Kalman Filter For State Estimation Research Articles

Design, Manufacturing and Testing of a Two-Wheeled Self-Balancing Robot

This work presents a self-balancing robot's modeling, manufacturing, and testing. This study aimed to design and construct a robot capable of maintaining balance while stationary and in motion, employing an effective control strategy. The robot's design incorporates sensors, microcontrollers, actuators, and control algorithms. The control strategy involved implementing an LQR (Linear Quadratic Regulator) controller and a Kalman filter for state estimation. The results demonstrate the robot's ability to effectively maintain balance and navigate flat terrain while controlled by a joystick. This study provides valuable insights into the design and control of self-balancing robots.

Power System Frequency Dynamics Modeling, State Estimation, and Control using Neural Ordinary Differential Equations (NODEs) and Soft Actor-Critic (SAC) Machine Learning Approaches

With the global energy transition of the electric power system, grid control, supervision, and protection is becoming more challenging. With the increasing integration of renewable energy sources (RES), the system dynamics are changing, causing traditional power system dynamic modeling with swing equation-based modeling approaches to fail. Additionally, the converter-dominated power grid is decreasing the system inertia, making the power system more fragile to the frequency swings. This paper first investigates and compares the application of a model-based Kalman filter state estimation approach with (i) a model-free machine learning approach --- neural ordinary differential equations (NODEs) --- and (ii) a data-driven system identification (SysId) approach to model and infer critical state values of the power system frequency dynamics. Then a model predictive control (MPC) framework is compared to a model-free Soft Actor-Critic (SAC) reinforcement learning (RL) control algorithm in providing efficient fast frequency response (FFR) to the power system frequency dynamics. The approaches are compared in terms of their performance goals as well as their per-timestep computational efficiency. The comparative study for state estimation shows that for the model-free requirement, both NODEs and SysId can provide accurate state estimates; however, with increasing model complexity, NODEs can be a better choice for model identification. Similarly, the results from the FFR comparative study show that the SAC RL-based FFR, once trained, outperforms MPC with better control signals and faster computation time, making the SAC RL-based FFR better option for providing FFR to the power system.

Observer‐based residual‐driven dynamic compensation strategy for performance improvement of grid‐forming inverter

SummaryGrid‐forming (GFM) inverters offer stable frequency support for microgrid systems, even in the absence of synchronous generators. However, the GFM inverters have low inertia and vulnerability to system uncertainties and external disturbances. The conventional dual‐loop proportional integral (PI) control strategy, while widely used for its simplicity and robustness, suffers from poor dynamic performance. Motivated by this, this paper presents an observer‐based residual‐driven dynamic compensation (RDDC) strategy based on the coprime factorization technic and the Youla parameterization theory to achieve the primary control of the GFM inverter. The observer‐based RDDC strategy comprises four components: a PI controller for tracking control, a linear quadratic regulator (LQR) controller for dynamic adjustment, a residual generator based on the Kalman filter for state estimation and residual generation, and a residual compensation controller designed using model matching theory and solved through linear matrix inequality (LMI) methods for disturbance suppression. Simulation and experiment results consistently demonstrate that the observer‐based RDDC strategy ensures system robustness, enhances the dynamic and steady‐state performance of the GFM inverter system, and strengthens the ability of the GFM inverter to suppress disturbances.

Development of the consider cubature Kalman filter for state estimation of hydrological models with parameter uncertainty

Filtering methods are popular for improving the forecast performances of hydrological models. Parameter uncertainty can result in poor state estimates; hence, augmented state-parameter estimation methods have gained much attention in the hydrologic literature. However, researchers have not treated the robustness of augmented methods in much detail. Another powerful approach to include the parameter uncertainty in the state estimation process is the consider Kalman filter (consider KF), which accounts for the parameter uncertainty without updating the parameters. The consider KF has received little to no attention in the hydrologic context. This paper investigates the usefulness of the consider KF by updating the states of a hydrological model (without updating the parameters), with the augmented state-parameter estimation method as the benchmark. Two nonlinear extensions for the augmented state-parameter estimation method (augmented cubature Kalman filter, ACKF) and consider Kalman filter (consider cubature Kalman filter, CCKF) are developed since hydrological models are mostly nonlinear. Both filters are tested with a lumped hydrological model. The results of a state update synthetic case show that both filters can improve the accuracy of the updated states by including the parameter uncertainty, while the ACKF may deteriorate the original performance when given perfect parameters. The results of a reforecast experiment show that both filters can improve the forecast performance by updating the model states based on the observed discharge, while the ACKF may suffer from potential divergence. Either the ACKF or CCKF can be considered for state estimation in the presence of parameter uncertainty in hydrological forecasting, with the CCKF being the more robust option.

Asynchronous and Adaptive State Estimation of Integrated Electricity–Gas Energy Systems

The combination of Internet of Things technology and power grid technology greatly strengthens the situational awareness in integrated energy systems. However, present development fails to consider the internal dynamic characteristics of the natural gas system. This is addressed in the present study by proposing an asynchronous distributed dynamic state estimation method for integrated electricity-gas energy systems. First, static state estimation based on a weighted least square method is applied to the electric power network, and an improved extended Kalman filter state estimation method based on a transient model is applied to the natural gas network. The asynchronous exchange of information between the two subsystems is addressed by applying an iterative update step correction to the dynamic state estimation of the natural gas network. Second, we propose an adaptive state estimation execution cycle adjustment method to track the state of the natural gas subsystem in real time under conditions of large disturbances in that subsystem. Application of the proposed algorithm to simulation case studies demonstrates that it can greatly improve the estimation accuracy and trajectory tracking performance of the state estimation process relative to conventional methods.

Modeling of a two-axle rail vehicle for state estimation

A two-axle rail vehicle having solid axle wheel-sets is developed by using Kalman filter state estimator. Approximately 18 states (from all the 8 inertial measurements) are observed by developing a Kalman filter. A total of five gyroscopes and three accelerometers calibrate the lateral acceleration, yaw velocity of the vehicle and two wheel-sets roll velocity. All the vehicle states are observed by formulating the Kalman filter, moreover, the Kalman filter is also used to calculate the different parameters like cant angle and radius of the curvature of the track. The design of the rail vehicle is validated by comparing the actual results with the same from computer simulations. The simulation also helps to develop an optimal and novel controller to assess the performance of the design of an active steering system.

Autonomous Underwater Vehicle Positioning Control - a Velocity Form LPV-MPC Approach*

In this work, the positioning control of an autonomous underwater vehicle (AUV) is considered for docking operations in the presence of varying tidal currents. The AUV model is described by a dynamic model and a kinetic model, both are linear parameter varying (LPV). A velocity form LPV model predictive control (LPV-MPC) scheme is proposed, in which the AUV dynamic model is used for the states and the kinetic model is used for the output. The interdependence of AUV kinematic model and dynamic model is exploited to avoid increased state dimension. The complete velocity form controller design enables the cancellation of disturbance effects through the use of the AUV's velocity vector increment for predicting the future evolution of the system. Compared to the original predictive control for the Naminow-D AUV that uses a time-varying Kalman filter for state estimation to approximate disturbances, the proposed algorithm does not require an estimator to eliminate unknown current disturbance, therefore simplifies design and implementation. Simulation studies show the merit of the proposed controller over the original Naminow-D predictive controller especially in terms of improved transient response and reduced sensitivity to time-varying external disturbances.

Multilayer Low-Cost Sensor Local-Global Filtering Fusion Integrated Navigation of Small UAV

Aimed at improving the nonlinear integrated navigation solution performance of multiple low-cost sensors fusion, this paper presents a multilayer loosely-coupled, local-global, and step-optimized MF5DCKF (Multisensor Federated fifth-degree Cubature Kalman filter) state estimation algorithm for the small unmanned aerial vehicle (UAV). This method establishes a multilayer nonlinear integrated navigation model composed of the nonlinear attitude and heading reference system (AHRS) error model, strapdown inertial navigation system/global positioning system (SINS/GPS) error model, and strapdown inertial navigation system/barometer (SINS/BARO) error model to enhance the robustness and richness of the navigation module. Further, based on the above navigation models, a loosely-coupled error state fusion frame is designed to obtain the local convergent state vector. Simultaneously, a three-layer fifth-degree Cubature Kalman filter is proposed to improve the local state estimation accuracy. Subsequently, to optimize the estimated local state, this paper presents a novel distributed MF5DCKF scheme fusing the local state vector to calculate the global optimal state parameters in a step-optimized process. The experimental flight test results show that the proposed algorithm achieves a higher state solution accuracy and a better convergent performance compared with some conventional multisensor fusion algorithms. The new algorithm framework can provide applicability and reliability for the small UAV during the flight.

Model Development for State-of-Power Estimation of Large-Capacity Nickel-Manganese-Cobalt Oxide-Based Lithium-Ion Cell Validated Using a Real-Life Profile

This paper investigates the model development of the state-of-power (SoP) estimation for a 43 Ah large-capacity prismatic nickel-manganese-cobalt oxide (NMC) based lithium-ion cell with a thorough aging investigation of the cells’ internal resistance increase. For a safe operation of the vehicle system, a battery management system (BMS) integrated with SoP estimation functions is crucial. In this study, the developed SoP model used for the estimation of power throughout the lifetime of the cell is coupled with a dual-polarization equivalent-circuit model (DP_ECM) for achieving the precise estimation of desired parameters. The SoP model is developed based on the pulse-trained internal resistance evolution approach, and hence the power is estimated by determining the rate of internal resistance increase. Hybrid pulse power characterization (HPPC) test results are used for extraction of the impedance parameters. In the DP_ECM, Coulomb counting and extended Kalman filter (EKF) state estimation methods are developed for the accurate estimation of the state of charge (SoC) of the cell. The SoP model validation is performed by using both dynamic Worldwide harmonized Light vehicles Test Cycles (WLTC) and static current profiles, achieving promising results with root-mean-square errors (RMSE) of 2% and 1%, respectively.

Online Kinematic Calibration for Legged Robots

This paper describes an online method to calibrate certain kinematic parameters of legged robots, including leg lengths, that can be difficult to measure offline due to dynamic deformation effects and rolling contacts. A kinematic model of the robot’s legs that depends on these parameters is used, along with measurements from joint encoders, foot contact sensors, and aninertial measurement unit (IMU) to predict the robot’s body velocity. This predicted velocity is then compared to another velocity measurement from, for example, a camera or motion capture system, and the difference between them is used to compute anupdate on the kinematic parameters. The method can be incorporated into both Kalman filter or sliding-window optimization-based state estimator. We provide a theoretical observability analysis of our method, as well as validation both in simulation and on hardware. Hardware experiments demonstrate that online kinematic calibration can significantly reduce position drift when relying on odometry.

Polynomial Wiener LQG Controllers based on Toeplitz Matrices

This paper re-examines the discrete-time Linear Quadratic Gaussian (LQG) regulator problem. The normal approach to this problem is to use a Kalman filter state estimator and Kalman control state feedback. Though quite successful, an alternative approach in the frequency domain was employed later. That method used z-transfer functions or polynomials in the z-domain. The transfer function approach is similar to the method used in Wiener filtering and requires the use of Diophantine equations (sometimes bilateral) to find the optimal controller. The contribution here uses a similar approach but uses lower triangular Toeplitz matrices instead of polynomials to gain advantage of eliminating the use of Diophantine equations. This is because the single Diophantine equation approach fails when the system has non-relative prime polynomials and the need for bilateral Diophantine equations is computationally far more complex.

Extended Kalman filter state estimation-based nonlinear explicit model predictive control design for blood glucose regulation of type 1 diabetic patient.

The computational burden of iterative online optimization-based model predictive control (MPC) process is solved by adapting off-line optimization-based nonlinear explicit model predictive control (NEMPC). In this paper, the application of NEMPC is verified to regulate blood glucose level in type 1 diabetes mellitus (T1DM) patients. The objective of glucose regulation is to avoid hyperglycemia (> 180 mg/dl) and hypoglycemia (< 50 mg/dl) by maintaining glucose level in the range of 70 to 180 mg/dl. It helps to avoid time complexity of iterative process during solution of optimization stage in MPC. In the nonlinear T1DM model, only the state dynamic of sugar is measurable with low complexity and high cost among other states of the model. Therefore, an extended Kalman filter (EKF) is used developed to estimate unavailable states. The information of the estimated states are used to develop the proposed control approach for the T1DM patients. The simulation results of the NEMPC along with EKF-based state estimator for T1DM model shows the regulation of blood glucose-level (BGL) within 70 to 180 mg/dl within 120 min. The robustness of the proposed scheme is also verified under the change in parameters and food disturbance. The control variability grid analysis (CVGA) of NEMPC for 50 numbers of virtual T1DM patients under random parametric changes and meal disturbances shows the avoidance of hypoglycemia and hyperglycemia in type 1 diabetic patients. The proposed control method is simple, robust and efficient to regulate glucose level in T1DM patients. Extended Kalman filter state estimator based nonlinear explicit model predictive control for blood glucose regulation in Type 1 diabetic patients.

A modified Sage-Husa adaptive Kalman filter for state estimation of electric vehicle servo control system

Kalman filter algorithm is widely used in state estimation of electric vehicles (EV) servo system as dynamic estimation of power system. In order to improve the estimation effect of noise interference in engineering practice, a scheme of Modified Sage-Husa adaptive Kalman filter (MSHAKF) combined with fuzzy clustering algorithm is proposed. The scheme processes the experimental data in a robust and adaptive way through the combination of constantly changing fuzzy clustering and MSHAKF, so as to better represent the dynamic characteristics of EV. The simulation results show that this method has higher estimation accuracy than the traditional method on the premise of unknown disturbance and noise characteristics in the system, and is more in line with the practical application of EV.

The Level Set Kalman Filter for State Estimation of Continuous-Discrete Systems

We propose a new extension of Kalman filtering for continuous-discrete systems with nonlinear state-space models that we name as the level set Kalman filter (LSKF). The LSKF assumes the probability distribution can be approximated as a Gaussian, and updates the Gaussian distribution through a time-update step and a measurement-update step. The LSKF improves the time-update step when compared to existing methods, such as the continuous-discrete cubature Kalman filter (CD-CKF) by reformulating the underlying Fokker-Planck equation as an ordinary differential equation for the Gaussian, thereby avoiding expansion in time. Together with a carefully picked measurement-update method, numerical experiments show that the LSKF has a consistent performance improvement over CD-CKF for a range of parameters, while also simplifies the implementation, as no user-defined timestep subdivision between measurements is required, and the spatial derivatives of the drift function are not explicitly needed.

Efficient and Simple Gaussian Process Supported Stochastic Model Predictive Control for Bioreactors using HILO-MPC

Model-based control of biotechnological processes is, in general, challenging. Often the processes are complex, nonlinear, and uncertain. Hence modeling tends to be complex and is often inaccurate. For this reason, non-model-based control strategies developed via fask, bench-scale, or pilot plant experiments are often applied in the biotechnology industry. Model-based control and optimization techniques can increase processes’ performance and automation level, thereby decreasing costs and guaranteeing the desired specifications. These rely on a model of the process to make predictions and optimize the inputs to the plant. To improve the quality of the models, it is often helpful to use combined first principle and data-driven models together in a hybrid modeling approach which increases the model prediction capabilities. The residual uncertainty of the hybrid model should be taken into account in the control level to satisfy the process specifications and constraints. This paper proposes to use a stochastic model predictive control scheme that exploits a hybrid, Gaussian processes-based model. We outline the effectiveness of the stochastic model-based approach in combination with a suitable Kalman filter for state estimation considering an example biotechnological process. Furthermore, we underline that appropriate tools exist that allow the simple application of such methods even for the novice user. To do so, we use an open-source Python package — HILO-MPC, which allows the simple yet efficient formulation and solution of machine learning-supported optimal control and estimation problems.

Extended Kalman Filter Sensor Fusion in Practice for Mobile Robot Localization

Self-driving vehicles and autonomously guided robots could be very beneficial to today's civilization. However, the mobile robot's position must be accurately known, which referred as the localization with the task of tracking the dynamic position, in order for the robot to be active and useful. This paper presents a robot localization method with a known starting location by a real-time reconstructed environment model that represented as an occupancy grid map. The extended Kalman filter (EKF) is formulated as a nonlinear model-based estimator for fuse Odometry and a LIDAR range finder sensor. Because the occupancy grid map for the area is provided, just the inaccuracies of the LIDAR range finder will be considered. The experimental results on the “turtlebot” robot using robot operating system (ROS) show a significant improvement in the pose of the robot using the Kalman filter compared with sample Odometry. This paper also establishes the framework for using a Kalman filter for state estimation, providing all relevant mathematical equations for differential drive robot, this technique can be used to a variety of mobile robots.

Ensemble kalman filter and particle filter-based state estimation on electrical power systems

In this article, a referential study of the sequential importance sampling particle filter with a systematic resampling and the ensemble Kalman filter is provided to estimate the dynamic states of several synchronous machines connected to a modified 14-bus test case, when a balanced three-phase fault is applied at a bus bar near one of the generators. Both are supported by Monte Carlo simulations with practical noise and model uncertainty considerations. Such simulations were carried out in MATLAB by the Power System Toolbox, whereas the evaluation of the Particle Filter and the Ensemble Kalman Filter by script files developed inside the toolbox. The results obtained show that the particle filter has higher accuracy and more robustness to measurement and model noise than the ensemble Kalman filter, which helps support the feasibility of the method for dynamic state estimation applications.

State Estimation for HALE UAVs With Deep-Learning-Aided Virtual AOA/SSA Sensors for Analytical Redundancy

High-altitudelong-endurance (HALE) unmanned aerial vehicles (UAVs) are employed in a variety of fields because of their ability to fly for a long time at high altitudes, even in the stratosphere. Two paramount concerns exist: enhancing their safety during long-term flight and reducing their weight as much as possible to increase their energy efficiency based on analytical redundancy approaches. In this letter, a novel deep-learning-aided navigation filter is proposed, which consists of two parts: an end-to-end mapping-based synthetic sensor measurement model that utilizes long short-term memory (LSTM) networks to estimate the angle of attack (AOA) and sideslip angle (SSA) and an unscented Kalman filter for state estimation. Our proposed method can not only reduce the weight of HALE UAVs but also ensure their safety by means of an analytical redundancy approach. In contrast to conventional approaches, our LSTM-based method achieves better estimation by virtue of its nonlinear mapping capability.

Estimation of flexible space tether state based on end measurement by finite element Kalman filter state estimator

This paper proposes a novel finite element Kalman filter to estimate the unmeasurable state of space tether systems based on the measured state at its ends only. The finite element method calculates the unmeasurable internal state as the virtual measurement based on the dynamic model of the system by imposing the input of measured state at the boundary to the model using the Lagrange multiplier method in the spatial space. Combining the real and virtual measurement into a hybrid measurement model of the system, the full state is reconstructed and propagated in the temporal space by the extended Kalman filter. Two state-space system models, the dynamics-based and kinematics-based state models, in the Kalman filter are explored. The observability and stability of the newly proposed finite element Kalman filter are examined and proved. The advantages of the proposed state estimator are (i) the singularity in the virtual measurement of state caused by the number of internal state greater than the number of state measured at the boundary is eliminated in the statistic meaning by the Kalman filter, and (ii) the effects of noises of the observation data and the uncertainties of model discretization are considered and minimized. The correctness and effectiveness of the proposed state estimator is demonstrated by the numerical analysis of a space tether system orbiting around the Earth. The results show the proposed state estimator with only measured state at the ends of the tether successfully provides an accurate time history estimation of geometric configuration and motion of the entire tether. Moreover, the results also show the difference caused by the dynamics-based and kinematics-based system models in the state estimator is negligible. The kinematics-based system model should be used in the state estimator due to its significantly low computational load. Finally, the proposed method can be easily applied for the state estimation process for other space tethered spacecraft systems.

Optimal closed-loop wake steering – Part 1: Conventionally neutral atmospheric boundary layer conditions

Abstract. Strategies for wake loss mitigation through the use of dynamic closed-loop wake steering are investigated using large eddy simulations of conventionally neutral atmospheric boundary layer conditions in which the neutral boundary layer is capped by an inversion and a stable free atmosphere. The closed-loop controller synthesized in this study consists of a physics-based lifting line wake model combined with a data-driven ensemble Kalman filter (EnKF) state estimation technique to calibrate the wake model as a function of time in a generalized transient atmospheric flow environment. Computationally efficient gradient ascent yaw misalignment selection along with efficient state estimation enables the dynamic yaw calculation for real-time wind farm control. The wake steering controller is tested in a six-turbine array embedded in a statistically quasi-stationary, conventionally neutral flow with geostrophic forcing and Coriolis effects included. The controller statistically significantly increases power production compared to the baseline, greedy, yaw-aligned control provided that the EnKF estimation is constrained and informed with a physics-based prior belief of the wake model parameters. The influence of the model for the coefficient of power Cp as a function of the yaw misalignment is characterized. Errors in estimation of the power reduction as a function of yaw misalignment are shown to result in yaw steering configurations that underperform the baseline yaw-aligned configuration. Overestimating the power reduction due to yaw misalignment leads to increased power over the greedy operation, while underestimating the power reduction leads to decreased power; therefore, in an application where the influence of yaw misalignment on Cp is unknown, a conservative estimate should be taken. The EnKF-augmented wake model predicts the power production in yaw misalignment with a mean absolute error over the turbines in the farm of 0.02P1, with P1 as the power of the leading turbine at the farm. A standard wake model with wake spreading based on an empirical turbulence intensity relationship leads to a mean absolute error of 0.11P1, demonstrating that state estimation improves the predictive capabilities of simplified wake models.