-sensor measurement in terms of reconstruction error and computational time.

The present study compares the performance of the DG and BDG methods with the random selection and DC methods under the conditions:

The present study is applied to randomly generated systems and practical datasets of flowfields around the airfoil and global climates. These results illustrate that the proposed DG and BDG methods extended to the vector-measurement-sensor-selection problem are superior to the random selection and DC methods in terms of the accuracy of the sensor selection and computational cost in the present study.

A D-optimal design corresponds to maximize the determinant of the Fisher information matrix. Therefore, maximization of the determinant of

Improved D-optimality-based greedy algorithm was presented in the previous study [

where

where

where

and

Then, the Bayesian estimation is derived with those prior information. Here, an a priori probability density function (PDF) of the POD mode amplitudes becomes

Here, the maximum a posteriori estimation on

A vector-measurement-sensor-selection problem in the undersampled and oversampled cases is considered by extending the previously explained DG and BDG methods. Again, the number of components of data is different for scalar and vector-sensor measurements: scalar and vector-sensor measurements obtain a single and multiple components of data at the sensor location, respectively. In the vector-sensor measurement, the following equation is considered:

Here,

where

The algorithm of the DG method for vector-measurement sensors in the undersampled and oversampled cases is summarized in Algorithm 4.

The step-by-step maximization of the determinant of

and therefore,

In the undersampled case, the DG method is the same as the vector-measurement sensor selection method [

The maximization of the determinant of

Therefore,

The complexity when searching all components of the vector becomes

The size of the matrix is

A fast implementation is considered as Saito et al. demonstrated in their determinant calculation using rank-one lemma [

where the covariance

and similarly, the variance of noise is defined as:

In addition,

where

The objective function of BDG is now considered based on the expressions above.

Because

Once the sensor is selected, _{2} and reduces the computational complexity [_{2} truncation in the previous study. Although the _{2} truncation will reduce the computational complexity in the vector-measurement-sensor-selection problem, the _{2} truncation is not addressed in the present study. This is because it additionally needs to conduct the parametric study for the reasonable _{2}, and the extensions of the vector-measurement-sensor selection of the DG and BDG methods are focused in the present study.

The numerical experiments are conducted and the proposed methods are validated. The random sensor, PIV, and NOAA-SST/ICEC problems are applied in the present study. The vector sensor-measurement matrix

Processor information | Intel(R) Core(TM) |
---|---|

5-8400@2.8 GHz | |

Random access memory | 64 GB |

System type | 64 bit operating system |

x64 base processor | |

Program code | Matlab R2020a |

Operating system | Windows 10 Pro |

The quality of the sensors is evaluated by considering the error between the original and estimated data. The error

Here, series of the estimation

Least squares estimation, LSE | |
---|---|

Bayesian estimation, BE |

The random data matrices,

_{2} truncation proposed in the previous study for the scalar-measurement-sensor-selection problem [_{2} truncation for the BDG (vector) method will reduce the computational time. The DG (vector) method which has the same error as the DC method is computationally fast, and the two methods extended in the present study are generally better methods in the random sensor problem.

The particle image velocimetry (PIV) for acquiring time-resolved data of velocity fields around an airfoil was conducted previously [^{4}. Time-resolved PIV measurement was conducted with a double-pulse laser. The sampling rate at which the velocity fields are acquired, the particle image resolution, and the total number of snapshots were 5000 Hz, 1024

The vector-measurement-sensor selection problem for the reconstruction of the lateral and vertical components of the velocities measured by PIV (

_{2} truncation [_{2} truncation is not addressed in the PIV problem. The DG (vector) method which has the same error as the DC (vector) method is faster than the BDG (vector) and DC (vector) methods. Therefore, the BDG (vector) and DG (vector) methods extended in the present study are better methods in the PIV problem obtained through real wind-tunnel tests.

The data set that we finally adopt is the NOAA OISST V2 mean sea surface dataset (NOAA-SST/ICEC), comprising weekly global sea surface temperature in ^{−1} times smaller than the maximum of those of the dataset, and

All the errors estimated by the BE method are smaller than those estimated by the LSE method as shown in ^{−1} times smaller than the maximum of the NOAA-SST/ICEC dataset, and _{2} truncation [_{2} truncation is not addressed in the NOAA-SST/ICEC problem. The DG (vector) method, which has the same error as the DC (vector) method, is faster than the BDG (vector) and DC (vector) methods in this NOAA-SST/ICEC problem. Therefore, the BDG (vector) and DG (vector) methods extended in the present study are better methods in the NOAA-SST/ICEC problem.

Problem | Data size |
Smallest error of the result using the LSE method | Smallest error of the result using the BE method | Shortest computational time |
---|---|---|---|---|

Random | DG | |||

DG | BDG | |||

BDG |
||||

DG |
||||

BDG |
||||

PIV | DG | |||

DG | BDG | |||

DG |
BDG |
|||

NOAA | DG | |||

DG | BDG | |||

BDG |
||||

DG |
||||

BDG |

Note: ^{*}In the NOAA-SST/ICEC problem, the locations are beforehand excluded from sensor candidate matrix for simplicity if their RMSs are 10^{−1} times smaller than the maximum of those of the dataset.

A vector-measurement-sensor-selection problem in the under and oversampled cases is considered by extending the previous novel approaches: a greedy method based on D-optimality (DG) and a noise-robust greedy method (BDG) in the present study. Extensions of the vector-measurement-sensor selection of the greedy algorithms are proposed and applied to randomly generated systems and practical datasets of flowfield around airfoil and global climates and the full states are reconstructed by the vector-sensor measurement. In all demonstrations, the random selection and convex approximation methods are evaluated as the references in addition to the proposed DG and BDG methods. The least squares and Bayesian estimation methods are employed as the state estimation method in the present study.

The results applied to randomly generated systems show the proposed DG and BDG methods select better the position of the sparse sensor than the random selection and convex approximation methods. The results applied to practical datasets of flowfield around the airfoil and global climates are similar to the results applied to randomly generated systems. In addition, the reconstructed fields from the selected sensor in the noise-robust greedy (BDG) method are closest to the original data in all demonstrations. These results illustrate that the proposed methods extended to the vector-sensor measurement are superior to the random selection and convex approximation methods in terms of the accuracy of the sensor selection and computational cost in the present study.

Although the vector-measurement-sensor-selection problem extended by the present study is a more realistic sensor placement problem than the traditional scalar-measurement-sensor-selection problem, there are gaps between ideal models and real-world practicalities. For example, which is the better whether to place a large number of cheap sensors having a low signal-to-noise level, a small number of expensive sensors having a high signal-to-noise level, or a mix of both. The sensor selection problem considering the cost problem [