A two-level estimator for time varying parameters

A crucial part in an adaptive control system is the estimation of the unknown parameters of the process. The estimation is often done using a Kalman filter or an Extended Kalman filter. These estimators give good results if the parameters are not varying too fast. When the parameters are varying fast there are difficulties for the estimator to follow the variations.This paper outlines a new approach to the estimation problem. The new estimator consists of two parts. One conventional Kalman filter for fine estimation and one estimator for coarse estimation. The coarse estimator consists of a finite number of fixed a priori models and a decision mechanism which points out the model which best fits the data.The paper describes the two-level estimator and discusses its properties. Some numerical examples illustrate the behavior of the estimator.     

A general approach to constructing parameter identification algorithms in the class of square root filters with orthogonal and J-orthogonal tranformations

We study modern implementations of the discrete Kalman filter, namely array square-root algorithms. An important feature of such algorithms is the use of orthogonal and J-orthogonal transformations on each filtering step. For the first time, we develop for this class of algorithms a simple universal approach that lets us generalize any numerically stable implementation of this type to the case of updates in sensitivity equations of the filter with respect to unknown system model parameters. An advantage of the resulting adaptive schemes is their numerical stability with respect to machine rounding errors. Estimation of the noisy state vector of the system and identification of unknown system parameters occur simultaneously. The proposed approach can be used for parameter identification problems, adaptive control problems, experiment planning, and others.     

Least squares estimation of nonstationary covariance parameters in linear systems

Least squares estimation techniques are employed to overcome previous difficulties encountered in accurately estimating the state and measurement noise covariance parameters in linear stochastic systems. In the past accurate and rapidly converging covariance parameter estimates have been achieved with complex estimation algorithms only after specifying the statistical nature of the noise in the system and constraining the time variation of the covariance parameters. Weighted least squares estimation allows these restrictions to be removed while achieving near optimal accuracy using a filter on the same order of complexity as a Kalman filter. Allowing the covariance parameters to vary in as general a manner in time as the state in a linear discrete time stochastic system, and assuming that a Kalman filter is applied to this system using incorrect knowledge of the a priori statistics, it is shown how a covariance system is developed similar to the original system. Unbiased least squares estimates of the covariance parameters and of the original state are obtained without the necessity of specifying the distribution on the noise in either system. The accuracy of these estimates approaches optimal accuracy with increasing measurements when adaptive Kalman filters are applied to each system.     

Exact analysis of the finite precision error generation and propagation in the FAEST and the fast transversal algorithms: A general methodology for developing robust RLS schemes

In this paper, an analysis for the actual and deeper cause of the finite precision error generation and accumulation in the FAEST-5p and the fast transversal filtering (FTF) algorithm is undertaken, on the basis of a new methodology and practice. In particular, it is proved that, in case where the input data in these algorithms is a white noise or a periodic sequence, then, out of all the formulas that constitute these two schemes, only four specific formulas generate an amount of finite precision error that consistently makes the algorithms fail after a certain number of iterations. If these formulas are calculated free of finite precision error, then all the results of the two algorithms are also computed error-free. In addition, it is shown that there is a very limited number of specific formulas that transmit the finite precision error generated by these four formulas. Moreover, a number of very general propositions is presented that allow for the calculation of the exact number of erroneous digits with which all the quantities of the FAEST and FTF schemes are computed, including the filter coefficients. Finally, a general methodology is introduced, based on the previous results, that allows for the development of new RLS algorithms that, intrinsically, suffer less of finite precision numerical problems and that therefore are, in practice, suitable for high quality fast Kalman filtering implementations.     

卡尔曼滤波计算方法研究进展

本文简要回顾了卡尔曼滤波研究的发展历程，重点对卡尔曼滤波及其在改善数值稳定性，提高计算效率等数值计算方面的研究与发展进行了综述，对ＱＲ分解，Ｕ－Ｄ分解，奇异值分解等在卡尔曼滤波，状态与偏差分离波及并行滤波与分散滤波等方面应用的新进展作了介绍。     

On-line identification and control of multivariablc discrete-time systems based on a transformed model

Recursive algorithms for on-line combined identification and control of linear discrete-time multivariable systems is presented. A variant of the observable canonical model with one-way coupling form of the state model is used to develop a solution of the problem. Exploiting the canonical structure of the model, the proposed solution turns out to be simpler than that obtained by El-Sherief (1983) and moreover, a combined decentralized state and parameter estimation based control scheme can be developed in three stages. In Stage I, the parameters of the system matrices are estimated by a recursive least-square algorithm or by a normalized stochastic approximation algorithm in decoupled manner. These parameters are then employed for state estimation in Stage 2 using a centralized conventional Kalman filter or by a decentralized adaptive Kalman filter which in turn reduces instrumentation and telemetry costs. Estimated parameters and states are then utilized in Stage 3 to implement the square-root based control law along with the good numerical behaviour of the control problem. The proposed algorithms are tested by considering an example of a third-order state-space model.     

Unscented Kalman filter with advanced adaptation of scaling parameter

The paper deals with state estimation of the nonlinear stochastic systems by means of the unscented Kalman filter with a focus on specification of the σ$\sigma$-points. Their position is influenced by two design parameters—the scaling parameter determining the spread of the σ$\sigma$-points and a covariance matrix decomposition determining rotation of the σ$\sigma$-points. In this paper, a choice of the scaling parameter is analyzed. It is shown that considering other values than the standard choice may lead to increased quality of the estimate, especially if the scaling parameter is adapted. Several different criteria for the adaptation are proposed and techniques to reduce computational costs of the adaptation are developed. The proposed algorithm of the unscented Kalman filter with advanced adaptation of the scaling parameter is illustrated in a numerical example.     

Maximum likelihood estimation via the extended covariance and combined square-root filters

The method of maximum likelihood is a general method for parameter estimation and is often used in system identification. To implement it, it is necessary to maximize the likelihood function, which is usually done using the gradient approach. It involves the computation of the likelihood gradient with respect to unknown system parameters. For linear stochastic system models this leads to the implementation of the Kalman filter, which is known to be numerically unstable. The aim of this work is to present new efficient algorithms for likelihood gradient evaluation. They are more reliable in practice and improve robustness of computations against roundoff errors. All algorithms are derived in measurement and time updates form. The comparison with the conventional Kalman filter approach and results of numerical experiments are given.     

Square root kalman algorithms in econometrics

This paper describes a sequential square root method which is aimed at solving the numerical problems affecting the conventional Kalman filter. Simple square root algorithms are derived for the Kalman covariance and information filters and for the smoothing equations. A comparison with other square root methods is also provided.     

Measurement updating using the U-D factorization

The discrete linear filtering problem is treated by factoring the filter error covariance matrix as P = UDU^T. Efficient and stable measurement updating recursions are developed for the unit upper triangular factor, U, and the diagonal factor, D. This paper treats only the parameter estimation problem; effects of mapping, inclusion of process noise and other aspects of filtering are treated in separate publications. The algorithm is simple and, except for the fact that square roots are not involved, can be likened to square root filtering. Like the square root filter our algorithm guarantees nonnegativity of the computed covariance matrix. As is the case with the Kalman filter, our algorithm is well suited for use in real time. Attributes of our factorization update include: efficient one point at a time processing that requires little more computation than does the optimal but numerically unstable conventional Kalman measurement update algorithm; stability that compares with the square root filter and the variable dimension flexibility that is enjoyed by the square root information filter. These properties are the subject of this paper.     

基于组合导航系统的非线性高斯滤波算法

由于组合导航系统具有强非线性和模型不确定性的特点, 工程中扩展卡尔曼滤波无法满足组合导航系统实际应用的要求. 为此, 针对贝叶斯框架下高斯类非线性滤波算法的估计性能给出具体分析. 首先, 在估计点处对非线性函数进行泰勒展开获得泰勒近似, 通过一阶矩和二阶矩分析滤波算法的近似精度; 然后, 通过数值稳定性对非线性滤波算法进行分析; 最后, 分别采用低维和高维模型对各滤波算法进行对比分析, 为组合导航系统的实践提供借鉴.

     

Experiments of one-dimensional soil moisture assimilation system based on ensemble Kalman filter

Ensemble Kalman filter is a new sequential data assimilation algorithm which was originally developed for atmospheric and oceanographic data assimilation. It can be applied to calculate error covariance matrix through Monte-Carlo simulation. This approach is able to resolve the nonlinearity and discontinuity existed within model operator and observation operator. When observation data are assimilated at each time step, error covariances are estimated from the phase-space distribution of an ensemble of model states. The error statistics is then used to calculate Kalman gain matrix and analysis increments. In this study, we develop a one-dimensional soil moisture data assimilation system based on ensemble Kalman filter, the Simple Biosphere Model (SiB2) and microwave radiation transfer model (AIEM, advanced integration equation model). We conduct numerical experiments to assimilate in situ soil surface moisture measurements and low-frequency passive microwave remote sensing data into a land surface model, respectively. The results indicate that data assimilation can significantly improve the soil surface moisture estimation. The improvement in root zone is related to the model bias errors at surface layer and root zone. The soil moisture does not vary significantly in deep layer. Additionally, the ensemble Kalman filter is predominant in dealing with the nonlinearity of model operator and observation operator. It is practical and effective for assimilating observations in situ and remotely sensed data into land surface models.     

A method for optimal control and filtering of multitime-scale linear singularly-perturbed stochastic systems

In this paper we introduce a transformation for the exact closed-loop decomposition of the optimal Kalman filter and the linear quadratic optimal controller of multi time scale continuous-time, linear, singularly-perturbed stochastic systems. The solution of the corresponding algebraic regulator and filter Riccati equations are obtained in terms of solutions of reduced-order subsystem, algebraic, Riccati equations corresponding to the system time scales. We have also obtained N completely independent reduced-order subsystem Kalman filters working in parallel in different time scales. This allows parallel processing of information with lower-order, different rates Kalman filters consistent with the system time scales.     

Discrete square root filtering: A survey of current techniques

The conventional Kalman approach to discrete filtering involves propagation of a state estimate and an error covariance matrix from stage to stage. Alternate recursive relationships have been developed to propagate a state estimate and a square root error covariance instead. Although equivalent algebraically to the conventional approach, the square root filters exhibit improved numerical characteristics, particularly in ill-conditioned problems. In this paper, current techniques in square root filtering are surveyed and related by applying a duality association. Four efficient square root implementations are suggested, and compared with three common conventional implementations in terms of computational complexity and precision. The square root computational burden should not exceed the conventional by more than 50 percent in most practical problems. An examination of numerical conditioning predicts that the square root approach can yield twice the effective precision of the conventional filter in ill-conditioned problems. This prediction is verified in two examples. The excellent numerical characteristics and reasonable computation requirements of the square root approach make it a viable alternative to the conventional filter in many applications, particularly when computer word length is limited, or the estimation problem is badly conditioned.     

An adaptive robustizing approach to kalman filtering

The performance of a linear Kalman filter will degrade when the dynamic noise is not Gaussian. A robust Kalman filter based on the m-interval polynomial approximation (MIPA) method for unknown non-Gaussian noise is proposed. Two situations are considered: (a) the state is Gaussian and the observation noise is non-Gaussian; (b) the state is non-Gaussian and the observation noise is Gaussian. It is shown, as compared with other non-Gaussian filters, the MIPA Kalman filter is computationally feasible, unbiased, more efficient and robust. For the scalar model, Monte Carlo simulations are given to demonstrate the ideas involved.     

Finite-frequency fault detection for two-dimensional Fornasini–Marchesini dynamical systems

This paper is concerned with the fault detection problem for two-dimensional (2-D) discrete-time systems described by the Fornasini–Marchesini local state-space model. The goal of the paper is to design a fault detection filter to detect the occurrence of faults in finite-frequency domain. To this end, a finite-frequency H_? index is used to describe fault sensitivity performance, and a finite-frequency H_∞ index is used to describe disturbance attenuation performance. In light of the generalised Kalman–Yakubovich–Popov lemma for 2-D systems and matrix inequality techniques, convex conditions are derived for this fault detection problem. Based on these conditions, a numerical algorithm is put forward to construct a desired fault detection filter. Finally, a numerical example and an industrial example are given to illustrate the effectiveness of the proposed algorithm.     

Extended state space recursive least squares

A new extended state space recursive least squares (ESSRLS) algorithm is proposed for state estimation of nonlinear systems. It is based on state space recursive least squares (SSRLS) approach and uses first order linearization of the system. It inherits the capability of obtaining state estimate without knowledge of process and measurement noise covariance matrices (Q and R respectively). The proposed approach is considered to provide new design option for scenarios where noise statistics and system dynamics vary. ESSRLS is initialized using delayed recursion method and a forgetting factor λ is employed to optimize the performance. The selection of λ can be problem specific as shown through experimental validations. However a value closer to and less than unity is generally recommended. Theoretical bases are validated by applying this algorithm to problems of tracking a non-conservative oscillator, a damped system with amplitude death and a signal modeled by mixture of Gaussian kernels. Simulation results show an MSE performance gain of 20 dB and 23 dB over extended Kalman filter (EKF) and unscented Kalman filter (UKF) while tracking van der Pol oscillator without knowledge about noise variances. The computational complexity of ESSRLS falls within that of EKF and UKF.     

An iterated fuzzy extended Kalman filter for nonlinear systems

The fuzzy extended Kalman filter (FEKF) for state estimation can be used to deal with fuzzy uncertainty effectively. However, the linearisation processing of the FEKF introduces truncation error, which degrades the estimation precision. In order to reduce the error, a new iterated fuzzy extended Kalman filter (IFEKF), based on the FEKF and the maximum a posteriori estimation, is proposed in this article. Compared with the FEKF, the proposed algorithm can be used not only to deal with the fuzzy uncertainty, but also to reduce the truncation error and to estimate the states more accurately. With an algebraic example and a passive location simulation, it is shown that the IFEKF has better estimation precision than that of the FEKF.     

Minimax state estimation for linear discrete-time differential-algebraic equations

This paper presents a state estimation approach for an uncertain linear equation with a non-invertible operator in Hilbert space. The approach addresses linear equations with uncertain deterministic input and noise in the measurements, which belong to a given convex closed bounded set. A new notion of a minimax observable subspace is introduced. By means of the presented approach, new equations describing the dynamics of a minimax recursive estimator for discrete-time non-causal differential-algebraic equations (DAEs) are presented. For the case of regular DAEs it is proved that the estimator’s equation coincides with the equation describing the seminal Kalman filter. The properties of the estimator are illustrated by a numerical example.     

Evolutionary-Programming-Based Kalman Filter for Discrete-Time Nonlinear Uncertain Systems

Some observations and improvements on the conventional Kalman filtering scheme to function properly are presented. The improvements can be achieved using the minimal principle evolutionary programming (EP) technique. A new linearization methodology is presented to obtain the exact linear models of a class of discrete-time nonlinear time-invariant systems at operating states of interest, so that the conventional Kalman filter can work for the nonlinear stochastic systems. Furthermore, a Kalman innovation filtering algorithm and such an algorithm based on the evolutionary programming optimal-search technique are proposed in this paper for discrete-time time-invariant nonlinear stochastic systems with unknown-but-bounded plant uncertainties and noise uncertainties to find a practically implementable “best” Kalman filter. The worst-case realization of the discrete-time nonlinear stochastic uncertain systems represented by the interval form with respect to the implemented “best” nominal filter is also found in this paper for demonstrating the effectiveness of the proposed filtering scheme.