Steady-state Kalman Filter Research Articles

On the equivalence between steady-state Kalman filter and DPLL

Fundamental results in the literature showed that a class of Kalman filters (KF) converges to a digital phase lock loop (DPLL) structure. Results were proved for second- and third-order KFs. We extend these results to KFs of any order and derive in closed form the equivalent loop filter constants as a linear combination of the steady-state Kalman gains. We also give the inverse relation. Both closed-form relations involve Stirling numbers of the first or second kind. We implement both filters in a global navigation satellite system (GNSS) receiver and illustrate their equivalence with real-world data. These extended theoretical results provide a deeper understanding of the equivalence between KF and DPLL and may be of practical interest in highly dynamic scenarios to design and tune tracking filters.

Review of DC Motor Modeling and Linear Control: Theory with Laboratory Tests

This review paper introduces the modeling, measurement, identification and control of direct current motors based on the state space modeling and the transfer function representation. These models are identified by real laboratory measurements, and the simulated results are compared with the measurements. Continuous-time and discrete-time PID (Proportional-Integral-Derivative) controllers, discrete-time state feedback and linear quadratic controllers are designed mathematically. The designed controllers are realized by the microcontroller Arduino UNO, and the behavior of the controllers is compared and analyzed. The noisy current signal has been measured by a discrete-time observer, steady-state Kalman filtering is also studied. The practical results of the implemented controllers support the theoretical results very well.

Integration of field-oriented and steady-state linear Kalman filter control in PMSG-based grid-connected system for improving voltage control and power balance operation

Integration of field-oriented and steady-state linear Kalman filter control in PMSG-based grid-connected system for improving voltage control and power balance operation

Information-Based Sensor Placement for Data-Driven Estimation of Unsteady Flows

Estimation of unsteady flowfields around flight vehicles may improve flow interactions and lead to enhanced vehicle performance. Although flowfield representations can be very high-dimensional, their dynamics can sometimes have low-order representations that may be estimated using a few, appropriately placed measurements. This paper presents a sensor-selection framework for the intended application of data-driven, flowfield estimation. This framework combines data-driven modeling, steady-state Kalman filter design, and sparse, sequential sensor selection. This paper also uses the sensor selection framework to design sensor arrays that can perform well for a collection of operating conditions. Flow estimation results on numerical data show that the proposed framework produces arrays that are highly effective at flowfield estimation for the flow behind and an airfoil at a high angle of attack using embedded pressure sensors. Analysis of the flowfields reveals that paths of impinging stagnation points along the airfoil’s surface during a shedding period of the flow are highly informative locations for placement of pressure sensors.

A Distributed Implementation of Steady-State Kalman Filter

This paper studies the distributed state estimation in sensor network, where <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$m$</tex></formula> sensors are deployed to infer the <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$n$</tex></formula> -dimensional state of a Linear Time-Invariant (LTI) Gaussian system. By a lossless decomposition of the optimal steady-state Kalman filter, we show that the problem of distributed estimation can be reformulated as that of the synchronization of homogeneous linear systems. Based on such decomposition, a distributed estimator is proposed, where each sensor node runs a local filter using only its own measurement, alongside with a consensus algorithm to fuse the local estimate of every node. We prove that the average of estimates from all sensors coincides with the optimal Kalman estimate, and under certain condition on the graph Laplacian matrix and the system matrix, the consensus error is bounded and the asymptotic error covariance is derived. As a result, the distributed estimator is stable for each single node. We further show that the proposed algorithm has a low message complexity of min( <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$m$</tex></formula> , <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$n$</tex></formula> n). Numerical examples are provided in the end to illustrate the efficiency of the proposed algorithm.

Parametric model identification of delta wing UAVs using filter error method augmented with particle swarm optimisation

AbstractFrom arsenal delivery to rescue missions, unmanned aerial vehicles (UAVs) are playing a crucial role in various fields, which brings the need for continuous evolution of system identification techniques to develop sophisticated mathematical models for effective flight control. In this paper, a novel parameter estimation technique based on filter error method (FEM) augmented with particle swarm optimisation (PSO) is developed and implemented to estimate the longitudinal and lateral-directional aerodynamic, stability and control derivatives of fixed-wing UAVs. The FEM used in the estimation technique is based on the steady-state extended Kalman filter, where the maximum likelihood cost function is minimised separately using a randomised solution search algorithm, PSO and the proposed method is termed FEM-PSO. A sufficient number of compatible flight data sets were generated using two cropped delta wing UAVs, namely CDFP and CDRW, which are used to analyse the applicability of the proposed estimation method. A comparison has been made between the parameter estimates obtained using the proposed method and the computationally intensive conventional FEM. It is observed that most of the FEM-PSO estimates are consistent with wind tunnel and conventional FEM estimates. It is also noticed that estimates of crucial aerodynamic derivatives ${C_{{L_\alpha }}},\;{C_{{m_\alpha }}},\;{C_{{Y_\beta }}},\;{C_{{l_\beta }}}$ and ${C_{{n_\beta }}}$ obtained using FEM-PSO are having relative offsets of 2.5%, 1.5%, 6.5%, 3.4% and 7.6% w.r.t. wind tunnel values for CDFP, and 1.4%, 1.9%, 0.1%, 9.6% and 7.5% w.r.t. wind tunnel values for CDRW. Despite having slightly higher Cramer-Rao Lower Bounds of estimated aerodynamic derivatives using the FEM-PSO method, the simulated responses have a relative error of less than 0.10% w.r.t. measured flight data. A proof-of-match exercise is also conducted to ascertain the efficacy of the estimates obtained using the proposed method. The degree of effectiveness of the FEM-PSO method is comparable with conventional FEM.

Possibility of Prediction Improvements for Atomic Clock Ensembles: Basis Selection in Undetectable Systems

To generate accurate time scales, such as national standard time in each country and network time in communication systems, a novel prediction algorithm is proposed for the atomic clock ensembles in which the system is unobservable. In the proposed prediction algorithm, to deal with unobservable state space, we consider a steady-state Kalman filter and an intuitive prediction algorithm to separately treat the observable and unobservable state spaces after Kalman canonical decomposition. It turns out that the basis selection plays an important role in affecting prediction performance. We compare the performance of the proposed algorithm with the conventional one and discuss the possibility of prediction improvements by altering the transformation matrix. A couple of numerical examples are presented to illustrate the efficacy of our results.

Design of Full-Order State Observer for Two-Mass Joint Servo System Based on the Fixed Gain Filter

This article mainly designs a novel full-order state observer for the two-mass elastic servo system. In order to observe all the states of the two-mass system without the aid of any additional sensors, the model of the observer is constructed that includes two relative parts. Based on the theory of steady-state Kalman filter (KF), the fixed gain filter (FGF) scheme is studied. To overcome the heavy computational processing burden from the KF, the mechanism of the only one adjustable parameter is studied for the FGF-based observer. The adjustable parameter can be further refined by stability analysis and control characteristics within a small range, and the exact value can be calculated according to the desired observation performance, by compromising both the noise sensitivity and tracking capability. The proposed full-order observer method is experimentally verified in a two-mass system to provide easy parameter tuning, decreased computational complexity, and flexible observation performance in all cases.

Application and Optimization of Luenberger Observer Phase-Locked Loop and Inductance-Free Vector Control Method for Aviation Three-Phase Converter in Wearable Equipment.

This work aims to strengthen the comprehensive performance of the Luenberger observer in the application of aviation three-phase converter and in physical exercise wearable devices to effectively detect human physiological signals. Firstly, the use status and characteristics of three-phase converters are discussed. Then, the Luenberger observer and its optimization process are described. Finally, the Luenberger observer is optimized through phase-locked loop technology and the vector control method. The experimental results indicate that the PLL of the steady-state linear Kalman filter is applicable to the multielectric aircraft converter for the aviation variable frequency power supply. The phase-locked loop of the steady-state linear Kalman filter is complicated, and the output angular frequency is inconsistent with the angular frequency of the actual voltage of the aircraft variable-frequency power supply. Consequently, it does not have the function of frequency locking. On the contrary, the Luenberger observer phase-locked loop designed here is suitable for the multielectric aircraft converter for the aircraft variable-frequency power supply. In addition, it is simpler than the steady-state linear Kalman filter phase-locked loop and realizes the frequency-locking function. In addition, the vector control method significantly improves the control performance of the Luenberger observer. The control error of the original observer is about 0.24°, and the control error of the optimized observer is about 0.18°. This work provides technical support for the performance optimization of the Luenberger observer and contributes to the performance improvement of the aviation three-phase converter.

Sequential Covariance Intersection Fusion Robust Time-Varying Kalman Filters with Uncertainties of Noise Variances for Advanced Manufacturing.

This paper addresses the robust Kalman filtering problem for multisensor time-varying systems with uncertainties of noise variances. Using the minimax robust estimation principle, based on the worst-case conservative system with the conservative upper bounds of noise variances, the robust local time-varying Kalman filters are presented. Further, the batch covariance intersection (BCI) fusion and a fast sequential covariance intersection (SCI) fusion robust time-varying Kalman filters are presented. They have the robustness that the actual filtering error variances or their traces are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved based on the proposed Lyapunov equations approach. The concepts of the robust and actual accuracies are presented, and the robust accuracy relations are proved. It is also proved that the robust accuracies of the BCI and SCI fusers are higher than that of each local Kalman filter, the robust accuracy of the BCI fuser is higher than that of the SCI fuser, and the actual accuracies of each robust Kalman filter are higher than its robust accuracy for all admissible uncertainties of noise variances. The corresponding steady-state robust local and fused Kalman filters are also presented for multisensor time-invariant systems, and the convergence in a realization between the local and fused time-varying and steady-state Kalman filters is proved by the dynamic error system analysis (DESA) method and dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and the correctness of the robust accuracy relations.

Real-time estimation of the heat transfer coefficient of pitot tubes undergoing freezing

PurposeThe purpose of this paper is to identify freezing in pitot tubes at real-time, by means of the estimated heat transfer coefficient (HTC) at the tip of the probe. The prompt identification of such freezing is paramount to activate and control mechanisms for ice removal, which in turn are essential for the safety of the aircraft and its passengers.Design/methodology/approachThe proposed problem is solved by means of an inverse analysis, performed within the Bayesian approach of inverse problems, with temperature measurements assumed available along the pitot probe over time. A heat conduction model is used for describing the average temperature of the pitot tube, which is then rewritten in the form of a state estimation problem. The model is linear and time invariant, so that the inverse problem can be solved using the steady-state Kalman filter (SSKF), a computationally efficient algorithm.FindingsThe results show that the SSKF is fully capable of recovering the HTC information from the temperature measurements. Any variation of the HTC – either smooth or discontinuous – is promptly detected with high accuracy. Computational effort is significantly lower than the physical time, so that the proposed methodology is fully capable of estimating the HTC at real-time.Originality/valueThe methodology herein solves the proposed problem not only by estimating the HTC accurately but also doing so with a very small computational effort, so that real-time estimation and freezing control become possible. To the best of the authors’ knowledge, no likewise publications have been found so far.

A Power-Efficient Brain-Machine Interface System With a Sub-mw Feature Extraction and Decoding ASIC Demonstrated in Nonhuman Primates.

Intracortical brain-machine interfaces have shown promise for restoring function to people with paralysis, but their translation to portable and implantable devices is hindered by their high power consumption. Recent devices have drastically reduced power consumption compared to standard experimental brain-machine interfaces, but still require wired or wireless connections to computing hardware for feature extraction and inference. Here, we introduce a Neural Recording And Decoding (NeuRAD) application specific integrated circuit (ASIC) in 180 nm CMOS that can extract neural spiking features and predict two-dimensional behaviors in real-time. To reduce amplifier and feature extraction power consumption, the NeuRAD has a hardware accelerator for extracting spiking band power (SBP) from intracortical spiking signals and includes an M0 processor with a fixed-point Matrix Acceleration Unit (MAU) for efficient and flexible decoding. We validated device functionality by recording SBP from a nonhuman primate implanted with a Utah microelectrode array and predicting the one- and two-dimensional finger movements the monkey was attempting to execute in closed-loop using a steady-state Kalman filter (SSKF). Using the NeuRAD's real-time predictions, the monkey achieved 100% success rate and 0.82 s mean target acquisition time to control one-dimensional finger movements using just 581 μW. To predict two-dimensional finger movements, the NeuRAD consumed 588 μW to enable the monkey to achieve a 96% success rate and 2.4 s mean acquisition time. By employing SBP, ASIC brain-machine interfaces can close the gap to enable fully implantable therapies for people with paralysis.

Comparative Analysis of Three Structures of Second-Order Generalized Integrator and Its Application to Phase-Locked Loop of Linear Kalman Filter

The present work explores the power quality problems of a microgrid in the aviation field, such as frequency offset or waveform distortion, caused by voltage imbalance and nonlinear load, to ensure the efficient and stable flight of aircraft, unmanned aerial vehicles, and other aircraft. There are many problems such as excessive harmonics, voltage imbalance, and direct current (DC) component in aviation variable frequency power supply voltage in microgrid under different conditions. Therefore, three kinds of second-order generalized integrators (SOGIs) with different structures are combined with linear Kalman filter phase-locked loops (PLLs). Besides, intelligent sensors are utilized for signal processing. Finally, simulation experiments are conducted to compare three SOGIs. The results show that the system is stable when the parameters k1 = 738.9553, k2 = 1092108.98405, k3 = 1477.91, and Ts = 0.000125 s and k ranges in [0.5, 3]. The angular frequency of PLL output is very low under the problems of many harmonics, three-phase voltage imbalance, and DC component ω g . The angular frequency ω g output by the PLL finally changes linearly with a change rate of 400 πrad/s2, so that the output phase angle θ g reaches a stable state. Thus, the proposed steady-state linear Kalman filter phase locking can accurately phase-lock the aviation variable frequency power supply. It provides an important reference for the power supply module in the microgrid to select the appropriate second-order generalized integrator to realize the accurate phase locking of the phase-locked loop under different conditions.

Distributed Periodic Steady State Kalman Filter

In this paper a distributed implementation for the periodic steady state Kalman filter is proposed. The distributed algorithm has parallel structure and can be implemented using processors in parallel without idle time. The number of processors is equal to the model period. The resulting speedup is also derived. The Finite Impulse Response (FIR) form of the periodic steady state Kalman filter is derived.

Semi-analytical assessment of the relative accuracy of the GNSS/INS in railway track irregularity measurements

An aided Inertial Navigation System (INS) is increasingly exploited in precise engineering surveying, such as railway track irregularity measurement, where a high relative measurement accuracy rather than absolute accuracy is emphasized. However, how to evaluate the relative measurement accuracy of the aided INS has rarely been studied. We address this problem with a semi-analytical method to analyze the relative measurement error propagation of the Global Navigation Satellite System (GNSS) and INS integrated system, specifically for the railway track irregularity measurement application. The GNSS/INS integration in this application is simplified as a linear time-invariant stochastic system driven only by white Gaussian noise, and an analytical solution for the navigation errors in the Laplace domain is obtained by analyzing the resulting steady-state Kalman filter. Then, a time series of the error is obtained through a subsequent Monte Carlo simulation based on the derived error propagation model. The proposed analysis method is then validated through data simulation and field tests. The results indicate that a 1 mm accuracy in measuring the track irregularity is achievable for the GNSS/INS integrated system. Meanwhile, the influences of the dominant inertial sensor errors on the final measurement accuracy are analyzed quantitatively and discussed comprehensively.

Steady state Kalman filter design for cases and deaths prediction of Covid-19 in Greece

In this work we study the applicability of the steady state Kalman filter in order to predict new cases and deaths of Covid-19. We use the actual observations of new cases and deaths. First, we deal with short term prediction, namely daily prediction. We propose the use of the golden steady state Kalman Filter, which is designed to have parameters related to the golden section. It was found that the proposed golden steady state Kalman Filter has a satisfactory behavior compared with the classical mean or average filter. Secondly, we deal with long term prediction, for example average prediction per quarantine period (14 days). We propose to process blocks of measurements of time window corresponding for example to the quarantine period in order to predict the average of cases and deaths using steady state Kalman Filter. It was found that the proposed golden steady state Kalman Filter produces more reliable predictions than the classical mean or average filter does. The use of steady state Kalman Filter for cases and deaths prediction of Covid-19 can be effective for resources and prevention measures planning.

A Novel Low Computational Burden Dual-Observer Phase-Locked Loop With Strong Disturbance Rejection Capability for More Electric Aircraft

In the past two decades, much research and industry effort has been dedicated to transportation electrification. In particular, aerospace industry, with the more electric aircraft (MEA) revolution, is shifting toward the usage of electrical in place of mechanical energy on board to save fossil fuel and reduce polluting emissions. For this reason, phase-locked loops (PLLs) systems have been well developed for synchronizing different power sources and loads within the aircraft electrical grid. Since one of the proposed solutions for MEA power grids is to operate at variable frequency from 360 to 800 Hz, a third-order model-based steady-state linear Kalman filter PLL (SSLKF-PLL) has been proposed in the literature to achieve fast tracking performance during such grid frequency variations. To suppress the potential disturbances due to harmonics in the grid, sensor scaling errors/unbalances, and dc offsets while maintaining low computational burden, this article aims to enhance the disturbance rejection ability of SSLKF-PLL by adding a repetitive observer (RO). Simulation and experimental tests show that RO allows stable and effective suppression of disturbances from all the abovementioned sources during variable frequency operation. The execution time of the proposed PLL is only 4.23 μs using a Xilinx Zynq-7000-based platform, where the added RO takes 2.664 μs to execute, and the SSLKF-PLL alone takes 1.566 μs.

Interference Prediction in Wireless Networks: Stochastic Geometry Meets Recursive Filtering

This article proposes and evaluates a technique to predict the level of interference in wireless networks. We design a recursive predictor that estimates future interference values by filtering measured interference at a given location. The predictor's parameterization is done offline by translating the autocorrelation of interference into an autoregressive moving average (ARMA) representation. This ARMA model is inserted into a steady-state Kalman filter enabling nodes to predict with low computational effort. Results show a good accuracy of predicted values versus true values for relevant time horizons. Although the predictor is parameterized for Poisson-distributed nodes, Rayleigh fading, and fixed message lengths, a sensitivity analysis shows that it also tends to work well in more general network scenarios. Numerical examples for underlay device-to-device communications, a common wireless sensor technology, and coexistence scenarios of Wi-Fi and LTE illustrate its broad applicability. The predictor can be applied as part of interference management to improve medium access, scheduling, and radio resource allocation.

Application of Model Predictive Control Based on Kalman Filter in Solar Collector Field of Solar Thermal Power Generation

There are two prominent features in the process of temperature control in solar collector field. Firstly, the dynamic model of solar collector field is nonlinear and complex, which needs to be simplified. Secondly, there are a lot of random and uncontrollable, measurable and unmeasurable disturbances in solar collector field. This paper uses Taylor formula and difference approximation method to design a dynamic matrix predictive control (DMC) by linearizing and discretizing the dynamic model of the solar collector field. In addition, the purpose of controlling the stability of the outlet solar field salt temperature is achieved by adjusting the mass flow of molten salt. In order to further improve the ability of the system to suppress unmeasured disturbances, a steady-state Kalman filter is designed to estimate state variables, so that the system has better stability and robustness. The simulation verification results show that the DMC control system based on Kamlan filtering has better control effect than the traditional DMC control system. In the case of large fluctuations in solar radiation intensity and consideration of undetectable interference, the overshoot of the system is reduced by 4% and the rise time remains unchanged.

Power Generation Prediction of an Open Cycle Gas Turbine Using Kalman Filter

The motivation for this paper is the enhanced role of power generation prediction in power plants and power systems in the smart grid paradigm. The proposed approach addresses the impact of the ambient temperature on the performance of an open cycle gas turbine when using the Kalman Filter (KF) technique and the power-temperature (P-T) characteristic of the turbine. Several Kalman Filtering techniques are tested to obtain improved temperature forecasts, which are then used to obtain output power predictions. A typical P-T curve of an open-cycle gas turbine is used to demonstrate the applicability of the proposed method. Nonlinear and linear discrete process models are studied. Extended Kalman Filters are proposed for the nonlinear model. The Time Varying, Time Invariant, and Steady State Kalman Filters are used with the linearized model. Simulation results show that the power generation prediction obtained using the Extended Kalman Filter with the piecewise linear model yields improved forecasts. The linear formulations, though less accurate, are a promising option when a power generation forecast for a small-term and short-term time window is required.