Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 349070, 18 pages

http://dx.doi.org/10.1155/2015/349070

## Mixed Estimators Variety for Model Order Reduction in Control Oriented System Identification

^{1}Université de Lyon, 42023 Saint-Étienne, France^{2}Université de Saint-Étienne (Jean Monnet), 42000 Saint-Étienne, France^{3}LASPI, IUT Roanne, 42334 Roanne, France^{4}Laboratoire des Sciences de l’Information et des Systèmes, UMR CNRS, ENSAM, 13617 Aix-en-Provence, France

Received 5 May 2014; Accepted 2 July 2014

Academic Editor: Guido Maione

Copyright © 2015 Christophe Corbier and Jean-Claude Carmona. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new family of MLE type estimators for model order reduction in dynamical systems identification is presented in this paper. A family of distributions proposed in this work combines () and () distributions which are quantified by four parameters. The main purpose is to show that these parameters add degrees of freedom (DOF) in the estimation criterion and reduce the estimated model complexity. Convergence consistency properties of the estimator are analysed and the model order reduction is established. Experimental results are presented and discussed on a real vibration complex dynamical system and pseudo-linear models are considered.

#### 1. Introduction

The choice of the norm is a fundamental problem in system identification. For some special cases, the least squares (LS) (), the least absolute deviation (LAD) (), and the Chebyshev estimator () methods have been proposed by many authors [1, 2]. It is well known that the norm estimators are efficient when the noise follows the Laplace distribution, the norm estimators are efficient when the noise follows the Gaussian distribution, and the norm estimators are efficient when the noise follows the uniform distribution. Furthermore, it has been shown that the norm estimator is the MLE if noise is a generalized -Gaussian [3]. Thus, we can always find a suitable exponent for the norm method. In the reference paper [4], the authors discuss smooth and sensitive norms for prediction error system identification when disturbances are magnitude-bounded. They show that a necessary condition for norm to be statistically robust with respect to the family of the -contaminated distribution model with support for some arbitrary is that its second derivative does not vanish on the support and must be strictly positive with respect to . This latter condition is fundamental in our framework by ensuring the parameter estimate variance convergence rate (see Proposition 4.2 and Corollary 4.3 [4]). However, many drawbacks come from the use of the norm. Firstly, is the unique DOF of the norm. Secondly, the support is a restricted condition to deal with the outliers [5]. Thirdly, the choice of may prevent robustness to the large data. Finally, most of the time, norm is used to the model reduction for robust control [6] with the restricted case where the model parameters number is limited. The norms are widely used in many domains [7, 8]. Even though the formal framework presents some difficulties, previous works show the interest to use these estimators. In system identification, Chen et al. in [9] propose an ARMA robust system identification using a generalized norm estimation algorithm. The authors use this norm to identify a system with a non-Gaussian noise, since the classical LSE is related to the situation when the noise distribution is white-Gaussian. Likewise, these authors in [10] propose the parameter estimation of linear systems with input-output noisy data, supposed to be corrupted by measurement noise unknown distributions.

Here, we deal with the problem of the estimation in system identification in the presence of atypical data (outliers) in the vibration dynamical system output signal, in order to reduce the model order for the controller design. In such a vibration dynamical system,* natural outliers* occur in the output signal, increasing considerably the model order in the estimation procedure.

To avoid this problem, a new variety of MLE type estimators are considered. We define a mixed distribution with multiple DOF containing and [11], parametrized by a threshold named* scaling factor*, denoted by . To ensure both the parameter estimate variance convergence and the model order reduction and apply conditions in [4] related to the second derivative of the norm on the support with respect to , we show that there exists a tradeoff between , , and . This reduction is efficient only with and , involving .

Model reduction has been subject to considerable interest in many domains. In [12], a method of the model reduction of the nonlinear complex Ginzburg-Landau equation is presented. In [13] the authors propose dimension reduction and dynamics of a spiking neural network model for decision making under neuromodulation. The model order reduction is a fundamental step, espacially in control design. To estimate a low order model of a system, several possibilities exist. The most obvious one is to directly estimate a lower order model. As known from, for example, Ljung [14], the prediction/output error estimate automatically gives models that are approximations of the* true system* in frequency-weight norm, determined by the input spectrum and noise model. See [14, chapter 8] for more details. A second possibility is to estimate a high order model which is then subjected to model reduction to the desired order. Indeed, some contributions that take into account that the high order model is obtained through an identification experiment when performing model reduction are Porat and Friedlander [15], Porat [16], Söderström et al. [17], Stoica and Söderström [18], Zhu and Backx [19, chapter 7] and Tjärnström and Ljung [20], and Tjärnström [21]. Porat and Friedlander study an ARMA parameter estimation via covariance estimates. Recently, in time domain, [22] presents a relevant paper on a new model order reduction algorithm based on general orthogonal polynomials. The following contributors deal with models having input signals. Söderström et al. [17] focus on model structures that can be embedded in larger structures which are easy to estimate, such as ARX structures. After estimating the high order structure, they reduce the estimate to the low order structure in a weighted nonlinear least-squares sense. The method is called* indirect prediction error method*. In [20], the authors focus on model reduction. They study FIR and OE models. The former is estimated directly from data and the latter is computed by reducing a high order model, by model reduction. For OE models, they show that the reduced model has the same variance as the directly estimated one, if the reduced model class used contains the true system. Recently, in [23], the authors illustrate procedures to identify a state-space representation of a lossless or dissipative system from a given noise-free trajectory. The idea is to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced order model. However, these different model order reduction methods are performed after estimation phases with high model orders. On the other hand, this reduction is made in favorable conditions, where no outliers are present in the dataset and the prediction errors. In practice, we all know that real conditions lead to model complexity strongly increased. The lack of robustness of these methods damages the estimation and the model order. Different works have been performed, using robust estimators. A first alternative is the use of the norm [24–26], as a possibility to reduce the model complexity to identify a complex dynamical system, for example, an acoustic duct for active noise control (ANC). The authors propose relevant OE (output error) models with complexity equal to . The second one is to use a symmetrix convex Huber’s function [27]. Recently, in [28, 29], the identification of the same plant, using estimators with low values of the scaling factor in Huber’s function, proposes OE- models with complexity equal to .

Here, we will tackle the distribution approach and, from this, we will propose a parameterized robust estimation criterion (PREC) using a symmetric convex function, offering to the user more* flexibility* in terms of an extra DOF, in order to improve the estimation balance robustness/performances. We will show that the number of DOF in this function facilitates the model order reduction. We will see that the reduced order models are provided by the estimation itself, where classical approaches separate the phases, estimation and then reduction. Indeed, we will demonstrate that the model order thus obtained remains lower than the corresponding estimate. Finally, experimental results on the acoustic duct will confirm these results.

We start in Section 2 by introducing the mathematical background related to definition and properties of the estimator. We develop the consistency and convergence properties of this estimator. The application to the system identification is presented in Section 3. Section 4 proposes the main results concerning the model order reduction. In Section 5, the improvement of the model order reduction with respect to the number of DOF is presented. Experimental results based on a vibration dynamical system are presented and discussed in Section 6. Conclusions and perspectives are drawn in Section 7.

#### 2. Definition and Properties of the Estimators

The new family of -distributions proposed in this paper combines and distributions, quantified by four parameters, (), (), (), and (). Hence, it can be viewed as a generalization of the* redescending* mixed distributions, which corresponds to the M-estimates and (see [27, chapter 4, page 84-85]). A formal definition of a probability density function (pdf) of a random variable is as follows.

*Definition 1. *Consider a function , such thatwhere withwhere and are, respectively, the complete and incomplete Euler’s gamma functions. We can easily verify that, for all , and , which ensure that is a probability density. The parameter is defined as a threshold named* scaling factor*. Its value depends on the degree of the contamination of the random variables . Moreover, a low value will ensure a good robustness and a high value a good efficiency of the estimation.

The pdf belongs to the maximum likelihood estimators (MLE) class. Then, there exists a symmetric function variety such that .

*Definition 2. *Let one define a new estimator variety as a estimator, denoted by . Let be a probability space and let be a sequence of* i.i.d.r.v’s* with values in . Let be a Borel subset in . Let be a symmetric function such that is measurable for each . The estimator is defined by a minimum of the formwherewith and for all .

In robust statistics, continuity conditions are necessary to ensure good estimations [27, chapter 2, page 24-25]. Therefore, these conditions must be applied for both and its first derivative. This latter must be a bounded continuous function. From these considerations, we define the dimensional continuity conditions (DCC) as follows:where and .

The dimensional is given from DCC and, after straightforward calculations, we obtain Equation (6) allows to show that, for the conditions and , we get . We now define the gross error model (GEM) in the mixed framework as a generalization of the GEM given in [27, chapter 1, page 12].

*Definition 3. *Let be a fixed probability distribution having a twice differentiable density , such that is convex on the convex support of . Let be the level of contamination and let be the set of all probability distributions arising from through the -contamination modelHere, is the set of all probability measures on the real line. The corresponding pdf of , contained in , is given by (1), and it puts all contamination outside . A suitable value of should efficiently reduce the propagation of outlier in the prediction errors. This is done by . Accordingly, the treatment of the estimated residuals is focused on the outlier occurrence, so the following errors rapidly decrease and they are advantageously treated by .

Now, our goal is to provide the relation between the level of contamination and the scaling factor . In practice, the robustness is defined by a suitable value of that fixes the -contamination model.

Theorem 4. *If one adjusts the scaling factor in , such that , from the condition , and are linked by**with **where , , is incomplete Euler’s gamma function.*

See Appendix A for the proof. Notice that in the limit case, where (M-estimates), we obtain, for , and . See [27, chapter 4, page 84].

Figure 1 shows an example of the level of contamination as a function of , , parameterized by , with . A large value of the scaling factor means that the probability distribution of the prediction errors is weakly disturbed, corresponding to (see Figure 1). Conversely, a small value corresponds to a pdf more disturbed. For , the curves have a minimum, meaning that a particular value of the scaling factor renders the corresponding pdf weakly contaminated.