Moving Horizon State Estimation Research Articles

Back-and-forth nudging moving horizon estimation for discrete-time linear systems

In this paper, we propose a novel moving horizon estimation algorithm for discrete-time linear systems with a limited number of measurements. Motivated by the idea of the back-and-forth nudging algorithm, we design the back-and-forth nudging in time moving-horizon estimation method by deploying two moving horizon estimators that move backward and forward iteratively. Based on a finite number of measurements, the proposed algorithm can be used for moving-horizon state estimation with guaranteed convergence. By using the proposed method, we show that the norm of the state estimation error is upper bounded by a sequence that converges to its steady-state in finite-time (i.e., using a finite number of measurements), provided that suitable parameters are selected. The unbiasedness properties and comparison results with the conventional (forward-in-time) moving horizon estimation are discussed. The effectiveness of the proposed approach is validated through a numerical example.

Data‐driven parallel Koopman subsystem modeling and distributed moving horizon state estimation for large‐scale nonlinear processes

AbstractIn 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.

Moving Horizon State Estimation for Linear System with Application to Autonomous Vehicle

AbstractThis paper proposes moving horizon estimation (MHE) to estimate the state variables of autonomous vehicle linear systems under measurement noises. To solve the MHE optimization problem, quadratic programming is employed. The steering angle, yaw angle, and global position constraints of an autonomous vehicle are considered in the estimation design. According to the simulation results, it can be observed that although the longer MHE step can give better results compared to the shorter MHE step, the difference in the MHE step only slightly affects the estimated results. However, the longer MHE step can increase the computational time. Additionally, the proposed MHE scheme is compared to the Kalman filter (KF) estimator. Based on the obtained results, the KF gives a better estimation than the MHE, but this notion must be verified for other case studies.Keywords: autonomous vehicle; Kalman filter; linear system; MHE; quadratic programming. AbstrakPaper ini mengusulkan moving horizon estimation (MHE) untuk mengestimasi variabel keadaan sistem linier kendaraan otonom karena pengaruh noise pengukuran. Untuk menyelesaikan masalah optimasi MHE, digunakan pemrograman kuadratik. Kendala sudut kemudi, sudut yaw dan posisi global dari kendaraan otonom dipertimbangkan dalam desain estimasi. Dari hasil simulasi dapat diketahui bahwa meskipun langkah MHE yang lebih panjang dapat memberikan hasil yang lebih baik dibandingkan dengan langkah MHE yang lebih pendek, perbedaan langkah MHE hanya sedikit mempengaruhi hasil estimasi. Namun, langkah MHE yang semakin panjang dapat meningkatkan waktu komputasi. Selain itu, skema MHE yang diusulkan dibandingkan dengan estimator Kalman filter (KF). Berdasarkan hasil yang diperoleh, KF memberikan estimasi yang lebih baik daripada MHE, tetapi gagasan ini harus diverifikasi untuk studi kasus lainnya.Kata Kunci: kendaraan otonom; Kalman filter; sistem linier; MHE; pemrograman kuadratik. 2020MSC: 62P35, 65D19

Nonlinear receding-horizon filter approximation with neural networks for fast state of charge estimation of lithium-ion batteries

An important component of electric vehicles is the Battery Management System (BMS), whose main objective is to monitor the state of charge (SoC) of the battery, which can significantly impact the vehicle’s autonomy. The SoC may be estimated using filtering algorithms; in this context, higher accuracy and computational complexity are of great importance. The present paper aims to propose receding-horizon strategies, namely Moving-Horizon State Estimation (MHSE) and Neural Network Moving-Horizon Estimation (NNMHE), for SoC estimation. MHSE is based on a constrained optimization problem, with information of measured samples along a larger observation window, which ensures high accuracy and robustness but requires a better processing capacity. Simulated results are obtained through this method, and it is demonstrated its capacity to jointly estimate the states and the unknown lumped parameters of the battery model using an augmented states formulation. The accuracy of MHSE in the process is high enough that its results may be used for training the NNMHE, so that a machine learning-based solution, with reduced processing time, is found. In this proposed method, a Neural Network is used to emulate the optimization problem solver, by which faster and approximate results are obtained. This approach is evaluated with an experimental dataset, achieving a coefficient of determination of almost 99% and about 20 times faster, which proves that it is effective and can be readily employed in an embedded systems application requiring less computational resources.

Anytime Proximity Moving Horizon Estimation: Stability and Regret

In this paper, we address the efficient implementation of moving horizon state estimation of constrained discrete-time linear systems. We propose a novel iteration scheme which employs a proximity-based formulation of the underlying optimization algorithm and reduces computational effort by performing only a limited number of optimization iterations each time a new measurement is received. We outline conditions under which global exponential stability of the underlying estimation errors is ensured. Performance guarantees of the iteration scheme in terms of regret upper bounds are also established. A combined result shows that both exponential stability and a sublinear regret which can be rendered smaller by increasing the number of optimization iterations can be guaranteed. The stability and regret results of the proposed estimator are showcased through numerical simulations.

Iterative distributed moving horizon estimation of linear systems with penalties on both system disturbances and noise

In this paper, partition-based distributed state estimation of general linear systems is considered. A distributed moving horizon state estimation scheme is developed via decomposing the entire system model into subsystem models and partitioning the global objective function of centralized moving horizon estimation (MHE) into local objective functions. The subsystem estimators of the distributed scheme that are required to be executed iteratively within each sampling period are designed based on MHE. Two distributed MHE algorithms are proposed to handle the unconstrained case and the case when hard constraints on states and disturbances, respectively. Sufficient conditions on the convergence of the estimates and the stability of the estimation error dynamics for the entire system are derived for both cases. A benchmark reactor-separator process example is introduced to illustrate the proposed distributed state estimation approach.

Moving Horizon Estimation for Digital Twins using Deep Autoencoders

Digital twins have emerged in recent years as black-box, software-based simulation tools that can mirror the behavior of complex dynamical systems. Digital twin simulations generate the same outputs as the target system using internal states; however, these states are not readily available online from the real system. In this paper, we develop a data-driven moving horizon estimation framework capable of using online noisy measurements of the real system in order to estimate digital twin states. Our framework combines the high expressiveness of deep autoencoders with a moving horizon state estimator that accurately predicts the internal state of the black-box digital twin without access to an analytical model of the system dynamics. We demonstrate that our approach outperforms extended and Koopman Kalman filter solutions on a benchmark reverse van der Pol oscillator example.

Partition-based distributed moving horizon state estimation with system disturbances and sensor noise penalties

In this article, partition-based distributed state estimation of general linear systems is considered. A distributed moving horizon state estimation algorithm is developed via partitioning the entire system model and the global objective function of centralized moving horizon estimation into subsystem models and local objective functions, respectively. Based on moving horizon estimation, we design unconstrained subsystem estimators of the distributed scheme. These estimators are required to be executed iteratively within each sampling period. The objective function of each estimator penalizes both the estimates of system disturbances and the estimate of output measurement noise. Convergence and stability of the estimation error dynamics are analyzed under the unconstrained setting. A benchmark chemical example is used to illustrate the proposed approach.

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.

Fault detection and isolation for uncertain linear systems: A robust moving horizon estimation scheme using LMIs

In this paper, a Fault Detection and Isolation (FDI) scheme based on moving horizon state estimation (MHE) ideas is analyzed for plants modeled as uncertain linear systems subject to actuator fault occurrences. The key point is to exploit the MHE approach to ameliorate the FDI capabilities of the diagnostic filter unit. The design is numerically accomplished via a semidefinite programming min-max optimization problem subject to H∞ requirements so that the robust MHE is in charge to cope with any source of uncertainties. The effectiveness of the proposed approach is then verified by considering a multi-area power system subject to anomalies on the power generation units.

Fast-track realization of reactive microemulsion systems – Systematic system analysis and tailored application of PSE methods

Despite the overarching necessity for sustainability, the realization of novel process concepts using innovative green reaction media, such as microemulsions is still hindered by their inherent complexity. To overcome this, a holistic guideline for the fast-track realization of microemulsion systems is presented and applied on a mini-plant for the hydroformylation of 1-dodecene in microemulsions. It combines a rigorous system analysis and identification of challenges for process design and operation. Here, critical unmeasurability of concentrations, small feasible operation regions, and a highly dynamic system behavior are identified. This is encountered by tailored PSE methods: a dynamic model for the complex three-phasic separation of microemulsions, the formulation of a concentration soft-sensor, as well as multi-rate moving horizon state estimation and subsequent dynamic real-time optimization. Applying those, stable operation of the mini-plant for up to 200 h is ensured together with a steady-state product yield of 38% and a product selectivity of 92%.

Distributed moving horizon state estimation for sensor networks with low computation capabilities

This paper focuses on distributed state estimation for sensor network observing a discrete-time linear system. The provided solution is based on a Distributed Moving Horizon Estimation (DMHE) algorithm considering a pre-estimating Luenberger observer in the formulation of the local problem solved by each sensor. This leads to reduce the computation load, while preserving the accuracy of the estimation. Moreover, observability properties of local sensors are used for tuning the weights related to consensus information fusion built on a rank-based condition, in order to improve the convergence of the estimation error. Results obtained by Monte Carlo simulations are provided to compare the performance with existing approaches, in terms of accuracy of the estimations and computation time.

Powder composition monitoring in continuous pharmaceutical solid-dosage form manufacturing using state estimation – Proof of concept

In continuous solid-dosage form manufacturing, the powder feeding system is responsible for supplying downstream the correct formulation of the drug product ingredients. The composition of the powder delivered by the feeding system is inferred from the measurements of powder mass flow from the system feeders. The mass flows are, in turn, inferred from the loss in weight measured in the feeder hoppers. Most loss-in-weight feeders post-process the mass flow signal to deliver a smoothed value to the user. However, such estimated mass flows can exhibit a low signal-to-noise ratio. As the feeders are critical elements of the control strategy of the manufacturing line, better instantaneous estimates of mass flow are desirable for improving the quality assurance. In this study, we propose a model-based approach for monitoring the composition of the powder fed to a continuous solid-dosage line. The monitoring system is based on a moving-horizon state estimator, which carries out model-based reconciliation of the feeder mass measurements, thus enabling accurate composition estimation of the powder mixture. Experimental datasets from a direct compression line are used to validate the methodology. Results demonstrate improvement with respect to current industrial solutions.

Simultaneous State and Parameter Estimation: The Role of Sensitivity Analysis

State and parameter estimation is essential for process monitoring and control. Observability plays an important role in both state and parameter estimation. In simultaneous state and parameter estimation, the parameters are often augmented as extra states of the original system. When the augmented system is observable, various existing state estimation approaches may be used to estimate the states and parameters simultaneously. However, when the augmented system is not observable, how we should proceed to maximally extract the information contained in the measured outputs is not clear. This paper concerns about simultaneous state and parameter estimation when the augmented system is not fully observable. Specifically, we first show how sensitivity analysis is related to observability of a dynamical system, and then illustrate how it may be used to select variables for simultaneous estimation. We also propose a moving horizon state estimation (MHE) design that can use the variable selection results in a natural way. Extensive simulations are carried out to show the efficiency of the proposed approach.

Towards adaptive high-power lasers: Model-based control and disturbance compensation using moving horizon estimators

Thermally-induced disturbances limit the beam quality of high-power lasers and evoke set point-dependent disturbances. As a result, the beam parameters and the laser’s output power of conventional high-power lasers cannot be chosen independently. To overcome this issue, we suggest spherical intra-cavity mirrors as actuators to dynamically compensate the impact of thermal lensing within the framework of automatic control. While the optical system is represented stationary as a Gaussian cavity with multiple transverse modes, the disturbance and the actuator are dynamically modeled by distributed parameters systems. A reliable estimation and prediction of the disturbance is crucial to achieve a proper rejection. To this end, an optimization-based moving horizon state estimator is designed to cope with model uncertainties, noisy measurements and bifurcations causing parasitic solutions of the nonlinear output equation. To maintain real-time capability, the relevant transient effects are lumped to an efficient discrete-time design model. The proposed compensation strategy explicitly considers input constraints and involves a reference governor to ensure the feasibility of the control objective. The performance of the individual components of the controller are validated by means of simulations and experiments. Eventually, we prove the overall concept of controlled adaptive high-power lasers by means of a scalable experimental setup with output lasers powers exceeding 500W.

Moving-Horizon Estimation for Linear Dynamic Networks With Binary Encoding Schemes

This article is concerned with moving-horizon state estimation problems for a class of discrete-time linear dynamic networks. The signals are transmitted via noisy network channels and distortions can be caused by channel noises. As such, the binary encoding schemes, which take advantages of the robustness of the binary data, are exploited during the signal transmission. More specifically, under such schemes, the original signals are encoded into a bit string, transmitted via memoryless binary symmetric channels with certain crossover probabilities, and eventually restored by a decoder at the receiver. Novel centralized and decentralized moving-horizon estimators in the presence of the binary encoding schemes are constructed by solving the respective global and local least-square optimization problems. Sufficient conditions are obtained through intensive stochastic analysis to guarantee the stochastically ultimate boundedness of the estimation errors. A simulation example is presented to verify the effectiveness of the proposed moving-horizon estimators.

A Decentralized Framework for Parameter and State Estimation of Infiltration Processes

The Richards’ equation is widely used in the modeling soil water dynamics driven by the capillary and gravitational forces in the vadose zone. Its state and parameter estimation based on field soil moisture measurements is important and challenging for field applications of the Richards’ equation. In this work, we consider simultaneous state and parameter estimation of systems described by the three dimensional Richards’ equation with multiple types of soil. Based on a study on the interaction between subsystems, we propose to use decentralized estimation schemes to reduce the complexity of the estimation problem. Guidelines for subsystem decomposition are discussed and a decentralized estimation scheme developed in the framework of moving horizon state estimation is proposed. Extensive simulation results are presented to show the performance of the proposed decentralized approach.

Efficient Hardware Implementation of Nonlinear Moving-horizon State Estimation with Artificial Neural Networks

In this contribution we investigate the application of radial basis functions artificial neural networks embedded in hardware for real-time moving-horizon state estimation. The solution of the optimal moving-horizon state estimation problem may be faced as the mapping from the inputs and outputs of the system to the state estimates according to the system model. This mapping may be solved offline with the optimal formulation and then approximated by any higher order function approximation algorithm, such as the ones from machine learning. An approximate version with radial basis functions neural networks is developed and implemented in a Field Programmable Gate Array (FPGA) showing good results in terms of accuracy and computational time. We show that the state estimate using the approximate version of the moving-horizon algorithm can be run using a laboratory scale kit of approximately 500 kHz for an inverted pendulum at a clock rate of about 110 MHz. The latency to provide an estimate can be further reduced when FPGAs with higher clocks are used as the artificial neural network architecture is inherently parallel.

Distributed monitoring of the absorption column of a post‐combustion CO2 capture plant

SummaryIn this work, we consider the monitoring of the absorption column of a typical post‐combustion capture plant within a distributed framework. The column is decomposed into a few subsystems, and a distributed moving horizon state estimator network is designed to estimate the state of the entire column. A state predictor is also designed for each subsystem to approximate the future evolution of the subsystem. These subsystem predictors form a distributed predictor network. Based on the distributed estimator and predictor networks, a fault diagnosis mechanism is proposed specifically for potential additive sensor faults. Extensive simulation results are provided to demonstrate the performance of the proposed approach.

Fast moving horizon state estimation for discrete‐time systems with linear constraints

SummaryFast moving horizon state estimation for nonlinear discrete‐time systems affected by disturbances is addressed by means of imperfect optimization at each time instant based on few iterations of the gradient, conjugate gradient, and Newton algorithms. Linear constraints on the state vector are taken into account through a projection on the subspace associated with such constraints. The stability of the estimation error for the resulting scheme is proved under suitable conditions. The effectiveness of the proposed approach is showcased via simulation results in comparison with moving horizon estimation based on complete optimization and extended Kalman filtering.