Moving Horizon Estimation Research Articles

Moving horizon estimation over relay channels: Dealing with packet losses

This paper is concerned with the moving horizon estimation problem for networked systems over relay channels. The relay channel under consideration is a wireless communication channel with a so-called relay node. Under the effects of relay node, the signal transmission between the plant and estimator is determined by a combination of three communication processes, namely the sensor-to-estimator, the sensor-to-relay and the relay-to-estimator communication processes. In order to characterize the impact of such kind of relay-channel-based communication, a three-terminal relay model is employed to describe the signal transmission behaviors between the plant and the estimator. Three Bernoulli-distributed white sequences are adopted to model the packet losses occurring in the communication processes. A full information estimation strategy is firstly developed in the sense of maximum a posteriori according to the system dynamics and transmission behaviors. Then, a moving horizon estimator is designed based on the derived full information estimation strategy. Furthermore, the solution of the proposed moving horizon estimation strategy is achieved by using the completing-the-square approach and, subsequently, sufficient conditions are derived to guarantee the mean-square ultimate boundedness of the resultant error dynamics of the state estimation. Two simulation examples are given to demonstrate the effectiveness of our developed estimation scheme.

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.

Simultaneous moving horizon estimation and control for nonlinear systems subject to bounded disturbances

SummaryIn this work, we address the output–feedback control problem for nonlinear systems under bounded disturbances using a moving horizon approach. The controller is posed as an optimisation‐based problem that simultaneously estimates the state trajectory and computes future control inputs. It minimizes a criterion that involves finite backward and forward horizons with respect to the unknown initial state, measurement noises and control input variables. The main novelty of this work relies on linking the lengths of the forward and backward windows with the closed‐loop stability, assuming detectability and decoding sufficient conditions to assure system stabilizability. It leads to a formulation that does not require to be a Control Lyapunov Function for the terminal cost of the controller. Simulation examples are carried out to compare the performance of solving simultaneously and independently the estimation and control problems. Furthermore, the examples show how the controller influences the length of the estimation window through its gain.

Moving horizon estimation of Amnioserosa cell dynamics during Drosophila dorsal closure

Dorsal closure is a morphogenetic process similar to wound healing, whereby a gap in the epithelium is closed through the coordination of the elongation of epidermal cells and contraction of Amnioserosa cells (AS). Throughout the early stages of this process, the AS cells exhibit fluctuations of their apical area which provides an ideal system to understand contractile epithelia, both in terms of the cellular mechanisms and how tissue behavior emerges from the activity of individual cells. In this work, we present a vertex-based model to explore the dynamics of Amnioserosa cells and provide insights into the regulation of dorsal closure. Towards this end, moving horizon estimation is used to predict the mechanics of AS cell oscillations and contractions modeled as a quasi-linear parameter-varying system with bounded unknown parameters. We perform estimation through the minimization of the least-squares moving horizon cost with respect to both state variables and unknown parameters simultaneously. For simulation purposes, data from the work published in [13] are used to evaluate the performance of moving-horizon estimation as compared to the extended Kalman filter. The comparative analysis reveals that the moving-horizon approach performs better than those provided by the Kalman filtering and with an increased robustness to measurement and process noises.

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 false data injection attack design against cyber–physical systems

Systematic attack design is essential to understanding the vulnerabilities of cyber–physical systems (CPSs), to better design for resiliency. In particular, false data injection attacks (FDIAs) are well-known and have been shown to be capable of bypassing bad data detection (BDD) while causing targeted biases in resulting state estimates. However, their effectiveness against moving horizon estimators (MHE) is not well understood. In fact, this paper shows that conventional FDIAs are generally ineffective against MHE. One of the main reasons is that the moving window renders the static FDIA recursively infeasible. This paper proposes a new attack methodology, moving-horizon FDIA (MH-FDIA), by considering both the performance of historical attacks and the current system’s status. Theoretical guarantees for successful attack generation and recursive feasibility are given. Numerical simulations on the IEEE-14 bus system further validate the theoretical claims and show that the proposed MH-FDIA outperforms state-of-the-art counterparts in both stealthiness and effectiveness. In addition, an experiment on a path-tracking control system of an autonomous vehicle shows the feasibility of the MH-FDIA in real-world nonlinear systems.

State and Parameter Estimation for Retinal Laser Treatment

Adequate therapeutic retinal laser irradiation needs to be adapted to local absorption. This leads to time-consuming treatments as the laser power needs to be successively adjusted to avoid undertreatment and overtreatment caused by too low or too high temperatures. Closed-loop control can overcome this burden by means of temperature measurements. To allow for model predictive control schemes, the current state and the spot-dependent absorption need to be estimated. In this article, we thoroughly compare moving horizon estimator (MHE) and extended Kalman filter (EKF) designs for joint state and parameter estimation. We consider two different scenarios, the estimation of one or two unknown absorption coefficients. For one unknown parameter, both estimators perform very similarly. For two unknown parameters, we found that the MHE benefits from active parameter constraints at the beginning of the estimation, whereas after a settling time, both estimators perform again very similarly as long as the parameters are inside the considered parameter bounds.

Trajectory Estimation of a Flying Robot With a Single Ranging Beacon and Derived Velocity Constraints

The performance of the trajectory estimation with a single beacon highly depends on the observability of the system, which is further determined by the excitation patterns of the robot. Because the desired excitation patterns regarding specific observability conditions cannot always be guaranteed in real-time, the trajectory estimation may casually fail with divergence. To tackle this issue, indirectly derived velocity constraints are first introduced in this work, and their effects on improving the performance of trajectory estimation are comparatively investigated. Considering the geometric evolution equation and using the state augmentation technique, seven different velocity constraints are evaluated. The comparative results indicate that the trajectory estimation divergence can be most effectively avoided with a pure velocity magnitude constraint. On this basis, a velocity-constrained moving horizon estimation algorithm is implemented and verified in an indoor experimental testbed to accurately estimate the trajectory of the flying robot with a single ultra-wideband beacon in a long time interval. The results also quantitatively illustrate that the overall performance of the proposed approach outperforms that of the latest benchmark approach in both estimation precision and computational efficiency. In particular, with approximately the same estimation precision, the computational time cost of the proposed approach is less than 1/10 of that of existing approaches.

Distributed robust moving horizon estimation for multisensor systems with stochastic and norm‐bounded uncertainties

SummaryIn this study, a distributed robust moving horizon estimation (DRMHE) approach is proposed for time‐varying multisensor systems affected by stochastic and norm‐bounded uncertainties. First, by considering the uncertainties in all model matrices, stochastic min–max optimization problems in local and collective forms are presented to achieve the proposed estimator. Then, by converting the problems to robust regularized least‐squares problems with uncertain parameters, closed‐form solutions are obtained for estimations in both local and collective forms. Furthermore, the stability of the proposed estimator is investigated under some appropriate conditions. At last, two different examples are employed to show the robust performance and superiority of the proposed DRMHE approach compared to the existing methods.

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.

Closed-loop operation of a simulated recirculating aquaculture system with an integrated application of nonlinear model predictive control and moving horizon estimation

High water recirculation combined with a low rate of water exchange can deteriorate water quality and affect fish welfare in recirculating aquaculture systems (RAS). A nonlinear model predictive controller (NMPC) integrated with moving horizon estimation (MHE) was implemented in this work to maintain water quality parameters (e.g., oxygen, ammonia, and nitrate concentration) in RAS. The performance of the proposed control scheme was then evaluated in the presence of process uncertainty and measurement noise with full and partial accessibility to the system’s states under several common scenarios that can arise during the operation of RAS. In all scenarios, the proposed framework showed acceptable performance in keeping the water quality parameters close to their setpoints, demonstrating the high potential of this strategy in closed-loop operation of RAS.

Influence of Estimators and Numerical Approaches on the Implementation of NMPCs

Advanced control strategies, together with state-estimation methods, are frequently applied to nonlinear and complex systems. It is crucial to understand which of these are the most efficient methods for the best use of these approaches in a chemical process. In the current work, nonlinear model predictive control (NMPC) approaches were developed that considered three numerical methods: single shooting (SS), multiple shooting (MS), and orthogonal collocation (OC). Their performance was compared against the Van de Vusse reactor benchmark while considering set-point changes, unreachable set-point, disturbances, and mismatches. The results showed that the NMPC based on OC presented less computational cost than the other approaches. The extended Kalman filter (EKF), constrained extended Kalman filter (CEKF), and the moving horizon estimator (MHE) were also developed. The estimators’ performance was compared for the same benchmark by considering the computational cost and the mean squared error (MSE) for the estimated variables, thereby verifying the CEKF as the best option. Finally, the performance of the nine combinations of estimators and control approaches was compared to consider the Van de Vusse reactor and the same scenarios, thereby verifying the best performance of the CEKF with the OC. The present work can help with choosing the numerical method and the estimator for controlling chemical processes.

Suboptimal Nonlinear Moving Horizon Estimation

In this paper, we propose a suboptimal moving horizon estimator for a general class of nonlinear systems. For the stability analysis, we transfer the "feasibility-implies-stability/robustness" paradigm from model predictive control to the context of moving horizon estimation in the following sense: Using a suitably defined, feasible candidate solution based on an auxiliary observer, robust stability of the proposed suboptimal estimator is inherited independently of the horizon length and even if no optimization is performed. Moreover, the proposed design allows for the choice between two cost functions different in structure: the former in the manner of a standard least squares approach, which is typically used in practice, and the latter following a time-discounted modification, resulting in better theoretical guarantees. We apply the proposed suboptimal estimator to a nonlinear chemical reactor process, verify the theoretical assumptions, and show that even a few iterations of the optimizer are sufficient to significantly improve the estimation results of the auxiliary observer. Furthermore, we illustrate the flexibility of the proposed design by employing different solvers and compare the performance with two state-of-the-art fast MHE schemes from the literature.

Distributed Moving Horizon Estimation Over Wireless Sensor Networks: A Matrix-Weighted Consensus Approach

This brief proposes a novel approach to distributed moving horizon estimation for linear discrete-time systems over a wireless sensor network. A distributed moving horizon estimator is presented by minimizing a cost function involving consensus steps on the prediction. A matrix-weighted rule for the consensus steps is designed by combining an orthogonal matrix with a stochastic matrix, where the orthogonal matrix is obtained from the observability decomposition rule. The proposed estimator only requires that each node transmits one state vector over the network, which reduces the communication burden. The estimation error of the proposed estimator is bounded by choosing an appropriate scalar parameter and a sufficiently large consensus step. Finally, a distributed target tracking example is presented to verify the performance of the developed results.

Distributed Moving Horizon Fusion Estimation for Nonlinear Constrained Uncertain Systems

This paper studies the state estimation of a class of distributed nonlinear systems. A new robust distributed moving horizon fusion estimation (DMHFE) method is proposed to deal with the norm-bounded uncertainties and guarantee the estimation performance. Based on the given relationship between a state covariance matrix and an error covariance matrix, estimated values of the unknown parameters in the system model can be obtained. Then, a local moving horizon estimation optimization algorithm is constructed by using the measured values of sensor nodes themselves, the measured information of adjacent nodes and the prior state estimates. By solving the above nonlinear optimization problem, a local optimal state estimation is obtained. Next, based on covariance intersection (CI) fusion strategy, the local optimal state estimates sent to the fusion center are fused to derive optimal state estimates. Furthermore, the sufficient conditions for the square convergence of the fusion estimation error norm are given. Finally, a simulation example is employed to demonstrate the effectiveness of the proposed algorithm.

A first-order generalized pseudo-Bayesian method based on moving horizon estimation for surrounding vehicle states estimation in complex environments

Predicting the motion of surrounding vehicles is important for autonomous vehicle trajectory planning. To ensure the accuracy of motion prediction, it is particularly important to obtain information about the state of surrounding vehicles at the current moment. For this purpose, a first-order generalized pseudo-Bayesian method based on moving horizon estimation (GPB1-MHE) is proposed here. First, for vehicles, pedestrians and obstacles in complex environments, we design several different intention-based motion models to describe multiple maneuvers of vehicles during driving. Second, we replace the Kalman filter (KF) in the first-order generalized pseudo-Bayesian (GPB1) method with moving horizon estimation (MHE) to obtain GPB1-MHE. Compared with GPB1, it makes full use of the historical state information of a vehicle to improve the estimation accuracy. Finally, the effectiveness of the estimation method is verified using four operating conditions, overtaking, overtaken, pedestrian approaching, and pedestrian staying away, in a SCANeR studio and Simulink joint simulation environment.

Plant-Inspired Soft Growing Robots: A Control Approach Using Nonlinear Model Predictive Techniques

Soft growing robots, which mimic the biological growth of plants, have demonstrated excellent performance in navigating tight and distant environments due to their flexibility and extendable lengths of several tens of meters. However, controlling the position of the tip of these robots can be challenging due to the lack of precise methods for measuring the robots’ Cartesian position in their working environments. Moreover, classical control techniques are not suitable for these robots because they involve the irreversible addition of materials, which introduces process constraints. In this paper, we propose two optimization-based approaches, combining Moving Horizon Estimation (MHE) with Nonlinear Model Predictive Control (NMPC), to achieve superior performance in point stabilization, trajectory tracking, and obstacle avoidance for these robots. MHE is used to estimate the entire state of the robot, including its unknown Cartesian position, based on available configuration measurements. The proposed NMPC approach considers process constraints by relying on the estimated state to ensure optimal performance. We perform numerical simulations using the nonlinear kinematic model of a vine-like robot, one of the newly introduced plant-inspired growing robots, and achieve satisfactory results in terms of reduced computation times and tracking error.

A regularized Moving Horizon Estimator for combined state and parameter estimation in a bioprocess experimental application

Due to the lack or high costs of measurement devices to monitor and control metabolites in microbial cultivation processes, state estimators are often required. These estimators depend on available on-line measurements and model dynamics. However, they are often characterized by simple models due to the lack of full knowledge on the process dynamics and high variability in the cell metabolism. This causes uncertainty in the model parameters and therefore the necessity of on-line model adaptation, for instance through simultaneous state and parameter estimation. However, these estimation problems are often ill conditioned. The Moving Horizon Estimator (MHE) is a good candidate in this context, since it easily allows enforcing hard constraints as well as regularization to address the ill-posedness. In this work, we present a method for simultaneous state and parameter estimation in the absence of full state measurements, with the aid of two regularization methods, in a microbial fed-batch cultivation.

State and covariance estimation of a semi-batch reactor for bioprocess applications

In this work, extended Kalman filter (EKF) and moving horizon estimation (MHE)-based approaches are introduced and applied to a Galacto-oligosaccharides (GOS) bioprocess system considering a recently developed enzymatic model. Using this model, plant data is simulated by varying the kinetic parameters and applying white noise to the resulting output. In terms of EKF applications, both the canonical and parametric (Dual EKF) formulations have their covariances specified using a modified direct optimization (DO) algorithm to reduce estimation error. This work also outlines the development and application of a novel Parameter-based Moving Horizon Estimation (P-MHE) approach and directly compares it to traditional MHE formulations. The proposed P-MHE method reduces both the estimation error and computational time of traditional MHE approaches. When compared to EKF-based approaches, P-MHE can compete in terms of estimation error while exhibiting excellent robustness characteristics such as guaranteed feasibility. Despite the increased computational time of P-MHE when compared to the EKF formulations, state estimates can be obtained in under 3 seconds once a measurement arrives, allowing this algorithm to be applied to real-time process monitoring and other process systems.

Model predictive control and moving horizon estimation for adaptive optimal bolus feeding in high-throughput cultivation of E. coli

We discuss the application of a nonlinear model predictive control (MPC) and moving horizon estimation (MHE) framework to achieve optimal operation of E. coli fed-batch cultivations with intermittent bolus feeding. 24 parallel experiments were considered in a high-throughput mini-bioreactor platform at a 10 mL scale. The robotic facility can run up to 48 fed-batch processes in parallel with automated liquid handling, online and at-line analytics. Three main challenges emerge in implementing the model-based monitoring and control framework: First, the inputs are given in an instantaneous pulsed form by bolus injections; second, online and at-line measurement frequencies are severely imbalanced; and third, optimization for the distinctive multiple reactors can be either parallelized or integrated. We address these challenges by incorporating the concept of impulsive control systems, formulating multi-rate MHE with identifiability analysis, and suggesting criteria for deciding the reactor configuration. In this study, we present the key elements and background theory of the implementation with in silico simulations for bacterial fed-batch cultivations.