Identification of Continuous-Time Systems

Frontmatter

Continuous-time models and approaches

Abstract

In this introductory chapter we distinguish between the discrete time (discrete- time) and continuous time approaches. We point out situations where the latter are preferable by drawing attention to certain problems associated with the conventional shift operator, z. Some viable alternatives for a continuous-time treatment are given. This is followed by an outline of the contents and organization of the various contributions to this book.

Ganti Prasada Rao, Naresh K. Sinha

Discrete-time modeling and identification of continuous-time systems: a general framework

Abstract

A general framework for modeling of a time-varying continuous-time SISO system from its sampled input and output while retaining the system parameters with their physical interpretation is presented. The theory can be specialized to the Poisson moment functional approach, the integrated sampling approach, the instantaneous sampling approach or the use of state variable filters. In all methods, the initial conditions can be removed by applying an appropriate discrete-time operator. Digital filtering is used to explicitly or implicitly reconstruct the time-derivatives of the sampled continuous-time signals involved. A thorough study of the approximations resulting from converting the continuous-time model to a discrete-time version is presented. It is shown how these errors can be controlled and that system parameter estimates can be obtained with an arbitrary accuracy. The digital filtering approach to integrated sampling and instantaneous sampling exhibit optimal properties among all methods that fit in the general framework. Moreover, using digital filter designs instead of numerical integration allows slower sampling. A maximum likelihood estimator is derived for time-invariant systems in an errors-in-variables stochastic framework. Finally, the theory is verified through simulations.

H. Van hamme, R. Pintelon, J. Schoukens

The Relationship Between Discrete Time and Continuous Time Linear Estimation

Abstract

We examine the problem of discrete time system estimation while not ignoring the underlying continuous time system. This leads to the use of a new discrete time operator, the S operator, which approximates the continuous time derivative operator \( \frac{d}{{dt}} \). We use this to formulate system estimation algorithms, and discuss their significantly superior numerical properties when compared to the equivalent shift operator formulated algorithms. We provide an overview of this new δ operator and also discuss some practical considerations in recursive least squares parameter estimation.

Brett M. Ninness, Graham C. Goodwin

Transformation of discrete-time models

Abstract

Several approaches to estimating the parameters of a continuous-time model of a linear multivariable system from an equivalent discrete-time model are presented. It is assumed that a suitable discrete-time model has been obtained from the samples of input and output observations using techniques that are already well established. Here, our emphasis is on transformations which will lead to a suitable continuous-time model from the identified discrete-time model. Several algorithms for such transformation are critically compared. Finally, a straightforward procedure for determining a continuous-time model from the δ-model is described.

N. K. Sinha, G. J. Lastman

Methods using Walsh functions

Abstract

Analytical expressions for integral functions of linear system output signals and their Walsh function coefficients are derived. Personal computer signal processing algorithms are then developed. Methods for Walsh coefficient evaluation of integral functions are discussed. Walsh function representation of zero-order hold digital controller outputs and application to system parameter identification are reviewed.

E. V. Bohn

Use of the block pulse operator

Abstract

A block pulse operator (BPO) and its applications to continuous model identification are introduced in this chapter. It includes some research works by the authors on BPO in the recent years. The applications of BPO to the identification of nonlinear and distributed parameter systems are given. The BPO method of optimal input design for identifying parameters in continuous dynamic systems are presented. Numerical examples is presented to illustrate the utility of this method.

Shien-Yu Wang

Recursive block pulse function method

Abstract

A recursive block pulse function method for continuous-time model identification is presented. Based on the block pulse difference equations corresponding to the differential equation models of single-input, single-output linear systems, recursive algorithms developed in discrete-time model identification can be applied directly to estimate the parameters of the original differential equations without much modification. The recursive block pulse function method developed in the single-input, single-output case can also be applied flexibly and conveniently to the identification of certain other continuous-time systems, e.g. multi-input, multi-output linear systems, linear systems with time delays and Hammerstein model nonlinear systems.

Z. H. Jiang, W. Schaufelberger

Continuous model identification via orthogonal polynomials

Abstract

The use of orthogonal polynomials as a class of potential functions for the estimation of parameters of a stipulated model of dynamical systems is outlined in this Chapter. In this process, the system’s initial conditions are also determined. Lumped and distributed parameter system models which are linear and time-invariant are considered for our study. The estimates of parameters and initial conditions obtained by orthogonal polynomials are compared with those determined by block-pulse and sine-cosine functions. Results of estimates with noisy output data are also included.

K. B. Datta

Use of numerical integration methods

Abstract

Identification of the continuous-time system model from samples of input-output data can be carried out directly by integrating the system dynamic equations over various sampling intervals and determining the model parameters which provide the best fit to the data in some sense. Using numerical integration methods, three methods are proposed for identification of continuous-time systems which are either single-input single-output systems or multi-input multi-output systems. Several methods of numerical integration are examined for their suitability, especially in the presence of noise. Robustness of the presented identification algorithms is also investigated.

H. Dai, N. K. Sinha

Application of digital filtering techniques

Abstract

Continuous-time system identification usually consists of two main parts: signal processing (or pre-filtering) and parameter estimation. Both analog and digital pre-filters for signal processing can be used, where analog pre-filters are implemented in a digital computer by using such techniques as the numerical integration and the bilinear transformation. As for parameter estimation, an emphasis is put on on-line identification algorithms. Using the pre-filters of digital form, a discrete-time identification model which retains the continuous-time model parameters is derived. Some fundamental identification methods such as the least squares method, bias-compensating methods and instrumental variable methods are reviewed. Finally the choice of the input signal is discussed with simulation experiments.

S. Sagara, Z. Y. Zhao

The Poisson moment functional technique — Some new results

Abstract

We present some generalisations of the classical (ordinary) Poisson moment functionals (PMF) approach towards improving the quality of estimation in continuous-time models of lumped linear single input single output(SISO) dynamical systems. Some results of an investigation on the influence of Poisson filter constant on the quality of estimation and useful guidelines for the proper choice of filter constant are discussed. This generalised PMF approach has been used for combined parameter and state estimation in a linear tuneinvariant system based on recursive least squares algorithm. This algorithm can be extended for identification of time-varying systems. A recursive instrumental variable (IV) algorithm based on PMFs of the process data is developed to reduce the bias in the estimation. Finally, an attempt has been made to apply the algorithm for the identification of parameters of a system operating under closed loop.

D. C. Saha, V. N. Bapat, B. K. Roy

Identification, Estimation and Control of Continuous-Time Systems Described by Delta Operator Models

Abstract

This Chapter outlines a unified approach to the identification, estimation and control of linear, continuous-time, stochastic, dynamic systems which can be described by delta (δ) operator models with constant or time-variable parameters. It shows how recursive refined instrumental variable estimation algorithms can prove effective both in off-line model identification and estimation, and in the implementation of self-tuning or self-adaptive True Digital Control (TDC) systems which exploit a special Non-Minimum State Space (NMSS) formulation of the δ operator models.

Peter C. Young, Arun Chotai, Wlodek Tych

Identification of multivariable continuous-time systems

Abstract

This chapter presents a parameter identification algorithm for multivariable, continuous-time systems. It is shown how to treat systems with (non-zero) initial conditions and measurement offset or bias. The system model structure used is a minimal order input-output representation. Differentiation of measured data is avoided by means of either multiple lowpass filtering or multiple finite time integration. Equations in unknown parameters are set up and then solved by means of the linear least-squares. The singular value decomposition has been used to solve the least-squares problem because of its numerical robustness and because of the extra data it provides for analysis. Examples of up to 14th order are discussed.

E Boje

Use of Pseudo-Observability Indices In Identification of Continuous-Time Multivariable Models

Abstract

A procedure for identifying linear continuous-time multivariable systems is presented. The scheme is based on a discrete identification method which requires no structural identification. This is accomplished by the use of pseudo-observable forms whose structure is defined by their corresponding pseudo-observability indices. A First-Order Hold transformation of this discrete-time model to an equivalent continuous-time model is then performed. First, the state transition matrix is transformed to an equivalent continuous-time system matrix using a procedure which is insensitive to the size of the spectral radii of the system. The remaining system matrices are then transformed using a generalized First-Order Hold transformation technique which places no restrictions on the singularity of the continuous-time system.

S. Bingulac, D. L. Cooper

SVD-based subspace methods for multivariable continuous-time systems identification

Abstract

Recently severed subspace methods have appeared in the literature for multivariable discrete-time state space identification, where state space models are computed directly from input/output data. These state space identification methods are viewed as the better alternatives to polynomial model identification, owing to the better numerical conditioning associated with state space models, especially for high-order multivariable systems.

In this contribution, a similar method is described for continuoustime state space identification. Here also, the key tool is the singular value decomposition (SVD), a numerical technique known to be very robust and accurate when dealing with noisy data. The noise coloring is compensated for by using a generalization of the SVD, namely the quotient SVD. The resulting identification scheme is then shown to give consistent results under certain conditions.

Marc Moonen, Bart De Moor, Joos Vandewalle

Identification of continuous-time systems using multiharmonic test signals

Abstract

The estimation of the parameters of continuous-time linear dynamic systems from error disturbed responses to periodic, multiharmonic test signals is discussed. Closed-form, linear instrumental variable estimators and weighted least squares estimators are proposed and the relation between these estimators is established. Expressions for the asymptotic covariance matrices of the estimators are presented. From these expressions optimal instrumental sequences and minimum variance least squares estimators are derived.

A. van den Bos

Adaptive Model Approaches

Abstract

In this Chaper algorithms for identification of continuous-time systems based on model reference principles will be presented. The main idea behind the model reference principle is to set up a procedure for adjustment, based on a comparison between the performance of two units, one of which serves as a reference for the other to follow. The adjustment is made on the unit which follows the reference unit in such a way that the discrepancy between the performance of the two units is minimized or reduced to zero. In the case of the identification problem the plant to be identified assumes the role of the reference model and the estimated model assumes the role of the adjustable unit. It is this dual character of the model reference principle that enables us to apply several techniques of the well known Model Reference Adaptive Control (MRAC) Systems (Landau 1979) to the identification problem. In order to assign distinct identity to this class of adaptive model techniques for identification, we will call them System Reference Adaptive Model (SRAM) techniques. These techniques may be further classified on the basis of the following features:

a)

Structure: In accordance with the disposition of the adjustable model with respect to the reference system the SRAM schemes can be classified as i) parallel, ii) series and iii) series-parallel schemes.

 

b)

Design method: There are mainly three design methods viz.

1)

Local optimization (e.g. gradient methods),

 

2)

Liapunov’s second method, and

 

3)

Hyperstable design method.

 

 

H. Unbehauen

Nonparametric approaches to identification

Abstract

Nonparametric identification methods, i.e. the methods for determining the frequency and the impulse responses from the measured data are presented. Based on a brief review of system and signal processing theory the Fourier analysis, the Spectral analysis, the Frequency response analysis and the Correlation analysis are described. Also bias and convergence analysis is given for all methods. Finally some problems with practical application of the methods on a digital computer are discussed.

D. Matko

From fine asymptotics to model selection

Abstract

A short survey of our recent results in the theory of identification of continuous time linear stochastic systems will be presented. The purpose of these investigations is to obtain fine asymptotic results for the parameter estimator process. A culmination of these results is a theorem on the almost sure asymptotics of a continuous-time stochastic complexity.

L. Gerencsér, Zs. Vágó

Real-time issues in continuous system identification

Abstract

In this chapter, the issues in real-time identification of continuous systems are considered. Starting with some aspects of plant model forms, measurement systems and preprocessing schemes, continuous model estimation algorithms are discussed. Further postprocessing techniques for the parameter estimates and residuals which may be necessary to satisfy the requirements of specific applications such as adaptive control, fault detection, condition monitoring etc. are indicated. Salient features of hardware and software important for practical implementations are also discussed briefly. A practical example of real-time parameter estimation is presented. The chapter concludes with a summarizing view and a look into the future in the light of emerging technology.

G. P. Rao, A. Patra, S. Mukhopadhyay

Backmatter