Variable Step Size Kalman Filter Using Event Handling Algorithm for Switching Systems

A new method based on the innovation of Kalman filter and extrapolation-accumulation is proposed to detect weak frequency anomalies in atomic clocks with short detection time and high detection probability. In this method, the detection statistics of the innovation extrapolation method in several epochs are accumulated. It avoids the influence of outliers, increases the noncentrality parameter of chi-square distribution, and realizes more effective detection of weak frequency anomalies. The simulation results show that compared with the innovation method and the innovation extrapolation method, the new method has a higher detection probability for micro frequency anomalies, and the detection time is shortened. The new method is used to analyze the real data of the cesium atomic clock, and the results are consistent with the simulation results.

Alignment is a critical pre-processing step before initiating inertial navigation operations. The extended Kalman filter (EKF), commonly employed for aligning strap-down inertial navigation systems (SINS), faces challenges due to linearization limitations. In response to some of these challenges, the invariant extended Kalman filter (IEKF) was developed. A key requirement for utilizing IEKF is that the system dynamics must exhibit a group affine property. However, considering inertial measurement unit (IMU) errors as state variables violates this condition. This paper introduces an improved version of IEKF, named the proportional-integral invariant Extended Kalman Filter (PI-IEKF). This approach initially ignores IMU bias and drift to preserve the group affine property. To compensate for these uncertainties, the Kalman filter's innovation term is reformulated into a proportional-integral (PI) structure. The PI gains are tuned based on Lyapunov stability criteria by solving a Linear Matrix Inequality (LMI). The PI-IEKF not only addresses the uncertainties but also enhances alignment accuracy. The proposed approach was evaluated through simulations and field tests. Simulations addressed the nonlinear alignment problem in marine environment with typical wave disturbances, while field tests used an open-source dataset from vehicle-mounted sensors. Comparative results with conventional IEKF demonstrated significant improvements in convergence speed and robustness.

Parameter estimation in a simple stochastic differential equation for phytoplankton modelling

The paper is concerned with the problem of distributed Kalman Filtering for discrete-time linear large-scale systems with decentralized sensors. Using the standard approach to the centralized Kalman filtering, the problem of distributed filtering is introduced, given the incidence of additive recurrence to realize such problem. The obtained solutions support the residual signal generation using Kalman filter innovations in the model-based fault detection design. The results, offering structures for fault detection filter realization, are illustrated with a numerical example to note the effectiveness of the approach.

A solution to the state estimation problem of systems with unmeasurable non-zero mean inputs/disturbances, which do not satisfy the disturbance decoupling conditions, is given using the Kalman filtering and Bayesian estimation theory. The proposed estimation algorithm, named Supervisory Kalman Filter (SKF), consists of a Kalman filter with an extra update step which is inspired by the particle filtering technique. The extra step, called supervisory layer, numerically solves the measurement equations for the portion of the state vector that cannot be estimated by the Kalman filter. First, it produces N randomly generated state vectors, the particles, which are distributed based on the Kalman filter’s last updated estimate. Then, the estimated measurement vector associated with each particle is compared to the actual measurement vector to identify the particle’s probability to be a solution. Finally, a so-called resampling stage is implemented to refine the particles with higher likelihoods. The effectiveness of the SKF is demonstrated by comparing its estimation performance with that of the standard Kalman Filter and the particle filter for a vehicle state estimation problem. The estimation results confirm that the SKF precisely estimates those states of the vehicle that cannot be estimated by either the Kalman filter or the particle filter, regardless of the unknown disturbances from the road. The filtering methodology offered in the article has a potential to improve performance of the systems presented in the patents WO2011115960, WO2010024751, US8073528, and US20110299730. Keywords: Bayesian state estimation, disturbance decoupling conditions, inputs/disturbances, particle filtering, state estimation, systems with unknown kalman filtering, unscented kalman filtering

For an implicitly defined discrete system, a new implicit form of the Kalman filter is developed and an efficient numerical approach for its implementation is given. Unlike the traditional explicit approach, the implicit filter can be applied to ill-conditioned systems. It is shown that when the system matrices of an implicit system are sparse, the implicit Kalman filter requires significantly less computer time and storage to implement than the traditional explicit Kalman filter. Simulation results are presented to illustrate and substantiate the theoretical developments.

We propose an adaptive Kalman phase locked loop (PLL) structure for ionospheric scintillation mitigation on global navigation satellite systems (GNSS) receivers, employing radial basis function (RBF) networks to model the scintillation phase and amplitude. The Kalman filter innovations are computed by the arctangent phase discriminator, and the state estimates are directly employed in the carrier replica generation. The nonlinear RBF provide more flexibility to capture nonlinear dynamics evolving with time, possibly present in the scintillation phase and amplitude. The weights of the RBF network and the covariance matrices of the process and measurement noise of the Kalman filter are estimated online, in the adaptive Kalman PLL structure. Simulations with synthetic severe scintillation data show the capability of the proposed Kalman PLL to improve robustness to scintillation effects in carrier synchronization, with performance similar to the corresponding structure employing autoregressive (AR) scintillation model. Simulations with real scintillation data collected by a commercial receiver highlight the learning and generalization capability of the RBF networks to cope with evolving scintillation characteristics over time with possibly nonlinear effects. The Kalman PLL structure employing the RBF network presents reduced state estimation errors when compared to the structure employing the AR model.

The dynamics of many systems are described by ordinary differential equations (ODE). Solving ODE's with standard methods (i.e. numerical integration) needs a high amount of computing time but only a small amount of storage memory. For some applications, e.g. short time weather forecast or real time robot control, long computation times are prohibitive. Is there a method which uses less computing time (but has drawbacks in other aspects, e.g. memory), so that the computation of ODE's gets faster? We will try to discuss this question for the method of a neural network which was trained on ODE dynamics and compare both methods using the same approximation error. In many cases, as for physics engines used in computer games, the shape of the approximation curve is important and not the exact values of the approximation. Therefore, we introduce as error measure the subjective error based on the Total Least Square Error (TLSE) which gives more consistent results than the standard error. Finally, we derive a method to evaluate where neural nets are advantageous over numerical ODE integration and where this is not the case.

The integration of GPS, PL and INS sensors can be implemented at three different levels. Compared with loose and tight integration, ultra-tight integration offers numerous advantages including increased robustness under high dynamics, and improved antijamming performance. In current ultra-tight integration scenarios, a centralised Kalman filter is commonly employed to fuse either In-phase (I) and Quadrature (Q) data from the tracking loop or the pseudorange measurement and Position, Velocity, Attitude (P, V, A) measurements from the Inertial Navigation System (INS). Though relatively simple, this centralised filter structure has some disadvantages. Firstly, to reduce the computational load, the filter only makes coarse estimates of Inertial Measurement Unit (IMU) random errors, which significantly degrades the system performance. Secondly, for more accurate estimates, the filter becomes much more complicated, resulting in a large increase in the computation time. All of these hinder the performance of ultra-tight integration considerably. This paper proposes a federated filter structure for the ultra-tight integration of GPS, PL and INS sensors. The new filter structure distributes the computing tasks to different Kalman filters, leading to reduced filter complexities and improved system performance. IMU random errors are estimated separately by the pre-filter at a high data rate, whilst the main filter has a simplified structure, i.e. no estimation of the IMU random errors, and operates at a relatively slow rate. This paper will discuss the dynamic modelling method based on the Walsh function transform for implementing the pre-filter and the simplification of the main filter. Simulation tests were performed to compare the performance of the federated filter with that of the usual centralised Kalman filter in the estimation of the IMU random errors. The results show that with the simplification of the Kalman filter structure, the federated filter design can achieve the almost equally precise estimates as the centralised Kalman filter does but with less computational burden. Hence the federated design is more suitable for implementing the ultra-tight integration for real-time applications. Finally, the simulated high dynamic flight test results of ultra-tight integration based on the federated Kalman filter are presented.

The article concerns remote measurement. Aiming to obtain current state variables of a mechanical system with time delay, we estimated current state variables from the delayed measurement signal by using a Kalman filter. In order to increase the stability of the estimation system, we have proposed a novel Kalman filter with dual mathematical models, and also have shown that it was very effective because it was more robust and precise than a conventional Kalman filter with a single model. We have examined an influence for the estimation error that was caused by modeling errors of the plant, and have examined one for the sampling time error. From these results it is safer to set the variance of the measurement noise and input noise higher if uncertain, because the increment of the estimation error is smaller. Moreover the increment of the estimation error caused by the sampling time error keeps almost constant from the experiment and the simulation, when the sampling time is set longer.

A new predictive control strategy for improving operating performance of a permanent magnet synchronous generator-based wind energy and superconducting magnetic energy storage hybrid system integrated with grid

Deterministic cell-growth models describe the growth of cell populations using fixed mathematical rules, assuming no randomness in the system. These models are often based on differential equations that account for the rates of cell division, death, and other biological processes. The solution to the system is obtained via numerical methods. Most of the developed approaches are based on fixed step sizes. However, fixed step size implementation failed to offer the optimal solutions when dealing with stiff challenges. Fixed step size methods can be unstable for stiff equations, where some components of the solution change much more rapidly than others. The step size, required to maintain stability can become impractically small. Thus, the adaptive step size method is required. Adaptive step size methods adjust the step size dynamically based on the behavior of the solution, aiming to maintain a desired level of accuracy while optimizing computational efficiency. These methods are particularly useful for solving ordinary differential equations (ODEs) where the solution can vary rapidly in some regions and slowly in others. This study is devoted to comparing the implementation of fixed step size and adaptive step size in solving ordinary differential equations (ODEs). The fixed step size and adaptive step size numerical method are solved in this study via the fourth order Runge-Kutta method (RK4) and Runge-Kutta Fehlberg 45 method (RKF45). The performance of both numerical methods used will be analyzed by comparing the numerical results approximated with the actual data. Subsequently, the absolute error, relative error, and rounding-off error will be calculated to compare both approaches. Based on the more precise findings, this work has shown that adaptive step size is predicted to be the optimal representation for solving ODEs. As a result, this may help mathematicians to choose the most effective numerical approach for solving ODEs.

High performance in the elbow joint angle estimation, using electromyography (EMG) signal, is essential in the devices which uses EMG as a control signal. The purpose of this study is to evaluate Kalman filter performance which is used to improve the linearity of the elbow joint angle estimation. The EMG signal is recorded from biceps muscle while the elbow performed a flexion and extension motion with two second periods of motion. In order to acquire the information which is associated with the elbow joint angle, the EMG signal is extracted for every 100 milliseconds by using the twelve time-domain features. Kalman filter is used to improve the elbow joint angle estimation after feature extraction process. The findings show that the maximum performance of the elbow joint angle estimation is 0.96±0.01 (correlation), 0.93±0.02 (R2), 9.02°±6.82° (intercept), 0.89±0.09 (slope) from sign slope change (SSC) feature. This proposed method has revealed the effectiveness of Kalman filter to improve the performance of the elbow joint angle estimation based on EMG signal.

Multibody dynamics comprises methodologies for the design and analysis of mechanical systems, with Kalman filters being the principal state estimation methods. The Kalman filters are generally formulated for unconstrained systems, the dynamics of which are described by ordinary differential equations in state-space models. However, multibody systems are constrained systems, and the commonly used method to describe their dynamics involves using differential algebraic equations (DAEs) comprising differential and algebraic equations. The differential equations also include the Lagrange multipliers. Hence, incorporating multibody systems, described by DAEs into the scheme of Kalman filters, cannot be achieved straightforwardly, which facilitates different strategies being addressed. This study develops a novel method for converting DAEs into a state-space model. A transition model of the time derivatives of Lagrange multipliers and a Lagrange multiplier constraint vector are devised and used in the state and output equations, respectively. The continuous- and discrete-time extended Kalman filters (CEKF and DEKF) are constructed using the proposed state-space model, and state estimations are simulated on the benchmark planar four- and five-bar linkages. Further, a demonstration of system observability is conducted, and sensitivity to the initial state estimates is studied. These tests demonstrate that the proposed state-space model achieves observable systems and that both the CEKF and DEKF, constructed using the proposed state-space model, can estimate the states with a wide range of initial conditions.

MATLAB-based general approach for square-root extended-unscented and fifth-degree cubature Kalman filtering methods