11 Output Only Functional Series Time Dependent AutoRegressive. . . 113 deterministic basis functions belonging to specific functional subspaces for estimation of model parameters [28]. Since TV parameters often change in a non-random way, deterministic methods are suitable for capturing deterministic parameter evolution. Based on our knowledge upon the survey of the related literature published since now, it can be said that there is no research work done by non-stationary output-only FS-TARMA method in the field of tool wear estimation. Also, most of the studies have focused on mathematical development of signal processing/modelling techniques and a method that considers the physics of tool/holder assembly during tool wear still needs to be developed. The aim of this paper is to develop a tool wear estimation algorithm in turning process that employs FS-TARMA method for identification of tool/holder system dynamics based on the vibration output. To this aim, wear sensitive features extracted using FS-TARMA models of signals will be related to major flank wear values. By the use of the foresaid method, higher accuracy is obtained in comparison to stationary signal modelling methods. 11.2 Functional Series TARMA Models of Signals and Their Estimation Estimation of non-stationary tool vibration signals based on Functional Series TARMA models is considered in this section. 11.2.1 FS-TARMA Models An FS-TARMA.na;nc/Œpa;pc;ps model, with na, nc indicating AR and MA orders respectively, pa, pc their corresponding functional basis dimensionalities, and ps dimensionality of the respective innovations variance, can be expressed in the following form [28]: x Œt C na XiD1 ai Œt :x Œt i DwŒt C nc XiD1 ci Œt :wŒt i ; wŒt NID 0; 2 wŒt ; t D1; : : : ;N; (11.1) inwhichx[t] designates the estimated non-stationary signal, w[t] an innovations sequence with zero mean and time-dependent variance 2 w[t]. Supposing that the system input is unobservable, it can be considered as to be a zero mean innovations (uncorrelated) sequence wŒt .f Œt wŒt / with time-dependent variance. 2 w[t]. ai[t] andci[t] represent AR and MA timedependent parameters respectively, that as well as the innovations variance 2 w[t] can be estimated by the related functional subspaces with respective bases: FAR ,˚Gba.1/ Œt ;Gba.2/ Œt ; : : : ;Gba.pa/ Œt ; FMA ,˚Gbc.1/ Œt ;Gbc.2/ Œt ; : : : ;Gbc.pc/ Œt ; F 2 w ,˚ Gbs.1/ Œt ;Gbs.2/ Œt ; : : : ;Gbs.ps/ Œt : (11.2) In the above expressions ”F00 designates functional subspace of the indicated quantity and ˚Gj Œt W j D0;1; : : : a set of orthogonal basis functions which are selected from a suitable family (such as Chebyshev, Legendre, polynomial, trigonometric, etc. functions). The indices ba.i/.i D1; : : : ;pa/, bc.i/.i D1; : : : ;pc/ and bs.i/.i D1; : : : ;ps/ designate the specific basis functions of a particular family that are included in each subspace. For an FS-TARMA model, the time-dependent AR and MA parameters and innovations variance, can be expressed based on basis functions as follows [28]: ai Œt , pa XjD1 ai;j:Gba.j/ Œt ; ci Œt , pa XjD1 ci;j:Gbc.j/ Œt ; 2 wŒt , pa XjD1 sj:Gbs.j/ Œt ; (11.3) whereai,j, ci,j andsj represent the AR, MA and innovations variance coefficients of projection, respectively. By the definitions of Eq. (11.3) the model parameters consist of projection coefficients ai,j, ci,j and sj, while a specific model structure, like M is defined by model orders na, nc and the functional subspaces FAR; FMA; F 2 w : M,˚na;nc; FAR; FMA; F 2 w : (11.4)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==