A distributed Kalman filter with symbolic zonotopes and unique symbols provider for robust state estimation in CPS

Abstract

Robust state estimation is addressed in a noisy environment and within a distributed and networked architecture. Both bounded disturbances and random noises are considered. A Distributed Zonotopic and Gaussian Kalman Filter (DZG-KF) is proposed where each network node implements a local state estimator using symbolic Zonotopes and Gaussian noise Mergers (s-ZGM), a class of Set-membership and Probabilistic Mergers (SPM). Each network node communicates its own state information only to its neighbours. The proposed system includes a dedicated service called Unique Symbols Provider (USP) giving unique identifiers. It also includes Matrices with Labelled Columns (MLC) featuring column-wise sparsity, and symbolic zonotopes (s-zonotopes). This significantly enhances the propagation of uncertainties and preserves global dependencies that would otherwise be lost (or impeded) by the peer-to-peer communication through the network. A number of other network-related constraints can be managed within this framework. Numerical simulations show significant improvements compared to a non-symbolic approach.

Keywords:

• Distributed uncertain systems
• unique identifiers and sparse structures
• set-based state estimation
• Kalman filter
• bounded-error intervals and random noises
• symbolic zonotopes

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Concretely, following the solution based on (32(32) $\begin{array}{rl}{\mathbf{v}}_i\in {\mathbb{V}}_i& ={\overline{\mathcal{M}}}_{\mathbf{s}}(0,E_{z,i}^{|!},Q_{v_g,i}),\end{array}$(32) )–(33(33) $\begin{array}{rl}{\mathbf{w}}_i\in {\mathbb{W}}_i& ={\overline{\mathcal{M}}}_{\mathbf{s}}(0,F_{z,i}^{|!},Q_{w_g,i}).\end{array}$(33) ) developed in the paragraph 8.1, the knowledge of ${\mathbb{V}}_i$ and ${\mathbb{W}}_i$ only requires the knowledge of time-varying matrix pairs $(E_{z,i},Q_{v_g,i})$ and $(F_{z,i},Q_{w_g,i})$, respectively.

2 The size $J(\cdot )$ of an s-ZGM is precisely formalised in the Definition 8.1.

3 Either globally or locally.

4 For all the CPS agents.

5 $I\mathrm{\setminus }J=\{x\in I|x\notin J\}$ is the relative complement of J in I.

6 MAC: Medium Access Control.

7 That is, under $I_j=!$ in (36(36) $\begin{array}{rl}& \left\{\begin{array}{l}{\mathbf{x}}_i\in {\mathbb{X}}_i={\overline{\mathcal{M}}}_{\mathbf{s}}(c_i,R_i^{|I_i},Q_i)\\ {\mathbf{x}}_j\in {\mathbb{X}}_j={\overline{\mathcal{M}}}_{\mathbf{s}}(c_j,R_j^{|I_j},Q_j),j\in N_i\end{array}\right.\\ & \Rightarrow {\mathbf{x}}_{i,+}\in {\mathbb{X}}_{i,+}\left(G_i\right)={\overline{\mathcal{M}}}_{\mathbf{s}}(c_{i,+},R_{i,+}^{|I_{i,+}},Q_{i,+}),\end{array}$(36) )–(40(40) $\begin{array}{rl}{\breve{A}}_i& =A_{ii}-G_i{C}_i,{\breve{B}}_i=B_i-G_i{D}_i.\end{array}$(40) ).

8 As formalised in definition 8.1.

9 Relative wrt steady position.

Additional information

Funding

This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the ‘Investments for the future’ Programme IdEx Bordeaux - SysNum (ANR-10-IDEX-03-02).