Modelling and Parameter Estimation of Dynamic Systems
Modelling and Systems Parameter Estimation for Dynamic Systems presents a detailed examination of the estimation techniques and modeling problems. The theory is furnished with several illustrations and computer programs to promote better understanding of system modeling and parameter estimation. The material is presented in a way that makes for easy reading and enables the user to implement and execute the programs himself to gain first hand experience of the estimation process.
Other keywords: dynamic systems; filtering methods; modelling; model error determination; output error method; parameter estimation; least squares method; model error concept; filter error method
 Book DOI: 10.1049/PBCE065E
 Chapter DOI: 10.1049/PBCE065E
 ISBN: 9780863413636
 eISBN: 9781849190374
 Page count: 404
 Format: PDF

Front Matter
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1 Introduction
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The present book aims at explicit determination of the numerical values of the elements of system matrices and evaluation of the approaches adapted for parameter estimation. The evaluation can be carried out by coding the algorithms in PC MATLAB and using them for system data analysis. The theoretical issues pertaining to the mathematical criteria and the convergence properties of the methods are kept to minimum. The emphasis in the present book is on the description of the essential features of the methods, mathematical representation, algorithmic steps, numerical simulation details and PC MATLAB generated results to illustrate the usefulness of these methods for practical systems.

2 Least squares methods
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In this chapter, we discuss the least squares/equation error techniques for parameter estimation, which are used for aiding the parameter estimation of dynamic systems (including algebraic systems), in general, and the aerodynamic derivatives of aerospace vehicles from the flight data, in particular. In the first few sections, some basic concepts and techniques of the least squares approach are discussed with a view to elucidating the more involved methods and procedures in the later chapters. Since our approach is modelbased, we need to define a mathematical model of the dynamic (or static) system.

3 Output error method
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The type of (linear or nonlinear) mathematical model, and the presence of process or measurement noise in data or both mainly drive the choice of the estimation method and the intended use of results. The equation error method has a cost function that is linear in parameters. It is simple and easy to implement. The output error method is more complex and requires the nonlinear optimisation technique (GaussNewton method) to estimate model parameters. The iterative nature of the approach makes it a little more computer intensive. The third approach is the filter error method which is the most general approach to parameter estimation problem accounting for both process and measurement noise. Being a combination of the Kalman filter and output error method, it is the most complex of the three techniques with high computational requirements. The output error method is perhaps the most widely used approach for aircraft parameter estimation and is discussed in this chapter, after discussing the concepts of maximum likelihood. The Gaussian least squares differential correction method is also an output error method, but it is not based on the maximum likelihood principle.

4 Filtering methods
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In the area of signal processing, we come across analogue and digital filtering concepts and methods. The reallife systems give rise to signals, which are invariably contaminated with the socalled random noise. A usual characteristic of such a random noise that affects the signal is Gaussian (normally distributed) noise with zero mean and some finite variance. This variance measures the power of the noise and it is often compared to the power of the signal that is influenced by the random noise. This leads to a measure called signal to noise ratio (SNR). The aim is then to maximise SNR by filtering out the noise from the signal/data of the dynamical system. There are mainly two approaches: model free and model based. In the model free approach, no mathematical model (equations) is presumed to be fitted or used to estimate the signal from the signal plus noise. These techniques rely upon the concept of the correlation of various signals, like inputoutput signals and so on. In the present chapter, we use the model based approach and especially the approach based on the statespace model of a dynamical system. Therefore, our major goal is to get the best estimate or prediction of the signal, which is buried, in the random noise. Estimation (of a signal) is a general term. One can make three distinctions in context of an estimate of a signal: filtered, predicted or smoothed estimate. We assume that the data is available up to the time 't'. Then, obtaining the estimate of a signal at the time 't' is called filtering. If we obtain an estimate, say at 't + 1', it is called prediction and if we obtain an estimate at 't1' by using data up to 't', it is called a smoothed estimate. In this chapter, we mainly study the problem of filtering and prediction using Kalman filtering methods.

5 Filter error method
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The filter error method is the most general approach to parameter estimation that accounts for both the process and the measurement noise. The method was first studied in Reference 1 and since then, various applications of the techniques to estimate parameters from measurements with turbulence (accounting for process noise) have been reported. As mentioned before, the algorithm includes a state estimator (Kalman filter) to obtain filtered data from noisy measurements. Three different ways to account for process noise in a linear system have been suggested. All these formulations use the modified GaussNewton optimisation to estimate the system parameters and the noise statistics. The major difference among these formulations is the manner in which the noise covariance matrices are estimated. A brief insight into the formulations for linear systems is provided next.

6 Determination of model order and structure
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The timeseries methods have gained considerable acceptance in system identification literature in view of their inherent simplicity and flexibility. These techniques provide external descriptions of systems under study and lead to parsimonious, minimum parameterisation representation of the process. The accurate determination of the dynamic order of the timeseries models is a necessary first step in system identification. Many statistical tests are available in the literature which can be used to find the model order for any given process. Selection of a reliable and efficient test criterion has been generally elusive, since most criteria are sensitive to statistical properties of the process. These properties are often unknown. Validation of most of the available criteria has generally been via simulated data. However, these order determination techniques have to be used with practical systems with unknown structures and finite data. It is therefore necessary to validate any model order criterion using a wide variety of data sets from differing dynamic systems. The aspects of timeseries/transfer function modelling are included here from the perspective of them being special cases of specialised representations of the general parameter estimation problems. The coefficients of timeseries models are the parameters, which can be estimated by using the basic least squares, and maximum likelihood methods discussed in Chapters 2 and 3. In addition, some of the model selection criteria are used in EBM procedure for parameter estimation discussed in Chapter 7, and hence the emphasis on model selection criteria in the present chapter.

7 Estimation before modelling approach
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The estimation before modelling (EBM) methodology is essentially a twostep approach. In the first step, the extended Kalman filter is used for state estimation. The filtered states or their derivatives/related variables are used in the next step of regression analysis. Thus, the parameter estimation is separated into two independent steps. This is unlike the output error method, where parameter estimation is accomplished in essentially onestep, though in an iterative manner. In the output error method, the model structure has to be defined a priori whereas in estimation before modelling, this is taken care of in the second step only. Often smoothing techniques are used in the first step to minimise errors from the extended Kalman filter. The main advantage of the EBM approach is that state estimation is accomplished before any modelling is done. For state estimation, usual system dynamics, which might have only a descriptive mathematical model, is used. In the second step of regression analysis, one can evolve the most suitable detailed mathematical model, the parameters of which are estimated using the least squares method. It is here that model selection criteria play an important role. Another advantage of the estimation before modelling approach is that it can be used to handle data from inherently unstable/augmented systems. In addition, this approach has great utility for aircraft parameter estimation.

8 Approach based on the concept of model error
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There are many real life situations where accurate identification of nonlinear terms (parameters) in the model of a dynamic system is required. In principle as well as in practice, the parameter estimation methods can be applied to nonlinear problems. We recall here that the estimation before modelling approach uses two steps in the estimation procedure and the extended Kalman filter can be used for joint state/parameter estimation. As such, the Kalman filter cannot determine the deficiency or discrepancy in the model of the system used in the filter, since it presupposes availability of an accurate statespace model. Assume a situation where we are given the measurements from a nonlinear dynamic system and we want to determine the state estimates. In this case, we use the extended Kalman filter and we need to have the knowledge of the nonlinear function f and h. Any discrepancy in the model will cause model errors that will tend to create a mismatch of the estimated states with the true state of the system. In the Kalman filter, this is usually handled or circumvented by including the process noise term Q. This artifice would normally work well, but it still could have some problems: i) deviation from the Gaussian assumption might degrade the performance of the algorithm; and ii) the filtering algorithm is dependent on the covariance matrix P of the state estimation error, since this is used for computation of Kalman gain K. Since the process noise is added to this directly, as GQG^{T} term, one would have some doubt on the accuracy of this approach. In fact, the inclusion of the 'process noise' term in the filter does not improve the model, since the model could be deficient, although the trick can get a good match of the states. Estimates would be more dependent on the current measurements. This approach will work if the measurements are dense in time, i.e., high frequency of measurements, and are accurate.

9 Parameter estimation approaches for unstable/augmented
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Parameter estimation for inherently unstable/augmented (control) systems has found major applications in modelling of aerospace vehicles. Many modern day high performance fighter aircraft are made inherently unstable or with relaxed static stability for gaining higher (lift/drag ratio) performance. However, such systems cannot fly without full authority control (laws) constantly working. Thus, the aircraft becomes a plant or system working within the closed loop control system. Several approaches for explicit parameter estimation of dynamic systems, in general, and aircraft in particular, have been elucidated in this chapter. Frequency domain methods could find increasing applications for such unstable/augmented systems/aircraft, if linear models are considered adequate.

10 Parameter estimation using artificial neural networks and genetic
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Research in the area of artificial neural networks has advanced at a rapid pace in recent times. The artificial neural network possesses a good ability to learn adaptively. The decision process in an artificial neural network is based on certain nonlinear operations. Such nonlinearities are useful: i) in improving the convergence speed (of the algorithm); ii) to provide more general nonlinear mapping between inputoutput signals; and iii) to reduce the effect of outliers in the measurements. One of the most successful artificial neural networks is the socalled feed forward neural network. In this chapter first the description of the feed forward neural network and its training algorithms is given. Next, parameter estimation using this approach is discussed. The presentation of training algorithms is such that it facilitates MATLAB implementation. Subsequently, recurrent neural networks are described. Several schemes based on recurrent neural networks are presented for parameter estimation of dynamical systems. Subsequently, the genetic algorithm is described and its application to parameter estimation considered.

11 Realtime parameter estimation
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For the online/realtime parameter estimation problem, several aspects are important: i) the estimation algorithm should be robust; ii) it should converge to an estimate close to the true value; iii) its computational requirements should be moderately low or very low; and iv) the algorithm should be numerically reliable and stable so that condition (i) is assured. It is possible to apply online techniques to an industrial process as long as transient responses prevail, since when these responses die out or subside, there is no activity and all inputoutput signals of the process (for identification) have attained the steady state and hence these signals are not useful at all for parameter estimation. Only the steady state gain of the plant/system can be determined.

Appendix A: Properties of signals, matrices, estimators and estimates
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A good estimator should possess certain properties in terms of errors in parameter estimation and/or errors in the predicted measurements or responses of the mathematical model thus determined. Since the measured data used in the estimation process are noisy, the parameter estimates can be considered to have some random nature. In fact, the estimates that we would have are the mean of the probability distribution, and hence the estimation error would have some associated covariance matrices. Thus, due to the stochastic nature of the errors, one would want the probability of the estimate being equal to the true value to be 1. We expect an estimator to be unbiased, efficient and consistent  not all of which might be achievable. In this appendix, we collect several properties of signals, matrices, estimators and estimates that would be useful in judging the properties and 'goodness of fit' of the parameter/state estimates and interpreting the results. Many of these definitions, properties and other useful aspects are used or indicated in the various chapters of the book and are compiled in this appendix.

Appendix B: Aircraft models for parameter estimation
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One of the important aspects of flighttesting of any aircraft is the estimation of its stability and control derivatives. Parameter estimation is an important tool for flight test engineers and data analysts to determine the aerodynamic characteristics of new and untested aircraft. The flightestimated derivatives are useful in updating the flight simulator model, improving the flight control laws and evaluating handling qualities. In addition, the flight determined derivatives help in validation of the predicted derivatives. These predicted derivatives are often based on one or more of the following: (i) wind tunnel; (ii) DATCOM (Data Compendium) methods; and (iii) some analytical methods. The aircraft dynamics are modelled by a set of differential equations (equations of motion already discussed). The external forces and moments acting on the aircraft are described in terms of aircraft stability and control derivatives. Using specifically designed control inputs, responses of the test aircraft and the mathematical model are obtained and compared. Appropriate parameter estimation algorithms are applied to minimise the response error by iteratively adjusting the model parameters. Thus, the key elements for aircraft parameter estimation are: manoeuvres, measurements, methods and models.

Appendix C: Solutions to exercises
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Back Matter
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Supplementary material

Supplementary files for "Modelling and Parameter Estimation of Dynamic Systems"

Supplementary files for "Modelling and Parameter Estimation of Dynamic Systems" can be downloaded here. The zipped folder contains Errata and files for chapters 2, 3, 4, 6, 7, 8, 9, 10 & 11

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