Orthogonal least-squares parameter estimation algorithms for non-linear stochastic systems

The derivations of orthogonal least-squares algorithms based on the principle of Hsia's method and generalized least-squares are presented. Extensions to the case of non-linear stochastic systems are discussed and the performance of the algorithms is illustrated with the identification of both simulated systems and linear models of an electric arc furnace and a gas furnace.     

Suboptimal sensors location in the state estimation problem for stochastic non-linear distributed parameter systems

The present paper deals with the minimal number sensor choice and their optimal location for the estimation in non-linear stochastic distributed parameter systems described by parabolic and hyperbolic partial differential equations. The necessary condition for the optimal sensor location by using the matrix minimum principle was obtained. In turn, the computational algorithm of the sensors location was determined on the basis of the necessary condition, applying the optimal control theory. The computational efficiency of this algorithm is defined by the suboptimal filtering algorithm which does not require solving of the matrix Riccati equation for the filter error covariance. Finally, one example is given to demonstrate the effectiveness of the present approach.     

Data-based mechanistic modelling of stochastic rainfall-flow processes by state dependent parameter estimation

Due to the inherent nonlinearity in the process of transformation of rainfall into river flow, a simple direct input-output transfer function (TF) model may not sufficiently capture the catchment's hydrological dynamics. This paper presents an application of state dependent parameter (SDP) models for nonlinear, stochastic dynamic system to identify the location and form of the nonlinearity in the rainfall-effective rainfall dynamics. The objective was to develop an effective rainfall input time series that was then used to improve the performance of an originally developed direct input-output TF model of daily rainfall-flow relationship. The CAPTAIN Toolbox in the MATLAB^® environment was used in the model identification in which the recursive filtering and smoothing procedures formulated within a stochastic state space setting were applied to the time series data in order to identify the location and form of nonlinearities within a generic TF model. The nonparametric estimation as well as the parametric optimisation of the resulting nonlinear models was done using the Curve Fitting Toolbox in MATLAB^®. The results showed an improved and more parsimonious TF model. The model improved from explaining only 13% of the data to 56% presenting an improvement of 43% in the model fit. The study demonstrates that simple stochastic but robust tools can be successfully applied to develop and improve applicable hydrological models.     

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.     

Inversion of non-linear stochastic models for the purpose of parameter estimation

Prediction error and maximum likelihood estimation of non-linear stochastic models requires inversion of the model, a step which may require substantial efforts, either in terms of manual calculations or through the use of software capable of symbolic computations. In this paper we show that model inversion can be easily implemented in numerical software such as, e.g. Simulink and Matrix X, by means of a feedback connection based on the model. It is further shown how the gradients, used for the optimization of the cost function, can be generated by a linear time-varying feedback system associated with the non-linear model. In addition, we derive sufficient conditions for the existence of a stable causal inverse as well as sufficient conditions for the initial transient to decay. These conditions are given in terms of properties for a linear time-varying system associated with the non-linear model. The method is illustrated on numerical examples.     

Linear state estimators for non-linear stochastic systems with noisy non-linear observations

The design of linear filters is considered for reconstructing the state of a class of discrete-time non-linear stochastic systems using noise-corrupted measurements. It is shown that for systems with mean-square stable dynamics, it is always possible to guarantee stable estimation schemes. This result is used to prove that a mean–square optimal one-step predictor has stable error dynamics and also to generate other stable predictors.     

State-dependent parameter modelling and identification of stochastic non-linear sampled-data systems

State-dependent parameter representations of stochastic non-linear sampled-data systems are studied. Velocity-based linearization is used to construct state-dependent parameter models which have a nominally linear structure but whose parameters can be characterized as functions of past outputs and inputs. For stochastic systems state-dependent parameter ARMAX (quasi-ARMAX) representations are obtained. The models are identified from input–output data using feedforward neural networks to represent the model parameters as functions of past inputs and outputs. Simulated examples are presented to illustrate the usefulness of the proposed approach for the modelling and identification of non-linear stochastic sampled-data systems.     

Numerical differentiation and parameter estimation in higher-orderlinear stochastic systems

For a linear time-invariant system of order d⩾2 with a white noise disturbance, the input and the output are assumed to be sampled at regular time intervals. Using only these observations, some approximate values of the first d-1 derivatives are obtained by a numerical differentiation scheme, and the unknown system parameters are estimated by a discretization of the continuous-time least-squares formulas. These parameter estimates have an error which does not approach zero as the sampling interval approaches zero. This asymptotic error is shown to be associated with the inconsistency of the quadratic variation estimate of the white noise local variance based on the sampled observations. The use of an explicit correction term in the least-squares estimates or the use of some special numerical differentiation formulas eliminates the error in the estimates     

On simultaneous on-line state and parameter estimation in non-linear state-space models

On-line estimation plays an important role in process control and monitoring. Obtaining a theoretical solution to the simultaneous state-parameter estimation problem for non-linear stochastic systems involves solving complex multi-dimensional integrals that are not amenable to analytical solution. While basic sequential Monte-Carlo (SMC) or particle filtering (PF) algorithms for simultaneous estimation exist, it is well recognized that there is a need for making these on-line algorithms non-degenerate, fast and applicable to processes with missing measurements. To overcome the deficiencies in traditional algorithms, this work proposes a Bayesian approach to on-line state and parameter estimation. Its extension to handle missing data in real-time is also provided. The simultaneous estimation is performed by filtering an extended vector of states and parameters using an adaptive sequential-importance-resampling (SIR) filter with a kernel density estimation method. The approach uses an on-line optimization algorithm based on Kullback–Leibler (KL) divergence to allow adaptation of the SIR filter for combined state-parameter estimation. An optimal tuning rule to control the width of the kernel and the variance of the artificial noise added to the parameters is also proposed. The approach is illustrated through numerical examples.     

Suboptimal state estimation algorithm for non-linear discrete-time distributed-parameter systems

A non-linear discrete-time distributed-parameter system may be described by stochastic partial differential equations. Some state variables are measured at selected points of the system space. For this system a suboptimal state estimation algorithm is proposed. The error covariance matrix is calculated by an approximate approach. This simplification considerably reduces computer calculations in comparison with an optimal algorithm. Finally, the digital simulation of a non-linear DPS demonstrates the effectiveness of the suboptimal estimator.     

Controllability of non-linear stochastic systems

Complete controllability of a semi-linear stochastic system assuming controllability of the associated linear sytem is studied. It is also shown that a non-linear stochastic system is locally null controllable provided that the corrsponding linearized system is controllable.     

Joint state and parameter estimation in particle filtering and stochastic optimization

In this paper, an adaptive estimation algorithm is proposed for non-linear dynamic systems with unknown static parameters based on combination of particle filtering and Simultaneous Perturbation Stochastic Approximation (SPSA) technique. The estimations of parameters are obtained by maximum-likelihood estimation and sampling within particle filtering framework, and the SPSA is used for stochastic optimization and to approximate the gradient of the cost function. The proposed algorithm achieves combined estimation of dynamic state and static parameters of nonlinear systems. Simulation result demonstrates the feasibility and efficiency of the proposed algorithm.     

Joint state and parameter estimation in particle filtering and stochastic optimization

In this paper, an adaptive estimation algorithm is proposed for non-linear dynamic systems with unknown static parameters based on combination of particle filtering and Simultaneous Perturbation Stochastic Approxi- mation （SPSA） technique. The estimations of parameters are obtained by maximum-likelihood estimation and sampling within particle filtering framework, and the SPSA is used for stochastic optimization and to approximate the gradient of the cost function. The proposed algorithm achieves combined estimation of dynamic state and static parameters of nonlinear systems. Simulation result demonstrates the feasibilitv and efficiency of the proposed algorithm     

Identification of non-linear rational systems using a prediction-error estimation algorithm

Identification of discrete–time non–linear stochastic systems which can be represented by a rational input–output model is considered. A prediction–error parameter estimation algorithm is developed and a criterion is derived using results from the theory of hypothesis testing to determine the correct model structure. The identification of a simulated system, and a heat exchanger are included to illustrate the algorithms.     

Dynamic state and parameter estimation applied to neuromorphic systems

Neuroscientists often propose detailed computational models to probe the properties of the neural systems they study. With the advent of neuromorphic engineering, there is an increasing number of hardware electronic analogs of biological neural systems being proposed as well. However, for both biological and hardware systems, it is often difficult to estimate the parameters of the model so that they are meaningful to the experimental system under study, especially when these models involve a large number of states and parameters that cannot be simultaneously measured. We have developed a procedure to solve this problem in the context of interacting neural populations using a recently developed dynamic state and parameter estimation (DSPE) technique. This technique uses synchronization as a tool for dynamically coupling experimentally measured data to its corresponding model to determine its parameters and internal state variables. Typically experimental data are obtained from the biological neural system and the model is simulated in software; here we show that this technique is also efficient in validating proposed network models for neuromorphic spike-based very large-scale integration (VLSI) chips and that it is able to systematically extract network parameters such as synaptic weights, time constants, and other variables that are not accessible by direct observation. Our results suggest that this method can become a very useful tool for model-based identification and configuration of neuromorphic multichip VLSI systems.     

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.     

A second-order method for state estimation of non-linear dynamical systems†

The paper is concerned with further elaboration of the concept of statistical approximation of a given nonlinear function and its application to the problem of state estimation of a non-linear noisy dynamical system from noise-corrupted observations. It is shown that under the assumption that the conditional probability density of the state variable is gaussian, it is possible to approximate a non-linear function of the state by a polynomial of arbitrary order. Using the second-order approximation, an algorithm is then developed for the problem of state estimation. Results obtained from this algorithm are compared with those obtained from a second-order algorithm based on Taylor's series expansion of the non-linear function.     

Multiobjective parameter estimation for non-linear systems: affine information and least-squares formulation

This paper defines a class of system information—affine information—that includes both the dynamic residuals and some types of auxiliary information that can be used in system parameter estimation as special cases. The types of information that can be cast under the affine information format give rise to quadratic functions that measure the extent to which a model fits such information, and that can be aggregated in a single weighted quadratic cost functional. This allows the definition of a multiobjective methodology for parameter estimation in non-linear system identification, which allows taking into account any type of affine information. The results are presented in terms of a set of efficient solutions of the multiobjective estimation problem—such a solution set is more meaningful than a single model. Since any affine information leads to a convex (quadratic) functional, the whole set of efficient solutions is exactly accessible via the minimization of the quadratic functional with different weightings, via a least-squares minimization (a non-iterative, computationally inexpensive procedure). The decision stage, in which a single model is chosen from the Pareto-set, becomes well-defined with a single global solution. Residual variance, fixed point location, static function and static gain are shown to fit in the class of affine information. A buck DC-DC converter is used as example.     

An improved fuzzy Kalman filter for state estimation of non-linear systems

The extended fuzzy Kalman filter (EFKF) of non-linear systems which can deal with fuzzy uncertainty effectively has been developed recently. But it seems to be inapplicable to the cases where the states change abruptly or there exist model mismatches in non-linear systems. Therefore, based on the EFKF, a new concept of the improved fuzzy Kalman filter (IFKF) is proposed in this article. Due to the introduction of the extension orthogonality principle given as a criterion to design the new algorithm, the IFKF can track the abrupt changes of the states and has definite robustness against the model mismatches. Finally, computer simulations with a MIMO non-linear model are presented, which illustrate that the proposed IFKF has the strong tracking ability and robustness against the model mismatches.     

Parameter estimation and control in non-linear systems with perfect measurements II. Non-linear parameter estimation

Three approximate non-linear parameter estimation algorithms are investigated numerically in the suboptimal control of a jacketed CSTR with perfect measurements. Depending upon the initial estimates and the noise covarianee, the algorithms may or may not be stable. A limiting analysis in a noise-free environment is developed to expose possible singularities. In this case the Bayesian-normal approach leads to serious instabilities.