Moving Horizon Estimation Research Articles

Analysis of Optimal Deep Learning Approach for Battery Health Condition Monitoring inElectric Vehicle

Compared with other commonly used batteries, lithium-ion batteries are featured by high energy density, high power density, long service life and environmental friendliness and thus have found wide application in the area of consumer electronics. The narrow area in which lithium-ion batteries operate with safety and reliability necessitates the effective control and management of battery management system. This present paper, through the analysis of literature and in combination with our practical experience, gives a brief introduction to the composition of the battery management system (BMS). First-principles models that incorporate all of the key physics that affect the internal states of a lithium-ion battery are in the form of coupled nonlinear PDEs. While these models are very accurate in terms of prediction capability, the models cannot be employed for on-line control and monitoring purposes due to the huge computational cost. A reformulated model is capable of predicting the internal states of battery with a full simulation running in milliseconds without compromising on accuracy. This paper demonstrates the feasibility of using this reformulated model for control-relevant real-time applications. The reformulated model is used to compute optimal protocols for battery operations to demonstrate that the computational cost of each optimal control calculation is low enough to be completed within the sampling interval in model predictive control (MPC). Observability studies are then presented to confirm that this model can be used for state-estimation-based MPC. A moving horizon estimator (MHE) technique was implemented due to its ability to explicitly address constraints and nonlinear dynamics. The MHE uses the reformulated model to be computationally feasible in real time. The feature of reformulated model to be solved in real time opens up the possibility of incorporating detailed physics-based model in battery management systems (BMS) to design and implement better monitoring and control strategies

Nonlinear full information and moving horizon estimation: Robust global asymptotic stability

In this paper, we propose time-discounted schemes for full information estimation (FIE) and moving horizon estimation (MHE) that are robustly globally asymptotically stable (RGAS). We consider general nonlinear system dynamics with nonlinear process and output disturbances that are a priori unknown. For FIE being RGAS, our only assumptions are that the system is time-discounted incrementally input–output-to-state-stable (i-IOSS) and that the time-discounted FIE cost function is compatible with the i-IOSS estimate. Since for i-IOSS systems such a compatible cost function can always be designed, we show that i-IOSS is sufficient for the existence of RGAS observers. Based on the stability result for FIE, we provide sufficient conditions such that the induced MHE scheme is RGAS as well for sufficiently large horizons. For both schemes, we can guarantee convergence of the estimation error in case the disturbances converge to zero without incorporating a priori knowledge. Finally, we present explicit converge rates and show how to verify that the MHE results approach the FIE results for increasing horizons.

A Hierarchical Estimation Method for Road Friction Coefficient Combining Single-Step Moving Horizon Estimation and Inverse Tire Model

To improve the real-time performance of the estimation method of road friction coefficient (RFC) based on moving horizon estimation (MHE), a hierarchical estimation method for RFC combining single-step MHE (S-MHE) and inverse tire model (ITM) based on lateral vehicle dynamics is proposed in this study. Firstly, a hierarchical estimation framework is designed to decouple vehicle and tire systems. Secondly, the S-MHE estimator is designed based on the nonlinear vehicle model to estimate the lateral tire force. Thirdly, the ITM is deduced based on the Pacejka model, and the estimator for the RFC based on the ITM is designed. Finally, the estimation accuracy, convergence speed, and real-time performance of the proposed method and the traditional MHE method are compared and discussed through different tests based on CarSim and Simulink. The results show that compared with the traditional MHE method, the proposed method reduces the average computation time to about 0.125 s and improves the real-time performance by more than 30% while ensuring the estimation accuracy and convergence speed.

A filter design for T-S fuzzy systems based on moving horizon estimator with measurement noise.

In this article, a filter based on moving horizon estimator is proposed with Takagi-Sugeno (T-S) fuzzy controllers for a kind of unknown discrete-time system. The T-S fuzzy control algorithm is employed to handle the unknown system dynamics, thus ensuring the property of input-to-state stability (ISS) of the system, which guarantees the boundedness of all states. Besides, the proposed filter and controller can significantly improve the robustness of the system with external disturbance, even if the disturbance has non-Gaussian characteristics. Finally, the effectiveness of the presented algorithm is demonstrated by simulation examples under two kind of noise situations.

Multi-vehicle localization by distributed MHE over a sensor network with sporadic measurements: Further developments and experimental results

This paper proposes a Distributed Moving Horizon Estimation (DMHE) approach performed by an external static Sensor Network (SN) composed of surveillance cameras and their associated low-cost computers. This approach allows to localize a non-cooperative Multi-Vehicle System (i.e. intruder vehicles which do not communicate with the SN) from sporadic measurements. In this context, measurements are available at time instants a priori unknown and the proposed DHME technique is designed to face this issue by resorting to time-dependent parameters in the problem formulation. Moreover, this technique is well-suited to better estimate the state of the intruder vehicles thanks to its capability to efficiently exploit environmental information via constraints. In fact, when dealing with sporadic measurements and biased noisy sensors data, the use of output constraints can contribute to locally enhance the estimation accuracy. In order to confirm its effectiveness, the proposed method is validated on an experimental setup (video presentation available at https://youtu.be/1CkSba2wVuI) within an indoor arena equipped with a motion capture system. Three scenarios are considered for the localization of a non-cooperative Multi-Vehicle System composed of five robots, where the proposed DMHE technique is performed using sporadic position measurements provided by an external static Sensor Network with low-cost cameras (webcams) and computers (Raspberry PI) connected to them. Rigorous comparisons in terms of computation time and accuracy of the estimates highlight the efficacy of the proposed approach.

Monitoring intracellular metabolite concentrations by moving horizon estimation based on kinetic modeling

The biotechnology industry can significantly benefit from new paradigms such as smart manufacturing, digitalization and quality-by-design to render more competitive and robust processes. Real-time monitoring of key process parameters and performance indicators can facilitate the transition toward smart biomanufacturing. Since cells are typically used to catalyze biotechnological processes, online monitoring of the cell's health and intracellular metabolic status is of interest to the bioprocess industry. However, intracellular monitoring is challenging due to the intrinsic physical limitation imposed by the cell wall/membrane and the long time required for analytical measurements. This work outlines how soft sensors based on moving horizon estimation allow inferring the intracellular metabolite concentrations in bioprocesses, even for advanced or complex metabolic systems. For example, it can be applied for monitoring metabolic cybergenetic systems, whereby metabolic pathways are dynamically regulated via genetic circuits and external inputs. The moving horizon estimator uses a kinetic model of the central carbon and energy metabolism that describes the dynamics of the intracellular metabolites. We underline the use of moving horizon estimation considering the anaerobic fermentation of Escherichia coli with optogenetic regulation of the adenosine triphosphate turnover. With the information on the extracellular substrate, product and biomass concentrations, we could reconstruct the internal cell's metabolic state with a good performance.

Real-time Battery State of Charge and parameters estimation through Multi-Rate Moving Horizon Estimator

For reliable and safe battery operations, accurate and robust State of Charge (SOC) and model parameters estimation is vital. However, the nonlinear dependency of the model parameters on battery states makes the problem challenging. We propose a Moving-Horizon Estimation (MHE)-based robust approach for joint state and parameters estimation. Dut to all the time scales involved in the model dynamics, a multi-rate MHE is designed to improve the estimation performance. Moreover, a parallelized structure for the observer is exploited to reduce the computational burden, combining both multi-rate and a reduced-order MHEs. Results show that the battery SOC and parameters can be effectively estimated. The proposed MHE observers are verified on a Simulink-based battery equivalent circuit model.

A Simple Suboptimal Moving Horizon Estimation Scheme With Guaranteed Robust Stability

We propose a suboptimal moving horizon estimation (MHE) scheme for a general class of nonlinear systems. To this end, we consider an MHE formulation that optimizes over the trajectory of a robustly stable observer. Assuming that the observer admits a Lyapunov function, we show that this function is an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {M}$ </tex-math></inline-formula> -step Lyapunov function for suboptimal MHE. The presented sufficient conditions can be easily verified in practice. We illustrate the practicability of the proposed suboptimal MHE scheme with a standard nonlinear benchmark example. Here, performing a single iteration is sufficient to significantly improve the observer’s estimation results under valid theoretical guarantees.

Command Value Determination Considering Generation Cost and Control Performance Using Model Predictive Control in Load Frequency Control

This paper deals with load frequency control (LFC) problem for power system. The control is conducted to retain power demand supply balance by controlling output of generators in power network. In order to realize LFC with low power generation cost while maintaining control performance, this paper proposes frequency control method using Moving Horizon Estimation (MHE) and Model Predictive Control (MPC). First, MHE is used to estimate the state of the system. Next, optimization calculation is performed using MPC to determine the command values for each generator. Finally, simulation of load frequency control is carried out to verify the effectiveness of the proposed method.

Frequency-Based Estimation of the B-H Curve of Steel Strips in a Continuous Induction Furnace*

A parameter estimation strategy for continuous inductive heating of ferromagnetic steel strips is developed and investigated. During strip processing in an induction furnace, the parameters of the magnetization curve (B-H curve) of the strip material are identified by a moving horizon estimator (MHE). This estimator uses measurements of the resonance frequency of the electric circuit. A tailored 2D harmonic model of the electromagnetic subsystem is used to calculate the complex coil impedance. Other model parameters are identified in a scenario, where the magnetization curve of the strip is known. The approach does not require a thermal model of the induction furnace in favor of the computing time and the achieved accuracy. For model validation, the simulated resonance frequency is compared with measurements. To avoid a high computational effort, the effective magnetization approach is applied to approximate the solution of the nonlinear Maxwell equations. The effective B-H curve is mathematically expressed in terms of a linear and an exponential function and parametrized by two parameters. The MHE is validated in a test scenario using a strip with a known magnetization curve.

Evaluation of Changes in Feed Particle Size within an Economic Model Predictive Control Strategy for Froth Flotation

This study presents the evaluation of the impact of feed particle size on an Economic Model Predictive Control (E-MPC) strategy for a flotation bank. The effect of particle size was assessed under two scenarios: (1) assuming constant floatability with no dependency on particle size, and (2) assuming variable floatability as a function of particle size. The E-MPC strategy uses a dynamic model that includes froth physics, which was previously calibrated and validated using experimental data. Two typical control variables were considered: air flowrate and pulp height setpoints. The proposed objective function depends on three flotation variables: (1) dynamic air recovery, a measurable variable used to quantify froth stability and is directly related to flotation performance, (2) metallurgical recovery at steady-state, and (3) dynamic concentrate grade. A moving horizon estimator (MHE) was implemented to estimate the model states in both scenarios. Simulation results showed that the estimation of metallurgical indicators (concentrate grade and recovery) was significantly affected by changes in the floatability parameter. A poor estimate of floatability is likely to lead to very different results for the control strategy. Future research should focus on estimating and updating the most significant parameters of the dynamic model, such as floatability, with an appropriate sampling time.

Robust Moving-Horizon Estimation for Quasi-LPV Discrete-Time Systems*

This paper deals with moving horizon estimation for a class of quasi-LPV systems. Under both observability condition and incremental exponential input output-to-state stability (i-EIOSS) assumption, novel stability conditions of the moving horizon estimator (MHE) are proposed. Such conditions guarantee exponential robust stability of the MHE based on a particular prediction step that is independent of the dynamic of the system. An application to vehicle motion estimation, using the kinematic model, is provided to show the validity and effectiveness of the proposed method, and to support the theoretical results.

On an Integral Variant of Incremental Input/Output-to-State Stability and Its Use as a Notion of Nonlinear Detectability

We propose a time-discounted integral variant of incremental input/output-to-state stability (i-iIOSS) together with an equivalent Lyapunov function characterization. Continuity of the i-iIOSS Lyapunov function is ensured if the system satisfies a certain continuity assumption involving the Osgood condition. We show that the proposed i-iIOSS notion is a necessary condition for the existence of a robustly globally asymptotically stable observer mapping in a time-discounted ``$L^2$-to-$L^\infty$'' sense. In combination, our results provide a general framework for a Lyapunov-based robust stability analysis of observers for continuous-time systems, which in particular is crucial for the use of optimization-based state estimators (such as moving horizon estimation).

Anderson Accelerated Feasible Sequential Linear Programming

This paper proposes an accelerated version of Feasible Sequential Linear Programming (FSLP): the AA(d)-FSLP algorithm. FSLP preserves feasibility in all intermediate iterates by means of an iterative update strategy based on repeated evaluation of zero-order information. This technique was successfully applied to techniques such as Model Predictive Control and Moving Horizon Estimation, but it can exhibit slow convergence. Moreover, keeping all iterates feasible in FSLP entails a large number of additional constraint evaluations. In this paper, Anderson Acceleration (AA(d)) is applied to the zero-order update strategy improving the convergence rate and therefore decreasing the number of constraint evaluations in the inner iterative procedure of the FSLP algorithm. AA(d) achieves an improved contraction rate in the inner iterations, with proven local linear convergence. In addition, it is observed that due to the improved zero-order update strategy, AA(d)-FSLP takes larger steps to find an optimal solution, yielding faster overall convergence. The performance of AA(d)-FSLP is examined for a time-optimal point-to-point motion problem of a parallel SCARA robot. The reduction of the number of constraint evaluations and overall iterations compared to FSLP is demonstrated.

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.

Bioptim, a Python Framework for Musculoskeletal Optimal Control in Biomechanics

Musculoskeletal simulations are useful in biomechanics to investigate the causes of movement disorders, estimate nonmeasurable physiological quantities, or study the optimality of human movement. We introduce Bioptim, an easy-to-use Python framework for biomechanical optimal control based on both direct multiple shooting and direct collocation, handling musculoskeletal models. Relying on algorithmic differentiation, Bioptim is fast and it interfaces several nonlinear solvers. The software is both computationally efficient (C++ core) and easily customizable, thanks to its Python interface. It allows to quickly define a variety of biomechanical problems, such as motion tracking/prediction, muscle-driven simulations, parameters optimization, multiphase problems, etc. It is also intended for real-time applications, such as moving horizon estimation and model predictive control.

Robust Convergence Analysis of Moving-Horizon Estimator for LPV Discrete-Time Systems*

This paper deals with the problem of robust stability analysis of Moving Horizon Estimator (MHE) for Linear Parameter Varying (LPV) systems. The main contribution of the paper lies in the introduction of novel stability analysis tools guaranteeing exponential robust convergence of the MHE under only the incremental Exponential Input-Output-to-State Stability (i-EIOSS) assumption. Indeed, the i-EIOSS property characterizes the detectability of a system, which is a less conservative assumption compared to the observability condition.

Nonlinear Moving Horizon Estimation for a High-Rise Adaptive Structure

The scarcity of resources and the rising population are becoming an increasing problem that require innovations in the construction sector. Adaptive structures enable the reduction of structural weight by being equipped with sensors and actuators to withstand external loads such as wind or earthquakes. As many buildings, these structures can have tension-only elements, which need to be modeled nonlinearly. A reliable control requires the knowledge of the system state. For this purpose, model-based estimators that account for the nonlinearities must be chosen. In this contribution, a nonlinear Moving Horizon Estimator (NMHE) is designed for a high-rise adaptive structure. In a simulation study with an excitation by an initial condition or an unknown wind disturbance, the NMHE is compared to linear filters. The NMHE outperforms the linear filters in all simulations. For a wind disturbance, it can be shown that linear filters are not sufficient to correctly reconstruct the state. In contrast to them, the estimation error of the NMHE is significantly reduced.

Variational Bayesian-Based Moving Horizon Estimation of Toolface for Rotary Steerable Drilling Tool Systems

This article is concerned with the estimation problem of Toolface for dynamic point-the-bit rotary steerable drilling tool systems. First, considering the unknown frequency and amplitude of vibration during the drilling process, the Toolface system is modeled as a time-varying stochastic system with unknown and time-varying noise covariance matrices. Under the assumption that the noises and their covariance matrices, respectively, obey the Gaussian distribution and the inverse Wishart distribution, the variational Bayesian-based moving horizon estimation algorithm is proposed. Then, the state and the noise covariance matrices are inferred by iteratively updating their approximate posterior probability distributions. Finally, simulations and experiments are provided to demonstrate the effectiveness and superiority of the developed estimation scheme.

Sub-Optimal Moving Horizon Estimation in Feedback Control of Linear Constrained Systems

Sub-Optimal Moving Horizon Estimation in Feedback Control of Linear Constrained Systems