# Sensor placement for optimal Kalman filtering: Fundamental limits, submodularity, and algorithms

@article{Tzoumas2016SensorPF, title={Sensor placement for optimal Kalman filtering: Fundamental limits, submodularity, and algorithms}, author={Vasileios Tzoumas and Ali Jadbabaie and George J. Pappas}, journal={2016 American Control Conference (ACC)}, year={2016}, pages={191-196} }

In this paper, we focus on sensor placement in linear dynamic estimation, where the objective is to place a small number of sensors in a system of interdependent states so to design an estimator with a desired estimation performance. In particular, we consider a linear time-variant system that is corrupted with process and measurement noise, and study how the selection of its sensors affects the estimation error of the corresponding Kalman filter over a finite observation interval. Our…

## 88 Citations

On the Complexity and Approximability of Optimal Sensor Selection for Kalman Filtering

- Computer Science, Mathematics2018 Annual American Control Conference (ACC)
- 2018

This paper shows that greedy algorithms can perform arbitrarily poorly for the problem of design-time sensor selection for Kalman filtering, and shows the stronger result that there is no constant-factor (polynomial-time) approximation algorithm for this problem.

Sensor selection for Kalman filtering of linear dynamical systems: Complexity, limitations and greedy algorithms

- Computer ScienceAutom.
- 2017

Resilient Sensor Placement for Kalman Filtering in Networked Systems: Complexity and Algorithms

- Computer Science, MathematicsIEEE Transactions on Control of Network Systems
- 2020

Given a linear dynamical system affected by noise, the resilient sensor placement problem is to find a sensor placement strategy to minimize the trace of the steady-state error covariance of the Kalman filter corresponding to the sensors that survive the attack.

Probabilistic Performance Bounds for Randomized Sensor Selection in Kalman Filtering

- Mathematics, Computer Science2021 American Control Conference (ACC)
- 2021

This work considers the problem of randomly choosing the sensors of a linear time-invariant dynamical system subject to process and measurement noise and derives probabilistic bounds on the steady-state estimation error covariance in the semi-definite sense for an arbitrary sampling distribution.

Optimal Sensor Precision for Multirate Sensing for Bounded Estimation Error

- Mathematics, Computer ScienceIEEE Transactions on Aerospace and Electronic Systems
- 2022

The presented theory is applied to realistic flight mechanics and astrodynamics problems to highlight its engineering value and determine redundant sensing architectures for linear time invariant systems, accurately estimate states with low-cost sensors, and optimally schedule sensors forlinear time-varying systems.

On the Complexity and Approximability of Optimal Sensor Selection and Attack for Kalman Filtering

- Computer ScienceIEEE Transactions on Automatic Control
- 2021

This work provides a specific example showing that greedy algorithms can perform arbitrarily poorly for the problem of design-time sensor selection for Kalman filtering, and shows that there is no polynomial-time constant-factor approximation algorithm for this problem.

Approximate Supermodularity of Kalman Filter Sensor Selection

- Computer ScienceIEEE Transactions on Automatic Control
- 2021

This article considers the problem of selecting sensors in a large-scale system to minimize the error in estimating its states, more specifically, the state estimation mean-square error (MSE) and…

Sensor Selection forKalmanFiltering of LinearDynamical Systems : Complexity , Limitations andGreedyAlgorithms ⋆

- Computer Science
- 2016

It is shown that certain typical objective functions are not submodular or supermodular in general, and while this makes it difficult to evaluate the performance of greedy algorithms for sensor selection, it is shown via simulations that these greedy algorithms perform well in practice.

Supermodular mean squared error minimization for sensor scheduling in optimal Kalman Filtering

- Business2017 American Control Conference (ACC)
- 2017

Empirical results confirm that random M-matrices lead to supermodular problems, while this is not the case for generic prior information matrices, and provide a practical application of the findings to an energy-constrained multi robot localization problem.

The mean square error in Kalman filtering sensor selection is approximately supermodular

- Computer Science2017 IEEE 56th Annual Conference on Decision and Control (CDC)
- 2017

This work leverages the concept of approximate supermodularity to derive near-optimality certificates for greedy solutions of this problem in the context of Kalman filtering, and shows that in typical application scenarios, these certificates approach the typical 1/e guarantee.

## References

SHOWING 1-10 OF 103 REFERENCES

Submodularity and greedy algorithms in sensor scheduling for linear dynamical systems

- Computer ScienceAutom.
- 2015

Geometric methods for optimal sensor design

- Computer Science, MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2016

This paper shows how to obtain optimal sensors for the Kalman filter by reducing the optimal sensor problem to an optimization problem on a Grassmannian manifold and proving that the function to be minimized is a Morse function with a unique minimum.

On efficient sensor scheduling for linear dynamical systems

- Computer ScienceProceedings of the 2010 American Control Conference
- 2010

Sensor Selection via Convex Optimization

- Computer ScienceIEEE Transactions on Signal Processing
- 2009

This paper describes a heuristic, based on convex optimization, that gives a subset selection as well as a bound on the best performance that can be achieved by any selection of k sensor measurements.

A Relaxation Approach to Dynamic Sensor Selection in Large-Scale Wireless Networks

- Computer Science2008 The 28th International Conference on Distributed Computing Systems Workshops
- 2008

The proposed dynamic sensor selection strategy is compared empirically to other dynamic and static sensor selection strategies with respect to state estimation performance of a convection-dispersion field arising from the problem of surface-based monitoring of CO2 sequestration sites.

On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage

- MathematicsAutom.
- 2006

Information acquisition with sensing robots: Algorithms and error bounds

- Computer Science2014 IEEE International Conference on Robotics and Automation (ICRA)
- 2014

This paper presents a non-greedy algorithm with suboptimality guarantees, which relies on concavity instead of submodularity and takes the sensor dynamics into account and can be used to generate adaptive policies for mobile sensors with non-linear sensing models.

Minimal Actuator Placement With Bounds on Control Effort

- Computer Science, MathematicsIEEE Transactions on Control of Network Systems
- 2016

An efficient algorithm is provided which approximates up to a multiplicative factor of O(log n), with n being the network size, any optimal actuator set that meets the same energy criteria; this is the best approximation factor one can achieve in polynomial time in the worst case.

Greedy sensor selection: Leveraging submodularity

- Computer Science49th IEEE Conference on Decision and Control (CDC)
- 2010

This work casts the sensor selection problem as the maximization of a submodular function over uniform matroids, and demonstrates that a greedy sensor selection algorithm achieves performance within (1 − 1 over e ) of the optimal solution.

Sensor Selection for Minimizing Worst-Case Prediction Error

- Computer Science2008 International Conference on Information Processing in Sensor Networks (ipsn 2008)
- 2008

The problem of choosing the "best" subset of k sensors to sample from among a sensor deployment of n > k sensors is studied, and it is shown that for any aggregate function satisfying certain concavity, symmetry and monotonicity conditions, the sensor selection problem can be modeled as a k-median clustering problem, and solved using efficient approximation algorithms designed for k- Medial clustering.