Optimally robust system identification of systems subject toamplitude-bounded stochastic disturbances

In this paper it is shown that log cos(πx/(2C)) is the optimally robust criterion function for prediction error methods with respect to amplitude-bounded stochastic disturbances. This criterion function minimizes the maximum asymptotic covariance matrix of the parameter estimates for the family of innovations of the systems which are amplitude bounded by the constant C. Furthermore, the stochastic worst case performance of the estimate corresponding to the criterion function log cos(πx/(2C)) is better than the worst case performance of the least squares estimate even if the constant C is chosen larger than the actual amplitude bound on the innovations. In addition to its favorable properties in a stochastic setting, this criterion function also generates estimates which are unfalsified in a deterministic framework     

Robust-nonsmooth Kalman Filtering for Stochastic Sandwich Systems with Dead-zone

In this paper, a robust-nonsmooth Kalman filtering approach for stochastic sandwich systems with dead-zone is proposed, which can guarantee the variance of filtering error to be upper bounded. In this approach, the stochastic sandwich system with dead-zone is described by a stochastic nonsmooth state-space function. Then, in order to approximate the nonsmooth sandwich system within a bounded region around the equilibrium point, a linearization approach based on nonsmooth optimization is proposed. For handling the model uncertainty caused by linearization and modeling, the robust-nonsmooth Kalman filtering method is proposed for state estimation of the stochastic sandwich system with dead-zones with model uncertainty. Finally, both simulation and experimental examples are presented for evaluating the performance of the proposed filtering scheme.

     

Filtering for a class of nonlinear MIMO uncertain time-delay stochastic systems

In this article, a high-gain nonlinear observer-based approach is proposed for filtering of nonlinear stochastic systems with time delays. The system under consideration contains parameter uncertainties, stochastic disturbances, time delays, as well as Lipschitz-like nonlinearities. The objective is to design a robust filter such that the dynamics of the estimation error is guaranteed to be stochastically exponentially ultimately bounded in a sense of mean square. A new high-gain filter idea is introduced to solve the stochastic filtering problem by using the Lyapunov method and stochastic analysis techniques. The design of the proposed filter does not necessitate the resolution of any dynamics systems and its expression is explicitly given, i.e. its calibration is achieved through the choice of a single parameter, and does not rely on solving any matrix inequalities by numerical computation. A simulation example is given to illustrate the performance of the proposed filter.     

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.     

Integrated fault estimation and fault‐tolerant control for stochastic systems with Brownian motions

This paper presents an integrated robust fault estimation and fault‐tolerant control technique for stochastic systems subjected to Brownian parameter perturbations. The augmented system approach, unknown input observer method, and optimization technique are integrated to achieve robust simultaneous estimates of the system states and the means of faults concerned. Meanwhile, a robust fault‐tolerant control strategy is developed by using actuator and sensor signal compensation techniques. Stochastic linear time‐invariant systems, stochastic systems with Lipschitz nonlinear constraint, and stochastic systems with quadratic inner‐bounded nonlinear constraint are respectively investigated, and the corresponding fault‐tolerant control algorithms are addressed. Finally, the effectiveness of the proposed fault‐tolerant control techniques is demonstrated via the drivetrain system of a 4.8 MW benchmark wind turbine, a 3‐tank system, and a numerical nonlinear model.     

State estimation via limited capacity noisy communication channels

The paper addresses a state estimation problem involving communication errors and capacity constraints. Discrete-time partially observed linear systems perturbed by stochastic unbounded additive disturbances are studied. Unlike the classic theory, the sensor signals are communicated to the estimator over a limited capacity noisy digital link modeled as a stochastic discrete memoryless channel. It is shown that the capability of the noisy channel to ensure state estimation with a bounded in probability error is identical to its capability to transmit information with as small probability of error as desired. In other words, the classic Shannon capacity of the channel constitutes the boundary of the observability domain. It is shown that whenever the Shannon capacity bound is met, a reliable observation can be ensured by means of a state estimator consuming a bounded (as time progresses) computational complexity and memory per unit time. The corresponding state estimator is constructed explicitly and is based on the classic block coding approach, so that traditional block encoding–decoding procedures can be employed for its implementation. This work was supported by the Australian Research Council and the Russian Foundation for Basic Research grant 06-08-01386.     

Stochastic bounded consensus tracking of second-order multi-agent systems with measurement noises based on sampled-data with general sampling delay

This paper investigates the stochastic bounded consensus tracking problems of second-order multi-agent systems, where the control input of an agent can only use the information measured at the sampling instants from its neighbours or the virtual leader with a time-varying reference state, the measurements are corrupted by random noises and the signal sampling process induces the general sampling delay. First, the stochastic bounded consensus tracking protocol based on sampled-data with the general sampling delay is presented by using the delay decomposition technique. Second, the augmented matrix method, the probability limit theory and some other techniques are employed to derive the necessary and sufficient conditions guaranteeing the mean square bounded consensus tracking. The theoretical results show that the convergence of the proposed protocol simultaneously depends on the constant feedback gains, the network topology, the sampled period and the sampling delay, and that the static consensus tracking error depends on not only the above-mentioned factors, but also the noise intensity and the upper bound of the velocity and the acceleration of the virtual leader. The obtained results cover no sampling delay and the small sampling delay as two special cases. Simulations are provided to demonstrate the effectiveness of the theoretical results.     

A Search Algorithm for Calculating Validated Reliability Bounds

The search algorithm presented allows the CDF of a dependent variable to be bounded with 100% confidence, and allows for a guaranteed evaluation of the error involved. These reliability bounds are often enough to make decisions, and often require a minimal number of function evaluations. The procedure is not intrusive, i.e. it can be equally applied when the function is a complex computer model (black box). The proposed procedure can handle input information consisting of probabilistic, interval-valued, set-valued, or random-set-valued information, as well as any combination thereof. The function as well as the joint pdf of the input variables can be of any type.     

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.     

Recursive state estimation for discrete‐time nonlinear systems with event‐triggered data transmission,norm‐bounded uncertainties and multiple missing measurements

In this paper, we consider the recursive state estimation problem for a class of discrete‐time nonlinear systems with event‐triggered data transmission, norm‐bounded uncertainties, and multiple missing measurements. The phenomenon of event‐triggered communication mechanism occurs only when the specified event‐triggering condition is violated, which leads to a reduction in the number of excessive signal transmissions in a network. A sequence of independent Bernoulli random variables is employed to model the multiple measurements missing in the transmission. The norm‐bounded uncertainties that could be considered as external disturbances which lie in a bounded set. The purpose of the addressed filtering problem is to obtain an optimal robust recursive filter in the minimum‐variance sense such that with the simultaneous presence of event‐triggered data transmission, norm‐bounded uncertainties, and multiple missing measurements; the filtering error is minimized at each sampling time. By solving two Riccati‐like difference equations, the filter gain is calculated recursively. Based on the stochastic analysis theory, it is proved that the estimation error is bounded under certain conditions. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

A neural network controller for systems with unmodeled dynamicswith applications to wastewater treatment

This paper considers the use of neural networks (NN's) in controlling a nonlinear, stochastic system with unknown process equations. The approach here is based on using the output error of the system to train the NN controller without the need to assume or construct a separate model (NN or other type) for the unknown process dynamics. To implement such a direct adaptive control approach, it is required that connection weights in the NN be estimated while the system is being controlled. As a result of the feedback of the unknown process dynamics, however, it is not possible to determine the gradient of the loss function for use in standard (backpropagation-type) weight estimation algorithms. In principle, stochastic approximation algorithms in the standard (Kiefer-Wolfowitz) finite-difference form can be used for this weight estimation since they are based on gradient approximations from available system output errors. However, these algorithms will generally require a prohibitive number of observed system outputs. Therefore, this paper considers the use of a new stochastic approximation algorithm for this weight estimation, which is based on a "simultaneous perturbation" gradient approximation. It is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite-difference gradient approximations. The approach is illustrated on a simulated wastewater treatment system with stochastic effects and nonstationary dynamics.     

A Partial-Nodes-Based Information fusion approach to state estimation for discrete-Time delayed stochastic complex networks

This paper is concerned with the information-fusion-based state estimation problem for a class of discrete-time complex networks with time-varying delays and stochastic perturbations. The measurement outputs available for state estimation are from a fraction of network nodes, and the addressed problem is therefore referred to as the so-called Partial-Nodes-Based (PNB) state estimation problem. By employing the Lyapunov stability theory, a novel framework is established to cope with the PNB state estimation problem by the measurement outputs collected from partial network nodes. By constructing specific Lyapunov-Krasovskii functionals, sufficient criteria are derived for the existence of the desired exponentially ultimately bounded state estimator in mean square for the complex networks. Moreover, a special case is considered where the complex network under investigation is free of stochastic perturbations and the corresponding analysis issue is discussed to ensure the existence of an exponential state estimator. In addition, the explicit expressions of the gains of the desired estimators are characterized. Finally, a numerical illustrative example is presented to demonstrate the effectiveness of the obtained theoretical results.     

H∞ control and filtering of discrete-time stochastic systems with multiplicative noise

Linear discrete-time systems with stochastic uncertainties in their state-space matrices are considered. The problems of finite-horizon filtering and output-feedback control are solved, taking into account possible cross-correlations between the uncertain parameters. In both problems, a cost function is defined which is the expected value of the relevant standard H_∞ performance index with respect to the uncertain parameters. A solution to the filtering problem is obtained first by applying the adjoint system and deriving a bounded real lemma for this system. This solution guarantees a prescribed estimation level of accuracy while minimizing an upper bound on the covariance of the estimation error. The solution of the filtering problem is also extended to the infinite-horizon case. The results of the filtering problem are used to solve the corresponding output-feedback problem. A filtering example is given where a comparison is made with the results obtained using bounded uncertainty design techniques.     

Lyapunov-based adaptive state estimation for a class of nonlinear stochastic systems

This paper is concerned with an adaptive state estimation problem for a class of nonlinear stochastic systems with unknown constant parameters. These nonlinear systems have a linear-in-parameter structure, and the nonlinearity is assumed to be bounded in a Lipschitz-like manner. Using stochastic counterparts of Lyapunov stability theory, we present adaptive state and parameter estimators with ultimately exponentially bounded estimator errors in the sense of mean square for both continuous-time and discrete-time nonlinear stochastic systems. Sufficient conditions are given in terms of the solvability of LMIs. Moreover, we also introduce a suboptimal design approach to optimizing the upper bound of the mean-square error of parameter estimation. This suboptimal design procedure is also realized by LMI computations. By a martingale method, we also show that the related Lyapunov function has a non-negative Lyapunov exponent.     

Optimal estimation of digital stochastic sequences

The paper considers the problem of the optimal on-line estimation of digital stochastic sequences. Specifically a calculus of variations approach is used to prove that for stationary stochastic sequences exponential and moving average algorithms are equally viable, both techniques representing close approximations to the theoretical optimum solution. For non-stationary sequences the moving average algorithm is shown to be the better choice, due to its symmetrical weighting function. The paper concludes with a discussion of the influence of induced correlation on the estimation process.     

Fast state estimation subject to random data loss in discrete-time nonlinear stochastic systems

This paper focuses on the design of the standard observer in discrete-time nonlinear stochastic systems subject to random data loss. By the assumption that the system response is incrementally bounded, two sufficient conditions are subsequently derived that guarantee exponential mean-square stability and fast convergence of the estimation error for the problem at hand. An efficient algorithm is also presented to obtain the observer gain. Finally, the proposed methodology is employed for monitoring the Continuous Stirred Tank Reactor (CSTR) via a wireless communication network. The effectiveness of the designed observer is extensively assessed by using an experimental tested-bed that has been fabricated for performance evaluation of the over wireless-network estimation techniques under realistic radio channel conditions.     

Weaker conditions for innovations informational equivalence in the independent Gaussian case

This paper deals with the problem of establishing conditions under which, in the independent Gaussian case, a stochastic process can be considered to be informationally equivalent to its innovations. In recent years this problem has been considered and, as a result, sufficient conditions implying informational equivalence are now available. On the other hand, these conditions are stronger than the ones implying whiteness of the innovations process. The aim of this paper is to fill the gap between conditions assuring whiteness of the innovations and the ones implying informational equivalence. More specifically, by considering the innovations problem in the context of multiplicity theory of stochastic processes and using the notion of a fully submitted process, a necessary and sufficient condition for informational equivalence of the innovations is established. This condition can be interpreted as a condition of nonsingularity in detection theory and turns out to be weaker than the measure-theoretic equivalence condition that has been used in essentially all of the most recent contributions to the innovations problem.     

Stochastic stability of cubature predictive filterCubature 预测滤波的随机稳定性分析

In this paper, the cubature predictive filter (CPF) is derived based on a third-degree spherical-radial cubature rule. It provides a set of cubature-points scaling linearly with the state-vector dimension, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear predictive filter (PF). In order to facilitate the new method, the algorithm CPF is given firstly. Then, the theoretical analyses demonstrate that the estimated accuracy of the model error and system for the proposed CPF is higher than that of the traditional PF. Moreover, the authors analyze the stochastic boundedness and the error behavior of CPF for general nonlinear systems in a stochastic framework. In particular, the theoretical results present that the estimation error remains bounded and the covariance keeps stable if the system’s initial estimation error, disturbing noise terms as well as the model error are small enough, which is the core part of the CPF theory. All of the results have been demonstrated by numerical simulations for a nonlinear example system.     

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。