The modified gain extended Kalman filter and parameter identification in linear systems

For a special class of systems, a general formulation and stochastic stability analysis of a new nonlinear filter, called the modified gain extended Kalman filter (MGEKF), is presented. Used as an observer, it is globally exponentially convergent. In the presence of uncertainties a nominal nonrealizable filter algorithm is developed for which global stochastic stability is proven. With respect to this nominal filter algorithm, conditions are obtained such that the effective deviations of the realizable filter are not destabilizing. In an appropriate coordinate frame, the parameter identification problem of a linear system is shown to be a member of this special class. For the example problems, the MGEKF shows superior convergence characteristics without evidence of instability.     

Convergence analysis of the extended Kalman filter used as anobserver for nonlinear deterministic discrete-time systems

In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly. The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems     

Performance analysis of multi-innovation stochastic Newton recursive algorithms

The stochastic Newton recursive algorithm is studied for system identification. The main advantage of this algorithm is that it has extensive form and may embrace more performance with flexible parameters. The primary problem is that the sample covariance matrix may be singular with numbers of model parameters and (or) no general input signal; such a situation hinders the identification process. Thus, the main contribution is adopting multi-innovation to correct the parameter estimation. This simple approach has been proven to solve the problem effectively and improve the identification accuracy. Combined with multi-innovation theory, two improved stochastic Newton recursive algorithms are then proposed for time-invariant and time-varying systems. The expressions of the parameter estimation error bounds have been derived via convergence analysis. The consistence and bounded convergence conclusions of the corresponding algorithms are drawn in detail, and the effect from innovation length and forgetting factor on the convergence property has been explained. The final illustrative examples demonstrate the effectiveness and the convergence properties of the recursive algorithms.     

Stochastic stability of the unscented Kalman filter with intermittent observations

In this paper, the stochastic stability of the discrete-time unscented Kalman filter for general nonlinear stochastic systems with intermittent observations is proposed. It is shown that the estimation error remains bounded if the system satisfies some assumptions. And the statistical convergence property of the estimation error covariance is studied, showing the existence of a critical value for the arrival rate of the observations. An upper bound on this expected state error covariance is given. A numerical example is given to illustrate the effectiveness of the techniques developed.     

基于LMI的一类非线性多时滞系统的模糊H∞滤波器设计

This paper develops fuzzy H1 filter for state estimation approach for nonlinear discrete-time systems with multiple time delays and unknown bounded disturbances. We design a stable fuzzy H1 filter based on the Takagi-Sugeno (T-S) fuzzy model, which assures asymptotic stability and a prescribed H1 index for the filtering error system. Sufficient condition for the existence of such a filter is established by solving the linear matrix inequality (LMI) problem. The LMI problem can be efficiently solved with global convergence using the interior point algorithm. Simulation examples are provided to illustrate the design procedure of the proposed method.     

Recursive identification of bilinear systems

Methods of identifying bilinear systems from recorded input-output data are discussed in this article. A short survey of the existing literature on the topic is given. ‘Standard’ methods from linear systems identification, such as least squares, extended least squares, recursive prediction error and instrumental variable methods are transferred to bilinear, input-output model structures and tested in simulation. Special attention is paid to problems of stabilizing the model predictor, and it is shown how a time-varying Kalman filter and associated parameter estimation algorithm can deal with this problem.     

Parameter convergence in nonlinearly parameterized systems

A large class of problems in parameter estimation concerns nonlinearly parametrized systems. Over the past few years, a stability framework for estimation and control of such systems has been established. We address the issue of parameter convergence in such systems in this paper. Systems with both convex/concave and general parameterizations are considered. In the former case, sufficient conditions are derived under which parameter estimates converge to their true values using a min-max algorithm. In the latter case, to achieve parameter convergence a hierarchical min-max algorithm is proposed where the lower level consists of a min-max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie. Using this hierarchical algorithm, a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization. In both cases, the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems. Explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions. A general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper.     

具有一般交互矩阵的多变量系统的随机直接自适应控制

本文使用系统的交互矩阵,提出了基于广义最小方差控制律的一般随机多变量系统的直 接自适应控制算法,并对该算法进行了稳定性和收敛性分析.该算法即使用于非最小相位系 统仍然具有全局收敛特性.     

Simultaneous state and input estimation of hybrid systems with unknown inputs

This paper addresses the problem of the simultaneous state and input estimation for hybrid systems when subject to input disturbances. The proposed algorithm is based on the moving horizon estimation (MHE) method and uses mixed logical dynamical (MLD) systems as equivalent representations of piecewise affine (PWA) systems. So far the MHE method has been successfully applied for the state estimation of linear, hybrid, and nonlinear systems. The proposed extension of the MHE algorithm enables the estimation of unknown inputs, or disturbances, acting on the hybrid system. The new algorithm is shown to improve the convergence characteristics of the MHE method by reducing the delay of convergent estimates, while assuring convergence for every possible sequence of input disturbances. To ensure convergence the system is required to be incrementally input observable, which is an extension to the classical incremental observability property.     

Robust filtering of process in the stationary difference stochastic system

A technique to construct the robust Kalman filter for process estimation in the difference linear stationary stochastic system with an unknown covariance observation error matrix was developed. Consideration was given to the algorithm of constructing the set of permissible covariance matrices from a priori statistical data. A numerical method for solution of the general minimax optimization problem was proposed; and on its basis an iterative algorithm to calculate the robust filter parameters was developed, and its convergence was proved. Results of the numerical experiment were presented.     

参数不确定系统的H∞估计问题的显式解和中心解

研究在连续时间情形下的具有部分参数不确定性的系统的H∞状态估计问题,它可 以被化简为带有一个自由可调参数对象的H∞状态估计,由此可得到滤波器的简洁通解显 式.并进一步研究了H∞估计的中心解,以及它与卡尔曼滤波器的关系.实例计算表明,对于 参数具有不确定性的系统,H∞滤波器的性能明显地优于卡尔曼滤波器.     

Optimal state estimation for stochastic systems: an informationtheoretic approach

In this paper, we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures. In this setting, the traditional minimum mean-square measure is compared with information theoretic measures, Kalman filtering theory is reexamined, and some new interpretations are offered. We show that for a linear Gaussian system, the Kalman filter is the optimal filter not only for the mean-square error measure, but for several information theoretic measures which are introduced in this work. For nonlinear systems, these same measures generally are in conflict with each other, and the feedback control policy has a dual role with regard to regulation and estimation. For linear stochastic systems with general noise processes, a lower bound on the achievable mutual information between the estimation error and the observation are derived. The properties of an optimal (probing) control law and the associated optimal filter, which achieve this lower bound, and their relationships are investigated. It is shown that for a linear stochastic system with an affine linear filter for the homogeneous system, under some reachability and observability conditions, zero mutual information between estimation error and observations can be achieved only when the system is Gaussian     

On adaptive Kalman filtering for the Luenberger canonical form

This work reports on simulation results obtained from an adaptive filter for systems in Luenberger canonical form. In particular the effects of errors in estimation of innovations, autocorrelations and initial conditions are examined. The scheme proves to be more efficient than a similar one based on measurement autocorrelations. For a certain class of systems new formulae are given for calculating the structural indices required by the algorithm from the state and measurement dimensions. Extension is made to the more general problem of coloured measurement noise.     

Iterative learning control: A survey and new results

Learning control is an iterative approach to the problem of improving transient behavior for processes that are repetitive in nature. In this article, we present some results on iterative learning control. A complete review of the literature is given first. Then, a general formulation of the problem is given. Next, we present a complete analysis of the learning control problem for the case of linear, time-invariant plants and controllers. This analysis offers: (1) insight into the nature of the solution of the learning control problem by deriving sufficient convergence conditions; (2) an approach to learning control for linear systems based on parameter estimation; and (3) an analysis that shows that for finite-horizon problems it is possible to design a learning control algorithm that converges, with memory, in one step. Finally, a time-varying learning controller is given for controlling the trajectory of a nonlinear robot manipulator. A brief simulation example is presented to illustrate the effectiveness of this scheme.     

Analysis of recursive stochastic algorithms

Recursive algorithms where random observations enter are studied in a fairly general framework. An important feature is that the observations my depend on previous "outputs" of the algorithm. The considered class of algorithms contains, e.g., stochastic approximation algorithm, recursive identification algorithm, and algorithms for adaptive control of linear systems. It is shown how a deterministic differential equation can be associated with the algorithm. Problems like convergence with probability one, possible convergence points and asymptotic behavior of the algorithm can all be studied in terms of this differential equation. Theorems stating the precise relationships between the differential equation and the algorithm are given as well as examples of applications of the results to problems in identification and adaptive control.     

Filtering on nonlinear time-delay stochastic systems

In this paper, we address the filtering problem for a general class of nonlinear time-delay stochastic systems. The purpose of this problem is to design a full-order filter such that the dynamics of the estimation error is guaranteed to be stochastically exponentially ultimately bounded in the mean square. Both filter analysis and synthesis problems are considered. Sufficient conditions are proposed for the existence of desired exponential filters, which are expressed in terms of the solutions to algebraic Riccati inequalities involving scalar parameters. The explicit characterization of the desired filters is also derived. A simulation example is given to illustrate the design procedures and performances of the proposed method.     

Design of an extended unknown input observer with stochastic robustness techniques and evolutionary algorithms

The paper deals with the problem of designing an unknown input observer for discrete-time non-linear systems. In particular, with the use of the Lyapunov method, it is shown that the proposed observer is convergent under certain, non-restrictive conditions. Based on the achieved results, a general solution for increasing the convergence rate is proposed and implemented with the use of stochastic robustness techniques. In particular, it is shown that the problem of increasing the convergence rate of the observer can be formulated as a stochastic robustness analysis task that can be transformed into a structure selection and parameter estimation problem of a non-linear function, which can be solved with the B-spline approximation and evolutionary algorithms. The final part of the paper shows an illustrative example based on a two phase induction motor. The presented results clearly exhibit the performance of the proposed observer.     

基于状态预测自适应一致滤波器的分布式估计融合算法

应用自适应滤波算法改进了基于一致滤波器的估计融合算法以加快节点估计的一致收敛速度,提出了 一种基于状态预测的自适应一致滤波器．在此算法中,节点采用状态预测值作为自适应滤波器的参考信号,应用自 适应算法修正一致滤波器的加权矩阵．仿真结果表明,本文提出的算法不仅能够加快节点估计的一致收敛速度,还 能减小收敛过程中节点的估计误差．     

New quasi-Newton iterative learning control scheme based on rank-one update for nonlinear systems

This paper develops an algorithm for iterative learning control on the basis of the quasi-Newton method for nonlinear systems. The new quasi-Newton iterative learning control scheme using the rank-one update to derive the recurrent formula has numerous benefits, which include the approximate treatment for the inverse of the system’s Jacobian matrix. The rank-one update-based ILC also has the advantage of extension for convergence domain and hence guaranteeing the choice of initial value. The algorithm is expressed as a very general norm optimization problem in a Banach space and, in principle, can be used for both continuous and discrete time systems. Furthermore, a detailed convergence analysis is given, and it guarantees theoretically that the proposed algorithm converges at a superlinear rate. Initial conditions which the algorithm requires are also established. The simulations illustrate the theoretical results.