Kalman Scheme Research Articles

System Monotonicity and Subspace Tracking: A Geometric Perspective of the Frisch–Shapiro Scheme

The Shapiro scheme, together with the closely related Frisch–Kalman scheme, has been an important approach to system identification and statistical analysis. A longstanding result on this scheme, known as the Shapiro theorem, is both informative and significant. This article imparts a geometric understanding to the Shapiro theorem and generalizes it to the asymmetric setting using the notion of cone-invariance. In particular, we establish the equivalence between two important properties of a real-valued square matrix—irreducible orthant-invariance and simplicity of its dominant eigenvalue under arbitrary diagonal perturbations. The result can be regarded as a converse Perron–Frobenius theorem. Furthermore, we investigate two applications of the proposed result in systems and control, namely, characterization of irreducibly orthant-monotone nonlinear systems and subspace tracking via decentralized control. We also extend the established result to accommodating polyhedral cones and obtain several insights.

Application Error Analysis of SOC Estimation of Pure Electric Vehicles Based on Kalman Signal Big Data Algorithm

The state of charge estimation of a pure electric vehicle power battery pack is one of the important contents of the battery management system. Improving the estimation accuracy of the battery pack’s SOC is conducive to giving full play to its performance and preventing overcharge and discharge of a single battery. At present, the open-circuit voltage ampere-hour integral method is traditionally used to estimate the SOC value of the battery pack; however, this estimation method is not accurate enough to correct the initial value of SOC and cannot solve the problem of current time integration error between this correction and the next correction. As for the battery performance and characteristics of electric vehicles, it is pointed out that the size of the model value will affect the estimation accuracy of the Kalman signal value. Based on the analysis of the factors to be referred to in the calculation and estimation of SOC by Kalman for pure electric vehicles, the scheme is improved considering the change of battery model value, and the Kalman scheme is proposed. The feasibility and accuracy of the scheme are proved by several battery simulation experiments.

Joint equalization of CD and RSOP using a time–frequency domain Kalman filter structure in Kramers–Kronig system

Kramers–Kronig (KK) system is a direct detection (DD) optical short-reach interconnection system which is suitable for high capacity data center interconnect (DCI) applications. For this kind of short-reach high capacity polarization-division multiplexing (PDM) optical transmission system, the fiber channel impairments contain chromatic dispersion (CD) and rotation of state of polarization (RSOP) will lead to a severe signal distortion and need to be equalized. This paper proposed a time–frequency domain Kalman filter scheme for joint equalization of CD and RSOP simultaneously in KK system. The proposed scheme is capable of tracking the fast RSOP in time domain and equalizing CD in frequency domain using a window-split structure. The performance of the Kalman scheme is investigated in a 28GBaud PDM 16QAM DD system with KK receivers. The simulation results prove that the proposed Kalman scheme exhibits a good performance of RSOP tracking ability up to speed of 2.5 Mrad/s, with a large accumulated CD tolerance as large as 2040 ps/nm at a low OSNR penalty. The results also show the proposed scheme has a comparable computational complexity compared with a generally used scheme which includes a CD compensation, combined with polarization demultiplexing algorithm constant modulus algorithm (CMA) and multiple modulus algorithm (MMA).

Параболическая модель в форме пространства состояний динамики накоплений

An important place in the theory of partial differential equations and its applications is occupied by the heat equation, a representative of the class of the so-called parabolic equations. It is known that to check the correctness of a mathematical model based on a parabolic equation, the existence of its solution is very important since a mathematical model is not always adequate to a specific phenomenon and the existence of a solution to a corresponding mathematical problem does not follow from the existence of a solution to a real applied problem. Therefore, methods for solving partial differential equations, both analytical and numerical, are always relevant. Nowadays, a computational experiment has become a powerful tool for theoretical research. It is carried out over a mathematical model of the object under study, but at the same time, other parameters are calculated using one of the parameters of the model and conclusions are drawn about the properties of the object or phenomenon under study. The problem of passive parametric identification of systems with distributed parameters for resource accumulation dynamics of many households using a stochastic distributed model in the form of a state space with regard to the white noise of the dynamics model of the object under study and the white noise of the model of a linear-type measuring system is considered in the paper. The use of the finite difference method allowed us to reduce the solution of partial differential equations of a parabolic type to the solution of a system of linear finite difference and algebraic equations represented by models in the form of a state space. It was also proposed to use a filtering algorithm based on the Kalman scheme for reliable estimation of the object behavior. Calculations were carried out using the Matlab mathematical system based on statistical data for five years, taken from the site “Agency for Strategic Planning and Reforms of the Republic of Kazakhstan Bureau of National Statistics”. Estimation of the coefficients of the equations for the household resource accumulation in the form of a state space using this technique is sufficiently universal and can be applied in other fields of science and technology.

Joint equalization of linear impairments using two-stage cascade Kalman filter structure in coherent optical communication systems

In this paper, instead of adopting different algorithms to deal with the different impairments separately based on different signal processing structures, we proposed a joint linear impairment equalization scheme using a two-stage cascade Kalman structure in which the first stage solves the residual chromatic dispersion (rCD), the rotation of state of polarization (RSOP) and polarization mode dispersion (PMD), and the second stage compensates for the carrier frequency offset (CFO) and carrier phase noise (CPN). The structures of the two stages are similar with a seamless connection and possess a clear logic architecture. The stages are compatible with each other and achieve excellent performance. We aim to use this two-stage Kalman scheme as a comprehensive DSP equalization scheme to replace the combined algorithms of the constant modulus algorithm (CMA), improved Mth-power algorithm (IMP), and blind phase search (BPS). Simulations based on a 28 Gbaud polarization division multiplexed quadrature-phase shift keying coherent optical communication system show that the proposed two-stage Kalman scheme comprehensively outperforms the CMA-IMP-BPS scheme in solving the problems of RSOP, PMD, rCD, CFO, and CPN. The experimental results confirm the simulation results. In addition, the complexity of the proposed Kalman scheme is relatively less than the CMA-IMP-BPS scheme.

Joint equalization scheme of ultra-fast RSOP and large PMD compensation in presence of residual chromatic dispersion.

Polarization demultiplexing is generally carried out by a multiple-input multiple-output (MIMO) based algorithm in polarization division multiplexing (PDM) coherent systems. However, in some extreme environments, the MIMO algorithm becomes inapplicable due to the ultra-fast rotation of the state of polarization (RSOP) and large polarization mode dispersion (PMD). In addition, the residual chromatic dispersion (RCD) is always present because of the mismatch of the compensated chromatic dispersion and real value induced in the optical fiber channel. According to the literature, the Kalman filter-based polarization demultiplexing algorithms possess very weak RCD tolerance. Faced with this dilemma, in this paper, a new Kalman filter structure is proposed, which can jointly compensate ultra-fast RSOP, large PMD and RCD. This Kalman filter structure enables the equalization of the RSOP in the time domain and compensation for RCD and PMD in the frequency domain. We verified the performance of the proposed Kalman scheme in the 28 Gbaud PDM-QPSK/16 QAM coherent system, with a comparison to constant modulus algorithm/multiple modulus algorithm (CMA/MMA). The simulation results confirm that, compared with CMA/MMA, the proposed Kalman scheme can provide a significant performance enhancement to cope with ultra-fast RSOP (up to 3 Mrad/s) and large PMD (more than 200 ps) with a large tolerance to RCD (over the range of ± 820 ps/nm in PDM-QPSK and ± 500 ps/nm in PDM-16 QAM).

An adaptive Kalman filter for extreme polarization effects equalization in coherent optical communication system

The specially designed Kalman filter was proved to be competent to complete the polarization equalization in some extreme environments with the large time-varying polarization mode dispersion (PMD) and the ultrafast rotation of state of polarization (RSOP). But the performance of this kind of Kalman filter is strongly restricted by the initial noise statistics represented by the process noise covariance Q and the measurement noise covariance R which will limit its effectiveness in an actual scene. To solve this problem, this paper proposes an adaptive Kalman filter structure by an adjustable way of estimating the process noise covariance Q and the measurement noise covariance R based on the measurement residual in order to be adaptive to the time-varying environments. The performance of the proposed adaptive Kalman scheme is checked both by the simulation and experiment on the 28 Gbaud PDM-QPSK/16QAM coherent optical communication system platforms. The simulation and experiment results demonstrate that the proposed adaptive Kalman filter scheme has a stable and better performance of the equalization for large PMD combined with ultrafast RSOP, with a greater tolerance the to time-varying extreme polarization environments than that using the extended Kalman filter scheme adopted in our previous work.

A Kalman-based tomographic scheme for directly reconstructing activation levels of brain function.

In functional near-infrared spectroscopy (fNIRS), the conventional indirect approaches first separately recover the spatial distribution of the changes in the optical properties at every time point, and then extract the activation levels by a time-course analysis process at every site. In the tomographic implementation of fNIRS, i.e., diffuse optical tomography (DOT), these approaches not only suffer from the ill-posedness of the optical inversions and error propagation between the two successive steps, but also fail to achieve satisfactory temporal resolution due to the requirement for a complete data set. To cope with the above adversities of the indirect approaches, we propose herein a direct approach to tomographically reconstructing the activation levels by incorporating a Kalman scheme. Dynamic simulative and phantom experiments were conducted for the performance validation of the proposed approach, demonstrating its potentials to improve the calculated images and to relax the speed limitation of the instruments.

Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system.

A window-split frequency domain Kalman scheme is proposed in this paper for the equalization of large polarization mode dispersion (PMD) and ultra-fast rotation of state-of-polarization (RSOP) which is an extreme environment due to the Kerr effect and the Faraday effect under the lightning strike near the fiber cables. In order to carry out the proposed Kalman scheme, we give a simplified and equivalent fiber channel model as a replacement for the general model of the polarization effect of the co-existence of PMD and RSOP. With this fiber channel model, we can conduct compensation for PMD in the frequency domain and tracking RSOP in time domain. A half analytical and half empirical theory for the initialization of the process and measurement noise covariance is also presented in theory and verified by the numerical simulation. The performance of the proposed Kalman scheme is checked in the 28Gbaud PDM-QPSK coherent system built on both simulation and experiment platforms. The simulation and experiment results confirm that compared with the generally used constant modulus algorithm (CMA), the proposed scheme provides excellent performance and stability to cope with large range DGD from 20ps to 200ps and RSOP from 200krad/s to 2Mrad/s, with less computational complexity.

A correction method for dynamic model calculations using observational data and its application in oceanography

A new data assimilation method for the correction of model calculations is developed and applied. The method is based on the least resistance principle and uses the theory of diffusion-type stochastic processes and stochastic differential equations. Application of the method requires solving a system of linear equations that is derived from this principle. The system can be considered as a generalization of the well-known Kalman scheme taking the model’s dynamics into account. The method is applied to the numerical experiments with the HYbrid Coordinate Ocean Model (HYCOM) and Archiving, Validating, and Interpolating Satellite Ocean (AVISO) data for the Atlantic. The skill of the method is assessed using the results of the experiments. The model’s output is compared with the twin experiments, namely, the model calculations without assimilation, which confirms the consistency and robustness of the proposed method.

Non-Parametric Signal Interpolation

This paper considers the problem of interpolation (smoothing) of a partially observable Markov random sequence. For the dynamic observation models, an equation for the interpolation of the posterior probability density is derived. The main goal of this paper is to consider the smoothing problem for the case of unknown distributions of an unobservable component of a random Markov sequence. Successful results were obtained for the strongly stationary Markov processes with mixing and for the conditional density belonging to the exponential family of densities. The resulting method is based on the empirical Bayes approach and kernel nonparametric estimation. The equation for the optimal smoothing estimator is derived in the form independent of unknown distributions of an unobservable process. Such form of the equation allows to use the nonparametric estimates for some conditional functionals in the equation given a set of dependent observations. To compare the nonparametric estimators with optimal mean square smoothing estimators in Kalman scheme, simulation results are given.

A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The general aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box (e.g. with commercial software). The proposed regularizing ensemble Kalman method can be derived as an approximation of the regularizing Levenberg–Marquardt (LM) scheme (Hanke 1997 Inverse Problems 13 79–95) in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad hoc fixes, such as covariance localization, are implemented. The aim of this paper is to provide a detailed numerical investigation of the regularizing and convergence properties of the proposed regularizing ensemble Kalman scheme; the proof of these properties is an open problem. By means of numerical experiments, we investigate the conditions under which the proposed method inherits the regularizing properties of the LM scheme of (Hanke 1997 Inverse Problems 13 79–95) and is thus stable and suitable for its application in problems where the computation of the Fréchet derivative is not computationally feasible. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method. The numerical investigation is carried out with synthetic experiments on two model inverse problems: (i) identification of conductivity on a Darcy flow model and (ii) electrical impedance tomography with the complete electrode model. We further demonstrate the potential application of the method in solving shape identification problems that arises from the aforementioned forward models by means of a level-set approach for the parameterization of unknown geometries.

Nonparametric interpolation of the Markov sequence

Consideration was given to the problem of interpolation (smoothing) of the nonobservable component of the composite Markov process within the framework of the conditional Markov scheme. In the case of the dynamic observation models such as autoregression, equations were derived for the a posteriori interpolation density of the probability of the state of the nonobservable component. The aim of the present paper was to construct a smoothing algorithm for an unknown family of the distributions of the nonobservable component of the partially observable random Markov sequence. The result was obtained for the strictly stationary random Markov processes with mixing and for the conditional densities in the observation model from the exponential family of distributions. Computer-aided modeling within the framework of the Kalman scheme demonstrated that the sampled root-mean-square error of the nonparametric smoothing algorithm constructed for an unknown state equation was situated between the errors of the optimal linear filtration and the optimal linear interpolation.

On the multiplicity of the observable process in the general kalman scheme

(1982). On the multiplicity of the observable process in the general kalman scheme. Stochastics: Vol. 6, No. 3-4, pp. 175-192.