Distribution function tracking filter design using hybrid characteristic functions

A new tracking filtering algorithm for a class of multivariate dynamic stochastic systems is presented. The system is expressed by a set of time-varying discrete systems with non-Gaussian stochastic input and nonlinear output. A new concept, such as hybrid characteristic function, is introduced to describe the stochastic nature of the dynamic conditional estimation errors, where the key idea is to ensure the distribution of the conditional estimation error to follow a target distribution. For this purpose, the relationships between the hybrid characteristic functions of the multivariate stochastic input and the outputs, and the properties of the hybrid characteristic function, are established. A new performance index of the tracking filter is then constructed based on the form of the hybrid characteristic function of the conditional estimation error. An analytical solution, which guarantees the filter gain matrix to be an optimal one, is then obtained. A simulation case study is included to show the effectiveness of the proposed algorithm, and encouraging results have been obtained.     

非高斯噪声下输出残差或状态估计误差的熵研究

基于非Gaussian噪声线性定常控制系统,通过控制滤波器输出残差或状态估计误差的条件概率密度函数形状来建立有效的滤波设计算法,创建滤波器输出残差或状态估计误差的条件概率密度函数的统一表现形式。利用复合概率密度函数的关系对残差或状态估计误差的条件概率密度函数的近似来实现非高斯残差的高斯化或相应的熵最小化。     

On minimal error entropy stochastic approximation

Information-theoretic concepts are developed and employed to obtain conditions for a minimax error entropy stochastic approximation algorithm to estimate the state of a non-linear discrete time system baaed on noisy linear measurements of the state. Two recursive suboptimal error entropy estimation procedures are presented along with an upper bound formula for the resulting error entropy. A simple example is utilized to compare the optimal and suboptimal error entropy estimators and the minimum mean Square error linear estimator.     

ON SYSTEM IDENTIFICATION WITH AND WITHOUT CERTAINTY

Abstract

Information-theoretic concepts are utilized to develop a procedure for identifying a parameter of a stochastic linear discrete time dynamic scalar system based on noisy linear measurements of the system's state. After various simplifying approximations, the derived error entropy identification algorithm reduces to an on-line adaptive identification algorithm that is similar in many respects to well-established identification techniques. Conditions under which the developed on-line adaptive algorithm identifies the system with certainty are presented. Using an error entropy estimation lower bound, which is independent of any estimation procedure, conditions for which identification cannot be made with certainty are also presented. Examples involving non-Gaussian statistics are used to illustrate the efficiency of the error entropy adaptive identification algorithm as well as to compare it with several other identification procedures.     

An extended orthogonal forward regression algorithm for system identification using entropy

In this paper, a fast identification algorithm for non-linear dynamic stochastic system identification is presented. The algorithm extends the classical orthogonal forward regression (OFR) algorithm so that instead of using the error reduction ratio (ERR) for term selection, a new optimality criterion, Shannon's entropy power reduction ratio (EPRR), is introduced to deal with both Gaussian and non-Gaussian signals. It is shown that the new algorithm is both fast and reliable and examples are provided to illustrate the effectiveness of the new approach.     

Deterministic and stochastic model based run-to-run control for batch processes with measurement delays of uncertain duration

A novel run-to-run control algorithm integrating deterministic and stochastic model based control is developed for batch processes with measurement delays of uncertain duration. This control algorithm is referred to as deterministic and stochastic model based control (DSMBC). The deterministic component responds quickly to deterministic changes while the stochastic component minimizes the effects arising from measurement delays of uncertain duration. The deterministic component uses a linear process model with parameters that are updated online. The stochastic component uses an error probability density function (PDF) to characterize the effects due to measurement delays and this error PDF is determined from deviations between the set-point and the available process output. To integrate the two control algorithms, the control input is determined by minimizing the weighted sum of the predicted error from the deterministic model and the information entropy of the error probability density distribution. Using a simulated setting where the rate of chemical vapor deposition is controlled, the performance of the proposed DSMBC is shown to be superior to that of EWMA.     

Entropy Optimization Filtering for Fault Isolation of Nonlinear Non-Gaussian Stochastic Systems

In this paper, the fault isolation (FI) problem is investigated for nonlinear non-Gaussian systems with multiple faults(or abrupt changes of system parameters) in the presence of noises. By constructing a filter to estimate the states, the FI problem can be reduced to an entropy optimization problem subjected to the non-Gaussian estimation error systems. The design objective for the FI purpose is that the entropy of the estimation error is maximized in the presence of diagnosed fault and is minimized in the presence of the nuisance faults or noises. It is shown that the error dynamics is represented by a nonlinear non-Gaussian stochastic system, for which new relationships are applied to formulate the probability density functions (PDFs) of the stochastic error in terms of the PDFs of the noises and the faults. The Renyi's entropy has been used to simplify the computations in the filtering for the recursive design algorithms. It is noted that the output can be supposed to be immeasurable (but with known stochastic distributions), which is different from the existing results where the output is always measurable for feedback. Finally, simulations are given to demonstrate the effectiveness of the proposed data-driven FI filtering algorithms.     

Minimum entropy based run-to-run control for semiconductor processes with uncertain metrology delay

A novel run-to-run control methodology for semiconductor processes with uncertain metrology delay which is developed by combining the minimum error entropy and the optimal control strategy is presented. In most semiconductor processes, the product quality data from the previous run are not often available before the start of the next run. Thus, the corrective step is often delayed by one batch or more, and the duration of the delay is uncertain with stochastic characteristics. Coupled with inaccurate process models, the delay may lead to significant variations of the process outputs even with the use of exponentially weighted moving average (EWMA) controllers. This paper proposes a new method of handling the uncertain metrology delay from a probability viewpoint. The fundamentals of the run-to-run control systems are first reexamined, and then an innovative performance index is given by incorporating the entropy (or information potential) and the mean value of tracking error with constraints on control input energy. The probability density function (PDF) based optimal control algorithm is proposed for processes where the disturbance and delay are non-Gaussian and the stability of the algorithm is analyzed. In addition, the methodology of the proposed control strategy is extended to include recursive PDF estimation and on-line real time implementation. The paper also includes a simulation example of minimum entropy control of a tungsten chemical-vapor deposition process to illustrate the methodology. Furthermore, comparisons between the conventional EWMA method and the proposed method are done to show the advantages of our newly proposed method.     

Fault diagnosis and fault-tolerant control for non-Gaussian nonlinear stochastic systems via entropy optimisation

ABSTRACT

In this paper, the fault diagnosis (FD) and fault tolerant control (FTC) problems are studied for non-linear stochastic systems with non-Gaussian disturbance and fault. Unlike classical FD algorithms, the minimum entropy FD is adopted to minimise the residual entropy and control the shape of the probability density function (PDF) of the residual signal. The observation error system can be proved to be locally and ultimately bounded in the mean square sense. Since entropy can be used to characteriSe the uncertainty of the tracking error for non-Gaussian stochastic systems, the FTC controller is obtained by minimising the performance function with regard to the entropy of the tracking error in this paper. The PDF of the output tracking error is approximated by the B-spline model. An illustrative example is utilised to demonstrate the effectiveness of the FD and FTC algorithm, and satisfactory results have been obtained.     

时变系统有限数据窗最小二乘辨识的有界收敛性

利用随机过程理论证明了有限数据窗最小二乘法的有界收敛性,给出了参数估计误差 上界的计算公式,阐述了获得最小均方参数估计误差上界时数据窗长度的选择方法.分析表明, 对于时不变随机系统,数据窗长度越大,均方参数估计误差上界越小;对于确定性时变系统,数 据窗长度越小,均方参数估计误差上界越小.因此,对于时变随机系统,一个折中方案是寻求一 个最佳数据窗长度,以使均方参数估计误差最小.该文的研究成果对于提高辨识算法的实际应 用效果有重要意义.     

基于随机牛顿算法的离散系统自适应参数估计

针对一类离散系统,提出一种基于随机牛顿算法的自适应参数估计新框架,相较于已有的参数估计算法,所提出方法仅要求系统满足有限激励条件,而非传统的持续激励条件.所提出算法的核心思想在于通过对原始代价函数的修正,在使用当前时刻误差信息的基础上融入历史误差信息,进而通过对历史信息和历史激励的复用使得持续激励条件转化为有限激励条件;然后,为了解决传统算法收敛速度慢的问题并避免潜在的病态问题,采用随机牛顿算法推导出参数自适应律,并引入含有历史信息的海森矩阵作为时变学习增益,保证参数估计误差指数收敛;最后,基于李雅普诺夫稳定性理论给出不同激励条件下所提出算法的收敛性结论和证明,并通过对比仿真验证所提出算法的有效性和优越性.     

基于最大熵原理的Web服务QoS概率分布获取

为了进行服务风险管理,需要了解服务质量（QoS）的随机特性,而描述QoS随机特性的一种有效手段是获得其准确的概率分布。为此,提出了一种基于最大熵原理在小样本情况下获取Web服务QoS概率分布的方法。方法采用最大熵原理将小样本情况下QoS概率分布获取的问题规约为一个由已知QoS数据确定约束条件的最优化问题进行求解,获得QoS概率密度函数的解析式,然后设计了对该概率密度函数解析式参数进行估计的算法。最后,以实际的Web服务QoS数据为基础,通过实验验证了该方法对不同QoS分布获取时的有效性和合理性,并验证了分布获取算法的效率和终止性。     

State estimation scheme for nonlinear dynamical systems based on the stochastic approximation

An algorithm for the state estimation of multivariable nonlinear dynamic systems with noisy nonlinear observation systems is investigated on the basis of stochastic approximation procedure.Using an extended version of Dvoretzky's theorem, we derive a sufficient condition that estimation error converges to zero, both in the mean square and with probability one for noise-free multivariable dynamical systems. We then show that our estimation procedure makes the estimation error bounded in the mean square norm for noisy dynamical systems. Some numerical examples are presented for the illustration of the approach mentioned above.     

时变系统最小均方算法的性能分析

在无过程数据平稳性假设和各态遍历等条件下,运用随机过程理论研究了最小方算法（LMS）的有界收敛性,给出了估计误差的上界,论述了LMS算法收敛因子或步长的选择方法,以使参数估计误差上界最小。这对于提高LMS算法的实际应用效果有着重要意义。LMS算法的收敛性分析表明：（1）对于确定性时不变系统,LMS算法是指数速度收敛的;（2）对于确定性时变系统,收敛因子等于1,LMS算法的参数估计误差上界最小;（3）对于时变或不变随机系统,LMS算法的参数估计误差一致有上界。     

Minimum entropy control of nonlinear ARMA systems over a communication network

In this paper, the entropy concept has been utilized to characterize the uncertainty of the tracking error for nonlinear ARMA stochastic systems over a communication network, where time delays in the communication channels are of random nature. A recursive optimization solution has been developed. In addition, an alternative algorithm is also proposed based on the probability density function of the tracking error, which is estimated by a neural network. Finally, a simulation example is given to illustrate the efficiency and feasibility of the proposed approach.     

Stochastic sensor activation for distributed state estimation over a sensor network

We consider distributed state estimation over a resource-limited wireless sensor network. A stochastic sensor activation scheme is introduced to reduce the sensor energy consumption in communications, under which each sensor is activated with a certain probability. When the sensor is activated, it observes the target state and exchanges its estimate of the target state with its neighbors; otherwise, it only receives the estimates from its neighbors. An optimal estimator is designed for each sensor by minimizing its mean-squared estimation error. An upper and a lower bound of the limiting estimation error covariance are obtained. A method of selecting the consensus gain and a lower bound of the activating probability is also provided.     

基于Youla参数化的动态随机系统最小误差熵控制

针对具有随机干扰的动态系统,提出一种最小误差熵控制方法。基本思想是应用Youla参数化公式构建具有闭环稳定性的反馈控制策略。其中Renyis熵被作为跟踪误差信息以测度闭环系统的不确定性,Youla参数被优化以使闭环系统误差熵最小,且一个仿真实例也表明了所提算法的有效性。     

Information-theoretical optimal smoothing estimators

The optimal smoothing estimators are derived based on information theory. For a continuous-time stochastic linear system it is proved that minimizing the mutual information between the smoothing estimate vector and the smoothing error vector is equivalent to minimizing the entropy of the smoothing estimation error vector. By minimizing the entropy of the smoothing estimation error vector we derive the basic equations for the optimal smoothing estimators. Using the basic equations, three kinds of optimal smoothing estimators are derived. Furthermore, the information structure of optimal-smoothing problems is clarified.     

基于2阶Renyi熵的自适应主动噪声控制

在前馈主动噪声控制中,基于均方误差准则的传统算法仅考虑了信号的2阶统计量,忽略了实际存在的非高斯信号,不能满足对非高斯噪声的控制要求.提出基于2阶Renyi熵的滤波X自适应有限脉冲响应 (finite impulse response,FIR)主动噪声控制算法,定义2阶Renyi熵作为性能指标,利用Parzen窗方法估计误差的概率密度函数,给出基于2阶Renyi熵的信息梯度下降算法,实现自适应FIR控制,同时分析了算法的收敛性和计算复杂度.对单频信号和实测宽带非高斯噪声的仿真结果表明该算法能很好地消除非高斯噪声.     

基于词汇化随机文法模型的RNA二级结构预测

针对经典的随机文法模型预测RNA二级结构存在精度不高的问题,本文给出了一种词汇化随机文法模型预测RNA二级结构的方法。首先,用最大熵模型获取RNA序列中的词条信息,通过Viterbi算法搜索每个词条被标注为某种二级结构类型的最大概率;然后,将这些词条信息作为先验信息在随机文法模型训练过程中引入,从而加快对二级结构的搜索过程,提高准确率。