Identification Of Stochastic Systems Research Articles

A General Framework for Nonparametric Identification of Nonlinear Stochastic Systems

In this article, nonparametric identification of nonlinear autoregressive systems with exogenous inputs (NARX) is considered; a general criterion function is introduced for estimating the value of the nonlinear function within the system at any fixed point. The criterion function is constructed using a kernel together with a convex objective function. Not only does this framework include the classical kernel-based weighted least-squares estimator but also the kernel-based L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> ,l ≥ 1 criteria as special cases. First, we prove that the minimizer of the general criterion function converges to the true function value with probability 1. Second, recursive algorithms are proposed to find the estimates, which minimize the criterion function, and it is shown that these estimates also converge to the true function value with probability 1. Numerical examples are given, justifying that the framework guarantees the strong consistency of the estimates and exhibits the robustness against outliers in the observations.

Backstepping control design for stochastic systems driven by Lévy processes

This paper presents a constructive method to design control inputs that almost surely globally -exponentially stabilise strict-feedback systems driven by Lévy processes, which consist of Wiener and compensated Poisson processes, at the origin. The control design is based on a recently developed Lyapunov-type theorem for the study of well-posedness and almost sure -exponential stability of SDEs driven by Lévy processes, the backstepping method, and new Lyapunov functions.

A new look at the statistical identification of nonstationary systems

The paper presents a new, two-stage approach to identification of linear time-varying stochastic systems, based on the concepts of preestimation and postfiltering. The proposed preestimated parameter trajectories are unbiased but have large variability. Hence, to obtain reliable estimates of system parameters, the preestimated trajectories must be further filtered (postfiltered). It is shown how one can design and optimize such postfilters using the basis function framework. The proposed solution to adaptive tuning of postfilter settings is based on parallel estimation and cross-validatory analysis. When compared with the classical solutions to the problem of parameter tracking, the new approach offers, without compromising good tracking performance, significant computational savings, higher numerical robustness and greater flexibility.

A fully probabilistic design for stochastic systems with input delay

This paper is concerned with the regulation problem of discrete-time stochastic systems involving input delays which is relevant to networked control systems. The problem is formulated in a fully probabilistic framework, and the control solution is obtained by minimising the Kullback–Leibler Divergence (KLD) between the actual and desired joint probability density functions of the system dynamics. A closed-form solution for the randomised controller is obtained for stochastic systems that can be described by arbitrary probability density functions. Furthermore, the analytic solution for a class of linear Gaussian stochastic systems is obtained. For this class of systems, the optimal randomised controller is shown to be a state feedback controller which is modified by an extra linear term that is related to the lagged and future control inputs. The developed method is demonstrated on a simulation example, and the results are compared with the standard fully probabilistic design control method.

Активная параметрическая идентификация стохастических линейных нерерывно-дискретных систем на основе планирования эксперимента при наличии аномальных наблюдении

Активная параметрическая идентификация стохастических линейных нерерывно-дискретных систем на основе планирования эксперимента при наличии аномальных наблюдении

Identification of non-linear stochastic systems using a new Hammerstein-Wiener neural network: a simulation study through a non-linear hydraulic process

Hammerstein-Wiener models have been proved to be suitable in modelling a class of typical non-linear dynamic systems. This paper aims at developing a Hammerstein-Wiener Neural Network (HWNN) which formulates a Hammerstein-Wiener mathematical model, in order to identify a nonlinear dynamic system operating in a stochastic environment. A central aspect is that a general situation has been considered including non-invertible non-linearity output and correlation of stochastic disturbances after the dynamic linear block. Different from the existing parameter identification methods, the model is developed to handle two types of learning algorithms that can directly obtain the parameters of the unknown time-varying nonlinear system. Firstly, all neural network weights in HWNN are adapted using a Back Propagation-based Gradient algorithm (BPG). The second method, namely Recursive Least Square Back Propagation based Gradient (RLSBPG) is derived from the BPG algorithm to achieve the parametric estimation of the Hammerstein scheme where the remaining parameters are estimated by the least-squares approach based on fuzzy technique to ameliorate the estimation quality. The convergence analysis of the algorithms is presented, and their performances are tested through a simulation study of a nonlinear hydraulic process.

Identification of non-linear stochastic systems using a new Hammerstein-Wiener neural network: a simulation study through a non-linear hydraulic process

Hammerstein-Wiener models have been proved to be suitable in modelling a class of typical non-linear dynamic systems. This paper aims at developing a Hammerstein-Wiener Neural Network (HWNN) which formulates a Hammerstein-Wiener mathematical model, in order to identify a nonlinear dynamic system operating in a stochastic environment. A central aspect is that a general situation has been considered including non-invertible non-linearity output and correlation of stochastic disturbances after the dynamic linear block. Different from the existing parameter identification methods, the model is developed to handle two types of learning algorithms that can directly obtain the parameters of the unknown time-varying nonlinear system. Firstly, all neural network weights in HWNN are adapted using a Back Propagation-based Gradient algorithm (BPG). The second method, namely Recursive Least Square Back Propagation based Gradient (RLSBPG) is derived from the BPG algorithm to achieve the parametric estimation of the Hammerstein scheme where the remaining parameters are estimated by the least-squares approach based on fuzzy technique to ameliorate the estimation quality. The convergence analysis of the algorithms is presented, and their performances are tested through a simulation study of a nonlinear hydraulic process.

Fuzzy Control with Pole Assignment and Variance Constraints for Continuous-time Perturbed Takagi-Sugeno Fuzzy Models: Application to Ship Steering Systems

The stability analysis and controller design of stochastic systems have become much more important because the stochastic behaviors usually exist in practical nonlinear systems. In this paper, a robust fuzzy controller design approach is proposed with multiple constraints, including state variance constraints, output variance constraints, and pole placement constraints. At first, nonlinear systems are expressed as the Takagi-Sugeno fuzzy model, and the parallel distributed compensation method is applied to design the robust fuzzy controllers. Next, considering the stability analysis and the performance constraints of perturbed Takagi-Sugeno fuzzy models, Lyapunov conditions are developed based on covariance control theory, pole placement theory and robust control theory. By constructing the stability conditions with multiple constraints, the proposed fuzzy control problem can be effectively transferred into the linear matrix inequality problem. It can be solved by the convex optimal programming algorithm. At last, a nonlinear ship steering system is selected to verify the effectiveness and applicability of the proposed robust fuzzy controller design method.

Over parameterisation and optimisation approaches for identification of nonlinear stochastic systems described by Hammerstein-Wiener models

This paper proposes two iterative procedures based on over-parameterisation and optimisation approaches for the identification of nonlinear systems which can be described by Hammerstein-Wiener stochastic models. In this case, the dynamic linear part of the considered system is described by ARMAX mathematical model. The static nonlinear block is approximated by polynomial functions. The first procedure is based on a combination of the prediction error method by using the recursive approximated maximum likelihood estimator (RAML), the singular value decomposition (SVD) approach and the fuzzy techniques in order to estimate the parameters of the considered process. As for the second procedure, it includes an appropriate representation named as generalised orthonormal basis filters (GOBF) in order to reduce the complexity of the considered system. The parametric estimation problem is formulated using the recursive extended least squares (RELS) algorithm incorporated with the singular value decomposition (SVD) and fuzzy techniques in order to segregate the coupled parameters and improve the estimation quality. The validity of the developed approaches is proved by considering a nonlinear hydraulic process simulation.

Over parameterisation and optimisation approaches for identification of nonlinear stochastic systems described by Hammerstein-Wiener models

This paper proposes two iterative procedures based on over-parameterisation and optimisation approaches for the identification of nonlinear systems which can be described by Hammerstein-Wiener stochastic models. In this case, the dynamic linear part of the considered system is described by ARMAX mathematical model. The static nonlinear block is approximated by polynomial functions. The first procedure is based on a combination of the prediction error method by using the recursive approximated maximum likelihood estimator (RAML), the singular value decomposition (SVD) approach and the fuzzy techniques in order to estimate the parameters of the considered process. As for the second procedure, it includes an appropriate representation named as generalised orthonormal basis filters (GOBF) in order to reduce the complexity of the considered system. The parametric estimation problem is formulated using the recursive extended least squares (RELS) algorithm incorporated with the singular value decomposition (SVD) and fuzzy techniques in order to segregate the coupled parameters and improve the estimation quality. The validity of the developed approaches is proved by considering a nonlinear hydraulic process simulation.

Nuclear norm subspace identification for continuous-time stochastic systems based on distribution theory method

A novel method for nuclear norm subspace identification of continuous-time stochastic systems based on distribution theory is proposed. The time-derivative problem of the system is solved by using random distribution theory, which is the key to obtain the input–output algebraic equation in the time-domain. Due to the fact that the system encounters the stochastic noise, we design a Kalman filter to achieve the state estimation and noise reduction. Nuclear norm minimization is constructed to optimize the system order in the process of subspace identification. Further, the optimization problem is solved by the alternating direction method of multipliers. Simulation results are provided to show the effectiveness of the proposed method.

Algorithms and Performance Analysis for Stochastic Wiener System Identification

We analyze the statistical performance of identification of stochastic dynamical systems with non-linear measurement sensors. This includes stochastic Wiener systems, with linear dynamics, process noise and measured by a non-linear sensor with additive measurement noise. There are many possible system identification methods for such systems, including the maximum likelihood (ML) method and the prediction error method. The focus has mostly been on algorithms and implementation, and less is known about the statistical performance and the corresponding Cramer–Rao lower bound (CRLB) for identification of such non-linear systems. We derive expressions for the CRLB and the asymptotic normalized covariance matrix for certain Gaussian approximations of Wiener systems to show how a non-linear sensor affects the accuracy compared to a corresponding linear sensor. The key idea is to take second order statistics into account by using a common parametrization of the mean and the variance of the output process. This analysis also leads to an ML motivated identification method based on the conditional mean predictor and a Gaussian distribution approximation. The analysis is supported by numerical simulations.

Online Identification of Nonlinear Stochastic Spatiotemporal System With Multiplicative Noise by Robust Optimal Control-Based Kernel Learning Method.

In this paper, we propose a novel kernel method for the online identification of stochastic nonlinear spatiotemporal dynamical systems using the robust control approach. By the difference method, the stochastic spatiotemporal (SST) systems driven by multiplicative noise are first transformed into a class of multi-input-multi-output-partially linear kernel models (PLKMs) with heterogeneous random terms. With the help of techniques established for reproducing kernel Hilbert space, the online learning problem is reasonably considered as an output feedback control problem for a group of time varying linear dynamical systems. We develop an effective algorithm to address the learning problem of PLKM and SST systems by employing the model predictive control theory. Compared with the existing learning methods, the new one can achieve adaptive, robust, and fast convergent online modeling performance for the spatiotemporal dynamics with multiplicative noise, which greatly facilitates the characterization of physical characteristics of the system. Moreover, this investigation for the first time addresses the learning problems for SST systems with novel robust control techniques, which can provide some novel insights into the design of kernel machine learning methods from the perspective of optimal control theory. Numerical studies for benchmark systems are presented to illustrate the effectiveness and efficiency of our new method.

A general convergence result for kernel-based nonparametric identification of nonlinear stochastic systems

In this paper we consider nonparametric identification of nonlinear autoregressive systems with exogenous inputs. Using a kernel function, a general criterion is introduced for estimating the values of the nonlinear function within the system at any fixed point. The criterion function includes the classical kernel based Ll, l > 1 criteria for nonparametric identification as special cases. By transforming the optimization of the criterion function into a root-finding problem, it is proved that the zero point of the root-finding function converges to the optimal value of the criterion function with probability one. A numerical example illustrating the convergence result is also given.

Активная идентификация стохастических систем при параметризации входного сигнала

Активная идентификация стохастических систем при параметризации входного сигнала

An offline approach for output-only Bayesian identification of stochastic nonlinear systems using unscented Kalman filtering

In this paper an offline approach for output-only Bayesian identification of stochastic nonlinear systems is presented. The approach is based on a re-parameterization of the joint posterior distribution of the parameters that define a postulated state-space stochastic model class. In the re-parameterization the state predictive distribution is included, marginalized, and estimated recursively in a state estimation step using an unscented Kalman filter, bypassing state augmentation as required by existing online methods. In applications expectations of functions of the parameters are of interest, which requires the evaluation of potentially high-dimensional integrals; Markov chain Monte Carlo is adopted to sample the posterior distribution and estimate the expectations. The proposed approach is suitable for nonlinear systems subjected to non-stationary inputs whose realization is unknown, and that are modeled as stochastic processes. Numerical verification and experimental validation examples illustrate the effectiveness and advantages of the approach, including: (i) an increased numerical stability with respect to augmented-state unscented Kalman filtering, avoiding divergence of the estimates when the forcing input is unmeasured; (ii) the ability to handle arbitrary prior and posterior distributions. The experimental validation of the approach is conducted using data from a large-scale structure tested on a shake table. It is shown that the approach is robust to inherent modeling errors in the description of the system and forcing input, providing accurate prediction of the dynamic response when the excitation history is unknown.

Global identification of stochastic dynamical systems under different pseudo-static operating conditions: The functionally pooled ARMAX case

The identification of a single global model for a stochastic dynamical system operating under various conditions is considered. Each operating condition is assumed to have a pseudo-static effect on the dynamics and be characterized by a single measurable scheduling variable. Identification is accomplished within a recently introduced Functionally Pooled (FP) framework, which offers a number of advantages over Linear Parameter Varying (LPV) identification techniques. The focus of the work is on the extension of the framework to include the important FP-ARMAX model case. Compared to their simpler FP–ARX counterparts, FP–ARMAX models are much more general and offer improved flexibility in describing various types of stochastic noise, but at the same time lead to a more complicated, non-quadratic, estimation problem. Prediction Error (PE), Maximum Likelihood (ML), and multi-stage estimation methods are postulated, and the PE estimator optimality, in terms of consistency and asymptotic efficiency, is analytically established. The postulated estimators are numerically assessed via Monte Carlo experiments, while the effectiveness of the approach and its superiority over its FP–ARX counterpart are demonstrated via an application case study pertaining to simulated railway vehicle suspension dynamics under various mass loading conditions.

A decoupled approach to filter design for stochastic systems

ABSTRACTThis paper presents a new theorem to guarantee the almost sure exponential stability for a class of stochastic triangular systems by studying only the stability of each diagonal subsystems. This result allows to solve the filtering problem of the stochastic systems with multiplicative noises by using the almost sure exponential stability concept. Two kinds of observers are treated: the full-order and reduced-order cases.

Controller Design for Stochastic Systems Based on Optimally Controlled Stationary States

In this paper, a method for controller design for stochastic systems based on optimally controlled stationary states is proposed. The Key idea in this method is invoking the optimally controlled system states to control the system states so as to the system states tend the controlled states close to desired system states. By computational experiments, it is shown that the proposed method is effective to control the nonlinear stochastic systems described by swing equation.

Functionally Pooled models for the global identification of stochastic systems under different pseudo-static operating conditions

The problem of identifying a single global model for stochastic dynamical systems operating under different conditions is considered within a novel Functionally Pooled (FP) identification framework. Within it a specific value of a measurable scheduling variable characterizes each operating condition that has pseudo–static effects on the dynamics. The FP framework incorporates parsimonious FP models capable of fully accounting for cross correlations among the operating conditions, functional pooling for the simultaneous treatment of all data records, and statistically optimal estimation. Unlike seemingly related Linear Parameter Varying (LPV) model identification leading to suboptimal accuracy in this context, the postulated FP model estimators are shown to achieve optimal statistical accuracy. An application case study based on a simulated railway vehicle under various mass loading conditions serves to illustrate the high achievable accuracy of FP modelling and the improvements over local models employed within LPV–type identification.