Elsevier

Information Sciences

Volume 637, August 2023, 118963
Information Sciences

Consensus-based distributed moving horizon estimation with constraints

https://doi.org/10.1016/j.ins.2023.118963Get rights and content

Abstract

This paper concerns distributed state estimation for linear systems over a sensor network. Two novel algorithms utilizing the framework of moving horizon estimation (MHE) are proposed, which are fully distributed, scalable and allow for taking into account constraints on the noises and the system states. The proposed methods estimate the state by minimizing a local quadratic objective function, which can be efficiently solved by quadratic programming (QP). Consensus technique is employed to fuse information to construct the local quadratic objective function for each node. The first algorithm, which minimizes a quadratic function involving consensus on measurement costs (CM), approaches the centralized MHE with a sufficiently large number of consensus steps. The second one involving consensus both on arrival costs (CA) and measurement costs, enjoys the benefits of both the CA and CM. To avoid directly running consensus on some functions, a novel consensus strategy for the CM is developed. The estimation errors of the proposed methods are proven to be stochastically ultimately bounded under certain conditions. Finally, numerical results are presented to verify the effectiveness of the developed algorithms.

Introduction

Moving horizon estimation (MHE) is a practical technique of state estimation, which is first put forward in dealing with estimation problems of uncertain systems by minimizing a quadratic objective function [1], [2]. The objective function usually consists of two quadratic functions involving a prior of the state and a fixed-size window of measurements. The first function with the prior, named arrival cost, summarizes the information of the past estimates, which has no explicit form in general but plays an important role in stability of the estimation error [3]. The second one, referred to as measurement cost in this work, evaluates the difference between the measurements and the estimates. MHE is an optimization-based method for state estimation and has advantages in handling systems subject to constraints on the noises (e.g., non-Gaussian noise) and the system states. This is different from the traditional Kalman filter (KF) framework, where the noises are assumed to be Gaussian. Thus, KF has limitations in practical applications in comparison to MHE since the noise statistics are sometimes unknown. On the other hand, if the constraints of MHE are expressed as a linear inequality, the optimal solution to the objective function can be effectively solved by quadratic programming (QP), which is a mathematical optimization technique involving quadratic functions e.g., [4], [5]. Furthermore, because of its guaranteed performance on estimating the state of linear and nonlinear systems, MHE has been widely applied in various kinds of occasions, such as nonlinear systems [6], time-delay systems [7], [8], Markov jump systems [9], quantization systems [10], [11], and so on.

With the rapid development of sensor networks, distributed architecture with sensors processing the capability of communication and computing has been introduced in modern monitoring and control systems [12], [13]. Sensor networks have the advantages of flexible architectures and low maintenance costs. However, there are still many theoretical challenges in this field, such as redesigning some algorithms of state estimation in a distributed way (i.e., each sensor of the network estimates the state without a fusion center by exchanging information only with their interconnected neighbors) while preserving estimation performance of their centralized counterparts as much as possible. In the past decade, distributed state estimation (DSE) over a sensor network has attracted a lot of attention in the literature [14], [15]. In general, DSE algorithms are required to satisfy scalability and robustness against network topology changes, i.e., the amount of local computation of each sensor is irrespective of the network size and the topology. One feasible scheme for designing a DSE algorithm while achieving scalability is to adopt the well-known consensus protocol for distributed implementation. Consensus has advantages in computing tasks in a distributed manner (without a fusion center) [16], [17]. Typically, there are lots of literature concerning with the consensus problem of multi-agent systems [18], [19] and some of them involve the issue of distributed optimization e.g., [20], [21]. In addition, plenty of results about consensus are available in the field of DSE. For example, a class of consensus-based distributed filters was presented in [22], which provides a general framework of distributed Kalman filter guaranteeing the stability even with a single consensus step. This framework was further extended to cases of network failure [23] and event-triggering communication [24].

Distributed MHE (DMHE) has become a hot topic recently. Since MHE is optimization-based (i.e., the estimator has no explicit form), the primary requirement of DMHE is to satisfy scalability. The early work in [25] presented a DMHE algorithm where each node solves a local MHE problem concerning a fused arrival cost. The fused arrival cost involves a fused prior of the state and a fused weighted matrix, which are respectively obtained by the consensus protocol in parallel. However, the consensus weight needs to be calculated in a centralized manner, which means that it is not fully distributed and does not satisfy scalability. To overcome this issue, the authors in [3] developed an improved algorithm (named CA-DMHE) with consensus steps on the arrival costs (CA), which is fully distributed, scalable and independent of any global information. Instead of directly running consensus steps on some functions (i.e., the arrival costs), the actual process of the CA-DMHE algorithm is also to compute a fused prior and a fused weighted matrix by the consensus protocol, which is easy to be implemented in practice. The works [3], [25] considered DMHE with constraints. On the contrary, a DMHE algorithm in the constraint-free case was presented [26], which performs consensus on measurements and local observability matrices. Due to the measurement spread over sensor networks, the results in [26] improved the estimation performance compared to the CA-DMHE algorithm under the unconstrained case.

Stability analysis is a fundamental issue of consensus-based DMHE [3]. In the context of DMHE, the arrival cost also serves an important role in stability of the estimation error. In [25], information can not be fully spread over the network since the fused prior and the fused weighted matrix of the arrival cost are calculated in parallel and do not affect each other during the consensus period. Its stability condition relies in computing the consensus weights by solving a linear matrix inequality involving global information at each time instant. Unlike [25], the fused terms of the CA-DMHE algorithm are related. To be specific, the fused prior is obtained by a similar fusing way of the covariance intersection fusion rule (the weighted matrix is regarded as the inverse of the estimated error covariance). This leads to a different stability condition. Under the requirement of strongly connected topology and collective observability, the estimation error of the CA-DMHE algorithm is stable with any number of consensus steps (even with a single consensus step). The method in [26] is a simplified one since it is unconstrained and the weighted matrix is replaced by a scalar weight. Thus, only a scalar parameter for the arrival cost is determined to ensure stability of estimation error. However, the number of consensus steps needs to be sufficiently large to guarantee requirements of observability since only measurements are spread over the network at each time instant.

This work deals with the problem of DMHE with constraints over a sensor network. Specifically, we focus on consensus-based DMHE since it is more likely to satisfy scalability. As discussed before, the objective function of MHE includes two quadratic functions i.e., the arrival cost and the measurement cost. The existing method in [3] pays more attention to the CA for developing a DMHE scheme which guarantees stability with a single consensus step. However, the measurement cost, obviously including more novel information than the arrival cost, should be taken into account, not just to guarantee stability, but to achieve better estimation performance. For this purpose, this work aims to develop a DMHE algorithm (named CM-DMHE) with consensus on the measurement costs (CM). The work in [26] reveals that consensus on the measurements does improve estimation performance. Different from the method in [26], the strategy in this work allows for estimating the system state with constraints. Notice that the measurement cost of each node is a function that is not directed to be transmitted over the sensor network. As thus, this work tries to find a feasible way to run CM in practice. On the other hand, although CM may improve the estimation performance, it cannot guarantee stability unless the number of consensus steps is large enough. That is, CA and CM show complementary features between estimation performance and minimal requirement of consensus steps. Motivated by that, this work further develops a hybrid method (named HCACM-DMHE) by combining the CA and the CM so as to preserve their benefits. The main contributions of this paper are as follows.

  • 1)

    Novel consensus-based DMHE algorithms: Instead of only performing the CA in [3], this work develops two novel DMHE algorithms taking into account the CM (i.e., the CM-DMHE and the HCACM-DMHE), which are scalable and fully distributed. For the CM part, a new consensus strategy is presented to construe two equivalent quadratic functions for running the CM in practice. Under inequity constraints, the proposed algorithms are formulated as two QP problems.

  • 2)

    Stochastically ultimately bounded estimation error: Through stochastic analysis, sufficient conditions are provided to guarantee the stochastically ultimate boundedness of the estimation error of both the CM-DMHE and the HCACM algorithms under the assumptions of strongly connected topology and the collective observability.

The rest of this paper is structured as follows: Centralized MHE and the CA-DMHE algorithm are given in Section 2, which are formulated as two QP problems. Finally, the problem of interest is introduced. In Section 3, two consensus-based DMHE algorithms are presented. Section 4 derives the stability condition for the proposed algorithms. Section 5 provides simulation results with different topologies of sensor networks to evaluate the estimation performance of the proposed algorithms. Finally, the conclusion is included in Section 6.

Notation: R, Rm, and Rv stand for the set of real numbers, real matrices, and real vectors, respectively. Denote the set of quadratic functions by Fq. The set of q-dimensional vectors and (p×q)-dimensional matrices are denoted by Rq and Rp×q, respectively. The mathematical expectation is denoted by E{}. The symbol P stands for the Euclidean norm weighted by a positive-definite matrix P. The symbol col() denotes the collecting vector. 1m represents the vector with m ones. The symbol ′ and ⊗ stand for the matrix transpose and the Kronecker product, respectively. For vectors a and b, ab means that each element of vector a is less than or equal to the corresponding element of vector b. For matrices A and B, AB (AB) means that the matrix AB is positive (semi-)definite. Im is the identity matrix with m-dimensions. Finally, we use diag{} to denote a block diagonal matrix.

Section snippets

Sensor network and system description

This work studies DMHE over a sensor network where nodes possess the capability of computation and communication. The sensor network is depicted by a static directed graph G=(N,E,W), where N={1,2,,m}, E=N×N, and W=[wij]Rm×m denote the node set, the edge set, and the adjacency matrix, respectively. In addition, the matrix W is row stochastic. The edge (j,i), associating with the weight value wij, belongs to the edge set E if there exists a directed edge such that node i can receive information

Proposed distributed methods

This section proposes two DMHE algorithms to minimize the objective functions (25) and (26), respectively. The first one performs consensus steps on the measurement costs, which is referred to as CM-DMHE. The other one performs consensus steps both on the arrival costs and the measurement costs, which is a hybrid one and is referred to as HCACM-DMHE.

Remark 2

The proposed methods are based on the consensus technique. Some basic concepts about consensus can be found in [29], [30]. It is known that the

Stable analysis

This section provides stable analysis of the proposed CM-DMHE and the HCACM-DMHE algorithms. To begin with, the estimation error of node i at time instant kN is denoted by ekNi=xkNxˆkNi. The following definition is introduced to describe the boundedness of the estimation error.

Definition 4

[32] The estimation error ekNi is said to be stochastically ultimately bounded if there exists a positive constant ϵ< such that the inequality limkE{ekNi2}<ϵ holds.

In addition, some assumptions are needed.

Assumption 2

Simulation

This section presents a performance comparison with different distributed state estimation methods, i.e., the CA-DMHE algorithm in [3], the proposed CM-DMHE and HCACM-DMHE algorithms, and the consensus-based distributed Kalman filters in [22], [34], which are, respectively, referred to as DKF1 and DKF2 in the next.

Conclusion

In this work, two consensus-based algorithms of distributed moving horizon estimation have been presented. The first algorithm (named CM-DMHE) with consensus steps on the local measurement costs has been proven to conditionally approaches the centralized counterpart. The second algorithm (named HCACM-DMHE) with consensus steps both on the local arrival costs and the local measurement costs has been shown to enjoy the complementary advantages of the CM-DMHE and the existing CA-DMHE algorithms.

The proof of Proposition 1

After g consensus steps in Eqs. (30) and (31), we haveτkN,gi=jNwgij(Gj)R¯jYkNj,ΦkN,gi=jNwgij(Gj)R¯jGj, and the fused measurement cost can be written asϒk,gi(X)=XΦkN,giX2XτkN,gi+(τkN,gi)(ΦkN,gi)1τkN,gi. Notice that wgij is the ith row and jth column entry of the matrix Wg. In addition, one derivesϒˆk,gi(X)=jNwgijYkNjGjXR¯j2=XjNwgij(Gj)R¯jGjX2XjNiwgij(Gj)R¯jYkNj+jNwgij(YkNj)R¯jYkNj. Finally, it becomes apparent that the measurement costs (59) and (60) are

CRediT authorship contribution statement

Zenghong Huang: Conceptualization, Methodology, Software, Writing – original draft. Zijie Chen: Conceptualization, Investigation. Chang Liu: Investigation, Visualization. Yong Xu: Supervision, Writing – review & editing. Peng Shi: Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (35)

  • Z. Wu et al.

    A distributed Kalman filtering algorithm with fast finite-time convergence for sensor networks

    Automatica

    (2018)
  • G. Battistelli et al.

    Kullback–Leibler average, consensus on probability densities, and distributed state estimation with guaranteed stability

    Automatica

    (2014)
  • G. Battistelli

    Distributed moving-horizon estimation with arrival-cost consensus

    IEEE Trans. Autom. Control

    (2018)
  • K. Bouyarmane et al.

    Quadratic programming for multirobot and task-space force control

    IEEE Trans. Robot.

    (2018)
  • Y. Qi et al.

    Complex-valued discrete-time neural dynamics for perturbed time-dependent complex quadratic programming with applications

    IEEE Trans. Neural Netw. Learn. Syst.

    (2019)
  • A. Alessandri et al.

    Fast moving horizon state estimation for discrete-time systems using single and multi iteration descent methods

    IEEE Trans. Autom. Control

    (2017)
  • L. Zou et al.

    Moving horizon estimation for networked time-delay systems under Round-Robin protocol

    IEEE Trans. Autom. Control

    (2019)
  • Cited by (0)

    This work was supported in part by the National Natural Science Foundation of China under Grants (62121004, 62006043, 61876041), the Local Innovative and Research Teams Project of Guangdong Special Support Program (2019BT02X353), Key Area Research and Development Program of Guangdong Province (2021B0101410005), and Guangdong Basic and Applied Basic Research Foundation (2021B1515420008).

    View full text