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Joint State and Sparse Input Estimation in Linear Dynamical Systems

Rupam Kalyan Chakraborty, Geethu Joseph, Chandra R. Murthy

TL;DR

This work presents a Bayesian approach that exploits the input sparsity to significantly improve estimation accuracy and outperform the state-of-the-art methods in terms of accuracy and time/memory complexities, especially in the low-dimensional measurement regime.

Abstract

Sparsity constraints on the control inputs of a linear dynamical system naturally arise in several practical applications such as networked control, computer vision, seismic signal processing, and cyber-physical systems. In this work, we consider the problem of jointly estimating the states and sparse inputs of such systems from low-dimensional (compressive) measurements. Due to the low-dimensional measurements, conventional Kalman filtering and smoothing algorithms fail to accurately estimate the states and inputs. We present a Bayesian approach that exploits the input sparsity to significantly improve estimation accuracy. Sparsity in the input estimates is promoted by using different prior distributions on the input. We investigate two main approaches: regularizer-based MAP, and {Bayesian learning-based estimation}. We also extend the approaches to handle control inputs with common support and analyze the time and memory complexities of the presented algorithms. Finally, using numerical simulations, we show that our algorithms outperform the state-of-the-art methods in terms of accuracy and time/memory complexities, especially in the low-dimensional measurement regime.

Joint State and Sparse Input Estimation in Linear Dynamical Systems

TL;DR

This work presents a Bayesian approach that exploits the input sparsity to significantly improve estimation accuracy and outperform the state-of-the-art methods in terms of accuracy and time/memory complexities, especially in the low-dimensional measurement regime.

Abstract

Sparsity constraints on the control inputs of a linear dynamical system naturally arise in several practical applications such as networked control, computer vision, seismic signal processing, and cyber-physical systems. In this work, we consider the problem of jointly estimating the states and sparse inputs of such systems from low-dimensional (compressive) measurements. Due to the low-dimensional measurements, conventional Kalman filtering and smoothing algorithms fail to accurately estimate the states and inputs. We present a Bayesian approach that exploits the input sparsity to significantly improve estimation accuracy. Sparsity in the input estimates is promoted by using different prior distributions on the input. We investigate two main approaches: regularizer-based MAP, and {Bayesian learning-based estimation}. We also extend the approaches to handle control inputs with common support and analyze the time and memory complexities of the presented algorithms. Finally, using numerical simulations, we show that our algorithms outperform the state-of-the-art methods in terms of accuracy and time/memory complexities, especially in the low-dimensional measurement regime.
Paper Structure (33 sections, 1 theorem, 121 equations, 5 figures, 1 table, 7 algorithms)

This paper contains 33 sections, 1 theorem, 121 equations, 5 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Estimation of sparse initial state
  4. Estimation of sparse states
  5. Joint estimation of states and sparse inputs
  6. Contributions
  7. Preliminaries: MAP Estimation of States and (Non-sparse) Inputs
  8. Regularized Robust Kalman Smoothing
  9. $\ell_1$-regularized Robust Kalman Smoothing
  10. Reweighed $\ell_2$-regularized Robust Kalman Smoothing
  11. Extension to Jointly Sparse Inputs
  12. Bayesian Robust Kalman Smoothing
  13. Sparse Bayesian Learning-based Robust Kalman Smoothing
  14. Variational Bayesian Robust Kalman Smoothing
  15. Comparison of Regularized RKS and Bayesian-RKS
  16. ...and 18 more sections

Key Result

Lemma 1

If $p\left(\boldsymbol{\beta} \mid \boldsymbol{\alpha}\right) = \mathcal{N}\left(\boldsymbol{A}\boldsymbol{\alpha}+\boldsymbol{c}, \boldsymbol{\Sigma}_{\boldsymbol{\beta}\mid \boldsymbol{\alpha}}\right)$ and $p\left(\boldsymbol{\alpha}\right) = \mathcal{N}\left(\boldsymbol{\mu}_{\boldsymbol{\alpha}}

Figures (5)

  • Figure 1: Performance comparison of our sparse recovery algorithms and RKS as a function of measurement dimension $p$ when the support of control inputs are time-varying support with $n=30$, $m=100$, $K=30$, $s=5$, and SNR $=20$ dB.
  • Figure 2: Performance comparison of our sparse recovery algorithms and RKS as a function of measurement dimension $p$ when the control inputs are jointly sparse with $n=30$, $m=100$, $K=30$, $s=5$, and SNR $=20$ dB.
  • Figure 3: Phase transition diagram for our sparse recovery algorithms with $n=30$, $m=100$, $K=30$, and SNR $=20$ dB.
  • Figure 4: Time domain tracking performance of SBL-RKS with $n=30$, $m=100$, $p=20$, $K=100$, $s=5$, and SNR $=20$ dB.
  • Figure 5: NMSE in input estimation of SBL-RKS and RKS as a function of measurement dimension when the control inputs have time-varying support, with $m=100$, $K=30$, and $s=5$.

Theorems & Definitions (3)

  • Lemma 1
  • proof
  • proof