State Estimation For Linear Systems Research Articles

Distributed State Estimation for Linear Systems in Networks With Antagonistic Interactions.

This article deals with the distributed state estimation problem for linear systems in networks with cooperative interactions and antagonistic interactions, where the cooperative interactions and the antagonistic interactions are characterized by the positive weights and the negative weights, respectively. Due to the coexistence of the cooperative interactions and the antagonistic interactions, the existing methods based on the non-negatively weighted graph become not applicable. First, a partition method of the nodes and a decomposition form of the system matrices are introduced. Then a new form of distributed observers is proposed in the network with cooperative interactions and antagonistic interactions. Necessary and sufficient conditions, which guarantee the existence of proposed distributed observers, are presented. In particular, when the communication network is cooperative, the proposed method is suitable for two cases, that is, the nodes in each independent strongly connected component are jointly detectable, or each node is jointly detectable with its neighbors. Finally, three examples are simulated to illustrate the effectiveness of the proposed methods.

Split-spectrum based distributed state estimation for linear systems

This paper studies a distributed state estimation problem for both continuous- and discrete-time linear systems. A simply structured distributed estimator {comprising interconnected local estimators} is first described for estimating the state of a continuous and multi-channel linear system whose sensed outputs are distributed across a fixed multi-agent network. The estimator is then extended to non-stationary networks whose graphs switch according to a switching signal. The estimator is guaranteed to solve the problem, provided a network-widely shared high gain condition achieving a form of spectrum separation is satisfied. As an alternative to sharing a common gain across the network, a fully distributed version of the estimator is also studied in which each agent adaptively adjusts a local gain, though the practicality of this approach is subject to a robustness issue common to adaptive control. A discrete-time version of the distributed state estimation problem is also studied, and a corresponding estimator based again on spectrum separation, but not high gain, is proposed for time-varying networks. For each scenario, it is explained how to construct the estimator so that the state estimation errors in the local estimators all converge to zero exponentially fast at a fixed but arbitrarily chosen rate, provided the network’s graph is strongly connected for all time. The proposed estimators are inherently resilient to abrupt changes in the number of agents and communication links in the inter-agent communication graph upon which the algorithms depend, provided the network is redundantly strongly connected and redundantly jointly observable.

Distributed exponential state estimation of linear systems over jointly connected switching networks

Recently, the distributed state estimation problem for continuous-time linear systems over jointly connected switching networks was solved. Due to the use of the generalized Barbalat’s Lemma, it can only show that the estimation errors asymptotically converge to the origin. By a completely different approach, this paper further studies the distributed state estimation problem with two new features. First, the asymptotic convergence is strengthened to the exponential convergence. This strengthened result not only offers a guaranteed convergence rate, but also renders the error system total stability and thus the capability to withstand small disturbances. Second, the coupling gains of our local observers can be different from each other and thus offers greater design flexibility, while the coupling gains in the existing result were required to be identical. These two new features are achieved by establishing exponential stability for two classes of linear time-varying systems, which may have other applications.

Interval state estimator for linear systems with known structure

Interval state estimator for linear systems with known structure

FE-aided Kalman Filter for nonlinear state estimation with unknown input

A Kalman Filter-Finite Element (KF-FE) framework for joint input-state estimation of nonlinear systems is proposed in the current study. The KF-FE framework has been developed aiming to estimate the nonlinear structural behavior induced by excessive loading such as earthquake-induced ground motions or strong waves, during which a limited number of responses can be measured. The proposed framework involves the use of a Bayesian recursive filter, namely the Kalman Filter, in conjunction with a nonlinear Finite Element (FE) model, which allows to update the state space model formulation at each time step. The proposed KF-FE framework builds its novelty for nonlinear systems on the basis of existing methods for online state estimation of linear systems. The application of such a framework can be suitable for nonlinear state estimation when only output measurements are available. A numerical testbed is adopted herein consisting of a 2D steel moment resisting frame structure subjected to earthquake ground motions. The KF-FE results from the nonlinear numerical analysis scheme indicate estimation performance of high accuracy for both global (e.g., nodal displacements and rotations) and local (moment-curvature relationships) demand parameters while the computational cost is kept low.

Consensus-based distributed moving horizon estimation with constraints

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.

Distributed secure state estimation for linear systems against malicious agents through sorting and filtering

This paper investigates the distributed secure state estimation problem for the cyber–physical systems monitored by a multi-agent network where partial agents are malicious. Through introducing a sort and filter approach, a novel distributed secure state estimation strategy, where the impact of malicious agents is mitigated through discarding partial extreme values from the received vectors, is proposed. Sufficient conditions on the graph topology and the system matrices to tolerant a bounded number of malicious agents are given. It is shown that by adopting the proposed secure state estimation strategy, normal agents can effectively generate correct state estimate in the presence of malicious agents. Besides, for efficiently updating the state estimate while new measurements are available, a distributed observer is also designed. Compared with the existing techniques searching an appropriate candidate from multiple ones, the proposed secure state estimation strategies are much more computationally efficient since reliable state estimates are generated without brute force search. Finally, simulation results are provided to illustrate the effectiveness of the proposed algorithms.

A Unified Approach to Communication Delay and Communication Frequency in Distributed State Estimation of Linear Systems

A Unified Approach to Communication Delay and Communication Frequency in Distributed State Estimation of Linear Systems

Revisiting State Estimation for Linear Systems with State and/or Filter Equality Constraints*

This article revisits optimal state estimation, in the mean squared error matrix sense, for linear systems with state and/or filter subject to linear equality constraints (LECs). First, it is shown that the conventional Wiener filter (WF) form incorporates any LECs on the state, thus yielding a filter subject to the same LECs. Conversely, an optimal linear filter subject to LECs (or linear equality gain constraints) in general does not exist. Therefore, adding LECs on the WF or WF gain matrix either leaves unchanged or degrades the constrained WF performance w.r.t. the unconstrained WF. Since the Kalman filter (KF) and Kalman predictor (KP) are recursive WF forms for linear discrete state-space (LDSS) systems, the same results hold for both estimators, which is in contradiction with several existing results in the literature. Actually, even if these existing results are mathematically correct, however, they have been derived for unsuitable assumed LDSS models where the state is surprisingly not compliant with the assumed LECs. Indeed, it is shown that for suitable assumed state models, both standard KF and KP forms satisfy the assumed LECs, making any additional projection step superfluous.

Event-Triggered Distributed Moving Horizon State Estimation of Linear Systems

In this article, an event-triggered distributed state estimation mechanism is proposed for general linear systems that comprise several subsystems. Two distributed moving horizon estimation (MHE) algorithms that can handle constraints on disturbances and noise are proposed. An event scheduler is exploited to govern the evaluation of the estimators and networked information exchange between the plant and the estimators, such that good estimates can be provided while both the usage of processors and networked communication frequency can be reduced. The estimation error provided by the event-triggered estimation mechanism is proven to be convergent and bounded. A numerical example and a chemical process example are used to verify the effectiveness and applicability of the proposed method.

Bounded Estimation Over Finite-State Channels: Relating Topological Entropy and Zero-Error Capacity

We investigate state estimation of linear systems over channels having a finite state not known by the transmitter or receiver. We show that similar to memoryless channels, zero-error capacity is the right figure of merit for achieving bounded estimation errors. We then consider finite-state, worst-case versions of the common erasure, and additive noise channels models, in which the noise is governed by a finite-state machine without any statistical structure. Upper and lower bounds on their zero-error capacities are derived, revealing a connection with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">topological entropy</i> of the channel dynamics. Separate necessary and sufficient conditions for bounded linear state estimation errors via such channels are obtained. These estimation conditions bring together the topological entropies of the linear system and the discrete channel.

A variational Bayes moving horizon estimation adaptive filter with guaranteed stability

This paper addresses state estimation of linear systems with special attention on unknown process and measurement noise covariances, aiming to enhance estimation accuracy while ensuring stability. To this end, the full information estimation problem over a finite interval is first addressed. Then, a novel adaptive variational Bayesian (VB) moving horizon estimation (MHE) method is proposed, exploiting VB inference, MHE, and Monte Carlo integration with importance sampling for joint estimation of the unknown process and measurement noise covariances, along with the state trajectory over a moving window of fixed length. Further, it is proved that the proposed adaptive VB MHE filter ensures mean-square boundedness of the estimation error with any number of importance samples and VB iterations, as well as for any window length. Finally, simulation results on a target tracking example demonstrate the effectiveness of the VB MHE filter with enhanced estimation accuracy and convergence properties compared to the conventional non-adaptive Kalman filter and other existing adaptive filters.

On sparsity‐inducing methods in system identification and state estimation

AbstractThe purpose of this article is to survey some sparsity‐inducing methods in system identification and state estimation. Such methods can be divided into two main categories: methods inducing sparsity in the parameters and those sparsifying the prediction error. In the last class we discuss in particular the least absolute deviation estimator and its robustness properties with respect to sparse noise in both cases of univariate and multivariate measurements. We also discuss the application of sparsity‐inducing methods to switched system identification and to state estimation for linear systems in the presence sparse and dense measurement noises. While the presentation focuses essentially on bridging some existing results, some technical refinements, and new features are also provided.

Performance evaluation and simulation of cubic observers

This paper investigates the performance of cubic observers in state estimation of linear systems. In particular, the proposed observer yields a smaller estimation error norm in comparison with a linear one. It is then shown that cubic observers can be designed to perform similar to linear observers in presence of disturbances and delays. It also compares a cubic observer with a nonlinear extended observer in a simulation example.

Two-stage gradient-based iterative algorithm for bilinear stochastic systems over the moving data window

For the bilinear stochastic system, the difficulty of identification lies in the product of the state vector and input in the system. This paper studies the iterative estimation of the parameters and states for the bilinear state-space systems in the observer canonical form. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. Therefore, this paper proposes a state filter (SF) for the bilinear systems based on the extremum principle. By means of the hierarchical principle, we decompose the identification model into two sub-identification models by introducing two fictitious output variables. Then an SF two-stage gradient-based iterative algorithm is proposed to achieve the combined parameter and state estimation according to the gradient search. For the purpose of improving the identification performance, an SF two-stage moving data window gradient-based iterative algorithm is derived by increasing the data utilization. The numerical example demonstrates the validity of the proposed algorithms.

A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling

The behavior of heterogeneous multi-agent systems is studied when the coupling matrices are possibly all different and/or singular, that is, its rank is less than the system dimension. Rank-deficient coupling allows exchange of limited state information, which is suitable for the study of multi-agent systems under output coupling. We present a coordinate change that transforms the heterogeneous multi-agent system into a singularly perturbed form. The slow dynamics is still a reduced-order multi-agent system consisting of a weighted average of the vector fields of all agents, and some sub-dynamics of agents. The weighted average is an emergent dynamics, which we call a blended dynamics. By analyzing or synthesizing the blended dynamics, one can predict or design the behavior of a heterogeneous multi-agent system when the coupling gain is sufficiently large. For this result, stability of the blended dynamics is required. Since stability of the individual agent is not asked, the stability of the blended dynamics is the outcome of trading off the stability among the agents. It can be seen that, under the stability of the blended dynamics, the initial conditions of the individual agents are forgotten as time goes on, and thus, the behavior of the synthesized multi-agent system is initialization-free and is suitable for plug-and-play operation. As a showcase, we apply the proposed tool to four application problems; distributed state estimation for linear systems, practical synchronization of heterogeneous van der Pol oscillators, estimation of the number of nodes in a network, and a problem of distributed optimization.

Distributed State Estimation for Linear Systems With Application to Full-Car Active Suspension Systems

In this article, we treat the problem of distributed state estimation for medium- to large-scale linear systems. We consider a system, which consists of several subsystems, and assume that there already exist linear local estimators for the individual subsystems. These existing local estimators could be designed using algorithms of different types. We propose a systematic approach to connect the local estimators through information flows, such that a distributed state estimation scheme is formed for the linear system. The convergence and boundedness of the estimation error for the developed distributed scheme are proven. The proposed method is applied to a full-car active suspension system. The entire system is decomposed into five subsystems interacting with each other. A distributed estimation architecture is established based on the proposed method. The effectiveness of the proposed approach for the full-car suspension system is illustrated using simulation results, which confirm good estimation performance.

Event-based distributed state estimation for linear systems under unknown input and false data injection attack

This paper investigates an event-based distributed state estimation problem for a linear system subject to unknown input and false data injection attack. The unknown input is treated as a process with a non-informative prior. An event-triggered transmission scheme is designed to reduce exchange of unreliable state estimate information between sensor nodes. In order to defend against false data injection attack, a novel event-based distributed state estimator is proposed by using state estimate information that comes from individual node and neighboring nodes, where filter gains are derived by minimizing an upper bound of estimation error covariance. Then sufficient conditions are established to ensure asymptotic boundedness of the designed estimation error covariance. Finally, effectiveness of the proposed technique is demonstrated by a numerical example.

Recursive state estimation for linear systems with lossy measurements under time-correlated multiplicative noises

This paper is concerned with the recursive state estimation problem for a class of linear discrete-time systems with lossy measurements and time-correlated multiplicative noises (TCMNs). The lossy measurements result from one-step transmission delays and packet dropouts. Different from the traditional white multiplicative noises, TCMNs are included in the measurement model in order to reflect engineering practice. Utilizing the state augmentation approach, the system under investigation is first converted into a stochastic parameter system, and some new recursive terms (including the estimation for the product of state and multiplicative noises) are introduced to handle the difficulties caused by the TCMNs. Then, by the well-known projection theorem, recursive state estimation algorithms are developed in the sense of minimum mean-square error, which facilitate the design of the filter, the multi-step predictor and the smoothers. The proposed algorithms are explicitly dependant on the key system parameters including the covariances of the TCMNs, the occurrence probabilities of the transmission delays and the packet losses. Finally, simulation results illustrate the effectiveness of the presented estimation algorithms.

Asymptotic state estimation for linear systems with sensor and actuator faults

This paper investigates the asymptotic state estimation problem for linear systems with sensor and actuator faults, where the faults are modeled via multiple modes. For the case of sensor faults, we first introduce a new notion of detectability, i.e., detectability of system against sensor faults. The notion helps to address the question of whether it is possible to asymptotically estimate the system state by using the control input and system output, irrespective of which mode the system is in and what values the fault signals are. A necessary and sufficient condition for the system to be detectable against sensor faults is given, and then two switched observers are proposed for asymptotic state estimation with the help of maximin strategy. For the system with l fault modes, we provide the explicit form of the switched observer, which is based on a bank of \(\frac{\ell(\ell+1)}{2}\) Luenberger-like or sliding-mode observers. Furthermore, extensions to the case of sensor and actuator faults are further studied. Finally, a simulation example of a reduced-order aircraft system is provided to show the effectiveness of the proposed approaches.