Distributed moving horizon state estimation: Simultaneously handling communication delays and data losses

In this work, we consider distributed moving horizon state estimation of nonlinear systems subject to communication delays and data losses. In the proposed design, a local estimator is designed for each subsystem and the distributed estimators communicate to collaborate. To handle the delays and data losses simultaneously, a predictor is designed for each subsystem estimator. A two-step prediction-update strategy is used in the predictor design in order to get a reliable prediction of the system state. In the design of each subsystem estimator, an auxiliary nonlinear observer is also taken advantage of to calculate a reference subsystem state estimate. In the local estimator, the reference state estimate is used to generate a confidence region within which the local estimator optimizes its subsystem state estimate. Sufficient conditions under which the proposed design gives decreasing and ultimately bounded estimation error are provided. The effectiveness of the proposed approach is illustrated via the application to a chemical process example.     

A homotopy-based moving horizon estimation

Moving horizon estimation (MHE) solves a constrained dynamic optimisation problem. Including nonlinear dynamics into an optimal estimation problem generally comes at the cost of tackling a non-convex optimisation problem. Here, a particular model formulation is proposed in order to convexify a class of nonlinear MHE problems. It delivers a linear time-varying (LTV) model that is globally equivalent to the nonlinear dynamics in a noise-free environment, hence the optimisation problem becomes convex. On the other hand, in the presence of unknown disturbances, the accuracy of the LTV model degrades and this results in a less accurate solution. For this purpose, some assumptions are imposed and a homotopy-based approach is proposed in order to transform the problem from convex to non-convex, where the sequential implementation of this technique starts with solving the convexified MHE problem. Two simulation studies validate the efficiency and optimality of the proposed approach with unknown disturbances.     

Distributed moving horizon state estimation for nonlinear systems with bounded uncertainties

In this work, we propose a distributed moving horizon state estimation (DMHE) design for a class of nonlinear systems with bounded output measurement noise and process disturbances. Specifically, we consider a class of nonlinear systems that are composed of several subsystems and the subsystems interact with each other via their subsystem states. First, a distributed estimation algorithm is designed which specifies the information exchange protocol between the subsystems and the implementation strategy of the DMHE. Subsequently, a local moving horizon estimation (MHE) scheme is designed for each subsystem. In the design of each subsystem MHE, an auxiliary nonlinear deterministic observer that can asymptotically track the corresponding nominal subsystem state when the subsystem interactions are absent is taken advantage of. For each subsystem, the nonlinear deterministic observer together with an error correction term is used to calculate a confidence region for the subsystem state every sampling time. Within the confidence region, the subsystem MHE is allowed to optimize its estimate. The proposed DMHE scheme is proved to give bounded estimation errors. It is also possible to tune the convergence rate of the state estimate given by the DMHE to the actual system state. The performance of the proposed DMHE is illustrated via the application to a reactor-separator process example.     

Two triggered information transmission algorithms for distributed moving horizon state estimation

In this work, we consider the reduction of information transmission frequency of distributed moving horizon estimation (DMHE) for a class of nonlinear systems in which interacting subsystems exchange information with each other through a shared communication network. Specifically, algorithms based on two event-triggered methods are proposed to reduce the number of information transmissions between the subsystems in a DMHE scheme. In the first algorithm, a subsystem sends out its current information when a triggering condition based on the difference between the current state estimate and a previously transmitted one is satisfied; in the second algorithm, the transmission of information from a subsystem to other subsystems is triggered by the difference between the current measurement of the output and its derivatives and a previously transmitted measurement. In order to ensure the convergence and ultimate boundedness of the estimation error, we also propose to redesign the local moving horizon estimator of a subsystem to account for the possible lack of state updates from other subsystems explicitly. A chemical process is utilized to demonstrate the applicability and performance of the proposed approaches.     

Observer-enhanced distributed moving horizon state estimation subject to communication delays

In this work, we focus on distributed moving horizon estimation (DMHE) of nonlinear systems subject to time-varying communication delays. In particular, a class of nonlinear systems composed of subsystems interacting with each other via their states is considered. In the proposed design, an observer-enhanced moving horizon state estimator (MHE) is designed for each subsystem. The distributed MHEs exchange information via a shared communication network. To handle communication delays, an open-loop state predictor is designed for each subsystem to provide predictions of unavailable subsystem states (due to delays). Based on the predictions, an auxiliary nonlinear observer is used to generate a reference subsystem state estimate for each subsystem. The reference subsystem state estimate is used to formulate a confidence region for the actual subsystem state. The MHE of a subsystem is only allowed to optimize its subsystem state estimate within the corresponding confidence region. Under the assumption that there is an upper bound on the time-varying delays, the proposed DMHE is proved to give decreasing and ultimately bounded estimation error. The theoretical results are illustrated via the application to a reactor–separator chemical process.     

Use of multiple models and qualitative knowledge for on-line moving horizon disturbance estimation and fault diagnosis

An integrated fault detection, fault isolation, and parameter estimation technique is presented in this paper. Process model parameters are treated as disturbances that dynamically affect the process outputs. A moving horizon estimation technique minimizes the error between process and model measurements over a finite horizon by calculating model parameter values across the estimation horizon. To implement qualitative process knowledge, this minimization is constrained such that only a limited number of different faults (parameters) may change during a specific horizon window. Multiple linear models are used to capture nonlinear process characteristics such as asymmetric response, variable dynamics, and changing gains. Problems of solution multiplicity and computational time are addressed. Results from a nonlinear chemical reactor simulation are presented.     

Moving horizon estimation for switching nonlinear systems

This paper is concerned with moving horizon estimation for a class of constrained switching nonlinear systems, where the system mode is regarded as an unknown discrete state to be estimated together with the continuous state. In this work, we establish the observability framework of switching nonlinear systems by proposing a series of concepts about observability and analyzing the properties of such concepts. By fully applying the observability properties, we prove the stability of the proposed moving horizon estimators. Simulation results are reported to verify the derived results.     

Distributed moving horizon estimation for nonlinear constrained systems

In this paper we consider a nonlinear constrained system observed by a sensor network and propose a distributed state estimation scheme based on moving horizon estimation (MHE). In order to embrace the case where the whole system state cannot be reconstructed from data available to individual sensors, we resort to the notion of MHE detectability for nonlinear systems, and add to the MHE problems solved by each sensor a consensus term for propagating information about estimates through the network. We characterize the error dynamics and provide conditions on the local exchanges of information in order to guarantee convergence to zero and stability of the state estimation error provided by any sensor. Copyright © 2010 John Wiley & Sons, Ltd.     

Inference of building occupancy signals using moving horizon estimation and Fourier regularization

We study the problem of estimating time-varying occupancy and ambient air flow signals using noisy carbon dioxide and flow sensor measurements. A regularized moving horizon estimation formulation is proposed that constrains time-varying signals to smooth Fourier expansions. We demonstrate that the regularization approach makes the estimator robust to high levels of noise. In addition, it requires minimal information about the shape of the signals. Computational experiments with simulated and real data demonstrate the effectiveness of the approach.     

Linear constrained moving horizon estimator with pre-estimating observer

In this paper, a constrained moving horizon estimation (MHE) strategy for linear systems is proposed. Recently, the use of a pre-estimating linear observer in the forward prediction equations in the MHE cost function has been proposed in order to reduce the effects of uncertainty. Here we introduce state constraints within this formulation and investigate stability properties in the presence of bounded disturbances and noise. The robustness and performance of the proposed observer is demonstrated with a simulation example.     

Online moving horizon estimation of fluxes in metabolic reaction networks

Using online state and parameter estimation, concentrations and fluxes in bioprocesses can be estimated for use in monitoring, optimization and control applications. Existing methodologies, however, either ignore the dynamic nature of the problem, or focus on the extracellular concentration states and pay less attention to accurate flux estimates. These estimates are useful for online monitoring of the flux state of an organism, or for developing novel flux-based strategies for online control of bioreactors.In this contribution, the dynamic metabolic flux analysis model structure is combined with two kinetic flux models: a linear flux model and a nonlinear, more mechanistic flux model. The parameters of these models are estimated online through a moving horizon estimation strategy. The resulting algorithm is illustrated on two simulated case studies: a small-scale network, to assess the influence of important algorithm parameters on the final estimates, and a medium-scale network for Escherichia coli, to empirically test the performance of the methodology in a more realistic situation.An important parameter in this estimation strategy is the chosen noise level on the estimated parameters. This choice is not trivial, but is observed to have a significant influence on the resulting estimates. Furthermore, also the effect of the choice of the null space basis for the stoichiometric matrix of the metabolic reaction network was assessed. In the small-scale case study, it was found that a linear flux model with a specific parameter noise level was performing well for both state and flux estimation. The influence of the choice of the null space basis matrix on the estimation performance was much lower. The resulting scenario was evaluated in the medium-scale case study and found to be performing very well also in that case.     

Computing arrival cost parameters in moving horizon estimation using sampling based filters

Moving horizon estimation (MHE) is a numerical optimization based approach to state estimation, where the joint probability density function (pdf) of a finite state trajectory is sought, which is conditioned on a moving horizon of measurements. The joint conditional pdf depends on the a priori state pdf at the start of the horizon, which is a prediction pdf based on historical data outside the horizon. When the joint pdf is maximized, the arrival cost is a penalty term based on the a priori pdf in the MHE objective function. Traditionally, the a priori pdf is assumed as a multivariate Gaussian pdf and the extended Kalman filter (EKF) and smoother are used to recursively update the mean and covariance. However, transformation of moments through nonlinearity is poorly approximated by linearization, which can result in poor initialization of MHE. Sampling based nonlinear filters completely avoid Taylor series approximations of nonlinearities and attempt to approximate the non-Gaussian state pdf using samples and associated weights or probability mass points. The performance gains of sampling based filters over EKF motivate their use to formulate the arrival cost in MHE. The a priori mean and covariance are more effectively propagated through nonlinearities and the resulting arrival cost term can help to keep the horizon small. It is also possible to find closed-form approximations to the non-Gaussian a priori pdf from the sampling based filters. Thus, more realistic nonparametric arrival cost terms can be included by avoiding the Gaussian assumption. In this paper the use of the deterministic sampling based unscented Kalman filter, the class of random sampling based particle filter and the aggregate Markov chain based cell filter are discussed for initializing MHE. Two simulation examples are included to demonstrate the benefits of these methods over the traditional EKF approach.     

基于滚动时域估计的带约束运动目标跟踪*

针对实际的运动目标跟踪问题中存在的各种物理约束,采用基于在线滚动优化原理的滚动时域估计方法,将跟踪滤波问题转换为带约束的有限时域优化问题,并通过引入到达代价函数,有效减少了优化问题求解所需的计算量。最后,对实际的目标跟踪问题进行了滚动时域估计仿真研究。Monte Carlo仿真结果表明,滚动时域估计能有效提高跟踪精度,并且能在采样周期之内完成求解,满足在线估计的需要。     

基于滚动时域估计的非线性目标跟踪*

实际的雷达跟踪问题大多属于非线性问题,存在着各种物理约束,采用基于在线滚动优化原理的滚动时域估计方法可以有效地处理带约束非线性目标跟踪问题。滚动时域估计通过引入到达代价函数,将非线性跟踪滤波问题转换为带约束的有限时域优化问题,可以有效减少优化问题求解的计算量,能够显著提高状态估计的准确度。针对实际的雷达跟踪问题,仿真结果表明,滚动时域估计能有效地提高非线性目标跟踪的精度。     

Humanoid state estimation using a moving horizon estimator

In this research, a new state estimator based on moving horizon estimation theory is suggested for the humanoid robot state estimation. So far, there are almost no studies on the moving horizon estimator (MHE)-based humanoid state estimator. Instead, a large number of humanoid state estimators based on the Kalman filter (KF) have been proposed. However, such estimators cannot guarantee optimality when the system model is nonlinear or when there is a non-Gaussian modeling error. In addition, with KF, it is difficult to incorporate inequality constraints. Since a humanoid is a complex system, its mathematical model is normally nonlinear, and is limited in its ability to characterize the system accurately. Therefore, KF-based humanoid state estimation has unavoidable limitations. To overcome these limitations, we propose a new approach to humanoid state estimation by using a MHE. It can accommodate not only nonlinear systems and constraints, but also it can partially cope with non-Gaussian modeling error. The proposed estimator framework facilitates the use of a simple model, even in the presence of a large modeling error. In addition, it can estimate the humanoid state more accurately than a KF-based estimator. The performance of the proposed approach was verified experimentally.     

整车主动悬架系统分布式滚动时域一致性估计

考虑整车主动悬架系统的约束状态估计问题,本文提出基于一致性原理的分布式滚动时域估计(DMHE)算法.首先,为了降低状态估计过程中的计算量,将整车主动悬架系统分解为若干降阶子系统.其次,为提高分布式状态估计效果,采用滚动时域估计(MHE)方法处理主动悬架系统的状态和噪声约束.考虑子系统与邻居估计状态的相关性,在采样间隔中执行多次一致性原理实现主动悬架系统状态的信息融合,进一步建立了算法的稳定性充分条件.最后,通过对比仿真实验验证算法的有效性和优越性.     

Constrained state estimation for stochastic jump systems: moving horizon approach

We discuss the state estimation advantages for a class of linear discrete-time stochastic jump systems, in which a Markov process governs the operation mode, and the state variables and disturbances are subject to inequality constraints. The horizon estimation approach addressed the constrained state estimation problem, and the Bayesian network technique solved the stochastic jump problem. The moving horizon state estimator designed in this paper can produce the constrained state estimates with a lower error covariance than under the unconstrained counterpart. This new estimation method is used in the design of the restricted state estimator for two practical applications.     

噪声方差不确定约束系统的滚动时域估计

针对噪声方差不确定的约束系统,讨论了一种鲁棒滚动时域估计(MHE)方法.首先,根据噪声方差不确定模型,找到满足所有不确定性的最小方差上界,在线性矩阵不等式(LMI)框架下求解优化问题,得到近似到达代价的表达形式;然后再融合预测控制的滚动优化原理,把系统的硬约束直接表述在优化问题中,在线优化性能指标,估计出当前时刻系统的状态.仿真时与鲁棒卡尔曼滤波方法进行比较,结果表明了该方法的有效性.     

Parameter optimization of moving horizon estimation based on hybrid intelligent algorithm

In the application of moving horizon estimation (MHE) algorithm, the window length will affect the estimation accuracy and the computing efficiency. For this kind of problem, a method of parameter optimization is proposed to obtain suitable window length. Firstly, in order to facilitate online solution, the optimization problem involved in the algorithm is transformed into a quadratic programming (QP) problem in matrix form. Secondly, for the time index and the estimated residual index that measure different properties, the normalization idea is adopted to incorporate them into the same dimension to design the fitness function, and a genetic optimization algorithm based on simulated annealing mechanism is given to search for the optimal window length. Finally, the proposed parameter optimization method is verified by two cases. The results show that the parameter optimization method has the advantages of excellent local search ability and sufficient convergence, and the window length obtained by this method can better take into account the two performance indexes of the MHE algorithm and improve the estimation performance.     

Bayesian estimation via sequential Monte Carlo sampling—Constrained dynamic systems

Nonlinear and non-Gaussian processes with constraints are commonly encountered in dynamic estimation problems. Methods for solving such problems either ignore the constraints or rely on crude approximations of the model or probability distributions. Such approximations may reduce the accuracy of the estimates since they often fail to capture the variety of probability distributions encountered in constrained linear and nonlinear dynamic systems. This article describes a practical approach that overcomes these shortcomings via a novel extension of sequential Monte Carlo (SMC) sampling or particle filtering. Inequality constraints are imposed by accept/reject steps in the algorithm. The proposed approach provides samples representing the posterior distribution at each time point, and is shown to satisfy the same theoretical properties as unconstrained SMC. Illustrative examples show that results of the proposed approach are at least as accurate as moving horizon estimation, but computationally more efficient and in addition, the approach indicates the uncertainty associated with these estimates.