Conditionally optimal estimation in stochastic differential systems

The theory of conditionally optimal estimation of the state and parameters and of conditionally optimal extrapolation of the state of a stochastic system described by differential equations is given. Previous results obtained by the author follow from this theory as special cases. Classes of admissible estimates in this new theory are constructed in such a way that any specific estimate defined as a result of some linear transform of the solution of a stochastic differential equation can be enclosed in the respective class of admissible estimates and compared with the conditionally optimal estimate in this class. This enables one to study the accuracy of any such estimate, in particular the accuracy of any known approximate method of nonlinear filtering, and to improve the accuracy without any complications by optimizing the coefficients in the respective differential equation. As a special case the filtering and prediction theory is developed for linear systems with white noises in the coefficients.     

Small-gain control method for stochastic nonlinear systems with stochastic iISS inverse dynamics

This paper investigates the small-gain type conditions on stochastic iISS (SiISS) systems and makes full use of these conditions in the design and analysis of the controller. The contributions are as follows: (1) A new proof of the stochastic LaSalle theorem is provided; (2) The small-gain type conditions on SiISS are developed and their relationship is discussed; (3) Based on the stochastic LaSalle theorem and SiISS small-gain type conditions, the adaptive controllers are designed to guarantee that all of the closed-loop signals are bounded almost surely and the stochastic closed-loop systems are globally (asymptotically) stable in probability.     

Global inverse optimal stabilization of stochastic nonholonomic systems

Optimality has not been addressed in existing works on control of (stochastic) nonholonomic systems. This paper presents a design of optimal controllers with respect to a meaningful cost function to globally asymptotically stabilize (in probability) nonholonomic systems affine in stochastic disturbances. The design is based on the Lyapunov direct method, the backstepping technique, and the inverse optimal control design. A class of Lyapunov functions, which are not required to be as nonlinearly strong as quadratic or quartic, is proposed for the control design. Thus, these Lyapunov functions can be applied to design of controllers for underactuated (stochastic) mechanical systems, which are usually required Lyapunov functions of a nonlinearly weak form. The proposed control design is illustrated on a kinematic cart, of which wheel velocities are perturbed by stochastic noise.     

Blind channel identification via stochastic approximation: constant step-size algorithms

This work is devoted to the study of a class of recursive algorithms for blind channel identification. Using weak convergence methods, the convergence of the algorithm is obtained and the rate of convergence is ascertained. The technique discussed can also be used in the analysis of rates of convergence for decreasing step-size algorithms.     

Parameter identification technique for multivariate stochastic systems

This paper considers the problem of obtaining accurate estimates of multivariate systems with reasonable computations. To avoid the structural identification problem which is associated with multivariate systems, we observe the system by a linear combination of the outputs. The two stage least square method is employed to estimate the model parameters. An optimum combination of the outputs is obtained such that the parameter estimates have the least asymptotic generalized variance. Computer simulations are provided to illustrate the usefulness of the proposed method.     

Detection techniques in least squares identification

Parameter estimation schemes based on least squares identification and detection ideas are proposed for ease of computation, reduced numerical difficulties, and bias reduction in the presence of colored noise correlated with the states of the signal generating system. The algorithms are simpler because in the calculations, the state vector is at one point replaced by a quantized version. This technique avoids to some extent numerical difficulties associated with ill-conditioning in least squares schemes and thus obviates the need for square root algorithms and the need for high order precision calculations. In recursive form, the schemes are designed to yield parameter estimates with negligible bias without the additional computational effort or instability risks associated with generalized and extended least squares, recursive maximum likelihood schemes, or the method of instrumental variables. Nonrecursive schemes are designed to minimize computational effort in a batch processing situation while at the same time giving some reduction of bias in the state dependent colored noise situation.

The novel algorithms have the limitation that they are suboptimal and there is thus a consequent reduction in the speed of convergence for some applications. The merits of the proposed schemes are assessed via simulation studies in this paper and an adaptive equalization application in a companion paper.     

A state predictor for continuous-time stochastic systems

This work investigates the state prediction problem for nonlinear stochastic differential systems, affected by multiplicative state noise. This problem is relevant in many state-estimation frameworks such as filtering of continuous-discrete systems (i.e. stochastic differential systems with discrete measurements) and time-delay systems. A very common heuristic to achieve the state prediction exploits the numerical integration of the deterministic nonlinear equation associated to the noise-free system. Unfortunately these methods provide the exact solution only for linear systems. Instead here we provide the exact state prediction for nonlinear system in terms of the series expansion of the expected value of the state conditioned to the value in a previous time instant, obtained according to the Carleman embedding technique. The truncation of the infinite series allows to compute the prediction at future times with an arbitrary approximation. Simulations support the effectiveness of the proposed state-prediction algorithm in comparison to the aforementioned heuristic method.     

Joint state and parameter robust estimation of stochastic nonlinear systems

Successful implementation of many control strategies is mainly based on accurate knowledge of the system and its parameters. Besides the stochastic nature of the systems, nonlinearity is one more feature that may be found in almost all physical systems. The application of extended Kalman filter for the joint state and parameter estimation of stochastic nonlinear systems is well known and widely spread. It is a known fact that in measurements, there are inconsistent observations with the largest part of population of observations (outliers). The presence of outliers can significantly reduce the efficiency of linear estimation algorithms derived on the assumptions that observations have Gaussian distributions. Hence, synthesis of robust algorithms is very important. Because of increased practical value in robust filtering as well as the rate of convergence, the modified extended Masreliez–Martin filter presents the natural frame for realization of the joint state and parameter estimator of nonlinear stochastic systems. The strong consistency is proved using the methodology of an associated ODE system. The behaviour of the new approach to joint estimation of states and unknown parameters of nonlinear systems in the case when measurements have non‐Gaussian distributions is illustrated by intensive simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Non-stationary stochastic embedding for transfer function estimation

This paper presents a consistent framework for the quantification of noise and undermodelling errors in transfer function model estimation. We use the, so-called, “stochastic embedding” approach, in which both noise and undermodelling errors are treated as stochastic processes. In contrast to previous applications of stochastic embedding, in this paper we represent the undermodelling as a multiplicative error characterised by random walk processes in the frequency domain. The benefit of the present formulation is that it significantly simplifies the estimation of the parameters of the embedded process yielding a closed-form expression for the model error quantification. Simulation and experimental examples illustrate how the random walks effectively capture typical cases of undermodelling found in practice, including underdamped modes. The examples also show how to use the method as a tool in the determination of model order and pole location in fixed denominator model structures.     

多新息随机梯度辨识方法

多新息随机梯度辨识方法是系统辨识和参数估计的一种基本方法.该方法由于采用了间断迭代,因此可以克服坏数据对参数估计的影响,且具有较强的鲁棒性,又可以跟踪时变参数.作者从理论上给出了多新息随机梯度辨识方法的推导过程,同时列出多新息随机梯度辨识方法的各种变形.数字仿真实验表明多新息随机梯度辨识方法具有良好的性能.     

Identification and estimation of discrete state-vector models with stochastic inputs

The study deals with the identification and estimation of the unknown parameters of an ‘extended’ state-vector model, in which stochastic input variables are treated as ‘state’-variables and the observed input-values as ‘output’-values of the model.A parameter identifiability criterion, based on Fisher's information matrix, is applied to the model and a general ML-estimation procedure is given. If a certain restriction on the covariance-matrix of the state-vector is placed, the ML-procedure simplifies and coincides with an operational method, called the Lisrel procedure. This procedure provides also a test for parameter identifiability.     

非均匀采样系统多新息随机梯度辨识性能分析

针对一类非均匀采样系统,提出了其输入输出表达的多新息随机梯度辨识方法．该方法将随机梯度算法中的新息项扩展为向量,有效利用了历史新息所包含的信息,从而提高辨识精度和算法的收敛速度,同时又保留了随机梯度算法计算量小的优点．仿真例子通过改变新息长度,验证了所提出辨识算法性能的优越性．     

Weighted averaging and stochastic approximation

We explore the relationship between weighted averaging and stochastic approximation algorithms, and study their convergence via a sample-path analysis. We prove that the convergence of a stochastic approximation algorithm is equivalent to the convergence of the weighted average of the associated noise sequence. We also present necessary and sufficient noise conditions for convergence of the average of the output of a stochastic approximation algorithm in the linear case. We show that the averaged stochastic approximation algorithms can tolerate a larger class of noise sequences than the stand-alone stochastic approximation algorithms.This research was supported by the National Science Foundation through Grants ECS-9410313 and ECS-9501652.This research was supported by the National Science Foundation through NYI Grant IRI-9457645.     

Adaptive divided difference filtering for simultaneous state and parameter estimation

A novel adaptive version of the divided difference filter (DDF) applicable to non-linear systems with a linear output equation is presented in this work. In order to make the filter robust to modeling errors, upper bounds on the state covariance matrix are derived. The parameters of this upper bound are then estimated using a combination of offline tuning and online optimization with a linear matrix inequality (LMI) constraint, which ensures that the predicted output error covariance is larger than the observed output error covariance. The resulting sub-optimal, high-gain filter is applied to the problem of joint state and parameter estimation. Simulation results demonstrate the superior performance of the proposed filter as compared to the standard DDF.     

Observer-based state tracking control of uncertain stochastic systems via repetitive controller

This paper develops the repetitive control scheme for state tracking control of uncertain stochastic time-varying delay systems via equivalent-input-disturbance approach. The main purpose of this work is to design a repetitive controller to guarantee the tracking performance under the effects of unknown disturbances with bounded frequency and parameter variations. Specifically, a new set of linear matrix inequality (LMI)-based conditions is derived based on the suitable Lyapunov–Krasovskii functional theory for designing a repetitive controller which guarantees stability and desired tracking performance. More precisely, an equivalent-input-disturbance estimator is incorporated into the control design to reduce the effect of the external disturbances. Simulation results are provided to demonstrate the desired control system stability and their tracking performance. A practical stream water quality preserving system is also provided to show the effectiveness and advantage of the proposed approach.     

Identification with stochastic sampling time jitter

This work investigates how stochastic sampling jitter noise affects the result of system identification, and proposes a modification of known approaches to mitigate the effects of sampling jitter, when the jitter is unknown and not directly measurable. By just assuming conventional additive measurement noise, the analysis shows that the identified model will get a bias in the transfer function amplitude that increases for higher frequencies. A frequency domain approach with a continuous-time model allows an analysis framework for sampling jitter noise. The bias and covariance in the frequency domain model are derived. These are used in bias compensated (weighted) least squares algorithms, and by asymptotic arguments this leads to a maximum likelihood algorithm. Continuous-time output error models are used for numerical illustrations.     

Detecting and diagnosing faults in dynamic stochastic distributions using a rational B-splines approximation to output PDFs

This paper presents a novel approach to detect and diagnose faults in the dynamic part of a class of stochastic systems . the Such a group of systems are subjected to a set of crisp inputs but the outputs considered are the measurable probability density functions (PDFs) of the system output, rather than the system output alone. A new approximation model is developed for the output probability density functions so that the dynamic part of the system is decoupled from the output probability density functions. A nonlinear adaptive observer is constructed to detect and diagnose the fault in the dynamic part of the system. Conver-gency analysis is performed for the error dynamics raised from the fault detection and diagnosis phase and an applicability study on the detection and diagnosis of the unexpected changes in the 2D grammage distributions in a paper forming process is included.     

Performance analysis of model-free PID tuning of MIMO systems based on simultaneous perturbation stochastic approximation

This paper addresses the performance comparison of simultaneous perturbation stochastic approximation (SPSA) based methods for PID tuning of MIMO systems. Four typical SPSA based methods, which are one-measurement SPSA (1SPSA), two-measurement SPSA (2SPSA), Global SPSA (GSPSA) and Adaptive SPSA (ASPSA) are examined. Their performances are evaluated by extensive simulation for several controller design examples, in terms of the stability of the closed-loop system, tracking performance and computation time. In addition, the performance of the SPSA based methods are compared to the other stochastic optimization based approaches. It turns out that the GSPSA based algorithm is the most practical in terms of the stability and the tracking performance.     

A stopped stochastic approximation algorithm

A stopping time problem for multidimensional stochastic approximation algorithms is studied in this paper. The stopping rule is so determined that the recursive procedure will be terminated if the unknown parameter θ is inside a desired ellipsoidal confidence region with high probability. The stopped process is shown to be asymptotically normal by means of weak convergence methods.