Recursive system identification in the presence of noise and model uncertainties

The main contribution of this paper is a recursive algorithm for parametric system identification in the presence of both noise and model uncertainties. The estimates provided by this algorithm are not invalidated, after a learning period, by the observed input-output data and the assumed system and uncertainty structures. A complementary off-line algorithm derived from the on-line algorithm is also presented.     

The asymptotic model quality assessment for instrumental variable identification revisited

In this paper the problem of computing uncertainty regions for models identified through an instrumental variable technique is considered. Recently, it has been pointed out that, in certain operating conditions, the asymptotic theory of system identification (the most widely used method for model quality assessment) may deliver unreliable confidence regions. The aim of this paper is to show that, in an instrumental variable setting, the asymptotic theory exhibits a certain “robustness” that makes it reliable even with a moderate number of data samples. Reasons for this are highlighted in the paper through a theoretical analysis and simulation examples.     

Kinematic model identification of industrial manipulators

The aim of the work presented in this paper is to improve the off-line programming capability of industrial robots by improving their accuracy. Rather than impose more strict manufacturing tolerances, it is widely accepted that a method of identifying kinematic parameters specific to each individual robot provides a cost effective way of improving accuracy. A procedure is presented for identification of actual kinematic parameters, which uses the plane of rotation and centre of rotation introduced by Stone. The procedure differs from that of Stone in that it makes use of the radius of rotation and also introduces a translation of the plane of rotation along the axis of rotation. This allows for the direct identification of the D–H model parameters which are more widely accepted and easier to interpret than the S model parameters. It is shown that, unlike the original method of Stone, the new procedure can also deal with the situation when two consecutive joint axes are parallel. The method is validated on both simulated data and real measured data for a Puma 560 robot, showing an improvement in positioning accuracy of around 80%.     

Computational system identification for Bayesian NARMAX modelling

In this contribution we derive a computational Bayesian approach to NARMAX model identification. The identification algorithm exploits continuing advances in computational processing power to numerically obtain posterior distributions for both model structure and parameters via sampling methods. The main advantage of this approach over other NARMAX identification algorithms is that for the first time model uncertainty is characterised as a byproduct of the identification procedure. The algorithm is based on the reversible jump Markov chain Monte Carlo (RJMCMC) procedure. Key features of the approach are (i) sampling of unselected model terms for testing for inclusion in the model (the birth move), which encourages global searching of the model term space, (ii) sampling of previously selected model terms for testing for exclusion from the model—a naturally incorporated pruning step (the death move), which leads to model parsimony, and (iii) estimation of model and parameter distributions, which are naturally generated in the Bayesian framework. We present a numerical example to demonstrate the algorithm and a comparison with a forward regression method: the results show that the RJMCMC approach is competitive and gives useful additional information regarding uncertainty in both model parameters and structure.     

Minimal required excitation for closed-loop identification: Some implications for data-driven,system identification

The development and effective use of all available data is extremely important. Previous work has shown that it is possible to identify process models using closed-loop data even if the reference signal was not being excited. However, such results require that the system have a sufficiently large time delay or alternatively a fast sampling time. Therefore, this paper seeks to examine and provide general results for identifiability of a process using closed-loop data with or without changes in the reference signal. Similarly to the previous case, it is shown that the complexity of the required reference signal depends strongly on the sampling time and time delay. However, since many fast processes without time delay can be modelled as first-order systems, they can indeed be identified when the excitation in the reference signal is a simple step function or sequence of such functions. Using numerical simulations as well as the Tennessee Eastman process, the effect on the continuous time parameters is investigated for different sampling times and excitations signals. It is shown that as expected an external reference signal can identify previously difficult-to-identifiable cases.     

Regularization aspects in continuous-time model identification

This paper presents an analysis of some regularization aspects in continuous-time model identification. The study particulary focuses on linear filter methods and shows that filtering the data before estimating their derivatives corresponds to a regularized signal derivative estimation by minimizing a compound criterion whose expression is given explicitly. A new structure based on a null phase filter corresponding to a true regularization filter is proposed and allows to discuss the filter phase effects on parameter estimation by comparing its performances with those of the Poisson filter-based methods. Based on this analysis, a formulation of continuous-time model identification as a joint system input-output signal and model parameter estimation is suggested. In this framework, two linear filter methods are interpreted and a compound criterion is proposed in which the regularization is ensured by a model fitting measure, resulting in a new regularization filter structure for signal estimation.     

Set Membership identification of nonlinear systems

In the paper the problem of identifying nonlinear dynamic systems, described in nonlinear regression form, is considered, using finite and noise-corrupted measurements. Most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities. A key problem in these methods is the proper choice of the model class, typically realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, based on a Set Membership framework is presented, not requiring assumptions on the functional form of the regression function describing the relations between measured input and output, but assuming only some information on its regularity, given by bounds on its gradient. In this way, the problem of considering approximate functional forms is circumvented. Moreover, noise is assumed to be bounded, in contrast with statistical methods, which rely on assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc., whose validity may be difficult to test reliably and is lost in presence of approximate modeling. In this paper, necessary and sufficient conditions are given for the validation of the considered assumptions. An optimal interval estimate of the regression function is obtained, providing its uncertainty range for any assigned regressor values. The set estimate allows to derive an optimal identification algorithm, giving estimates with minimal guaranteed L_p error on the assigned domain of the regressors. The properties of the optimal estimate are investigated and its worst-case L_p identification error is evaluated. The presented approach is tested and compared with other nonlinear methods on the identification of a water heater, a mechanical system with input saturation and a vehicle with controlled suspensions.     

Validity of the standard cross-correlation test for model structure validation

In the standard prediction error framework of system identification, statistical properties of estimated models are typically derived under the assumption that the true system is in the model class. The standard model structure validation test for plant models is the sample cross-correlation test between the residuals of the model and the input. It turns out that the standard test itself is valid only under exactly those assumptions it is meant to verify, i.e. the system is in the model class. It is shown that for reliable results of the validation test a vector-valued test is required and that accurate noise modelling is indispensable for reliable model structure validation. This shows the limitation of separate validation of plant and noise model structures. Improvements of the test are presented, and it is motivated by the fact that reserving data only to be used for model validation is not efficient.     

一种多输入多输出系统模型辨识方法及其应用

目前的辨识方法一般需要在系统输入端加入激励信号，而且多输入多输出系统的在线辨识仍很困难。本文提出一种基于牛顿迭代法的多输入、多输出对象模型迭代辨识方法，模型参数更新的依据是使模型预测输出与全部采样时刻的对象实际输出之间的均方差递减，直到收敛。这种基于全局数据迭代的辨识方法可进行闭环辨识，无需外加激励信号，适用于多输入多输出对象的在线辨识。对一个两输入、两输出对象模型的仿真研究和某电厂300MW机组负荷被控对象的计算结果表明，辨识效果令人满意。     

Parameter identification for models with bounded errors in all variables

In this paper, the problem of parameter identification for models with bounded measurement errors both on the input and on the output is addressed and some corrections to previously published results are presented. In particular, it is shown that only parameter overbounds can in general be computed for systems of the form y = (φ + δφ)θ + δy when the bounded measurement errors δφ and δy are correlated. Since ARMAX and bilinear systems can be represented in this form, it turns out that tight parameter bounds are in general not available for these systems. Finally, we show that it is possible to check a posteriori whether the obtained bounds are tight or not.     

Bilinear black-box identification and MPC of the activated sludge process

In this paper the activated sludge process, which is a process for biological nitrogen removal in municipal wastewater treatment plants, is modeled as a discrete-time bilinear system by application of a recursive prediction error method system identification technique. A novel bilinear model predictive control algorithm is also derived and applied on a simulation model of the activated sludge process. For discrete-time bilinear systems, a quadratic cost on the predicted outputs and inputs, together with input/state constraints, results in a nonlinear non-convex optimization problem. An investigation is performed where the suggested control algorithm is compared with a linear counterpart. The results reveals that even though the identified bilinear black-box model describes the dynamics of the activated sludge process better than linear black-box models, bilinear model predictive control only gives moderate improvements of the control performance compared to linear model predictive control laws.     

System identification method inheriting steady-state characteristics of existing model

In this paper, we propose an identification method to construct a state-space model that inherits steady-state characteristics from an existing model. It is assumed that in prior to an identification experiment, a designer has a model which accurately expresses steady-state characteristics of an actual system responding to certain inputs. The characteristics are extracted and inherited to a reconstructed state-space model via the combination with a subspace identification method. By applying a change-of-variable technique, the combined identification problem, which is formulated as nonlinear optimisation, is reduced to a least squares problem. Finally, we show the effectiveness of the proposed method in three different numerical simulations.     

On the accuracy in errors-in-variables identification compared to prediction-error identification

Errors-in-variables estimation problems for single-input–single-output systems with Gaussian signals are considered in this contribution. It is shown that the Fisher information matrix is monotonically increasing as a function of the input noise variance when the noise spectrum at the input is known and the corresponding noise variance is estimated. Furthermore, it is shown that Whittle’s formula for the Fisher information matrix can be represented as a Gramian and this is used to provide a geometric representation of the asymptotic covariance matrix for asymptotically efficient estimators. Finally, the asymptotic covariance of the parameter estimates for the system dynamics is compared for the two cases: (i) when the model includes white measurement noise on the input and the variance of the noise is estimated, and (ii) when the model includes only measurement noise on the output. In both cases, asymptotically efficient estimators are assumed. An explicit expression for the difference is derived when the underlying system is subject only to measurement noise on the output.     

Selection of input variables for model identification of static nonlinear systems

System identification can be divided into structure and parameter identification. In most system-identification approaches the structure is presumed and only a parameter identification is performed to obtain the coefficients in the functional system. Yet, often there is little knowledge about the system structure. In such cases, the first step has to be the identification of the decisive input variables. In this paper a black-box input variable identification approach using feedforward neural networks is proposed.     

Decoupling the linear and nonlinear parts in Hammerstein model identification

The main result of this paper is to show that the linear part can be made decoupled from the nonlinear part in Hammerstein model identification. Therefore, identification of the linear part for a Hammerstein model becomes a linear problem and accordingly enjoys the same convergence and consistency results as if the unknown nonlinearity is absent.     

Non-asymptotic confidence regions for model parameters in the presence of unmodelled dynamics

This paper deals with the problem of constructing confidence regions for the parameters of truncated series expansion models. The models are represented using orthonormal basis functions, and we extend the ‘Leave-out Sign-dominant Correlation Regions’ (LSCR) algorithm such that non-asymptotic confidence regions for the parameters can be constructed in the presence of unmodelled dynamics. The constructed regions have guaranteed probability of containing the true parameters for any finite number of data points. The algorithm is first developed for FIR models and then extended to models with generalized orthonormal basis functions. The usefulness of the developed approach is demonstrated for FIR and Laguerre models in simulation examples.     

An improved bias-compensation approach for errors-in-variables model identification

Parametric estimation of the dynamic errors-in-variables models is considered in this paper. In particular, a bias compensation approach is examined in a generalized framework. Sufficient conditions for uniqueness of the identified model are presented. Subsequently, a statistical accuracy analysis of the estimation algorithm is carried out. The asymptotic covariance matrix of the system parameter estimates depends on a user chosen filter and a certain weighting matrix. It is shown how these can be tuned to boost the estimation performance. The numerical simulation results suggest that the covariance matrix of the estimated parameter vector is very close to the Cramér-Rao lower bound for the estimation problem.