Recursive Identification Method Research Articles

A Recursive and Robust Identification Method for a Class of Nonlinear Systems

Abstract This paper presents a recursive and robust identification method for a class of non linear systems in presence of unknown but bounded errors. Based on the formulation of the input output difference equation of Multi-Input Single-Output (MISO) Wiener Hammerstein model, a modified Weighted Extended-Recursive Least Squares (WE-RLS) procedure is employed to estimate separately parameters of the linear sub-systems and the static non linear elements related to each input when the disturbances upper bound is supposed to be known. The proposed algorithm is performed by minimizing a cost function where a time varying factor is introduced such as the estimated parameters are consistent with the measurements and the noise constraints and, to improve convergence properties. Sufficient conditions to ensure asymptotic convergence are established. Efficiency of the proposed algorithm is shown through a numerical example of high nonlinearities.

Recursive Identification Methods

Since the dynamics of real systems are either time varying or nonlinear in nature, the self-tuning controller design is appropriate where a recursive identification algorithm is utilized to monitor or track the process behaviour. The systems identification algorithms are utilized to estimate the parameters representing the dynamics of the underlying process in linear, nonlinear or both forms, depending upon the required analysis. Therefore, the controller parameters can be tuned on line by reference to the periodically updated plant model. Ideally, the controller synthesis should optimize plant performance taking the uncertainty in the model into account as well as the performance criterion. However, for simplicity, the plant model is usually considered accurate during control synthesis yielding the certainty equivalence self-tuner. Self-tuning control combines the design of feedback controllers based on plant system models with the on-line estimation of the model parameters or the controller parameters using input and output data measurements. The robustness or alertness of self-tuning is totally dependent upon the robustness of the system identification algorithm and its ability to detect and track rapid changes in system performance. The identification techniques find its way into aerospace applications through identifying the flight parameters, adaptive control or adaptive signal and image processing. For example, it could be used to identify the attitude of an aircraft or a missile, engine parameters like engine load variations, fluctuations in fuel supply, and aerodynamic effects from the airstream that change considerably,... etc. The subject of system identification has extremely diverse approaches depending upon the purpose of identifier and the underlying process to be identified. The choice between them depends on factors among among them are: model complexity, noise-to-signal ratio, convergence rate, and computational expenses. The most popular and easiest method for system identification is the least squares (LS) approach. Therefore, this paper presents some of these algorithms in a simplified way leading to the equations that must be solved through the specified algorithm, without going deeply into the derivation of each algorithm. Then, one of them is numerically implemented/evaluated for some examples. In addition, it highlights the identifiability which is a joint property of an identification experiment and a model and establishes the environment for adequate parameters estimates from the injected signal point of view yielding what is called the persistency of excitation. The results showed good convergence, fast response and good tracking utilizing some of the mechanisms or techniques that improve the numerical robustness of any identification algorithm

A Robust and Recursive Identification Method for the Hammerstein Model

In this note, a robust identification method of non linear systems using the Hammerstein model is presented. The main property of the proposed approach is that parameters of the linear and nonlinear parts are estimated in a recursive way while the global convergence is guaranted. The proposed method consists first to transform the non linear representation into an input-output model linear in parameters, after, a regular transformation based on the pseudo-inverse technique allows us to estimate in the least squares sense parameters vector of the original realization. For the case of correlated noise, to model measurement disturbances, we propose a simple technique based on the pseudo-linear regression method and we investigate all parameters dependencies to obtain a consistent solution. Sufficient conditions for global convergence are derived. Two numerical examples with different correlation structures and different Signal to Noise Ratio values are provided.

APPLICATION OF OFF-LINE AND ON-LINE IDENTIFICATION TECHNIQUES TO BUILDING SEISMIC RESPONSE DATA

The objectives of this paper are to present a comparison of the dynamic characteristics of a seven-storey reinforced concrete building (Van Nuys-Holiday Inn) identified from four recorded strong-motion response data (Whittier earthquake, Landers earthquake, Big Bear earthquake and Northridge earthquake). In the analysis, time-domain methods for estimating the system parameters and the modal properties of the building are studied. Both off-line and on-line identification algorithms are applied to these seismic response data. Under the assumption of a linear time-invariant system the ARX model and ARMAX model are used. Comparison of the identification results using different models are made. In addition, recursive procedures are adapted as on-line identification and the time-varying modal parameters are estimated. For structural systems under strong earthquake excitation, a recursive identification method, adaptive forgetting through multiple models (AFMM), is introduced to identify systems with rapidly changing parameters. Through the analysis of the seismic response data of the building subjected to four earthquakes the identification algorithm and the identification results are discussed.

APPLICATION OF OFF‐LINE AND ON‐LINE IDENTIFICATION TECHNIQUES TO BUILDING SEISMIC RESPONSE DATA

The objectives of this paper are to present a comparison of the dynamic characteristics of a seven-storey reinforced concrete building (Van Nuys–Holiday Inn) identified from four recorded strong-motion response data (Whittier earthquake, Landers earthquake, Big Bear earthquake and Northridge earthquake). In the analysis, time-domain methods for estimating the system parameters and the modal properties of the building are studied. Both off-line and on-line identification algorithms are applied to these seismic response data. Under the assumption of a linear time-invariant system the ARX model and ARMAX model are used. Comparison of the identification results using different models are made. In addition, recursive procedures are adapted as on-line identification and the time-varying modal parameters are estimated. For structural systems under strong earthquake excitation, a recursive identification method, adaptive forgetting through multiple models (AFMM), is introduced to identify systems with rapidly changing parameters. Through the analysis of the seismic response data of the building subjected to four earthquakes the identification algorithm and the identification results are discussed.

Identification of a class of nonlinear deterministic systems with application to manipulators

This paper presents a recursive identification method to estimate the minimal parameters of a class of nonlinear deterministic system. The result can be applied to a manipulator with unknown torque constants. A procedure to identify the torque constants, the friction, and the gravity parameters at the same time is then proposed. This is worthwhile since the PD feedback control with the gravity and friction compensation ensures a zero steady-state response for the set-point control. A theory addresses the minimal parameters composed of the three groups of parameters. The identification procedure for the minimal parameters is conducted by moving one joint with a constant velocity at a time. The persistently exciting trajectories for the identification are also suggested. The experiment on the PUMA 560 illustrates the identification method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An Improved Recursive Identification Method For Bilinear Systems

This paper discusses the identification of a special kind of nonlinear systems: bilinear systems. A new formulation of discrete bilinear systems is developed, and an improved recursive identificati...

Modelling of the Puma Robot for Control of Grinding Applications

In this paper, the identification problem of the PUMA 762 robot for grinding applications is considered by using several different recursive identification methods. A detailed procedure of modelling the robot is described. Various experiments have been performed using the PUMA 762/VAL II industrial robot equipped with a 4 Hp pneumatic grinder and a JR 3 force sensor. The PUMA 762 robot has been successfully modelled by a linear third order model with reliable accuracy. The nonlinearities of the robot have been investigated by utilizing different pseudorandom binary sequence (PRBS) signals. Clearly, these linear models, representing the input-output dynamics of the robot, will make it possible to design and develop adaptive controllers for the robotic grinding process.

Identification of continuous systems using digital low-pass filters

Direct methods for recursive identification of continuous systems from sampled input–output data using digital low–pass filters are discussed. The digital low–pass filters are introduced to avoid direct approximations of system signal derivatives from sampled data. Using a pre–designed digital low–pass filter, an approximated discrete–time estimation model is constructed easily. Thus the system parameters can be identified directly by recursive identification algorithms. Numerical results, show that the parameter estimates are not so sensitive to the pass-band of the filter, and if the filter is designed so that its pass-band matches that of the system closely and thus the noise effects are sufficiently reduced, accurate estimates can be obtained by recursive identification algorithms. Two classes of filters (FIR digital filter and IIR digital filter) are considered. It is shown that some other methods can be unified to be either the IIR or the FIR filtering approach.

Convergence Analysis of a Robust Recursive Identification Method for Autoregressive Models

Recently the concept of robustness has attracted considerable attention of control engineers for several control problems. One of this direction is robust identification. Polyak and Tsypkin (1979, 1980) proposed a robust recursive identification method analogous to the recursive least squares method. It follows the idea of Huber's maximum-likelihood-type estimator (M-estimator), which seeks for a minimax solution, i.e., an estimator minimizing the maximum asymptotic variance over some prescribed convex class to which the underlying distribution of innovations belongs. Its convergence properties for dynamic system identification have not been discussed thoroughly because of the introduction of some approximations in the identification method and correlation between observations. In this paper, convergence analysis of the robust recursive identification method is discussed for autoregressive models. Two theorems on convergence properties of the robust identification method are presented and proved by using the ODE approach and martingale convergence theory.

Identification of ARX models with markovian parameters

ABSTRACT Many different recursive identification methods for time-varying systems have been suggested in the literature. An assumption that the variations in the system parameters are slow is common for all the methods. When using the methods on systems with faster variations, one is forced to compromise between alertness to parameter variations on one hand and noise sensitivity on the other. The topic of this paper is to investigate if this compromise can be avoided for a special class of systems. The systems considered are such that their dynamic changes between some different typical modes. The philosophy behind the approach taken in the paper is to separate the observations into different sets corresponding to the different modes. The parameters of the different modes can then be estimated using the separated data sets. Technically, this parallel modelling is achieved by describing the system parameters as the realizations of a Markov chain. A parameter-identification algorithm for time-varying ARX mo...

Adaptive forgetting in recursive identification through multiple models†

Abstract A new recursive identification method, adaptive forgetting through multiple models (AFMM) is presented and evaluated using computer simulations. AFMM is especially suited for identification of systems with jumping or rapidly changing parameters. It can be viewed as a particular way of implementing adaptive gains or adaptive forgetting factors for recursive identification. The new method essentially consists of multiple recursive least-squares (RLS) algorithms running in parallel, each with a corresponding weighting factor. The simulations indicate that AFMM is able to track rapidly changing parameters well, and that the method is robust in several respects.

Recursive methods for off-line identification

Many recursive identification methods have been developed for on-line estimation of parameters in dynamical systems. Such methods can also be applied to data for off-line problems, usually by passing the data through the recursive identifier a few times. In this paper such an approach is discussed. Suppose that a recursive identification method is applied to a data string obtained by concatenation of the original finite-data record. It is proved that the recursive identification method will then converge to a (local) minimum of the off-line finite-data identification criterion. We also provide some arguments as to why recursive identification may be a competitive way of performing the minimization. When using the data more than once it is a priori not clear how to treat the initial values and the forgetting factor. The problem is investigated and comparisons are made with an off-line prediction error method. Practical experiments show that the execution time for a repeated recursive algorithm is less than or equal to that of the off-line algorithm, and that the obtained accuracies are comparable.

Convergence and rate of convergence of a recursive identification and adaptive control method which uses truncated estimators

A stochastic approximation-like method is used for the recursive identification of the coefficients in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y_{n}=\sum\min{1}\max{l_{1}}a_{i}y_{n-i}+\sum\min{0}\max{l_{2}}b_{i}u_{n-i}+ \sum\min{1}\max{l_{3}}c_{i}w_{n-i}+w_{n}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{w_{n}}</tex> is a sequence of mutually independent and bounded random variables, and is independent of the bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{u_{n}}</tex> . Such methods normally require the recursive estimation of the "residuals" or the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{w_{n}}</tex> , and the algorithms for doing this can be unstable if the parameter estimates are not close enough to their true values. The problem is solved here by use of a simple truncated estimator, which is probably what would be used in implementation in any, case. Then, under a stability, and strict positive real type condition, with probability 1 (w.p.1) convergence is proved and the rate of convergence is obtained. An associated adaptive control problem is also treated.

Recursive Identification Methods for Off-Line Identification Problems

Many recursive identification methods have been developed for online estimation of parameters in dynamical systems. Such methods can also be applied to data for off-line problems, usually by passing the data through the recursive identifier a couple of times. In this paper such an approach is discussed.Suppose that a recursive identification method is applied to a data string obtained by concatenation of the original finite data record. It is proved that the recursive identification method will then converge to a (local) minimum of the off-line, finite data, identification criterion. We also provide some arguments why recursive identification may be a competitive way of performing the minimization. The results are illustrated with simulations.

Analysis of a general recursive prediction error identification algorithm

A general class of algorithms for recursive identification of (stochastic) dynamical systems is studied. In this class, the discrepancy between the measured output and the output, predicted from previous data according to a candidate model (‘the prediction error’) is minimized over the model set using a stochastic approximation approach. It is proved that this class of methods has the same convergence properties as its off-line counterparts under mild and general assumptions. These assumptions do not, for example, include stationary conditions or conditions that the true system can be exactly represented within the model set. The considered class of methods contains as special cases several well-known algorithms. Other common recursive identification methods, like the extended least squares method and the extended Kalman filter can be interpreted as approximate prediction error methods with simplified gradient calculations. Therefore, the approach taken here, may also serve as a basis for unified description of many recursive identification methods.

Self-Tuning Load-Following Control of a Nuclear Power Plant

The paper describes an application on self-tuning (adaptive) control to the nuclear/thermal part of a pressurized water reactor (PWR) nuclear power station. The control design is based on a rather simple mathematical low-order model with unknown, possibly time-varying parameters. To ensure that the low-order model describes the complex real dynamics well enough for control purposes, its parameters are updated in real time with a recursive identification method. The algorithm includes a self-tuning feedforward compensation toaccount for load (disturbance) changes. The controller was developed to handle load changes of rather large amplitude. Its purpose was to hold the reactor temperature as near as possible to a desired but changing setpoint-value. The position of the control rods is an appropriate control variable. The application was very successful : the control behaviour was far beyond the specifications. This points to the possibilities of these modem controlmethods for implementation in actual plants.

On the Relationship between Some Well-Known Recursive Identification Methods - The Recursive Tally Principle

Starting fran a general set of equations for one-step paramreter estination methods, four recursive identification methods are derived. It is indicated how scme well-known recursive methods can be found to be a special member of this general class. A recursive tally principle method, which seems to be new in the literature, also appears as a special case. Comparisons with the other methods point to a powerful recursive parameter estimator.

Experiments with a Software Package for Process Identification and Control

A software package for process identification will be described with emphasis on its application to laboratory pilot plants and full scale industrial production units. Methods for off-line parameter estimation as well as real-time recursive versions are evaluated. The recursive identification methods are extensively used in connection with control algorithms. The resulting adaptive controllers are also dealt with in this paper.

Convergence of Recursive Estimators

This paper surveys the convergence analysis and the convergence properties of recursive identification methods for dynamical, stochastic systems. For the survey, two classes of recursive estimators are recognized, to which most of the commonly used methods belong. These are recursive prediction error methods and pseudo linear regressions, respectively. The former possess the general property that they converge to a local minimum of a certain criterion, while convergence of the latter class is tied to positive realness of certain transfer functions associated with the true system.