Structured Set Membership identification of nonlinear systems with application to vehicles with controlled suspension

This paper considers an iterative algorithm for the identification of structured nonlinear systems. The systems considered consist of the interconnection of a MIMO linear systems and a MIMO nonlinear system. The considered interconnection structure can represent as particular cases Hammerstein, Wiener or Lur’e systems. A key feature of the proposed method is that the nonlinear subsystem may be dynamic and is not assumed to have a given parametric form. In this way the complexity/accuracy problems posed by the proper choice of the suitable parametrization of the nonlinear subsystem are circumvented. Moreover, the simulation error of the overall model is shown to be a nonincreasing function of the number of algorithm iteration. The effectiveness of the algorithm is tested on the problem of identifying a model for vertical dynamics of vehicles with controlled suspensions from both simulated and experimental data.     

CAD package for industrial process identification

A CAD system for off-line industrial process identification is presented. The system has two modules. In the first data acquisition is performed. The second performs the process identification procedure. The functions and facilities of these methods are described, which include: the configuration of the data acquisition procedure to be performed, preliminary data analysis, model structure and parameter estimation method selection, time varying parameter estimation and model validation. An industrial application in petrochemical process identification is presented.     

Variable selection algorithm for the construction of MIMO operatingpoint dependent neurofuzzy networks

An input variable selection procedure is introduced for the identification and construction of multi-input multi-output (MIMO) neurofuzzy operating point dependent models. The algorithm is an extension of a forward modified Gram-Schmidt orthogonal least squares procedure for a linear model structure which is modified to accommodate nonlinear system modeling by incorporating piecewise locally linear model fitting. The proposed input nodes selection procedure effectively tackles the problem of the curse of dimensionality associated with lattice-based modeling algorithms such as radial basis function neurofuzzy networks, enabling the resulting neurofuzzy operating point dependent model to be widely applied in control and estimation. Some numerical examples are given to demonstrate the effectiveness of the proposed construction algorithm     

Closed loop hierarchical control for river pollution

In this paper a feedback solution is obtained for a two polluter river pollution control problem using a hierarchial method. The hierarchial optimisation algorithm uses the continuous time ‘No Delay’, ‘Pure Delay’ and ‘Distributed Delay’ models of the river Cam near Cambridge. All the calculations are done off-line within a decentralised computational structure. The resulting constant gains provide optimal feedback control for any initial condition and this control can be implemented on-line. The method is also demonstrated on the three polluter ‘Distributed Delay’ model.     

Reduced order optimization for model predictive control using principal control moves

In order to reduce the computational complexity of model predictive control (MPC) a proper input signal parametrization is proposed in this paper which significantly reduces the number of decision variables. This parametrization can be based on either measured data from closed-loop operation or simulation data. The snapshots of representative time domain data for all manipulated variables are projected on an orthonormal basis by a Karhunen-Loeve transformation. These significant features (termed principal control moves, PCM) can be reduced utilizing an analytic criterion for performance degradation. Furthermore, a stability analysis of the proposed method is given. Considerations on the identification of the PCM are made and another criterion is given for a sufficient selection of PCM. It is shown by an example of an industrial drying process that a strong reduction in the order of the optimization is possible while retaining a high performance level.     

Practical aspects of process identification

After the presentation of various identification and parameter estimation methods in the previous papers, some selected practical aspects of process identification are discussed. This includes, for a given identification method, the steps from the design of experiments to the verification of the final model. Therefore a general procedure of process identification, the selection of input signals, the selection of the sampling time, off-line and on-line identification, comparison of parameter estimation methods, model order testing and model verification is presented. A short discussion on program packages for process identification follows.     

Automated nonlinear system modelling with multiple neural networks

This article discusses the identification of nonlinear dynamic systems using multi-layer perceptrons (MLPs). It focuses on both structure uncertainty and parameter uncertainty, which have been widely explored in the literature of nonlinear system identification. The main contribution is that an integrated analytic framework is proposed for automated neural network structure selection, parameter identification and hysteresis network switching with guaranteed neural identification performance. First, an automated network structure selection procedure is proposed within a fixed time interval for a given network construction criterion. Then, the network parameter updating algorithm is proposed with guaranteed bounded identification error. To cope with structure uncertainty, a hysteresis strategy is proposed to enable neural identifier switching with guaranteed network performance along the switching process. Both theoretic analysis and a simulation example show the efficacy of the proposed method.     

Optimal cooperation between external criterion and data division in GMDH

The problem of cooperation between external criterion and data division is important for application and development of group method of data handling (GMDH) theory. Based on research of ‘critical noise level’ and ‘noise immunity’, this article presents the concept of ‘expected critical noise level’. The relationship between ‘expected critical noise level’ and both prediction set and model structure is expressed by a formula. According to ‘expected critical noise level’, a selection method is designed, which is able to obtain an optimal cooperation between external criterion and data division in GMDH. Finally, the corresponding algorithm and an example are given.     

Real-time design of an adaptive nonlinear predictive controller

Based on real-time identification and using the concept of NARX (Nonlinear AutoRegressive with exogenous inputs) models, a new adaptive nonlinear predictive controller (ANPC) design is proposed. NARX models represent a natural way to describe the input-output relationship of severely nonlinear systems. From an initial batch of input-output data, a parsimonious NARX model is obtained using the Modified Gram-Schmidt (MGS) orthogonalization algorithm. Following this initial off-line identification and model reduction procedure, the control loop is closed. The ANPC directly uses the obtained structure and initial parameter estimates, which are updated each time step using recursive identification. The controller is designed similar to a typical linear predictive controller based on solving a nonlinear programming (NLP) problem. This paper shows how to solve this NLP problem on-line without the knowledge of the NARX model structure. The design is given for the multi-input multi-output (MIMO) case.     

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.     

M-estimator and D-optimality model construction using orthogonal forward regression.

This correspondence introduces a new orthogonal forward regression (OFR) model identification algorithm using D-optimality for model structure selection and is based on an M-estimators of parameter estimates. M-estimator is a classical robust parameter estimation technique to tackle bad data conditions such as outliers. Computationally, The M-estimator can be derived using an iterative reweighted least squares (IRLS) algorithm. D-optimality is a model structure robustness criterion in experimental design to tackle ill-conditioning in model structure. The orthogonal forward regression (OFR), often based on the modified Gram-Schmidt procedure, is an efficient method incorporating structure selection and parameter estimation simultaneously. The basic idea of the proposed approach is to incorporate an IRLS inner loop into the modified Gram-Schmidt procedure. In this manner, the OFR algorithm for parsimonious model structure determination is extended to bad data conditions with improved performance via the derivation of parameter M-estimators with inherent robustness to outliers. Numerical examples are included to demonstrate the effectiveness of the proposed algorithm.     

Comments on `Identification of non-linear parametrically varying models using separable least squares’ by F. Previdi and M. Lovera: black-box or open box?

This note compares and contrasts the non-linear parameter varying (NLPV) and state-dependent parameter (SDP) model classes. It shows that, while they have similarities, the two-stage SDP modelling procedure, involving non-parametric identification, followed by parametric estimation, is quite different from the single stage NLPV procedure. In particular, the SDP procedure allows for the identification of the model structure and the nature of the non-linearities, prior to the estimation of the parameters that characterize this identified model structure. In contrast to NLPV modelling, therefore, SDP estimation opens up the ‘black box’ and reveals the inner nature of the non-linear system.     

Minimal parametrizations of single-input linear dynamical systems

The parametrization of the set of all controllable single-input time-invariant linear dynamical systems is defined as a map π from a parameter set M_π onto a given set of canonical forms 𝒸_π. A parametrization is called ‘minimal’ if it induces a continuous canonical map and the number of parameters is minimal. Some of its general properties, necessary and sufficient conditions, and several new concrete forms suitable for identification are also given.     

An adaptive recurrent fuzzy system for nonlinear identification

This paper describes the architecture and training procedure of a recurrent fuzzy system (RFS). The RFS is composed of a fuzzy inference system (FIS) and a delayed feedback connection. The recurrent property comes from feeding the FIS output back to the FIS input via an adjustable feedback parameter. Both the on-line and off-line training procedures based on the backpropagation-through-time (BPTT) algorithm have been investigated. The adjoint model of the RFS is obtained and used to compute the gradients. It is shown that the off-line training is insufficient to adapt to changes in system dynamics. So, an on-line training procedure is derived. In this procedure, a first in first out stack is used to store a certain history of the input–output data to perform a truncated BPTT algorithm. A quasi-Newton optimization method with a line search algorithm is used to adjust the RFS parameters. The performance of the developed RFS is demonstrated by applying to the identification of nonlinear dynamic systems. The simulation studies show that the proposed identification model has the ability to learn dynamics of highly nonlinear systems and compensate system uncertainties. The results are promising for the further application in the area of control and modeling.     

Derivation of a ‘designer's guide’ for interactive ‘grey-box’ identification of nonlinear stochastic objects

The problem of using an ‘identification tool box’ for the design of ‘grey-box’ models for nonlinear dynamic objects is non-trivial. The design is an interactive process, and it is not given a priori in what order to execute the various subtasks that the tool box supports what design parameters to manipulate and how to interpret the intermediate results. The difficulties are enhanced when the uncertainty of the designer's a-priori information and the quality of the experiment are such that a model contains other stochastic elements than measurement error.

This paper derives a systematic procedure for design of such models, assuming a generic tool box. The origin is the basic procedure commonly used in natural sciences, namely that of repeated refinement and falsification of hypotheses. The derivation is based on statistical decision theory and leads to the specification of a ‘designer's guide’ for grey-box identification. The procedure has been tested on two industrial processes, using the IdKit tool box, and a prototype of the guide has been implemented as a Unix shell. A simulated example illustrates the procedure.     

Dedicated sensor and classifier of rail head defects

This paper presents an original system based on a specialized eddy-current sensor for the inspection of railway tracks. The device aims at the detection of broken rails and large head spalls of the rail, and it has been designed to be mounted underneath automatic driving railway vehicles. In fact, less important types of defects can also be detected. Another use of the sensor would be for predictive management of the rail. All these applications need a procedure for parametrization of the output signals. Then a classification procedure is performed by a set of neural networks which is able to assign each ‘defect’ into one particular class.     

Structural identification of linear multivariable systems using overlapping forms: a new parametrization

The choice of a parametrization for the representation of a linear multivariable control system amounts to the selection of a basis of the rows of the Hankel matrix of Markov elements. The so-called ‘ overlapping ’ or ‘ pseudo-canonical ’ forms are traditionally obtained by imposing two selection rules : a block selection rule and a chain selection rule. In this paper, these constraints are relaxed to requiring only a chain selection rule. This allows for more flexibility in selecting numerically well-conditioned parametrizations.     

Robust identification and interpolation in H∞

We consider system identification in H_∞ in the framework proposed by Helmicki, Jacobson and Nett. An algorithm using the Jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems. It is shown that this, and similar identification algorithms, can be successfully combined with a model reduction procedure to produce low-order models. Connections with the Nevanlinna-Pick interpolation problem are explored, and an algorithm is given in which the identified model interpolates the given noisy data. Some numerical results are provided for illustration. Finally, the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions.     

An adaptive orthogonal search algorithm for model subset selection and non-linear system identification

A new adaptive orthogonal search (AOS) algorithm is proposed for model subset selection and non-linear system identification. Model structure detection is a key step in any system identification problem. This consists of selecting significant model terms from a redundant dictionary of candidate model terms, and determining the model complexity (model length or model size). The final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system. In the new AOS algorithm, a modified generalized cross-validation criterion, called the adjustable prediction error sum of squares (APRESS), is introduced and incorporated into a forward orthogonal search procedure. The main advantage of the new AOS algorithm is that the mechanism is simple and the implementation is direct and easy, and more importantly it can produce efficient model subsets for most non-linear identification problems.