Distributed moving horizon estimation for nonlinear constrained systems

MHE Based State and Parameter Estimation for Systems subjected to Non-Gaussian Disturbances

Nonlinear state estimation is a pre-requisite for advanced process control and fault diagnosis tasks. In literature, various recursive nonlinear filtering techniques have been proposed and used for state estimation of nonlinear systems. Over the last few years, Moving Horizon Estimation (MHE) is increasingly being used for state estimation of nonlinear systems. Moving horizon estimation works with a window of data and hence requires additional online computation compared to recursive nonlinear filters. However, MHE performs both smoothing and filtering and thus has the potential to obtain more accurate state estimates as compared to the recursive filters. Most of the available comparisons of MHE with recursive filters are based on simulation case studies where the true states and parameters, as well as noise processes are exactly known. In this work, we apply MHE to a three-tank experimental setup and compare its performance with various nonlinear filters available in literature such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Gaussian Sum EKF (GSEKF). We present some of the challenges in experimental implementation of state estimation approaches. These include presence of unknown disturbances, non-whiteness of noise signals as well as lack of accurate measurements. We then discuss the approach followed for obtaining model parameters and noise characterization to make the models amenable for filter implementation. It is found that EKF, GSEKF and UKF perform as well as MHE as far as accuracy is concerned, but require significantly lower computational efforts.

NONLINEAR MODEL PREDICTIVE CONTROL BASED ON SEQUENTIAL MONTE CARLO STATE ESTIMATION

A distributed state estimation scheme based on quantized observations in wireless sensor network (WSN) is proposed. Unlike the SOI-KF approach, we address the state estimation problem in two steps: firstly, local sensors and the fusion center apply the decentralized estimation scheme (DES) to design the quantized message function and the fusion function; secondly, the fusion center constructs a linear filter to obtain the state estimation by minimizing an upper bound of the error power norm. The optimal filter is computed by the linear matrix inequality (LMI) solver. An example is presented to demonstrate the efficiency of our method.

A Comparison of Moving Horizon and Bayesian State Estimators with an Application to a pH Process

International audience

MHE is an optimization based strategy for state estimation that explicitly allows for nonlinear models and inequality constraints. In this work we investigate strategies to guarantee the stability of moving horizon estimation (MHE). We begin our discussion by analyzing the stability of the abstract MHE problem. Sufficient conditions for asymptotic stability are established. Using forward dynamic programming to analyze MHE, we introduce the fundamental concept of arrival cost to bound the size of the estimation problem. The key result is if one can construct a global lower bound for the arrival cost, then asymptotic stability is guaranteed. In the second part of our discussion, we propose a strategy that circumvents the need for a global lower bound for the arrival cost. In particular, by including bounds to guarantee the existence of a decreasing nominal sequence, we circumvent the need for a global lower bound for the arrival cost. This result is significant, because, in general, we cannot calculate nor bound the arrival cost. Notable exceptions are linear systems, where the Kalman filter covariance generates a global lower bound for the arrival cost.

State estimation is an important topic in the study of dynamical systems. The problem of estimation can be structured into three categories: 1 centralised scheme; 2 decentralised scheme; 3 distributed scheme. Distributed estimation is a compromise between completely centralised and decentralised versions of estimation. In this paper, we will provide an assessment of distributed estimation based on Kalman filtering techniques for large-scale or sensor networks. In simulation, a second order dynamical system is employed in a scenario of ten sensor nodes. The sensor nodes attempt to estimate the states of the dynamical system with embedded consensus filters. The results show that the distributed estimation algorithm effectively approximates the central Kalman filter. It is concluded that the distributed estimation techniques for distributed dynamical system requires further extensive research.

We present a distributed observer for nonlinear systems, which can be used to estimate the state of a central plant with unknown parameters using a network of sensors that measure noisy linear combinations of the state. Our state estimator is based on the unscented Kalman filter and the dual Kalman filter method, which runs two Kalman filters in parallel for both state and parameter estimation. This allows us to improve estimates of the central plants parameters in order to improve state estimates, and vice versa. Furthermore, using results from stochastic stability analysis, we prove that under certain conditions the expected value of the proposed algorithm's estimation error is bounded. This result also shows us that as t goes to infinity, the error bound scales as a function of the largest eigenvalue of the sensor noise covariance matrix. Lastly, we demonstrate the ability of our state estimator to accurately track the state of nonlinear systems through simulations.

COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability of the SARS-CoV-2 virus, vaccination coverage, and mobility restriction measures change epidemic dynamics over time. A model-based control strategy requires reliable predictions to be efficient on a long-term basis. In this paper, a SEIR-based model is proposed considering dynamic feedback estimation. State and parameter estimations are performed on state estimators using augmented states. Three methods were implemented: constrained extended Kalman filter (CEKF), CEKF and smoother (CEKF & S), and moving horizon estimator (MHE). The parameters estimation was based on vaccine efficacy studies regarding transmissibility, severity of the disease, and lethality. Social distancing was assumed as a measured disturbance calculated using Google mobility data. Data from six federative units from Brazil were used to evaluate the proposed strategy. State and parameter estimations were performed from 1 October 2020 to 1 July 2021, during which Zeta and Gamma variants emerged. Simulation results showed that lethality increased between 11 and 30% for Zeta mutations and between 44 and 107% for Gamma mutations. In addition, transmissibility increased between 10 and 37% for the Zeta variant and between 43 and 119% for the Gamma variant. Furthermore, parameter estimation indicated temporal underreporting changes in hospitalized and deceased individuals. Overall, the estimation strategy showed to be suitable for dynamic feedback as simulation results presented an efficient detection and dynamic characterization of circulating variants.

Bayesian Estimation of Unconstrained Nonlinear Dynamic Systems

Combined nonlinear model predictive control and moving horizon estimation for a copolymerization process

Constrained state estimation in nonlinear/non-Gaussian processes has been the domain of optimization based methods such as moving horizon estimation (MHE). MHE has a Bayesian interpretation, but it is not practical to implement a recursive MHE without assumptions of Gaussianity and linearized dynamics at various stages. This paper presents the constrained cell filter (CCF) as an alternative to MHE, requiring no linearization, jacobians, or nonlinear program. The CCF computes a piecewise constant approximation of the state probability density function with support defined by constraints; thus, all point estimates are constrained. The CCF can be more accurate and orders of magnitude faster than MHE for problems of a size as investigated in this work.

In this article, we consider a state estimation problem for large‐scale nonlinear processes in the absence of first‐principles process models. By exploiting process operation data, both process modeling and state estimation design are addressed within a distributed framework. By leveraging the Koopman operator concept, a parallel subsystem modeling approach is proposed to establish interactive linear subsystem process models in higher‐dimensional subspaces, each of which correlates with the original nonlinear subspace of the corresponding process subsystem via a nonlinear mapping. The data‐driven linear subsystem models can be used to collaboratively characterize and predict the dynamical behaviors of the entire nonlinear process. Based on the established subsystem models, local state estimators that can explicitly handle process operation constraints are designed using moving horizon estimation. The local estimators are integrated via information exchange to form a distributed estimation scheme, which provides estimates of the unmeasured/unmeasurable state variables of the original nonlinear process in a linear manner. The proposed framework is applied to a chemical process and an agro‐hydrological process to illustrate its effectiveness and applicability. Good open‐loop predictability of the linear subsystem models is confirmed, and accurate estimates of the process states are obtained without requiring a first‐principles process model.

MHE is an optimization based strategy for state estimation that explicitly allows for nonlinear models and inequality constraints. In this work we investigate strategies to guarantee the stability of moving horizon estimation (MHE). We begin our discussion by analyzing the stability of the abstract MHE problem. Sufficient conditions for asymptotic stability are established. Using forward dynamic programming to analyze MHE, we introduce the fundamental concept of arrival cost to bound the size of the estimation problem. The key result is if one can construct a global lower bound for the arrival cost, then asymptotic stability is guaranteed. In the second part of our discussion, we propose a strategy that circumvents the need for a global lower bound for the arrival cost. In particular, by including bounds to guarantee the existence of a decreasing nominal sequence, we circumvent the need for a global lower bound for the arrival cost. This result is significant, because, in general, we cannot calculate nor bound the arrival cost. Notable exceptions are linear systems, where the Kalman filter covariance generates a global lower bound for the arrival cost.