Continuous-discrete Kalman Filter Research Articles

Kalman Filter Fusion With Smoothing for a Process With Continuous-Time Integrated Sensor

Integrated Sensor is a well-known methodology of slow sampling rates in chemical processes, which gradually collects material samples within a finite period. This type of infrequent measurement specified for quality variables is a function of states over sampling time. Our objective is to solve the problem of estimating states in fast sampling rates for a continuous-time system in the presence of a slow-rate integrated sensor. For this purpose, two reformulated models are suggested for describing the system, while they do not explicitly include the integral term. Integrated Measurement Kalman Filter (IMKF) for these two models are presented by the extension of the classical Continuous-Discrete (CD) Kalman filter to the integrated systems. Then, a novel sliding window smoothing algorithm is proposed for integrated systems using pseudo-estimations of the output, past time information of the fast-rate estimates and the last available slow-rate measurement. Also, the optimal fusion algorithm employing the estimation results of two reformulated models is extended for the integrated system. Eventually, the observability condition and exponential convergence of observation error are proved for the proposed algorithm. Simulation and experimental implementations are employed to demonstrate the effectiveness of the proposed method through a drum boiler model and a laboratory-scale pressure control process, respectively.

Accounting for simulation errors in continuous-discrete Kalman filtering

In continuous-discrete Kalman filter implementations there is a tradeoff between the computational requirements for real time implementation, and errors incurred by selection of step size and method order in the simulation of the continuous time system model between sampling instances. Because the Kalman filter corrects errors in the state vector using output measurements, a systematic approach to account for the effects of simulation errors in the continuous-discrete Kalman filter allows the filter to be designed to deal with simulation errors in addition to conventional process (state) and measurement (output) noise.

Data assimilation as a key step towards the implementation of an efficient management of dissolved oxygen in land-based aquaculture

A data assimilation (DA) methodology, e.g. the continuous-discrete Kalman filter (CD-KF), was applied to the assimilation of dissolved oxygen data, in order to obtain a dynamic estimation of the oxygen demand in a land-based aquafarm. The CD-KF was implemented on a dynamic model, which included as state variables the concentration of dissolved oxygen (DO) and fish respiration rate: the latter was considered as a non-observable stochastic variable. The model was applied to a 1-month long set of observations collected at a raceway rainbow trout farm, including (1) hourly time series of water temperature and dissolved oxygen concentration in the raceway influent and effluent and (2) a daily time series of fish number and fish weight distribution. The results show that the assimilation of DO data led to a dynamic estimate of DO demand which showed changes in the daily mean and the daily pattern: these were related to changes in the feeding regime. Furthermore, the methodology provided accurate short-term predictions of the DO concentration also in the presence of short-term fluctuations, which would be very difficult to relate to external forcings in a mechanistic model. These findings indicate that DA could be effectively used to design and implement efficient and robust control systems for optimizing the oxygen supply, thus contributing to the implementation of Precision Fish Farming in land-based aquafarms.

UAV Attitude Angle Measurement Method Based on Magnetometer-Satellite Positioning System

To improve the measurement accuracy of unmanned aerial vehicles (UAVs), in this study, we proposed a method for measuring the attitude angle of a UAV by the combination of a magnetometer and satellite positioning system. Based on the measurement principle, we established a combined measurement error model, analyzed the root cause of the measurement error, and calibrated the magnetometer using the least-squares algorithm. For handling external or system interference effects, we designed a compensation algorithm that does not depend on environmental information. For handling measurement errors, a “current” statistical model was established, and the continuous-discrete Kalman filter algorithm was used to eliminate measurement errors. The test results showed that after using the compensation algorithm, the measurement error was reduced from 17.80% to 6.86% compared to the average absolute error of the local magnetic field intensity, and the absolute measurement error was reduced by 61.49%. The absolute measurement error was compared after using the compensation + filtering algorithm, and the error after compensation was further reduced by 42.36%. Hence, we proved the feasibility and effectiveness of our method.

Overall hyperbolic-singular-value-decomposition-based square-root solutions in Kalman filters with deterministically sampled mean and covariance for state estimation in continuous-discrete nonlinear stochastic systems

The paper aims at square-rooting continuous-discrete Kalman filters with deterministically sampled mean and covariance. These cover all methods devised within the quadrature, cubature and unscented Kalman filtering frameworks. The main problem addressed is a potential negativity of some weights used in calculations of the sampled expectations and covariances, which precludes from an orthogonal square-rooting procedure to be designed and applied. However, square-root solutions are often requested because of their exceptional robustness to round-off and other errors committed. These also preserve the symmetry and positivity of the covariances computed in automatic mode. Here, we apply the J-orthogonal square-rooting approach devised earlier to designing our overall square-root solutions in the realm of quadrature, cubature and unscented Kalman filtering techniques, which are adjusted easily to any state estimator in use by setting its weights and deterministically selected nodes. Our J-orthogonal square-rooting procedure is grounded on the hyperbolic singular value decomposition. It leads to two algorithms covering all Itô-Taylor-approximation-based Kalman filters with deterministically sampled mean and covariance. Performances of such square-root solutions are assessed and compared to their non-square-root counterparts in a simulated flight control scenario with ill-conditioned measurements.

Reevaluation of Algorithmic Basics for ZUPT-Based Pedestrian Navigation

During the last 10 to 15 years, pedestrian navigation based on zero velocity updates (ZUPT) has become very popular. One of the main reasons for this is the increasing availability of small, low-cost inertial measurement units. However, the processing of the data from these units for pedestrian navigation is almost exclusively based on algorithmic features that originate from classical inertial navigation with high-grade sensors. In addition, the historical background of the ZUPT approach presupposes also sensors with high accuracy. This leads to the problem that neither the ZUPT approach nor the algorithmic features mentioned are consistent with the accuracy level of the inertial measurement units used normally for pedestrian navigation. Therefore, the question arises of whether the usual algorithmic basics and numerical procedures employed by ZUPT-based pedestrian navigation are adequate. Supported by a literature review showing the state of the art, this study investigates the effect of basics such as the system states and the usage of Runge-Kutta method on the navigation accuracy from pedestrian inertial data aided by ZUPT. To this end, a comparative processing of a well-known, published dataset of a short walk and of own data from the authors using different data fusion algorithms was employed. The important results of this study are that the often-used omission of inertial sensor biases cannot be recommended, that a continuous-discrete Kalman filter in combination with a Runge-Kutta algorithm performs better than traditional filter configurations, and that total navigation states are an interesting alternative to classical navigation error-states.

Input-to-state Stability for Hybrid Kalman Filters

In this article, we propose a Lyapunov-based characterization of exponential input-to-state stability (ISS) for hybrid systems. We propose a new formulation of the continuous-discrete Kalman filter algorithm Särkkä (2006) using the hybrid systems framework Goebel et al. (2012) where we consider continuous-time system dynamics and discrete-time observations. We prove exponential input-to-state stability on the estimation error of the hybrid Kalman filter under mild assumptions on the system's model.

Optimal operation of an ice-tank for a supermarket refrigeration system

The increasing proportion of renewable energy sources in power grids leads to challenges concerning balancing production and consumption. One solution to this grid challenge is to utilize demand-side flexibility. To use the full potential of demand-side flexibility, dynamical models and optimal control methods must be used. This paper demonstrates how demand-side flexibility can be enabled for a refrigeration system using an add-on ice-tank module to actively curtail the refrigeration system and thereby leveraging time-varying power prices. The operation of the ice-tank is the solution to an optimal control problem that minimize the integrated electricity costs. This optimal control problem is solved numerically and the performance of the strategy is successfully tested in a real experiment where cost-savings of approximately 20% are observed (compared to not having an ice-tank available). The proposed optimal control strategy is tested in real-life experiments spanning over 24 h. The dynamical relation between the operation of the ice-tank and the power consumption (the compressor capacity) is modelled using stochastic differential equations. This differential equation model is calibrated on 13 h of training data using the continuous–discrete Kalman filter and the maximum likelihood framework.

Underwater mobile gravity measurement data processing using continuous-discrete Kalman filter

To solve the data fusion problem in underwater mobile gravity measurement, a direct estimation method based on pose information is proposed. First, the state model is established using the continuous evolution equation of the state. Then, to obtain the space model of the state for data fusion of the measured underwater mobile gravity, the measurement model is constructed according to the external measurement equation. To solve the continuous-discrete space model of the state, the numerical integration method for continuous time updates and the virtual sampling iterative method for discrete observation updates are proposed, which can be used for the time update case and the observation update case, respectively. In this way, the continuous-discrete iterative extended Kalman filter for direct pose estimation is formed. The proposed methods are compared and analyzed based on the motion simulation data of different carrier and inertial measurement unit simulation data for different accuracy levels. The method proposed in this paper has been validated by carrying out the experiments on a lake. The results show that the method proposed in this paper can minimize the effect of accuracy of initial alignment on strapdown gravimetric measurement and shows a better dynamic estimation performance.

A Hybrid, Coupled Approach to the Continuous-Discrete Kalman Filter

In this letter we present a novel approach to continuous-discrete (CD) Kalman filtering. Unlike the EKF or other hybrid UKF-EKF filters, the novel approach does not require direct calculation of system Jacobians and instead uses unscented transforms (UTs) to extract a pair of matrices, each made up of a linear combination of derivatives (with respect to the state), that are used in its place. More specifically, they are used (1) in parts of the filtering process, and (2) in the linearly implicit numerical integration scheme of the filter's state propagation stage. Extracting these matrices from the filter's UTs and using them in both the filtering process and model simulation, or what we refer to as coupling the filter to model simulation, avoids having to calculate further function evaluations for standard implicit methods, making the process of state estimation for stiff systems more efficient. Another benefit of the proposed approach is that it offers UKF accuracy levels but improved numerical stability over the UKF. This is because it uses symmetric and positive-definite (PD) representations of the state covariance propagation and measurement update equations.

Networked State Estimation With Delayed and Irregularly Spaced Time-Stamped Observations

We analyze the performance of a networked multisensor continuous-discrete Kalman filter in the presence of random observation delay and irregular sampling. The primary cause for irregular intersample intervals is the occasional loss of information packets. Additional causes include lack of synchrony between sensors, difference in their sampling rates, and the superposition of multiple sampling patterns. We rely on time stamping of all sensor packets to accurately determine, at the estimation hub, both transmission delay and sensor sampling instants. We relate the average (steady-state) error covariance to two moment-generating functions: 1) we provide lower and upper bounds on the average error covariance in the presence of irregular sampling, expressed in terms of the moment-generating function of the sampling intervals; 2) we obtain a stability condition that depends on the region of convergence of this moment-generating function; and 3) we derive an expression for the added error caused by transmission delay, which depends on the moment-generating function of the delay. We also demonstrate that the average error covariance depends primarily on the average sampling interval, with a very minor dependence on the variance and higher order moments of the multisensor intersample interval.

Event-Triggered Continuous-Discrete Kalman Filter With Controllable Estimation Error

In this paper, we first study the event-triggered continuous-discrete Kalman filtering problem to control the MSE under a given triggering threshold, where if the mean squared error (MSE) is smaller than this threshold, then an event is triggered to let the system take a new measurement to reduce the MSE. Previously, since the accurate relationship between the sampling period and the MSE is unknown, it is difficult to control the MSE under a certain system requirement by tuning the sampling period. With our proposed event-triggered continuous-discrete Kalman filter, we are able to upper bound the MSE as expected. Specifically, we strictly prove that whenever a measurement is taken, and the decrease in MSE is lower bounded by a constant determined by the system parameters as well as the triggering threshold if the output matrix has no zero columns. Based on this important property, we provide a rigorous proof that the MSE is finally upper bounded by the triggering threshold, as time goes to infinity. Also, we prove that the event-triggered continuous-discrete Kalman filter can be implemented in a self-triggered manner. Finally, our simulation results corroborate the effectiveness and accuracy of the theoretical results.

Distributed mixed continuous‐discrete receding horizon filter for multisensory uncertain active suspension systems with measurement delays

This study presents a new robust filtering method in modelling an active multisensory suspension system with measurement delays and parameteric uncertainties in a state-space dynamical model. To achieve good performance of the system, a new distributed fusion receding horizon filtering frameworks are constructed to couple the continuous dynamics with the multisensory discrete measurements, and to coordinately deal with the parametric uncertainty and time-delays. The novel filtering algorithm is proposed based on the receding horizon strategy, standard mixed continuous-discrete Kalman filtering and discrete Kalman filtering for systems with time-delays in order to achieve high estimation accuracy and stability under parametric uncertainties. The key theoretical contributions of this study are the derivation of the error cross-covariance equations between the local receding horizon filters in order to compute the optimal matrix fusion weights. The high accuracy and efficiency of the new filter are demonstrated through its implementation and performance and then compared to the existing vehicle active suspension system.

Rejection of Aliased Disturbances in a Production Pulsed Light Source

We study a disturbance rejection problem in a production pulsed light source, used in semiconductor photolithography, to yield quantifiable and guaranteed improved performance over existing control techniques. The disturbances of interest include an offset with reset properties and sinusoids which appear aliased in the measured data which is available only at pulse events. The light source is pulsed at varying rates yet actuators move in continuous-time, yielding a system which blends aspects of continuous-time and variable-data-rate discrete-time. We employ novel modifications to standard continuous-discrete Kalman filtering ideas for disturbance state estimation and establish and solve a non-standard regularized minimum variance control problem within a disturbance rejection framework. The controller as discussed is now in production in semiconductor lithography manufacturing lines. We analyze data from these production light sources and show the controller has the capacity to remove aliased sinusoids from the measured output and yields operational performance levels provably close to optimal for the hardware.

Limited view PET reconstruction of tissue radioactivity maps

This paper proposes a state space method for limited view PET reconstruction. Due to the high-level of noise and data-incompletion, prior knowledge is required to guide PET recovery. The compartmental model is used as an evolution equation to regularize the dynamic reconstruction. The continuous–discrete Kalman filter is adopted to calculate the radioactivity value recursively. With tracer kinetic information as prior, the state space approach can obtain a better result compared with the MLEM algorithm. The identifiability of this method is proved by computer synthetic simulation and real phantom experiment on the Hamamatsu SHR-22000PET scanner.

Sampled-Data Filtering Framework for Cardiac Motion Recovery: Optimal Estimation of Continuous Dynamics From Discrete Measurements

Quantitative and noninvasive estimation of cardiac kinematics has significant physiological and clinical implications. In this paper, a sampled-data filtering framework is presented for the recovery of cardiac motion and deformation functions from periodic medical image sequences. Cardiac dynamics is a continuously evolving physical/physiological process, whereas the imaging data can provide only sampled measurements at discrete time instants. Given such a hybrid paradigm, stochastic multiframe filtering frameworks are constructed to couple the continuous dynamics with the discrete measurements, and to coordinately deal with the parameter uncertainty of the biomechanical constraining model and the noisy nature of the imaging data. The state estimates are predicted according to the continuous-time biomechanically constructed state equation between observation time points, and then updated with the new imaging-derived measurements at discrete time instants, yielding physically more meaningful and more accurate estimation results. Both continuous-discrete Kalman filter and sampled-data Hinfinity filter are applied for motion recovery. While Kalman filter is the optimal estimator under Gaussian noises, the Hinfinity scheme can give robust estimation results when the types and levels of model uncertainties and data disturbances are not available a priori. The strategies are validated through synthetic data experiments to illustrate their advantages and on canine MR phase contrast images and human MR tagging data to show their clinical potential.

Continuous-Discrete Observer Design for a CHO-K1 Cell Culture in Suspension

Bioprocess models are often uncertain due to the lack of experimental data for structure selection and parameter estimation. Maximum-likelihood parameter estimation techniques, which take the measurement errors into account, allow confidence intervals in the identified parameters to be evaluated. In turn, this information can be advantageously exploited in the design of (more) robust state estimators (or software sensors) for process monitoring and control. In this paper, the formulation of continuous-discrete Kalman filters and receding horizon observers are extended to include a posteriori knowledge on model parameter uncertainties. These extended observers/filters are then successfully applied to a real-case study, i.e., animal cell cultures in perfusion mode.

Nonlinear control of a batch crystallizer

A nonlinear geometric feedback controller with and without state estimation (the extended continuous-discrete Kalman filter) is developed and applied to a 0.027 m 3 potash alum batch crystallizer. The manipulated variable isthetemperature of the inlet cooling water supplied to the jacket of the crystallizer, and the controlled variable is the supersaturation. It is shown that the controller eliminates the large initial peak in the supersaturation (which results in excessive nucleation) and maintains the supersaturation at its set-point, provided that themanipulated variable does not reach its constraints. The controller performs well with only two measured states (the crystallizer temperature and the soluteconcentration) and results in larger terminal crystal mean size in com-parison with natural cooling and linear cooling policies with fines dissolution.

Isolating cyclical patterns in irregular time-series data

Abstract Times-series data which are observed at irregular time intervals often arise in economics and the bio-sciences. Existing methods for modelling these data have focused on the discretisation of continuous processes. A method is proposed for fitting cyclical components to irregular time-series data based on the continuous-discrete Kalman filter which incorporates numerical integration of the differential equations describing the model. The method is applied to seawater temperature data and empirical sampling distributions for parameter estimators are enumerated. The supporting sampling distributions suggest that the method yields estimates which have satisfactory statistical properties.