Reduced-order Extended Kalman Filter Research Articles

Reduced-Order Extended Kalman Filter for Deformable Tissue Simulation

Modelling of soft tissue deformation is a key issue in surgical simulation. Despite extensive research studies on this issue, accurate modelling of soft tissue deformation in run time still remains challenging. This paper proposes a new reduced-order nonlinear Kalman filter to emulate nonlinear behaviors of biological deformable tissues. This approach defines the deformable modelling problem as a reduced-order filtering problem to accurately calculate soft tissue deformation in real time. Soft tissue deformation is discretized in space using nonlinear finite element method based on hyperelasticity and further formulated as a nonlinear state-space equation for filtering estimation. Subsequently, the order of this nonlinear state-space equation is reduced using proper orthogonal decomposition to reduce the computational cost. Upon this reduced-order state-space equation, an extended Kalman filter is constructed to online calculate nonlinear behaviors of tissue physical deformation. Simulation results and comparison analysis prove the effectiveness of the suggested method for accurate simulation of tissue physical deformation in real time.

A Dynamic High-Order Equivalent Modeling of Lithium-Ion Batteries for the State-of-Charge Prediction Based on Reduced-Order Extended Kalman Filtering Algorithm

Detection of battery power has always been the core of the battery management system of electric vehicles, and the fast and accurate estimation of charged state can guarantee the safe operation of electric vehicles. The key to improving accurate state-of-charge estimation is an appropriate model establishment coupled with a suitable estimation algorithm. This research seeks to adopt and accomplish a lithium-ion battery state-of-charge estimation based on the Gaussian function to build up the open-circuit voltage algorithm. A reduced-order extended Kalman filtering algorithm is proposed with hybrid pulse power characterization parameter identification to estimate the battery characterization state-of-charge. The model’s parameters in different state-of-charge points are calculated through the lithium-ion battery’s charge and discharge process; the 2RC modeling correction method and Reduced-order extended Kalman filter method are used separately based on the High-order equivalent 2RC modeling. The Experimental results show that the above method can achieve state-of-charge estimation more accurately and conveniently, providing a certain reference value for the rational management and distribution of power lithium-ion batteries. The maximum error of state-of-charge estimation based on the established high-order equivalent 2RC model using the Reduced-order extended Kalman filtering algorithm is less than 1.85%. The REKF algorithm achieved a maximum voltage error of 0.0409V and an average error of 0.0299V and therefore can satisfy the accuracy of the battery management system application needs. Keywords: Lithium-ion battery; state-of-charge; high-order equivalent 2RC modeling; open-circuit voltage; parameter identification; reduced-order extended Kalman filtering algorithm DOI: 10.7176/JETP/11-3-03 Publication date: June 30 th 2021

The Regional Ice Ocean Prediction System v2: a pan-Canadian ocean analysis system using an online tidal harmonic analysis

Abstract. Canada has the longest coastline in the world and includes diverse ocean environments, from the frozen waters of the Canadian Arctic Archipelago to the confluence region of Labrador and Gulf Stream waters on the east coast. There is a strong need for a pan-Canadian operational regional ocean prediction capacity covering all Canadian coastal areas in support of marine activities including emergency response, search and rescue, and safe navigation in ice-infested waters. Here we present the first pan-Canadian operational regional ocean analysis system developed as part of the Regional Ice Ocean Prediction System version 2 (RIOPSv2) running in operations at the Canadian Centre for Meteorological and Environmental Prediction (CCMEP). The RIOPSv2 domain extends from 26∘ N in the Atlantic Ocean through the Arctic Ocean to 44∘ N in the Pacific Ocean, with a model grid resolution that varies between 3 and 8 km. RIOPSv2 includes a multivariate data assimilation system based on a reduced-order extended Kalman filter together with a 3D-Var bias correction system for water mass properties. The analysis system assimilates satellite observations of sea level anomaly and sea surface temperature, as well as in situ temperature and salinity measurements. Background model error is specified in terms of seasonally varying model anomalies from a 10-year forced model integration, allowing inhomogeneous anisotropic multivariate error covariances. A novel online tidal harmonic analysis method is introduced that uses a sliding-window approach to reduce numerical costs and allow for the time-varying harmonic constants necessary in seasonally ice-infested waters. Compared to the Global Ice Ocean Prediction System (GIOPS) running at CCMEP, RIOPSv2 also includes a spatial filtering of model fields as part of the observation operator for sea surface temperature (SST). In addition to the tidal harmonic analysis, the observation operator for sea level anomaly (SLA) is also modified to remove the inverse barometer effect due to the application of atmospheric pressure forcing fields. RIOPSv2 is compared to GIOPS and shown to provide similar innovation statistics over a 3-year evaluation period. Specific improvements are found near the Gulf Stream for all model fields due to the higher model grid resolution, with smaller root mean squared (rms) innovations for RIOPSv2 of about 5 cm for SLA and 0.5 ∘C for SST. Verification against along-track satellite observations demonstrates the improved representation of mesoscale features in RIOPSv2 compared to GIOPS, with increased correlations of SLA (0.83 compared to 0.73) and reduced rms differences (12 cm compared to 14 cm). While the RIOPSv2 grid resolution is 3 times higher than GIOPS, the power spectral density of surface kinetic energy provides an indication that the effective resolution of RIOPSv2 is roughly double that of the global system (35 km compared to 66 km). Observations made as part of the Year of Polar Prediction (2017–2019) provide a rare glimpse at errors in Arctic water mass properties and show average salinity biases over the upper 500 m of 0.3–0.4 psu in the eastern Beaufort Sea in RIOPSv2.

A novel FPGA‐Based Bi input‐reduced order extended Kalman filter for speed‐sensorless direct torque control of induction motor with constant switching frequency controller

This study proposes an FPGA-based hardware in the loop (HIL) emulator for speed-sensorless of induction motor (IM) constant switching frequency controller-based direct torque control (CSFC-DTC) with a novel bi input-reduced order extended Kalman filter (BI-ROEKF). The full precision single floating point numbers in the IEEE 754 standard are used during the implementation of the HIL emulator which contains closed-loop speed-sensorless drive system of IM on the Xilinx Virtex XC5VLX-110T ML506 FPGA board. In this HIL emulator of speed-sensorless IM drive system, stator stationary axis components of stator flux, rotor mechanical angular speed, load torque, stator and rotor resistances are estimated with the novel BI-ROEKF which is proposed for the first time in the literature. The proposed BI-ROEKF is created by applying two different non-linear and linear system input functions obtained from two different IM models to the single reduced order extended Kalman filter (ROEKF) algorithm. Thus, the order and the computational burden of the EKF are reduced. The HIL emulator of the speed-sensorless drive system of IM is implemented on FPGA using the advantage of hand-written VHDL on getting an optimal logical design to reduce the sampling time which directly effects the estimation performance of the model-based estimator like the novel BI-ROEKF and hence the control performance of drive system. The estimation performance of the novel BI-ROEKF is tested with speed-sensorless CSFC-DTC IM drive system under different challenging scenarios in HIL emulator. Thus, the control and the implementation performances of digitalised emulator are tested. Finally, the estimation and control performance results and the execution time of the each part of the proposed HIL emulator of the speed-sensorless BI-ROEKF-based CSFC-DTC of the IM are presented.

An Extended Kalman Filter, with pre-adjusted covariance matrices, applied to the sensorless speed control of three-phase induction motors

With the emergence of sensorless control methods, there was a need for the use of estimators and/or state observers to give it the robustness and precision required in the drive of induction motors. This work deals with the application of the Extended Kalman Filter (EKF) in the estimation of rotor speed and position, aiming at the implementation of the indirect vector control technique in a sensorless speed control system for three-phase induction motors. The mathematical development of the system state variables associated with the EKF stochastic process is presented in this study, and point out its application under variable speed and load conditions, which are imposed on these motors in everyday life. The sensorless control strategy was tested through routine lines in the Matlab® software, simulating operating conditions of this type of engine, being proven its performance, as well as the convergence times consistent with the usual requirements of high performance systems. The main contributions of this work are the use of a reduced-order EKF (ROEKF) and the preset of covariance matrices to accelerate convergence in speed and position estimates for future implementations in currently accessible digital signal processors.

Estimation of Spatial Concentration Profiles in a Plug Flow Reactor using Reduced Dimensional Models

Efficient monitoring and control of a plug flow reactor system requires knowledge of temporal variations of spatial profiles of concentrations. However, estimating the state profiles in spatial domain requires discretization of the governing PDEs at a large number of spatial grid points, which results in high order DAE systems. Computational cost associated with solving these DAEs makes them unattractive candidates for development of advanced on-line monitoring and control schemes. In this work, a novel method for reconstructing spatial profiles of state variables is proposed, which exploits the fact that orthogonal collocation based discretization of PDEs employs time-dependent interpolation polynomials as approximate solutions. A reduced-order approximation of PDE system is developed using Lagrange interpolation polynomials and further used to develop a reduced dimensional extended Kalman filter (EKF). States estimated using the reduced-order EKF, in combination with Lagrange polynomials, are used to construct estimates of spatial profiles of state variables. The proposed approach facilitates handling of measurements available from sensors placed at arbitrary spatial locations other than the collocation points. The performance of proposed reduced-order observer is evaluated via simulation studies performed on a benchmark counter-current plug flow reactor system. Analysis of the simulation results reveals that the proposed approach is capable of generating state profile estimates that are comparable to profile estimates obtained using a high dimensional observer with a significant reduction in the computational cost.

A Method to Improve the Response of a Speed Loop by Using a Reduced-Order Extended Kalman Filter

In servo systems, encoders are usually used to measure the position and speed signals of electric machines. But in a low speed range, the traditional M/T method has a larger time delay, which will cause an increase of the speed loop order and degradation of the speed loop performance. A method employed to reduce the delay of speed feedback by using a reduced-order Extended Kalman Filter (EKF) is introduced in this paper. The speed of the permanent magnet synchronous motor is estimated by the reduced-order EKF in a low speed range, which reduces the delay of speed feedback and extends the cutoff frequency of the speed loop to improve the dynamic performance of the servo system. In order to solve the issues that the traditional full-order EKF is sensitive to the inertia of the system and computationally complex, a composite load torque observer (CLTO) is proposed in this paper. The load torque and the friction torque are simultaneously observed by the CLTO. Additionally, the CLTO is used to reduce the order of the EKF, which reduces the sensitivity of EKF on inertia to enhance the robustness of the algorithm and simplifies the computational complexity. The feasibility and effectivity of the above method are verified by simulations and experiments.

FPGA-Based Sensorless PMSM Speed Control Using Reduced-Order Extended Kalman Filters

This paper presents the design and implementation of a field-programmable gate array (FPGA)-based architecture for the speed control of sensorless permanent-magnet synchronous motor (PMSM) drives. For the reduction of computation resources, as well as accuracy improvement in the rotor position estimation, a parallel reduced-order extended Kalman filter (EKF) is proposed in this work. Compared with an EKF, the system order is reduced and the iteration process is greatly simplified, resulting in significant savings of resource utility, while maintaining high estimation performance. The whole control system includes a current-control-and-coordinate-transformation unit, a proportional-integral (PI) speed controller, and other accessory modules, all implemented in a single FPGA chip. A hardware description language is adopted to describe advantageous features of the proposed control system. Moreover, the finite-state-machine method is applied with the purpose to reduce logic elements used in the FPGA chip. The validity of the approach is verified through simulation based on the Modelsim/Simulink cosimulation method. Finally, experimental results are obtained on an FPGA platform with an inverter-fed PMSM to show the feasibility and effectiveness of the proposed system-on-programmable-chip for PMSM drives.

Covariance and State Estimation of Weakly Observable Systems: Application to Polymerization Processes

Physical models for polymerization may be overly complex considering the available measurements, and they may contain many unobservable and weakly observable modes. Overly complex structures lead to ill-conditioned or singular problems for disturbance variance estimation. Ill conditioning leads to unrealistic data demands for reliable covariance estimates and state estimates. The goal of this paper is to build nonlinear state estimators for weakly observable systems, with focus on polymerization processes. State estimation requires knowledge about the noise statistics affecting the states and the measurements. These noise statistics are usually unknown and need to be estimated from operating data. We introduce a linear time-varying autocovariance least-squares (LTV-ALS) technique to estimate the noise covariances for nonlinear systems using autocorrelations of the data at different time lags. To reduce or eliminate the ill-conditioning problem, we design a reduced-order extended Kalman filter (EKF) to estimate only the strongly observable system states. This reduced filter, which is based on the Schmidt-Kalman filter, is used to perform the estimation of noise covariances by the LTV-ALS technique. Results of the implementation of the proposed method on a large-dimensional ethylene copolymerization example show that better conditioned state and covariance estimation problems can be obtained. We also show that high-quality state estimates can be obtained after the specification of the noise statistics of EKF estimators by ALS.

Research on Anti-Error Performance of Speed and Flux Estimator for Induction Motor Using Robust Reduced-Order EKF

A novel speed and flux estimator of induction motor based on robust reduced-order extended Kalman filter (RROEKF) is proposed in this paper, and the effect of gross error on estimation accuracy of reduced-order EKF (ROEKF) was analyzed. Whether RROEKF can give better estimation performance and robust ability than ROEKF for speed and flux estimation of induction machines is explored. The speed estimated error and the flux fluctuation of RROEKF by gross external disturbance and internal estimated error is compared with ROEKF. The simulation and experimental results show that RROEKF has the better estimation performance of anti-gross error than ROEKF and EKF.

Hardware Design and Simulation of Reduced-Order Extended Kalman Filter Estimator and Speed Fuzzy Controller for Sensorless PMSM Drives

The design and co-simulation of a sensorless control for permanent magnet synchronous motor (PMSM) drive is presented in this paper. Firstly, the block diagram of vector control and fuzzy algorithm are designed. Secondly, a flux position and speed of rotor are estimated through the reduced-order extended kalman filter (EKF). Thirdly, the very-high-speed IC hardware description language (VHDL) is adopted to describe the behavior of all functions of the system. Fourthly, the simulation work is performed by MATLAB/Simulink and ModelSim co- simulation mode. The PMSM, inverter and speed command are performed in Simulink and the sensorless speed control of PMSM drive is executed in ModelSim. Finally, the comparison results between PI and Fuzzy controller validate the effectiveness of the sensorless PMSM speed control system. It has fast reacting speed and good dynamic performance. 

Speed sensorless torque control of an IPMSM drive with online stator resistance estimation using reduced order EKF

Speed sensorless control of an interior permanent magnet synchronous motor (IPMSM) based on direct torque control (DTC) is proposed in this paper. The rotor speed and position of the IPMSM are estimated based on an active flux concept, where, the active flux vector position is identical to the rotor position. The proposed algorithm does not require neither high frequency injection signal nor complicated schemes even at vary low speed operation. Torque/ flux sliding mode controller (SMC) combined with space vector modulation is proposed to improve the performance of the classical DTC. Stator resistance value is required for a stator flux and electromagnetic torque estimation. Its variation due to temperature or frequency degrades the scheme performance, especially, at low speed operation. To overcome this problem, a reduced order extended Kalman filter (EKF) is proposed to update online the stator resistance. The advantages of the direct torque control, sliding mode controller, and speed sensorless control are incorporated in the proposed scheme. Simulation works are carried out to show the ability of the proposed scheme at different operating conditions. The results demonstrate the activity of the scheme at wide range speed operation with load disturbance and parameters variation.

Predictive Speed Control of a Two-Mass System Driven by a Permanent Magnet Synchronous Motor

This paper presents a predictive strategy for the speed control of a two-mass system driven by a permanent magnet synchronous motor (PMSM). The proposed approach allows to manipulate all the system variables simultaneously, including mechanical and electrical variables in a single control law. The state feedback is achieved with a reduced order extended Kalman filter, which observes the non-measured variables as well as reduces the impact of measurement noise. The performance of the control strategy is shown through simulation and experimental results in a 4 [kW] laboratory prototype.

병렬형 저감 차수 칼만 필터를 이용한 매입형 영구자석 동기전동기의 센서리스 제어

This paper proposes a novel sensorless control scheme for a Permanent Magnet Synchronous Motor (PMSM) by using a parallel reduced-order Extended Kalman Filter. The proposed scheme can obtain rotor position and speed by back-EMF that is estimated by reduced-order EKF and save computation time greatly due to using a parallel structure that works by turns every sampling time. Therefore, proposed scheme has merits of conventional EKF, and problems of parameter sensitivity are partially overcome. And proposed scheme can safely estimate rotor speed and position by using new algorithms according to driving regions. Experimental results show the validity of the proposed estimation technique, and to verify the merit of the proposed scheme, a comparison of a new reduced-order EKF algorithm with a conventional EKF algorithm has been also made in terms of computation time.

Improved sensorless operation of interior PM BLAC motor drives with reduced-order EKF

It is a challenge to realise an EKF-based sensorless controller for an Interior Permanent Magnet (IPM) brushless ac motor drive due to the influence of non-linearities and rotor saliency. An improved reduced-order EKF method is presented, reducing the rotor position estimation error of an IPM motor, as evident in [1] due to the neglect of saliency. The effectiveness of the proposed method is verified experimentally.

An adaptively reduced-order extended Kalman filter for data assimilation in the tropical Pacific

The reduced-order extended Kalman (ROEK) filter has been introduced by Cane et al. (J. Geophys. Res. 101(1996) 599) as a means to reduce the cost of the extended Kalman filter. It essentially consists of projecting the dynamics of the model onto a low dimensional subspace obtained via an empirical orthogonal functions (EOF) analysis. However, the choice of the dimension of the reduced-state space (or the number of EOFs to be retained) remains a delicate question. Indeed, Cane et al. found that increasing the number of EOFs does not improve, and even sometimes worsens, the performance of the ROEK filter. We speculate that this is probably due to the optimal character of the EOF analysis that is optimal in a time-mean sense only. In this respect, we develop a simple efficient adaptive scheme to tune, according to the model mode, the dimension of the reduced-state space, which would be therefore variable in time. In a first application, twin experiments are conducted in a realistic setting of the Ocean Parallèlisè (OPA) model in the tropical Pacific. The observations are assumed to be synthetic altimeter data sampled according to the Topex/Poseidon mission features. The adaptive scheme is shown to improve the performance of the ROEK filter especially during model unstable periods.

High performance IPMSM drives without rotational position sensors using reduced-order EKF

An extended Kalman filter (EKF) based approach for position sensor elimination in interior permanent magnet synchronous motor (IPMSM) drives is presented in this paper. The EKF is capable of estimating system parameters and state variables for the IPMSM by eliminating virtually all influences of structural noises in the vector control scheme. This paper presents a design method of a reduced-order EKF. Position and angular speed of the rotor are obtained through the reduced-order EKF only by measuring stator currents. Also, due to an angle modification scheme with error tracking, the sensorless drive system is robust to parameter variations. Simulation and experimental results are provided to verify the proposed approach based on the reduced-order EKF.

Observer-based speed estimation method for sensorless vector control of induction motors

The method is based on an adaptive flux observer in the rotor-speed reference frame, in which a second-order Kalman filter is employed to modify the estimated rotor flux to improve the performance of speed estimation. The Kalman filter modifies the estimated rotor flux based on the measured stator currents. The estimated speed is used in the speed feedback for vector control and in the co-ordinate transformation for current controller. The proposed method has the advantage of saving much computation time in comparison with the reduced-order extended Kalman filter. Compared with the conventional adaptive observer, the proposed method has the advantage of better accuracy to follow the speed command under heavy loads. Experimental results show the effectiveness of the proposed method.

A reduced order extended Kalman filter for sequential images containing a moving object

The extended Kalman filter (EKF) is applied to the reduction of noise in sequential images containing a moving object and to the estimation of the object's velocity. A computationally tractable approximation of the EKF, called the parallel extended Kalman filter (PEKF), is generated. The PEKF consists of a parallel bank of third-order EKFs, operating on the Fourier coefficients of the image, followed by a finite impulse response filter. The PEKF is shown to converge to the optimal (in the mean square sense) algorithm in the limit as the velocity estimation errors approach zero. The performance of the PEKF is demonstrated for very low signal-to-noise ratio (SNR) images. The PEKF also provides a natural setting for tracking slow changes in the object (real or apparent) and its velocity, since these variations are included in the model. The relation of the PEKF to another frequency domain algorithm for velocity estimation is discussed. The algorithm is illustrated by application to an example and its performance is demonstrated in the presence of velocity estimation errors.

A Reduced Order Extended Kalman Filter Algorithm for Parameter and State Estimation of an Induction Motor 1

Abstract It is known that the application of the field-oriented control strategy to induction motors needs the knowledge of the rotor flux components and/or the rotor time constant. The aim of this paper is to give contributions to the solution of the above problem by means of real-time algorithms. The first part of the paper gives a short introduction about Kalman Filter theory. The second part is dedicated to the development of a suitable deterministic reduced order model with only the unmeasurable rotor fluxes as states, from which two reduced-order Kalman filters are obtained. The former one is called Reduced Order Linear Kalman Filter (RKF) and represents an effective algorithm for the estimation of the rotor flux components. The second filter, called Reduced Order Extended Kalman Filter (REKF) is obtained first developing a non linear model where the three state variables are the rotor flux components and the rotor time constant and then applying the extended Kalman Filter theory. The last part of the paper is related to the simulations results and the implementation tests both for RKF and REKF algorithms to verify their effectiveness in the problems of real-time applications.