Moving Horizon Estimation Research Articles

A novel robust moving horizon estimator for discrete-time linear systems subject to measurement outliers

State estimation is a crucial problem in modern industries and has been widely applied across various fields. The performance of the estimator depends on the quality of the measurement data. Measurements being corrupted by outliers is becoming an unavoidable phenomenon that leads to degradation of estimator performance. It is critical to develop estimators with outlier suppression capabilities to mitigate the adverse impact of measurement outliers. In this paper, we propose an effective outlier suppression technique for discrete-time linear systems within the framework of moving horizon estimation (MHE). The proposed estimator solves the issues of poor estimation accuracy and low computational efficiency among the existing MHE-based outlier-robust estimators. Moreover, the proposed method allows us to not only achieve robust state estimation but also detect outliers. Specifically, we propose a set of least-squares cost functions and an outlier identification mechanism to implement the estimation process. Subsequently, the stability of the estimation error of the proposed estimator is demonstrated. The estimation error can achieve exponential convergence by choosing appropriate design parameters. Lastly, the proposed estimator is applied to target tracking simulations and compared with state-of-the-art outlier-robust estimation methods, confirming the effectiveness and superiority of the proposed estimator.

A model-based framework for controlling activated sludge plants

This work presents a general framework for the advanced control of a common class of activated sludge plants (ASPs). Based on a dynamic model of the process and plant sensors and actuators, we design and configure a highly customisable Output Model-Predictive Controller (Output MPC) for the flexible operation of ASPs as water resource recovery facilities. The controller consists of a i) Moving-Horizon Estimator for determining the state of the process, from plant measurements, and ii) a Model-Predictive Controller for determining the optimal actions to attain high-level operational goals. The Output MPC can be configured to satisfy the technological limits of the plant equipment, as well as operational desiderata defined by plant personnel. We consider exemplary problems and show that the framework is able to control ASPs for tasks of practical relevance, ranging from wastewater treatment subject to normative limits, to the production of an effluent with varying nitrogen content, and energy recovery.

Flexible development and evaluation of machine‐learning‐supported optimal control and estimation methods via HILO‐MPC

AbstractModel‐based optimization approaches for monitoring and control, such as model predictive control and optimal state and parameter estimation, have been used successfully for decades in many engineering applications. Models describing the dynamics, constraints, and desired performance criteria are fundamental to model‐based approaches. Thanks to recent technological advancements in digitalization, machine‐learning methods such as deep learning, and computing power, there has been an increasing interest in using machine learning methods alongside model‐based approaches for control and estimation. The number of new methods and theoretical findings using machine learning for model‐based control and optimization is increasing rapidly. However, there are no easy‐to‐use, flexible, and freely available open‐source tools that support the development and straightforward solution to these problems. This article outlines the basic ideas and principles behind an easy‐to‐use Python toolbox that allows to solve machine‐learning‐supported optimization, model predictive control, and estimation problems quickly and efficiently. The toolbox leverages state‐of‐the‐art machine learning libraries to train components used to define the problem. Machine learning can be used for a broad spectrum of problems, ranging from model predictive control for stabilization, set point tracking, path following, and trajectory tracking to moving horizon estimation and Kalman filtering. For linear systems, it enables quick generation of code for embedded model predictive control applications. HILO‐MPC is flexible and adaptable, making it especially suitable for research and fundamental development tasks. Due to its simplicity and numerous already implemented examples, it is also a powerful teaching tool. The usability is underlined, presenting a series of application examples.

Dynamic Event-Triggered Intermittent Fault Detection for Time-Varying Stochastic Systems

In this paper, the problem of intermittent fault (IF) detection is investigated for stochastic linear time-varying (LTV) systems using dynamic event-triggered methods. Using the nonuniform sampling approach, the event-triggered system is transformed into a time-varying system with varying sampling periods. Using the moving horizon estimation strategy, a new IF detection filter is designed to generate residual signals which are decoupled from event-triggered transmission errors and estimation errors. Moreover, an event-triggered IF detection algorithm is proposed such that the appearance time and disappearance time of IFs can be detected quickly for stochastic LTV systems. In order to analyze the detectability of IFs for systems with/without event-triggered cases, the concept of distinguishability is introduced for IFs. Sufficient conditions are derived to guarantee the detectability of IFs for LTV systems. Finally, an experiment concerning the rotary steerable drilling tool system is provided to illustrate the effectiveness of the proposed IF detection method.

Robust stability of moving horizon estimation for continuous-time systems

Abstract We consider a moving horizon estimation (MHE) scheme involving a discounted least squares objective for general nonlinear continuous-time systems. Provided that the system is detectable (incrementally integral input/output-to-state stable, i-iIOSS), we show that there exists a sufficiently long estimation horizon that guarantees robust global exponential stability of the estimation error in a time-discounted L 2-to-L ∞ sense. In addition, we show that i-iIOSS Lyapunov functions can be efficiently constructed by verifying certain linear matrix inequality conditions. In combination, we propose a flexible Lyapunov-based MHE framework in continuous time, which particularly offers more tuning possibilities than its discrete-time analog, and provide sufficient conditions for stability that can be easily verified in practice. Our results are illustrated by a numerical example.

Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems

Abstract Motion planning and controller design are challenging tasks for highly coupled and nonlinear dynamical systems such as autonomous vehicles and robotic applications. Nonlinear model predictive control (NMPC) is an emerging technique in which sampling-based methods are used to synthesize the control and trajectories for complex systems. In this study, we have developed the sampling-based motion planning algorithm with NMPC through Bayesian estimation to solve the online nonlinear constrained optimization problem. In the literature, different filtration techniques have been applied to extract knowledge of states in the presence of noise. Due to the detrimental effects of linearization, the Kalman filter with NMPC only achieves modest effectiveness. Moving horizon estimation (MHE), on the other hand, frequently relies on simplifying assumptions and lacks an effective recursive construction. Additionally, it adds another optimization challenge to the regulation problem that has to be solved online. To address this problem, particle filtering is implemented for Bayesian filtering in nonlinear and highly coupled dynamical systems. It is a sequential Monte Carlo method that involves representing the posterior distribution of the state of the system using a set of weighted particles that are propagated through time using a recursive algorithm. For nonlinear and strongly coupled dynamical systems, the novel sampling-based NMPC technique is effective and simple to use. The efficiency of the suggested method has been assessed using simulated studies.

Generic Stability Implication From Full Information Estimation to Moving-Horizon Estimation

Optimization-based state estimation is useful for handling of constrained linear or nonlinear dynamical systems. It has an ideal form, known as full information estimation (FIE) which uses all past measurements to perform state estimation, and also a practical counterpart, known as moving-horizon estimation (MHE) which uses most recent measurements of a limited length to perform the estimation. This work reveals a generic link from robust stability of FIE to that of MHE, showing that the former implies at least a weaker robust stability of MHE which implements a long enough horizon. The implication strengthens to strict robust stability of MHE if the corresponding FIE satisfies a mild Lipschitz continuity condition. The revealed implications are then applied to derive new sufficient conditions for robust stability of MHE, which further reveals an intrinsic relation between the existence of a robustly stable FIE/MHE and the system being incrementally input/output-to-state stable.

Moving horizon estimation based on distributionally robust optimisation

This paper presents a novel moving horizon estimation approach based on distributionally robust optimisation to tackle the state estimation problem of non-linear systems with missing noise distribution information. The proposed method adopts a fuzzy set to mitigate the impact of uncertainties on state estimation. Specifically, the method derives an empirical distribution within the prediction window using a priori data and constructs a fuzzy sphere set using the Wasserstein metric with the empirical distribution as the sphere centre. This enables the estimation of the state sequence under the worst probability distribution of the fuzzy set. To demonstrate the effectiveness of the proposed method, a simple simulation example is conducted to compare its performance with that of traditional moving horizon estimation. The results provide evidence of the feasibility and superiority of the proposed approach.

Physics-Informed Online Learning by Moving Horizon Estimation: Learning Recurrent Neural Networks in Gray-box Models*

In Model Predictive Control (MPC) closed-loop performance heavily depends on the quality of the underlying prediction model, where such a model must be accurate and yet simple. A key feature in modern MPC applications is the potential for online model adaptation to cope with time-varying changes, part-to-part variations, and complex features of the system dynamics not caught by models derived from first principles. In this paper, we propose to use a physics-informed, or gray-box, model that extends the physics-based model with a data-driven component, namely a Recurrent Neural Network (RNN). Relying on physics-informed models allows for a rather limited size of the RNN, thereby enhancing online applicability compared to pure black-box models. This work presents a method based on Moving Horizon Estimation (MHE) for simultaneous state estimation and learning of the RNN sub-model, a potentially challenging issue due to limited information available in noisy input-output data and lack of knowledge of the internal state of the RNN. We provide a case study on a quadruple tank benchmark showing how the method can cope with part-to-part variations.

Moving Horizon Estimation With Variable Structure Interacting Multiple Model for Surrounding Vehicle States in Complex Environments

Moving Horizon Estimation With Variable Structure Interacting Multiple Model for Surrounding Vehicle States in Complex Environments

Estimation and Prediction of Glucose Appearance Rate for Use in a Fully Closed-Loop Dual-Hormone Intraperitoneal Artificial Pancreas.

A fully automated artificial pancreas requires a meal estimator and predictions of blood glucose levels (BGL) to handle disturbances during meal times, all without relying on manual meal announcements and user interventions. This study introduces a technique for estimating the glucose appearance rate (GAR) and predicting BGL in people with type 1 diabetes and insulin and glucagon administration. It is demonstrated for intraperitoneal insulin and glucagon delivery but may be adapted to other delivery sites. The estimator is designed based on the moving horizon estimation (MHE) approach, where the underlying cost function incorporates prior statistical information on the GAR in subjects over the course of a day. The proposed prediction scheme is developed to predict GAR using estimated states and an intestinal model, which is then used to predict BGL with the help of an animal glucose metabolic model. The intraperitoneal dual-hormone estimator was evaluated on three anesthetized animals, achieving a 21.8% mean absolute percentage error (MAPE) for GAR estimation and a 10.0% MAPE for BGL prediction when the future GAR is known. For a 120-minute prediction horizon, the proposed predictor achieved an 18.0% MAPE for GAR and a 28.4% MAPE for BGL. The findings demonstrate the effectiveness and reliability of the proposed estimator and its potential for use in a fully automated artificial pancreas and reducing user interventions. This study represents advancements toward the development of a fully automated artificial pancreas, ultimately enhancing the quality of life for people with type 1 diabetes.

Maximum-likelihood arrival cost for moving-horizon estimation – Application to mammalian cell culture

Moving-horizon estimation performance depends on the arrival-cost approximation. Different methods exist to approximate the arrival cost with different trade-offs, such as limited performance, complex implementation, or abusive computational cost. In particular, we can distinguish two different methods based on using an external filter or the optimality conditions to calculate the covariance of the estimations. This paper presents a simpler alternative for the arrival-cost approximation based on maximum-likelihood estimation. The performance of this approximation is assessed using a mammalian cell culture case study where biomass and glutamine concentrations are estimated.

A Recursive Approach to Approximate Arrival Costs in Distributed Moving Horizon Estimation

In this paper, we present a novel approach to distributed moving horizon estimation for nonlinear processes. The method involves approximating the arrival costs of local estimators through a recursive framework. First, we formulate distributed full-information estimation for linear unconstrained systems, which serves as the foundation for deriving the analytical expression of the arrival cost. Subsequently, a recursive arrival cost design for linear distributed moving horizon estimation is developed. Then, we extend the arrival cost design obtained for linear systems to account for the nonlinear context, and a partition-based constrained distributed moving horizon estimation algorithm for nonlinear systems is formulated. A chemical process is introduced to illustrate the effectiveness of the proposed method.

PID and Model Predictive Control Approach for Drug Dosage in Anesthesia During Induction: a Comparative Study

In this paper, a PID controller is compared to an extended moving horizon estimator coupled with a model predictive control approach for the problem of dosing propofol and remifentanil during the induction of anesthesia. For the PID controller, taken from the literature, a fixed ratio is considered between propofol and remifentanil flow rates and an anti-windup strategy is used to prevent integration wind-up. The optimal control approach uses an MHE to estimate both the states and the pharmacodynamic parameters of the system, followed by a non-linear model predictive controller to compute the optimal drug rates according to the model. Both controllers are tuned using the same criterion, and are compared by simulating 500 uncertain patient models for the induction phase.

Towards fast embedded moving horizon state-of-charge estimation for lithium-ion batteries

For state-of-charge (SOC) estimation with high precision, moving horizon estimation (MHE) has recently emerged as a competitive alternative to various Kalman filters or observers due to its significantly improved accuracy and robustness. However, MHE demands considerably higher computational complexity, preventing its deployment on a low-cost embedded microcontroller. To address this challenge, we propose a fast moving horizon SOC estimation algorithm to retain the benefits of MHE at a vastly reduced computational cost. In particular, we consider joint SOC and parameter estimation for an SOC-dependent equivalent circuit model subject to model mismatch. The proposed fast joint MHE (jMHE) algorithm performs a fixed number of Gauss–Newton (GN) iterations to approximate the fully converged solution. At each iteration, the GN Hessian matrix is factorized by exploiting its block tridiagonal structure to construct computationally efficient forward–backward recursions. To further speed up computations, another fast jMHE algorithm wtih an Event-Triggered Relinearization strategy (jMHE-ETR) is proposed to avoid refactorizing the GN Hessian matrix at each iteration. Using experimental datasets under different operating conditions, it is verified on both laptop and Raspberry Pi microcontroller that compared to the conventional optimal MHE solved with an off-the-shelf solver, the proposed fast jMHE and jMHE-ETR algorithms both have a slight loss of performance (still significantly better than extended Kalman filtering) while reducing its computational cost by an order of magnitude.

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.

Decentralized Moving Horizon Estimation for Large-Scale Networks of Interconnected Unconstrained Linear Systems

This paper addresses the problem of designing a decentralized state estimation solution for a large-scale network of interconnected unconstrained linear time invariant (LTI) systems. The problem is tackled in a novel moving horizon estimation (MHE) framework, while taking into account the limited communication capabilities and the restricted computational power and memory, which are distributed across the network. The proposed design is motivated by the fact that, in a decentralized setting, a Luenberguer-based framework is unable to leverage the full potential of the available local information. A method is derived to solve a relaxed version of the resulting optimization problem. It can be synthesized offline and its stability can be assessed prior to deployment. It is shown that the proposed approach allows for significant improvement on the performance of recent Luenberger-based filters. Furthermore, we show that a state-of-the-art distributed MHE solution with comparable requirements underperforms in comparison to the proposed solution.

Reliable Estimation of Automotive States Based on Optimized Neural Networks and Moving Horizon Estimator

Accurate estimation of vehicle sideslip angle and attitude angles are essential for the safety control and lateral behaviour of driving performance. In this article, the variation of wheels cornering stiffness is considered for sideslip estimation and addressed by introducing a recursive least squares approach. Based on the nonlinear vehicle dynamic model and the investigated coupling effect between lateral and longitudinal velocity, an optimized moving horizon estimator is proposed to obtain the vehicle sideslip angle, in which an iteration decent algorithm is integrated. Furthermore, a framework, consisting of inertial navigation system measurements, a dual neural network and a square-root cubature Kalman filter, is designed, such that the influence of sensor noise and varied maneuvers are alleviated when estimating the system states. Finally, extensive simulation and field experiments are carried out on different driving scenarios to verify the effectiveness of the developed method. The obtained results clearly indicate the satisfactory estimation accuracy of the designed strategy, superior to the existing estimation methods, such as sole neural networks methods and Kalman-based filters.

Distributed Moving-Horizon Estimation With Event-Triggered Communication Over Sensor Networks

This paper is concerned with the distributed moving-horizon estimation problem for linear discrete-time systems over wireless sensor networks. To reduce the communication burden, an event-triggered communication mechanism is proposed to govern the transmission of one-step predictions and measurements in each node. Under such a mechanism, a novel event-triggered distributed moving-horizon estimator is designed to deal with the state estimation problem subject to noise and state constraints. Each node computes its local state estimate by minimizing a cost function, which includes the event-triggered one-step predictions and measurements from the neighbors. It is proved that the developed distributed estimator is stable with the uniformly bounded estimation error in each node, provided by network connectivity, regional observability and small enough triggering parameters. A numerical simulation is given to verify the effectiveness of the proposed estimation algorithm.

Anomaly detection and multi-step estimation based remaining useful life prediction for rolling element bearings

The prediction of the Remaining Useful Life (RUL) of rotating machinery targets the avoidance of sudden machine failures and the optimization of the maintenance schedules as well as the management of repairs and replacements. A number of statistical model-based prognostics methodologies have been proposed in recent literature but a number of open challenges still exist, blocking their integration in the industry: (i) The extraction of a high-quality Health Indicator (HI) is critical for bearings’ prognostics but till now the trend of many classic HIs is often disturbed; deviating from a monotonical increase or decrease. (ii) The detection of the Start Prediction Time (SPT) on the HI is usually required in order to improve the accuracy of the RUL estimation but it can be incorrect especially when distinct random outliers occur. (iii) The widely applied RUL estimators, e.g. the Kalman filter (KF) and the Particle filter (PF), belong to the category of single-step estimation techniques that assume that the current step estimation depends only on the state of the previous step. However, the occurrence of bearings’ damage is usually not a sudden change but an accumulated process. Therefore a novel prognostic strategy for Rolling Element Bearings (REB) is proposed in this paper, which integrates a robust anomaly detection technique, the Support Vector Data Description (SVDD), with a multi-step estimation method, the Moving Horizon Estimation (MHE). Various advanced entropy and sparsity HIs are extracted by signals filtered at different frequency bands. By comparing all the extracted HIs based on a proposed criterion, a premium prognostic HI is chosen from a specific frequency band. Moreover, the SVDD, deployed and combined with several proposed constraints, can robustly detect the occurrence of a bearing fault and decide the proper SPT. Furthermore, the MHE is combined with two popular exponential statistical models and a second-order polynomial model for bearings’ prognostics. Experimental results from fifteen degraded bearings demonstrate that the MHE outperforms the classic Kalman filter and Particle filter. Moreover, the polynomial model achieves a better RUL estimation compared to the two classic exponential models in combination with the MHE.