# Efficient Sensor Node Selection for Observability Gramian Optimization

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## Abstract

**:**

## 1. Introduction

## 2. LTI State Space and the Observability Gramian

## 3. Sensor Selection Strategies

#### 3.1. Convex Relaxation Methods

#### 3.1.1. Semidefinite Programming-Based Selection (SDP)

#### 3.1.2. Newton Method for Approximate Convex Relaxation, and Its Customized Algorithm with Randomized Subspace Sampling (Approximate Convex Relaxation)

Algorithm 1 Newton algorithm for Equation (13) |

Input: $\mathbf{C}\in {\mathbb{R}}^{n\times r},\phantom{\rule{0.166667em}{0ex}}\mathbf{A}\in {\mathbb{R}}^{r\times r},\phantom{\rule{0.166667em}{0ex}}p\in \mathbb{N}$ |

Output: Indices of chosen p sensor positions ${\mathcal{I}}_{p}$ |

Set an initial weight $\mathbf{s}\leftarrow \mathbf{1}p/n$ |

while convergence condition not satisfied do |

Calculate $\nabla f$ by Equation (15) and ${\nabla}^{2}f$ by Equation (16) |

Calculate $\delta \mathbf{s}$ by Equation (14) |

Obtain step size t by backtracking line search |

Set $\mathbf{s}\leftarrow \mathbf{s}+t\delta \mathbf{s}$ |

end while |

Return the indices of the p-largest components of $\mathbf{s}$ as ${\mathcal{I}}_{p}$ |

Algorithm 2 Customized algorithm of Algorithm 1 (BRS-Newton) |

Input: $\mathbf{C}\in {\mathbb{R}}^{n\times r},\phantom{\rule{0.166667em}{0ex}}\mathbf{A}\in {\mathbb{R}}^{r\times r},\phantom{\rule{0.166667em}{0ex}}p>0,\phantom{\rule{0.166667em}{0ex}}\tilde{n}>0$ |

Output: Indices of chosen p sensor positions ${\mathcal{I}}_{p}$ |

Set $\mathbf{s}\leftarrow \mathbf{1}p/n$ |

while convergence condition not satisfied do |

Select ${\mathcal{I}}_{\phantom{\rule{0.166667em}{0ex}}\tilde{n}}$ [Equation (19)] and set ${\mathbf{S}}_{\tilde{n}}$ |

Calculate subsampled derivatives $\nabla \tilde{f}$ and ${\nabla}^{2}\tilde{f}$ |

Calculate $\delta \tilde{\mathbf{s}}$ |

Obtain step size t by backtracking line search |

Set $\mathbf{s}\leftarrow \mathbf{s}+t\left({\mathbf{S}}_{\tilde{n}}^{\top}\delta \tilde{\mathbf{s}}\right)$ |

end while |

Return the indices of the p-largest components of $\mathbf{s}$ as ${\mathcal{I}}_{p}$ |

#### 3.2. Greedy Algorithms

#### 3.2.1. Greedy Selection with Simple Evaluation (Pure Greedy)

Algorithm 3 Determinant-based greedy algorithm (pure greedy) |

Input: $\mathbf{C}\in {\mathbb{R}}^{n\times r},\phantom{\rule{0.166667em}{0ex}}\mathbf{A}\in {\mathbb{R}}^{r\times r},\phantom{\rule{0.166667em}{0ex}}p\in \mathbb{N}$ |

Output: Indices of chosen p sensor positions ${\mathcal{I}}_{p}$ |

${\mathcal{I}}_{n}\leftarrow \left\{1,\dots ,n\right\},\phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{0}\leftarrow \varnothing ,\phantom{\rule{0.166667em}{0ex}}$ |

for $q=1,\dots ,p$ do |

${\mathcal{I}}_{*}\leftarrow \left\{i:i\in \underset{i\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{n}\phantom{\rule{0.166667em}{0ex}}\backslash \phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{q-1}}{\mathrm{argmax}}\mathrm{rank}{\mathbf{W}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{O}}\left({\mathcal{I}}_{q-1}\cup \left\{i\right\}\right)\right\}$ |

${i}_{q}\leftarrow \underset{i\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{*}}{\mathrm{argmax}}\mathrm{log}\mathrm{det}{\widehat{\mathbf{W}}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{O}}\left({\mathcal{I}}_{q-1}\cup \left\{i\right\}\right)$ |

${\mathcal{I}}_{q}\leftarrow {\mathcal{I}}_{q-1}\cup \left\{{i}_{q}\right\}$ |

end for |

#### 3.2.2. Greedy Selection with Gradient Approximation (Gradient Greedy Proposed in This Study)

Algorithm 4 Determinant-based gradient greedy algorithm (Gradient greedy) |

Input: $\mathbf{C}\in {\mathbb{R}}^{n\times r},\phantom{\rule{0.166667em}{0ex}}\mathbf{A}\in {\mathbb{R}}^{r\times r},\phantom{\rule{0.166667em}{0ex}}p\in \mathbb{N},\phantom{\rule{0.166667em}{0ex}}\delta >0$ |

Output: Indices of chosen p sensor positions ${\mathcal{I}}_{p}$ |

${\mathcal{I}}_{n}\leftarrow \left\{1,\dots ,n\right\},\phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{0}\leftarrow \varnothing ,\phantom{\rule{0.166667em}{0ex}}$ |

for $q=1,\dots ,p$ do |

${\mathbf{W}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{O}}\left({\mathcal{I}}_{q-1}\right)\leftarrow {\mathbf{W}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{O}}\left({\mathcal{I}}_{q-1}\right)+\mathrm{diag}\left[\delta ,\phantom{\rule{0.166667em}{0ex}}\delta ,\dots ,\phantom{\rule{0.166667em}{0ex}}\delta \right]\in {\mathbb{R}}^{r\times r}$ |

Find $\mathbf{M}$ s.t. $\mathbf{A}\mathbf{M}{\mathbf{A}}^{\top}-\mathbf{M}+{\mathbf{W}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{O}}{\left({\mathcal{I}}_{q-1}\right)}^{-1}=\mathbf{0}$ |

${i}_{q}\leftarrow \underset{i\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{n}\phantom{\rule{0.166667em}{0ex}}\backslash \phantom{\rule{0.166667em}{0ex}}{\mathcal{I}}_{q-1}}{\mathrm{argmax}}{\mathbf{c}}_{i}\mathbf{M}{\mathbf{c}}_{i}^{\top}$ |

${\mathcal{I}}_{q}\leftarrow {\mathcal{I}}_{q-1}\cup \left\{{i}_{q}\right\}$ |

Calculate ${\mathbf{W}}_{\phantom{\rule{-0.166667em}{0ex}}\mathrm{O}}\left({\mathcal{I}}_{q}\right)$ |

end for |

#### 3.3. Expected Computational Complexity

## 4. Comparison and Discussion

`dlyap`function was used for the solutions of the discrete-time Lyapunov equations, such as Equations (9) and (18), adopting the subroutine libraries from the Subroutine Library in Control Theory (SLICOT) [87,88,89,90]. It should also be noted that there is another approach to solving the equation, such as [91]. The MATLAB programs are available through the GitHub repository of the present authors [92].

#### 4.1. Results of the Randomly Generated System

#### 4.2. Results for Data-Driven System Derived from Real-World Experiment

#### 4.3. Results for Data-Driven System Derived from Weather Observation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. CVX Implementation for Equation (12)

`[r,~] = size(A);`

`[n,~] = size(C);`

`cvx_solver(’MOSEK’)`

`cvx_begin sdp`

`variable z(n) nonnegative`

`variable X(r,r) symmetric semidefinite`

`maximize( det_rootn(X) )`

`subject to`

`% matrix inequality`

`A’*X*A-X+C’*(repmat(z,1,r).*C) >= zeros(r,r);`

`% bounds selection variables`

`z <= 1;`

`sum(z) == p;`

`cvx_end`

## References

- Sakiyama, A.; Tanaka, Y.; Tanaka, T.; Ortega, A. Eigendecomposition-Free Sampling Set Selection for Graph Signals. IEEE Trans. Signal Process.
**2019**, 67, 2679–2692. [Google Scholar] [CrossRef] [Green Version] - Nomura, S.; Hara, J.; Tanaka, Y. Dynamic Sensor Placement Based on Graph Sampling Theory. arXiv
**2022**. [Google Scholar] [CrossRef] - Sun, C.; Yu, Y.; Li, V.O.K.; Lam, J.C.K. Multi-Type Sensor Placements in Gaussian Spatial Fields for Environmental Monitoring. Sensors
**2019**, 19, 189. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, B.; Liu, H.; Wang, R. Efficient Sensor Placement for Signal Reconstruction Based on Recursive Methods. IEEE Trans. Signal Process.
**2021**, 69, 1885–1898. [Google Scholar] [CrossRef] - Natarajan, M.; Freund, J.B.; Bodony, D.J. Actuator selection and placement for localized feedback flow control. J. Fluid Mech.
**2016**, 809, 775–792. [Google Scholar] [CrossRef] [Green Version] - Inoue, T.; Ikami, T.; Egami, Y.; Nagai, H.; Naganuma, Y.; Kimura, K.; Matsuda, Y. Data-Driven Optimal Sensor Placement for High-Dimensional System Using Annealing Machine. arXiv
**2022**. [Google Scholar] [CrossRef] - Inoba, R.; Uchida, K.; Iwasaki, Y.; Nagata, T.; Ozawa, Y.; Saito, Y.; Nonomura, T.; Asai, K. Optimization of sparse sensor placement for estimation of wind direction and surface pressure distribution using time-averaged pressure-sensitive paint data on automobile model. J. Wind. Eng. Ind. Aerodyn.
**2022**, 227, 105043. [Google Scholar] [CrossRef] - DeVries, L.; Paley, D.A. Observability-based optimization for flow sensing and control of an underwater vehicle in a uniform flowfield. In Proceedings of the 2013 American Control Conference, Washington, DC, USA, 17–19 June 2013; IEEE: Piscataway, NJ, USA, 2013; pp. 1386–1391. [Google Scholar] [CrossRef] [Green Version]
- Kanda, N.; Nakai, K.; Saito, Y.; Nonomura, T.; Asai, K. Feasibility study on real-time observation of flow velocity field using sparse processing particle image velocimetry. Trans. Jpn. Soc. Aeronaut. Space Sci.
**2021**, 64, 242–245. [Google Scholar] [CrossRef] - Kaneko, S.; Ozawa, Y.; Nakai, K.; Saito, Y.; Nonomura, T.; Asai, K.; Ura, H. Data-Driven Sparse Sampling for Reconstruction of Acoustic-Wave Characteristics Used in Aeroacoustic Beamforming. Appl. Sci.
**2021**, 11, 4216. [Google Scholar] [CrossRef] - Carter, D.W.; De Voogt, F.; Soares, R.; Ganapathisubramani, B. Data-driven sparse reconstruction of flow over a stalled aerofoil using experimental data. Data-Centric Eng.
**2021**, 2, e5. [Google Scholar] [CrossRef] - Tiwari, N.; Uchida, K.; Inoba, R.; Saito, Y.; Asai, K.; Nonomura, T. Simultaneous measurement of pressure and temperature on the same surface by sensitive paints using the sensor selection method. Exp. Fluids
**2022**, 63, 171. [Google Scholar] [CrossRef] - Li, S.; Li, W.; Noack, B.R. Least-order representation of control-oriented flow estimation exemplified for the fluidic pinball. J. Phys. Conf. Ser.
**2022**, 2367, 012024. [Google Scholar] [CrossRef] - Kanda, N.; Abe, C.; Goto, S.; Yamada, K.; Nakai, K.; Saito, Y.; Asai, K.; Nonomura, T. Proof-of-concept Study of Sparse Processing Particle Image Velocimetry for Real Time Flow Observation. Exp. Fluids
**2022**, 63, 143. [Google Scholar] [CrossRef] - Inoue, T.; Ikami, T.; Egami, Y.; Nagai, H.; Naganuma, Y.; Kimura, K.; Matsuda, Y. Data-driven optimal sensor placement for high-dimensional system using annealing machine. Mech. Syst. Signal Process.
**2023**, 188, 109957. [Google Scholar] [CrossRef] - Alonso, A.A.; Frouzakis, C.E.; Kevrekidis, I.G. Optimal sensor placement for state reconstruction of distributed process systems. AIChE J.
**2004**, 50, 1438–1452. [Google Scholar] [CrossRef] - Ren, Y.; Ding, Y.; Liang, F. Adaptive evolutionary Monte Carlo algorithm for optimization with applications to sensor placement problems. Stat. Comput.
**2008**, 18, 375. [Google Scholar] [CrossRef] [Green Version] - Hoseyni, S.M.; Di Maio, F.; Zio, E. Subset simulation for optimal sensors positioning based on value of information. Proc. Inst. Mech. Eng. Part O J. Risk Reliab.
**2022**, 1748006X221118432. [Google Scholar] [CrossRef] - Castro-Triguero, R.; Murugan, S.; Gallego, R.; Friswell, M.I. Robustness of optimal sensor placement under parametric uncertainty. Mech. Syst. Signal Process.
**2013**, 41, 268–287. [Google Scholar] [CrossRef] - Krause, A.; Leskovec, J.; Guestrin, C.; VanBriesen, J.; Faloutsos, C. Efficient sensor placement optimization for securing large water distribution networks. J. Water Resour. Plan. Manag.
**2008**, 134, 516–526. [Google Scholar] [CrossRef] [Green Version] - Lee, E.T.; Eun, H.C. Optimal Sensor Placement in Reduced-Order Models Using Modal Constraint Conditions. Sensors
**2022**, 22, 589. [Google Scholar] [CrossRef] - Bates, R.; Buck, R.; Riccomagno, E.; Wynn, H. Experimental design and observation for large systems. J. R. Stat. Soc. Ser. B Methodol.
**1996**, 58, 77–94. [Google Scholar] [CrossRef] - Tzoumas, V.; Xue, Y.; Pequito, S.; Bogdan, P.; Pappas, G.J. Selecting sensors in biological fractional-order systems. IEEE Trans. Control. Netw. Syst.
**2018**, 5, 709–721. [Google Scholar] [CrossRef] - Yildirim, B.; Chryssostomidis, C.; Karniadakis, G. Efficient sensor placement for ocean measurements using low-dimensional concepts. Ocean Model.
**2009**, 27, 160–173. [Google Scholar] [CrossRef] [Green Version] - Kraft, T.; Mignan, A.; Giardini, D. Optimization of a large-scale microseismic monitoring network in northern Switzerland. Geophys. J. Int.
**2013**, 195, 474–490. [Google Scholar] [CrossRef] [Green Version] - Nagata, T.; Nakai, K.; Yamada, K.; Saito, Y.; Nonomura, T.; Kano, M.; Ito, S.i.; Nagao, H. Seismic Wavefield Reconstruction based on Compressed Sensing using Data-Driven Reduced-Order Model. Geophys. J. Int.
**2022**, 322, 33–50. [Google Scholar] [CrossRef] - Nakai, K.; Nagata, T.; Yamada, K.; Saito, Y.; Nonomura, T.; Kano, M.; Ito, S.i.; Nagao, H. Observation Site Selection for Physical Model Parameter Estimation toward Process-Driven Seismic Wavefield Reconstruction. arXiv
**2022**, arXiv:2206.04273. [Google Scholar] [CrossRef] - Doğançay, K.; Hmam, H. On optimal sensor placement for time-difference-of-arrival localization utilizing uncertainty minimization. In Proceedings of the 2009 17th European Signal Processing Conference, Glasgow, UK, 24–28 August 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 1136–1140. [Google Scholar]
- Yeo, W.J.; Taulu, S.; Kutz, J.N. Efficient magnetometer sensor array selection for signal reconstruction and brain source localization. arXiv
**2022**, arXiv:2205.10925. [Google Scholar] - Brunton, S.L.; Kutz, J.N. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Kutz, J.N.; Brunton, S.L.; Brunton, B.W.; Proctor, J.L. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems; SIAM: Philadelphia, PA, USA, 2016; Volume 149. [Google Scholar]
- Baddoo, P.J.; Herrmann, B.; McKeon, B.J.; Kutz, J.N.; Brunton, S.L. Physics-informed dynamic mode decomposition (piDMD). arXiv
**2021**, arXiv:2112.04307. [Google Scholar] - Scherl, I.; Strom, B.; Shang, J.K.; Williams, O.; Polagye, B.L.; Brunton, S.L. Robust principal component analysis for modal decomposition of corrupt fluid flows. Phys. Rev. Fluids
**2020**, 5, 054401. [Google Scholar] [CrossRef] - Jovanović, M.R.; Schmid, P.J.; Nichols, J.W. Sparsity-promoting dynamic mode decomposition. Phys. Fluids
**2014**, 26, 024103. [Google Scholar] [CrossRef] [Green Version] - Iwasaki, Y.; Nonomura, T.; Nakai, K.; Nagata, T.; Saito, Y.; Asai, K. Evaluation of Optimization Algorithms and Noise Robustness of Sparsity-Promoting Dynamic Mode Decomposition. IEEE Access
**2022**, 10, 80748–80763. [Google Scholar] [CrossRef] - Zhang, X.; Ji, T.; Xie, F.; Zheng, H.; Zheng, Y. Unsteady flow prediction from sparse measurements by compressed sensing reduced order modeling. Comput. Methods Appl. Mech. Eng.
**2022**, 393, 114800. [Google Scholar] [CrossRef] - Berkooz, G.; Holmes, P.; Lumley, J.L. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech.
**1993**, 25, 539–575. [Google Scholar] [CrossRef] - Schmid, P.J. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech.
**2010**, 656, 5–28. [Google Scholar] [CrossRef] [Green Version] - Rowley, C.; Williams, M.; Kevrekidis, I. Dynamic Mode Decomposition and the Koopman Operator: Algorithms and Applications; IPAM, UCLA: Los Angeles, CA, USA, 2014; p. 47. [Google Scholar]
- Manohar, K.; Brunton, B.W.; Kutz, J.N.; Brunton, S.L. Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns. IEEE Control Syst. Mag.
**2018**, 38, 63–86. [Google Scholar] [CrossRef] [Green Version] - Saito, Y.; Nonomura, T.; Yamada, K.; Nakai, K.; Nagata, T.; Asai, K.; Sasaki, Y.; Tsubakino, D. Determinant-based fast greedy sensor selection algorithm. IEEE Access
**2021**, 9, 68535–68551. [Google Scholar] [CrossRef] - Paris, R.; Beneddine, S.; Dandois, J. Robust flow control and optimal sensor placement using deep reinforcement learning. J. Fluid Mech.
**2021**, 913, A25. [Google Scholar] [CrossRef] - Fukami, K.; Maulik, R.; Ramachandra, N.; Fukagata, K.; Taira, K. Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning. Nat. Mach. Intell.
**2021**, 3, 945–951. [Google Scholar] [CrossRef] - Yoshimura, R.; Yakeno, A.; Misaka, T.; Obayashi, S. Application of observability Gramian to targeted observation in WRF data assimilation. Tellus A Dyn. Meteorol. Oceanogr.
**2020**, 72, 1–11. [Google Scholar] [CrossRef] [Green Version] - Misaka, T.; Obayashi, S. Sensitivity Analysis of Unsteady Flow Fields and Impact of Measurement Strategy. Math. Probl. Eng.
**2014**, 2014, 359606. [Google Scholar] [CrossRef] [Green Version] - Mons, V.; Marquet, O. Linear and nonlinear sensor placement strategies for mean-flow reconstruction via data assimilation. J. Fluid Mech.
**2021**, 923, A1. [Google Scholar] [CrossRef] - Fisher, R.A. Theory of Statistical Estimation. Math. Proc. Camb. Philos. Soc.
**1925**, 22, 700–725. [Google Scholar] [CrossRef] [Green Version] - Martin, M.; Perez, J.; Plastino, A. Fisher information and nonlinear dynamics. Phys. A Stat. Mech. Its Appl.
**2001**, 291, 523–532. [Google Scholar] [CrossRef] - Kay, S.M. Fundamentals of Statistical Signal Processing: Estimation Theory; Prentice-Hall, Inc.: Hoboken, NJ, USA, 1993. [Google Scholar]
- Joshi, S.; Boyd, S. Sensor selection via convex optimization. IEEE Trans. Signal Process.
**2009**, 57, 451–462. [Google Scholar] [CrossRef] [Green Version] - Nakai, K.; Yamada, K.; Nagata, T.; Saito, Y.; Nonomura, T. Effect of Objective Function on Data-Driven Greedy Sparse Sensor Optimization. IEEE Access
**2021**, 9, 46731–46743. [Google Scholar] [CrossRef] - Nemhauser, G.L.; Wolsey, L.A.; Fisher, M.L. An analysis of approximations for maximizing submodular set functions. Math. Program.
**1978**, 14, 265–294. [Google Scholar] [CrossRef] - Lovász, L. Submodular functions and convexity. In Mathematical Programming the State of the Art; Springer: Berlin/Heidelberg, Germany, 1983; pp. 235–257. [Google Scholar] [CrossRef]
- Krause, A.; Golovin, D. Submodular Function Maximization. In Tractability; Cambridge University Press: Cambridge, UK, 2014; pp. 71–104. [Google Scholar] [CrossRef] [Green Version]
- Mirzasoleiman, B.; Badanidiyuru, A.; Karbasi, A.; Vondrák, J.; Krause, A. Lazier than lazy greedy. In Proceedings of the AAAI Conference on Artificial Intelligence, Austin, TX, USA, 25–30 January 2015; Volume 29. [Google Scholar]
- Hashemi, A.; Ghasemi, M.; Vikalo, H.; Topcu, U. Randomized greedy sensor selection: Leveraging weak submodularity. IEEE Trans. Autom. Control
**2020**, 66, 199–212. [Google Scholar] [CrossRef] [Green Version] - Krause, A.; Singh, A.; Guestrin, C. Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies. J. Mach. Learn. Res.
**2008**, 9, 235–284. [Google Scholar] - Chepuri, S.P.; Leus, G. Sparse sensing for distributed detection. IEEE Trans. Signal Process.
**2015**, 64, 1446–1460. [Google Scholar] [CrossRef] - Clark, E.; Askham, T.; Brunton, S.L.; Kutz, J.N. Greedy sensor placement with cost constraints. IEEE Sens. J.
**2018**, 19, 2642–2656. [Google Scholar] [CrossRef] [Green Version] - Yamada, K.; Saito, Y.; Nankai, K.; Nonomura, T.; Asai, K.; Tsubakino, D. Fast greedy optimization of sensor selection in measurement with correlated noise. Mech. Syst. Signal Process.
**2021**, 158, 107619. [Google Scholar] [CrossRef] - Liu, S.; Chepuri, S.P.; Fardad, M.; Maşazade, E.; Leus, G.; Varshney, P.K. Sensor selection for estimation with correlated measurement noise. IEEE Trans. Signal Process.
**2016**, 64, 3509–3522. [Google Scholar] [CrossRef] [Green Version] - Dhingra, N.K.; Jovanović, M.R.; Luo, Z.Q. An ADMM algorithm for optimal sensor and actuator selection. In Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, CA, USA, 15–17 December 2014; IEEE: Piscataway, NJ, USA, 2014; pp. 4039–4044. [Google Scholar] [CrossRef]
- Nagata, T.; Nonomura, T.; Nakai, K.; Yamada, K.; Saito, Y.; Ono, S. Data-driven sparse sensor selection based on A-optimal design of experiment with ADMM. IEEE Sens. J.
**2021**, 21, 15248–15257. [Google Scholar] [CrossRef] - Nagata, T.; Yamada, K.; Nonomura, T.; Nakai, K.; Saito, Y.; Ono, S. Data-driven sensor selection method based on proximal optimization for high-dimensional data with correlated measurement noise. IEEE Trans. Signal Process.
**2022**, 70, 5251–5264. [Google Scholar] [CrossRef] - Wouwer, A.V.; Point, N.; Porteman, S.; Remy, M. An approach to the selection of optimal sensor locations in distributed parameter systems. J. Process Control
**2000**, 10, 291–300. [Google Scholar] [CrossRef] - Summers, T.H.; Cortesi, F.L.; Lygeros, J. On Submodularity and Controllability in Complex Dynamical Networks. IEEE Trans. Control. Netw. Syst.
**2016**, 3, 91–101. [Google Scholar] [CrossRef] [Green Version] - Summers, T.; Shames, I. Convex relaxations and Gramian rank constraints for sensor and actuator selection in networks. In Proceedings of the 2016 IEEE International Symposium on Intelligent Control (ISIC), Buenos Aires, Argentina, 19–22 September 2016; IEEE: Piscataway, NJ, USA, 2016. [Google Scholar] [CrossRef]
- DeVries, L.; Majumdar, S.J.; Paley, D.A. Observability-based optimization of coordinated sampling trajectories for recursive estimation of a strong, spatially varying flowfield. J. Intell. Robot. Syst.
**2013**, 70, 527–544. [Google Scholar] [CrossRef] - Montanari, A.N.; Freitas, L.; Proverbio, D.; Gonçalves, J. Functional observability and subspace reconstruction in nonlinear systems. Phys. Rev. Res.
**2022**, 4, 043195. [Google Scholar] [CrossRef] - Shamaiah, M.; Banerjee, S.; Vikalo, H. Greedy sensor selection: Leveraging submodularity. In Proceedings of the 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA, 15–17 December 2010; IEEE: Piscataway, NJ, USA, 2010; pp. 2572–2577. [Google Scholar] [CrossRef]
- Zhang, X.; Tian, Y.; Cheng, R.; Jin, Y. An Efficient Approach to Nondominated Sorting for Evolutionary Multiobjective Optimization. IEEE Trans. Evol. Comput.
**2015**, 19, 201–213. [Google Scholar] [CrossRef] [Green Version] - Zhang, H.; Ayoub, R.; Sundaram, S. Sensor selection for Kalman filtering of linear dynamical systems: Complexity, limitations and greedy algorithms. Automatica
**2017**, 78, 202–210. [Google Scholar] [CrossRef] - Ye, L.; Roy, S.; Sundaram, S. On the complexity and approximability of optimal sensor selection for Kalman filtering. In Proceedings of the 2018 Annual American Control Conference (ACC), Milwaukee, WI, USA, 27–29 June 2018; IEEE: Piscataway, NJ, USA, 2018; pp. 5049–5054. [Google Scholar] [CrossRef] [Green Version]
- Tzoumas, V.; Jadbabaie, A.; Pappas, G.J. Sensor placement for optimal Kalman filtering: Fundamental limits, submodularity, and algorithms. In Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA, 6–8 July 2016; IEEE: Piscataway, NJ, USA, 2016; pp. 191–196. [Google Scholar]
- Manohar, K.; Kutz, J.N.; Brunton, S.L. Optimal Sensor and Actuator Selection Using Balanced Model Reduction. IEEE Trans. Autom. Control
**2021**, 67, 2108–2115. [Google Scholar] [CrossRef] - Clark, E.; Kutz, J.N.; Brunton, S.L. Sensor selection with cost constraints for dynamically relevant bases. IEEE Sens. J.
**2020**, 20, 11674–11687. [Google Scholar] [CrossRef] - Zhou, K.; Salomon, G.; Wu, E. Balanced realization and model reduction for unstable systems. Int. J. Robust Nonlinear Control
**1999**, 9, 183–198. [Google Scholar] [CrossRef] - Nankai, K.; Ozawa, Y.; Nonomura, T.; Asai, K. Linear Reduced-order Model Based on PIV Data of Flow Field around Airfoil. Trans. Jpn. Soc. Aeronaut. Space Sci.
**2019**, 62, 227–235. [Google Scholar] [CrossRef] [Green Version] - Drmač, Z.; Saibaba, A.K. The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces. SIAM J. Matrix Anal. Appl.
**2018**, 39, 1152–1180. [Google Scholar] [CrossRef] [Green Version] - Zare, A.; Mohammadi, H.; Dhingra, N.K.; Georgiou, T.T.; Jovanovic, M.R. Proximal algorithms for large-scale statistical modeling and sensor/actuator selection. IEEE Trans. Autom. Control
**2019**, 65, 3441–3456. [Google Scholar] [CrossRef] [Green Version] - Boyd, S.; Boyd, S.P.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Gower, R.; Koralev, D.; Lieder, F.; Richtarik, P. RSN: Randomized Subspace Newton. In Advances in Neural Information Processing Systems 32; Wallach, H., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E., Garnett, R., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2019; pp. 616–625. [Google Scholar]
- Nonomura, T.; Ono, S.; Nakai, K.; Saito, Y. Randomized Subspace Newton Convex Method Applied to Data-Driven Sensor Selection Problem. IEEE Signal Processing Lett.
**2021**, 28, 284–288. [Google Scholar] [CrossRef] - Feige, U.; Mirrokni, V.S.; Vondrák, J. Maximizing non-monotone submodular functions. SIAM J. Comput.
**2011**, 40, 1133–1153. [Google Scholar] [CrossRef] - CVX Research, I. CVX: Matlab Software for Disciplined Convex Programming, (Version 2.0); CVX Research, Inc.: Austin, TX, USA, 2012; Available online: http://cvxr.com/cvx (accessed on 29 April 2023).
- Grant, M.; Boyd, S. Graph implementations for nonsmooth convex programs. In Recent Advances in Learning and Control; Blondel, V., Boyd, S., Kimura, H., Eds.; Lecture Notes in Control and Information Sciences; Springer: Berlin/Heidelberg, Germany, 2008; pp. 95–110. Available online: http://stanford.edu/~boyd/graph_dcp.html (accessed on 29 April 2023).
- Benner, P.; Mehrmann, V.; Sima, V.; Van Huffel, S.; Varga, A. SLICOT—A Subroutine Library in Systems and Control Theory: Bucharest 1, Romania. 1999. Available online: http://www.slicot.org/ (accessed on 29 April 2023).
- Van Huffel, S.; Sima, V.; Varga, A.; Hammarling, S.; Delebecque, F. High-performance numerical software for control. IEEE Control Syst. Mag.
**2004**, 24, 60–76. [Google Scholar] - Barraud, A.Y. A numerical algorithm to solve AXA - X = Q. In Proceedings of the 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications, New Orleans, LA, USA, 7–9 December 1977; pp. 420–423. [Google Scholar] [CrossRef]
- Hammarling, S.J. Numerical solution of the stable, non-negative definite lyapunov equation lyapunov equation. IMA J. Numer. Anal.
**1982**, 2, 303–323. [Google Scholar] [CrossRef] [Green Version] - Kitagawa, G. An algorithm for solving the matrix equation X = FXF
^{⊤}+ S. Int. J. Control**1977**, 25, 745–753. [Google Scholar] [CrossRef] - Yamada, K. Sensor Placement Based on Observability Gramian. 2023. Available online: https://github.com/Aerodynamics-Lab/Sensor-Placement-Based-on-Observability-Gramian (accessed on 29 April 2023).
- Strassen, V. Gaussian elimination is not optimal. Numer. Math.
**1969**, 13, 354–356. [Google Scholar] [CrossRef] - Luo, Z.Q.; Ma, W.K.; So, A.; Ye, Y.; Zhang, S. Semidefinite Relaxation of Quadratic Optimization Problems. IEEE Signal Process. Mag.
**2010**, 27, 20–34. [Google Scholar] [CrossRef] - Nonomura, T.; Nankai, K.; Iwasaki, Y.; Komuro, A.; Asai, K. Quantitative evaluation of predictability of linear reduced-order model based on particle-image-velocimetry data of separated flow field around airfoil. Exp. Fluids
**2021**, 62, 112. [Google Scholar] [CrossRef] - Nonomura, T.; Nankai, K.; Iwasaki, Y.; Komuro, A.; Asai, K. Airfoil PIV Data for Linear ROM. 2021. Available online: https://github.com/Aerodynamics-Lab/Airfoil-PIV-data-for-linear-ROM (accessed on 29 April 2023).
- Saito, Y.; Yamada, K.; Kanda, N.; Nakai, K.; Nagata, T.; Nonomura, T.; Asai, K. Data-Driven Determinant-Based Greedy Under/Oversampling Vector Sensor Placement. CMES-Comput. Model. Eng. Sci.
**2021**, 129, 1–30. [Google Scholar] [CrossRef] - Yamada, K.; Saito, Y.; Nonomura, T.; Asai, K. Greedy Sensor Selection for Weighted Linear Least Squares Estimation under Correlated Noise. IEEE Access
**2022**, 10, 79356–79364. [Google Scholar] [CrossRef] - Reynolds, R.W.; Rayner, N.A.; Smith, T.M.; Stokes, D.C.; Wang, W. An improved in situ and satellite SST analysis for climate. J. Clim.
**2002**, 15, 1609–1625. [Google Scholar] [CrossRef] - NOAA. NOAA Optimal Interpolation (OI) Sea Surface Temperature (SST) V2; NOAA: Boulder, CO, USA, 2023. Available online: https://www.esrl.noaa.gov/psd/data/gridded/data.noaa.oisst.v2.html (accessed on 29 April 2023).

**Figure 1.**Brief description of this manuscript. (

**Left**) The representative data point is revealed from “rich” measurement data by use of a data-driven model and optimization procedure. (

**Right**) A data-driven method constructs linear reduced-order models before the optimization is conducted by approximate algorithms, including our novel methods, denoted by bold types.

**Figure 2.**Schematic of the linear approximation of the gradient greedy algorithm for a convex function $f\left(\mathbf{s}\right)$.

**Figure 3.**Computation times and optimization results for randomly generated systems, for (

**a**,

**b**) varying the number of state variables r, i.e., the size of $\mathbf{A}$ matrix (average of 100 times trial, $n=1024,\phantom{\rule{3.33333pt}{0ex}}p=10$); (

**c**,

**d**) varying the number of sensor candidates n, i.e., the rows of $\mathbf{C}$ matrix (average of 20 times trial, $p=10,\phantom{\rule{3.33333pt}{0ex}}r=10$); (

**e**,

**f**) varying the number of sensors selected p, (average of 100 times trial, $r=10,\phantom{\rule{3.33333pt}{0ex}}n=1024$).

**Figure 5.**Optimization results for a system based on the experimental data of fluid dynamics. (

**a**) Relative determinant values for various r $(p=20,n=9353)$. (

**b**) Relative determinant values for various p $(r=10,n=9353)$.

**Figure 6.**Mean value distribution of the global sea surface temperature (from 31 December 1989 to 29 January 2023).

**Figure 7.**Optimization results for a system based on the experimental data. (

**a**) Relative determinant values for various r (p = 10, n = 44,219). (

**b**) Relative determinant values for various p (r = 10, n = 44,219).

**Table 1.**Selection algorithms for each relaxed problem and the expected arithmetic complexity order based on the basic matrix operations.

**Several approaches**are proposed in the presented paper.

Problem | Algorithm | Expected Complexity |
---|---|---|

Linear relaxation SDP [67] | Path-following method | $\left(\mathcal{O}\left({n}^{4}\right)+\mathcal{O}\left({n}^{2}{r}^{2}\right)+\mathcal{O}\left(n{r}^{3}\right)\right)$/iter. |

Approximate convex relaxation | BRS-Newton [Algorithm 2] | $\left(\mathcal{O}\left({n}^{3}\right)+\mathcal{O}\left({n}^{2}{r}^{2}\right)+\mathcal{O}\left(n{r}^{3}\right)\right)$/iter. |

Greedy [66] | Pure greedy [Algorithm 3] | $\mathcal{O}\left(pn{r}^{3}\right)$ |

Greedy | Gradient greedy [Algorithm 4] | $\mathcal{O}\left(pn{r}^{2}\right)+\mathcal{O}\left(p{r}^{3}\right)$ |

**Table 2.**Practical orders of the computation times of selection methods, investigated with respect to each parameter individually.

Selection Method | r | n | p |
---|---|---|---|

SDP | ${r}^{\left[4\right]}$ | n | ${p}^{0}$ |

Approx. conv. [Algorithm 2] | ${r}^{\left[3\right]}$ | ${n}^{\left[2\right]}$ | ${p}^{\left[1\right]}$ |

Pure greedy [Algorithm 3] | ${r}^{3}$ | n | ${p}^{1}$ |

Gradient greedy [Algorithm 4] | N/A | n | ${p}^{1}$ |

**Table 3.**Iteration numbers of the convex relaxation approaches, investigated with respect to each parameter individually. Averages are rounded to integers.

(a) Various r for Randomly Generated Systems, Average of 100 Trials | |||||||

r | 10 | 20 | 30 | 40 | 50 | 60 | |

SDP | 18 | 13 | 11 | 12 | 12 | 12 | |

Approx. conv. | 203 | 293 | 348 | 373 | 380 | 388 | |

(b) Various n for randomly generated systems, average of 20 trials | |||||||

n | ${2}^{10}$ | ${2}^{11}$ | ${2}^{12}$ | ${2}^{13}$ | ${2}^{14}$ | ${2}^{15}$ | ${2}^{16}$ |

SDP | 18 | 21 | 23 | 27 | 28 | 31 | 33 |

Approx. conv. | 209 | 167 | 147 | 141 | 139 | 139 | 141 |

(c) Various p for randomly generated systems, average of 100 trials | |||||||

p | 1 | 2 | 4 | 8 | 10 | 20 | 40 |

SDP | 14 | 13 | 15 | 17 | 18 | 18 | 17 |

Approx. conv. | 171 | 168 | 172 | 198 | 206 | 273 | 411 |

(a) Wind Tunnel Experiment | |

Airfoil | NACA0015 (Chord length 100 mm) |

Wind tunnel | Recirculating low-speed wind tunnel |

Flow speed | 10 m/s |

Angle of attack | ${18}^{\xb0}$ |

(b) Acquisition condition of velocity distribution | |

Sampling rate | 5000 Hz |

Spatial sample | 9353 points |

Snapshot sample | 2000 snapshots × 5 sets |

Data type | Weekly means from 31 December 1989 to 29 January 2023 |

Grid scale | 1.0 degree lat × 1.0-degree long grid (180 × 360) |

Spatial sample | 44,219 |

Temporal sample | 1727 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yamada, K.; Sasaki, Y.; Nagata, T.; Nakai, K.; Tsubakino, D.; Nonomura, T.
Efficient Sensor Node Selection for Observability Gramian Optimization. *Sensors* **2023**, *23*, 5961.
https://doi.org/10.3390/s23135961

**AMA Style**

Yamada K, Sasaki Y, Nagata T, Nakai K, Tsubakino D, Nonomura T.
Efficient Sensor Node Selection for Observability Gramian Optimization. *Sensors*. 2023; 23(13):5961.
https://doi.org/10.3390/s23135961

**Chicago/Turabian Style**

Yamada, Keigo, Yasuo Sasaki, Takayuki Nagata, Kumi Nakai, Daisuke Tsubakino, and Taku Nonomura.
2023. "Efficient Sensor Node Selection for Observability Gramian Optimization" *Sensors* 23, no. 13: 5961.
https://doi.org/10.3390/s23135961