Journal of Control Science and Engineering

Volume 2018, Article ID 3138149, 10 pages

https://doi.org/10.1155/2018/3138149

## Identification of Two-Time Scaled Systems Using Prefilters

^{1}Department of Computation and Automation, Federal University of Rio Grande do Norte, Natal, RN, Brazil^{2}Division of Electronic Engineering, Technological Institute of Aeronautics, São José dos Campos, SP, Brazil^{3}Department of Automatic Control & Systems Engineering, University of Sheffield, Sheffield, UK

Correspondence should be addressed to Anderson L. O. Cavalcanti; rb.nrfu.acd@nosredna

Received 15 April 2018; Revised 24 July 2018; Accepted 17 September 2018; Published 3 October 2018

Academic Editor: Mario Russo

Copyright © 2018 Anderson L. O. Cavalcanti et al. 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

This paper deals with the identification of two-time scale linear dynamic systems, which are an important class of multiscale systems. Classical identification processes may fail to yield accurate parameters for systems of this class and, for this reason, the authors propose two different techniques to estimate the system parameters. The first technique utilizes two prefilters that are iteratively tuned. The second one considers wavelet filters that are tuned based on the results of the first iterative algorithm. Identification and analysis results for a dynamical aircraft model are shown to demonstrate the algorithm’s performance.

#### 1. Introduction

The development of system models is an essential task to any branch of science. Being a complex task, the construction of a model based on observations of the modelled system is most often developed from a single view or observation scale. Taking the example of the human body, models are developed considering scales of observation on the level of organs, tissues, cells, and molecules. The human body system works from the harmonious integration of subsystems observed on these various levels. However, the construction of a model that integrates all these scales of observation is not a trivial task. As exemplified with the human body system, complex dynamic systems can be modelled per scale of observation. The construction of several models that consider their various scales of observation and subsequent integration is what is called multiscale modelling [1].

Complex systems are characterized by a hierarchical multiscale nature with respect to not only space but also time [1]. Considering this nature of the systems, methodologies of study and modelling are in demand. Any multiscale methodology must consider the following issues: correlation between phenomena at different scales, trade-off between different dominant mechanisms, coupling between spatial and temporal structural changes, and critical phenomena occurring in complex systems.

Two-time scaled systems (TTSS) are a particular and important case of multiscale systems because several physical systems such as batteries [2], aircraft longitudinal dynamical model [3], and thermal building models [4] present two-scale behaviour. In this particular case, the system has two well defined time scales (one slow and the other fast) and one of the objectives is to identify these dynamics [5, 6]. Spatial scales are not considered in this paper.

Works related to two-time scaled systems have particular importance because traditional prediction error methods (PEMs) tends to overemphasize dynamical modes that affect the overall model response more heavily [2, 7–9].

A two-time scaled system identification procedure presented in [7] is one of the main related works. The authors of that paper show that the classical least squares (LS) method typically fails to give an accurate estimation because the data of a two-time scaled system is scattered in the frequency domain. In that paper, the authors propose a new technique to identify this kind of system and present a second-order two-time scaled system as example. The approach proposed by the authors is compared with the classic least squares method.

In [8, 9], the authors show that measurement noise can result in further model inaccuracies. The application considered therein was a model for a battery system (applied in electric vehicles) that exhibits two-time scales. In batteries, the main two phenomenological effects [2] are charge transfer and diffusion, two of the most important effects in battery dynamics. Charge transfer occurs on a time scale of 0.1–100 Hz, while diffusion occurs from around 1 Hz down to 0.001 Hz. The authors in [2] also compare their technique with the LS method and show that the proposed method produced better estimated models.

Both [2, 7] consider a prefiltering data mechanism to separate the slow and fast dynamics. In both papers, the prefilter design is carried out based on previous knowledge of the system’s characteristics. The cut-off frequencies of the filters are calculated offline and kept fixed during the estimation procedure.

The contribution of this paper is to present techniques that estimate two-time scaled models. The first one is an iterative algorithm that computes cut-off frequencies for classical prefilters and these frequencies are corrected during the iterations. The second one takes the result (cut-off frequencies) of the mentioned iterative algorithm and uses it as parameter of a second estimation stage. This second estimation stage is based on wavelets. A Lockheed F104G aircraft model example is shown to demonstrate the efficiency of the proposed procedures. This paper is organized as follows: the Materials and Methods section summarizes a theoretical background on two-time scaled system identification, presents the algorithm based on classic filters, and shows the framework that considers wavelet filters using the results of the iterative algorithm; the section Application Example and Results shows results obtained for an example, and the last section brings the conclusions.

#### 2. Materials and Methods

In this section, we will briefly define TTSS. The definition is presented in continuous time because of the simplicity of the formulation and its easy extension to discrete time.

##### 2.1. Definition of Two-Time Scaled System

TTSS is a system that can be characterized by two well defined scales, one fast and the other slow. Mathematical descriptions of this kind on system may be considered in time domain (singular perturbation theory) or in frequency domain. In frequency domain, the transfer function is written as a function of a small parameter called scale parameter. Reference [5] gives a complete characterization of two-frequency scale transfer functions. In that paper, its authors describe a transfer function matrix (MIMO) as a function of that may be characterized as two-time scaled given certain conditions. In that paper, the class of systems that can be described by that approach is very wide. In this paper, the mathematical formulation is based on [5], but is more restricted and will be described in the following.

Consider the following model in transfer function form [7, 10]:where models the slow system dynamics scaled by the parameter and corresponds to the fast dynamics. Considering the frequency response of the system, for high frequencies we can assume thatand for low frequencies

To keep these approximations, we will assume that the static gain of is nonzero and that the high-frequency gain of* T*_{s} is neither zero nor infinity. Adopting these assumptions, we shall further consider that* T*_{s} is biproper. From a numerical point of view,* T*_{s} must contain both slow zeros and poles, with some of these still left in* T*_{f}. This is to allow near cancellation in* T*_{s} for higher frequencies.

The described system (TTSS) has a frequency response whose Bode plot is monotonic over a “wide” frequency bandwidth and such that the approximations (2) and (3) are valid. In [5] the mathematical description of TTSS is not restricted to the monotonic case.

##### 2.2. Prefiltering in the Identification Process

As mentioned before, prefilters are a classic solution to avoid inaccurate results when identifying TTSS. The use of prefilters may be illustrated by Figure 1.