Parametric Model

A generalized model structure of the LTI system, which may give rise to different model sets, is given by Equation (1) where five polynomials have either of the forms like Equation (2) where A, C, D and F have leadings 1‘s (Soderstrom, 2001).

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The Equation (1) is illustrated in Figure 1. The polynomial A corresponds to poles that are common between the dynamic model and the noise model. This polynomial is appropriate if the noise enters the system close to the input. Likewise, F determines the poles that are unique for the dynamics from input and D the poles that are unique for the noise. This structure is too generalized for most purposes and one or several of the polynomials can be set to unity. This gives rise to more common special black box SISO (single input – single output) models listed in Table 1. A parametric model is a model whose parameters are adjusted to fit the data and do not reflect physical considerations in the system. The polynomial models in Table 1 have four different structures known as auto-regressive with external input (ARX), auto- regressive moving average with external input (ARMAX), box-jenkins (BJ) and output error (OE) (Ljung, 2010).

Figure 1 Generalized model structure LTI systems Table 1: Some common parametric models Polynomial used Name of model structure

AB ARX

ABC ARMAX

BF OE

BFCD BJ

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An auto-regressive model with independent inputs (ARX) model is presented as Equation (3) (Ljung, 2010). The noise model is and the noise is coupled to the dynamic model.

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An auto-regressive moving average with external input (ARMAX) model is presented as Equation (4) (Ljung, 2010). The ARMAX model is an extension of the ARX structure by providing more flexibility for modeling noise using the C parameters (a moving average of white noise).

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( ) ( ) ( ) An output error (OE) is presented by Equation (5) (Ljung, 2010).

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A box-jenkins (BJ) is presented by Equation (6) (Ljung, 2010). The BJ model provides completely independent parameterization for the dynamics and the noise using rational polynomial functions.

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( ) ( ) ( ) The General Procedure

A general procedure of the parametric system identification has been given by Ljung (1999). This procedure is illustrated in Figure 2. Note that the restart after the model validation gives an iterative scheme. There are three principles of the parametric system identification: least square method, the gradient correction method and the maximum likelihood method. The recognition system described in this article is the least square offline parametric system identification. Due to the complexity and diversity of the real system, the actual modeling problem from data acquisition to model establishment is difficult to complete by manual labor because it needs repeatedly questing and the amount of calculation is quite huge. The System Identification has simplified the calculation process and improved the efficiency of the parametric system identification. The process is as follow: data pre-processing, choosing structural parameters, parameter estimation, and model checking. In the system, an important issue is to choose rational structural parameters of the model.

Considering ARX model structure, the first step is to generate ARX structure

parameter, and the second step is to compute loss function, that is, the normalized

quadratic sum of output prediction error. The last step is to select structural

parameters based on loss function. After the identification results are obtained, it also

needs to verify whether this model is applicable. If not, the model structure should be

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changed and the parameters should be re-estimated. There are five ways for model validation and simulation, i.e.: to compare the predicted output with the measured output; computes and tests the residuals of the model; computes prediction errors;

computes the k-step ahead prediction; simulates a given dynamic system.

Figure 2 the flow charts parametric system identification procedure (Ljung, 1999) Validation Model

A best model order will be finding should be fulfilling that it is flexible enough and not too complex. The other said, the model structure should be large enough to cover the true system. This structure can be chosen from two or more candidates. This means that for a good model, ̂( ) should resemble the measured output as given in Equation (7) (Soderstorm, 2001).

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The model can be equivalently expressed as Equations (8) and (9) (Soderstorm,

2001).

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( )

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) ( ) The model as Equation (8) and (9) have exactly the same form as considered linear regression. The estimated parameter in Equation (8) is known as the least squares (LS) estimation method. The name equation error method also appears in the literature. The criterion which maps the sequence of prediction errors into a scalar can be chosen in many ways. Concerning the first approach, it is often useful to plot the measured data and the model output. The model output ̂( ) is defined as the output of the model excited by the measured input without disturbances added. The deviation of ̂( ) from ( ) is due both to modeling errors and to the disturbances.

It is therefore important to realize that if the data are noisy then ̂( ) should differ from ( ). In other words, it should only describe the part of the output that is due to the input signal. The term ( ) denotes the equation error. Here, the following class of criteria is adopted. ( ) is frequently called a loss function as shown in Equation (10).

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( ) A more general form of the loss function is, for example as given by Equation (11) where the scalar-valued function ( ) must satisfy some regularity conditions. It is also possible to apply the prediction error approach to nonlinear models. The only requirement is, naturally enough, that the models provide a way of computing the prediction errors ( ) from the data (Soderstorm, 2001).

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( ) Figure 3 provides an illustration of the prediction error method (Soderstorm, 2001).

Figure 3 Block diagram prediction error method

A Final Prediction Error (FPE) criterion provides a measure of the quality of the model by simulating a situation where the model is tested on a different data set.

After computing the number of different models, compare models using these

criteria. According to this theory, the most accurate model has the smallest FPE. If it

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uses the same data set for both model estimation and validation, the fit always improves as the order of the model increases. There is a flexibility of the model structure such as in ARX model. High order causes the model to have larger FPE.

The Final Prediction Error (FPE) is defined by Equation (12) (Ljung, 2010).

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If VN is the loss function, d is the number of estimated parameters, and N is the number of estimation data set. Toolbox assumes that the final prediction error is asymptotic for d<<N, and Equation (13) is used to compute the estimated FPE. The loss function VN is defined by Equation (14) where θN represents the estimated parameter.

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) ( ) Polynomial models are created using ARX, ARMAX, OE and BJ models. The parametric model estimated from the data as a linear polynomial. To get a linear model is estimated data are continuous time. In order to estimate and verify the polynomial model, it must provide input delay and order model. If already insight into the physics of the system, it can determine the number of poles and zeros. In most cases, the order of the prior model is not known. In order to obtain an initial model order and delay of the system, one can estimate and verify several ARX models with various orders and delays, and then compare the performance of these models. The most suitable model that has the best performance is then chosen, and the order is then used as an initial guess for another model. Linear discretized models prepared on the base of the measurements various presentations as the parametric and process models. Their precision in physical phenomenon description has been valuated using two general criterions: Final Prediction Error (FPE) and FIT (represented in window Model Output). The values of the FIT and of the FPE are calculated from a formulae as given by Equation (15), where: – are values of the measurements,

– are the values calculated from the model, – is the number of the experimental points, and – is the number of the model coefficients (Ljung, 2010).

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The character of the reaction of the model to the step function has been investigated for the best models (with the maximum of the FIT and the minimum of the FPE).

The transfer function or/and the step function are generally used in automatic fashion

in order to investigate the dynamics of the elements. However, the difficulties in their

interpretation have caused the necessity to examine them in different measuring

positions. Residuals are actually representation of misfit between the data and the

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model. Any information remaining in the residuals is a clue that the model might be

insufficiently complex; otherwise, it is appropriate (Johansson, 1993).