A Distributed Observer for a Time-Invariant Linear System

Wang, L.; Morse, A. Stephen

doi:10.1109/tac.2017.2768668

Cited by 145 publications

(102 citation statements)

References 16 publications

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“…To proceed, we introduce orthogonal transformation T i 1 that yields observable decomposition for the pairs ( A , C i ), where

\begin{align} A_{i} & = T_{i 1}^{T} A T_{i 1} = ([], \begin{matrix} center center \\ array A_{i 1} & array 0 \\ array A_{i 2} & array A_{i 3} \end{matrix}), C_{i} T_{i 1} = ([], \begin{matrix} center center \\ array C_{i 1} & array 0 \end{matrix}), \\ T_{i 1} B & = B_{i}, C_{i 1} \in {normalR}^{p_{i} prefix\times r_{i}}, \end{align}

where

T_{i 1} = [\begin{array}{cc} T_{i 11} & T_{i 12} \end{array}]

, and T i 11 consists of the first r i columns of T i 1 . Then the unobservable subspace is given by

i m (T_{i 12}) \in \ker O_{C_{i}}

, where

O_{C_{i}} = {[C_{i}^{T}, {(A C_{i})}^{T}, \dots, {(C_{i} A^{n - 1})}^{T}]}^{T}

. Note that

i m

…”

Section: Preliminaries and Problem Formulationmentioning

confidence: 99%

“…The proposed local observer i under the distributed framework is in a form of:

\begin{align} z_{i} ((), k + 1) & = G_{i} z_{i} ((), k) + H_{i} u ((), k) + L_{i} y_{i} ((), k) + M_{i} true \sum_{j = 1}^{h} ((), {\overset{x}{true}}_{j} ((), k) prefix- {\overset{x}{true}}_{i} ((), k)), \\ T_{i} {\overset{x}{true}}_{i} ((), k + 1) & = z_{i} ((), k + 1), \end{align}

where it utilizes available local output

y_{i} (k) = C_{i} x (k) + D_{i} u (k) \in R^{p_{i}}

, received information from neighbors

{\hat{x}}_{j} (k) \in R^{n}

, and the coefficient matrix M i to estimate

x (k + 1) \in R^{n}

. The distributed observer structure has been utilized in several papers such as References . However, in these references, the observer designs along with the convergence conditions require the explicit system model and thus cannot be directly extended to data‐driven observer design.…”

Section: Preliminaries and Problem Formulationmentioning

confidence: 99%

“…The following lemma proposed in Reference 6 gives a solution to the Luenberger equations specified in (5).…”

Section: Preliminaries Of Luenberger Observer Designmentioning

confidence: 99%

“…Traditional distributed state estimation methods generally assume that the process model is known and the processes can be decomposed using process knowledge (see References and the references therein). However, accurate process knowledge is not always available.…”

Section: Introductionmentioning

confidence: 99%

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Distributed data‐driven observer for linear time invariant systems

Alipouri

Zhao

Huang

2020

Adaptive Control & Signal

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Summary This paper is concerned with distributed data‐driven observer design problem. The existing data‐driven observers rely on a common assumption that all the information about the system, and the calculations based upon this information are centralized. Therefore the resulting algorithms cannot be applied to the distributed systems in which each local observer receives only a part of the output signal. On the other hand, traditional model‐based distributed state estimation methods generally assume that the processes are decomposed according to the known process models, while in data‐driven approaches there is no such information available. The main goal of this paper is to extend the centralized data‐driven observer design approach to the distributed framework. The stability of the proposed data‐driven distributed observer is also proved analytically. A quadruple‐tank process is simulated to demonstrate the performance of the proposed scheme.

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“…To proceed, we introduce orthogonal transformation T i 1 that yields observable decomposition for the pairs ( A , C i ), where

\begin{align} A_{i} & = T_{i 1}^{T} A T_{i 1} = ([], \begin{matrix} center center \\ array A_{i 1} & array 0 \\ array A_{i 2} & array A_{i 3} \end{matrix}), C_{i} T_{i 1} = ([], \begin{matrix} center center \\ array C_{i 1} & array 0 \end{matrix}), \\ T_{i 1} B & = B_{i}, C_{i 1} \in {normalR}^{p_{i} prefix\times r_{i}}, \end{align}

where

T_{i 1} = [\begin{array}{cc} T_{i 11} & T_{i 12} \end{array}]

, and T i 11 consists of the first r i columns of T i 1 . Then the unobservable subspace is given by

i m (T_{i 12}) \in \ker O_{C_{i}}

, where

O_{C_{i}} = {[C_{i}^{T}, {(A C_{i})}^{T}, \dots, {(C_{i} A^{n - 1})}^{T}]}^{T}

. Note that

i m

…”

Section: Preliminaries and Problem Formulationmentioning

confidence: 99%

“…The proposed local observer i under the distributed framework is in a form of:

\begin{align} z_{i} ((), k + 1) & = G_{i} z_{i} ((), k) + H_{i} u ((), k) + L_{i} y_{i} ((), k) + M_{i} true \sum_{j = 1}^{h} ((), {\overset{x}{true}}_{j} ((), k) prefix- {\overset{x}{true}}_{i} ((), k)), \\ T_{i} {\overset{x}{true}}_{i} ((), k + 1) & = z_{i} ((), k + 1), \end{align}

where it utilizes available local output

y_{i} (k) = C_{i} x (k) + D_{i} u (k) \in R^{p_{i}}

, received information from neighbors

{\hat{x}}_{j} (k) \in R^{n}

, and the coefficient matrix M i to estimate

x (k + 1) \in R^{n}

Section: Preliminaries and Problem Formulationmentioning

confidence: 99%

“…The following lemma proposed in Reference 6 gives a solution to the Luenberger equations specified in (5).…”

Section: Preliminaries Of Luenberger Observer Designmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Distributed data‐driven observer for linear time invariant systems

Alipouri

Zhao

Huang

2020

Adaptive Control & Signal

View full text Add to dashboard Cite

show abstract

“…This problem is originally studied in [1]- [3] through a consensus-based Kalman filter assuming that data fusion can be achieved in finite time. In [4], [5], [11], this problem is solved by recasting this as a decentralized control problem. The method allows to freely assign the spectrum of the estimators under the condition that the network is strongly connected and fixed.…”

Section: Introductionmentioning

confidence: 99%

A Distributed Observer for a Continuous-Time Linear System

Wang

Liu

Morse

2019

2019 American Control Conference (ACC)

Self Cite

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A simply structured distributed observer is described for estimating the state of a continuous-time, jointly observable, input-free, linear system whose sensed outputs are distributed across a time-varying network. It is explained how to design a gain g in the observer so that their state estimation errors all converge exponentially fast to zero at a fixed, but arbitrarily chosen rate provided the network's graph is strongly connected for all time. A linear inequality for g is provided when the network's graph is switching according to a switching signal with a dwell time or an average dwell time, respectively. It has also been shown the existence of g when the stochastic matrix of the network's graph is chosen to be doubly stochastic under arbitrarily switching signals. This is accomplished by exploiting several well-known properties of invariant subspaces and properties of perturbed systems.

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Distributed observers design for a class of nonlinear systems to achieve omniscience asymptotically via differential geometry

Wang

et al. 2021

Intl J Robust & Nonlinear

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Distributed observers are the framework of fusing observed data by constructing a cooperative observer network. This article renders a scheme of distributed observers for a class of nonlinear system. With this scheme, each local nonlinear observer in the cooperative network can estimate the whole states of the underlying system by their own output measurements and information obtained from their neighbor via communication network. The proposed distributed observers are constructed with Partial Observable Canonical Form, which is a technology studied in differential geometry. The common observable subspace pruning and the weight matrix with spatial transformation are innovatively developed to solve the problem that each local observer is not in the same space. The sufficient conditions are proposed to guarantee the achievement of asymptotical omniscience of distributed nonlinear observers. Furthermore, our distributed observers can also be applied to linear systems and systems with some zero outputs. In addition, comparing with some existing methods of distributed nonlinear observers, our method admits some unbounded systems. Four simulations including multi‐Wen‐bridge oscillator systems, multi‐manipulators systems, unbounded nonlinear system and a linear system are employed to show the validity of the main result.

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A Distributed Observer for a Time-Invariant Linear System

Cited by 145 publications

References 16 publications

Distributed data‐driven observer for linear time invariant systems

Distributed data‐driven observer for linear time invariant systems

A Distributed Observer for a Continuous-Time Linear System

Distributed observers design for a class of nonlinear systems to achieve omniscience asymptotically via differential geometry

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