Simple Method of a Dynamical System Asymmetric Transients Detection

Transkriptio

1 AUTOMATYKA 2011 Tom 15 Zeszy 3 Pior Osalczyk*, Kaarzyna Maciaszczyk**, Anna Pajor** Simple Mehod of a Dynamical Sysem Asymmeric Transiens Deecion 1. Inroducion In his paper a simple mehod of a dynamical sysem ransien characerisics asymmery deecion is presened. One considers a non-linear muli-inpu single oupu dynamical sysem periodic orbi achieved by an appropriae inpu signals or non-zero iniial condiions. The sysem asymmery is modelled by an asymmeric saic sauraion elemen. Adding an addiional sae one can easily check he rajecory asymmery. The paper is organised as follows. Firs a dynamical sysem and asymmerical sauraion elemen descripion is given. In Secions 3 and 4 a linear second order oscillaor wih asymmerical sauraion is analysed. Invesigaions lead o he sysem asymmery measure which is applied o he analyse of a voice signal Mahemaical preliminaries One considers a muli-inpu one oupu (in general) (MISO) coninuous-ime non- -linear dynamical sysem. I can be described by a se of he firs-order non-linear differenial equaions [3] f x& = x, u, and a se of non-linear algebraic oupu equaions g x u y =,, where: u() he sysem inpu vecor, x() he sysem sae vecor, y() he sysem oupu signal, f [x(), u(), ] he sae funcion, g[x(), u(), ] he oupu funcion. * Compuer Engineering Deparmen, Technical Universiy of Lodz, Poland ** Deparmen of Oolaryngology, Medical Universiy of Lodz, Barlicki Universiy Hospial, Lodz, Poland (1) (2) 455

2 456 Pior Osalczyk, Kaarzyna Maciaszczyk, Anna Pajor A block diagram of a sysem is presened in Figure 1. Noe ha pahs of he vecors u() and x() are depiced by wide whereas he oupu signal by narrow one, respecively. One should menion ha formulae (1) and (2) can describe open and closed-loop dynamical sysems. Moreover, i is mainly assumed ha he descripion concerns a proper dynamic sysem operaion. u() f(x, u, ) x() g(x, u, ) y() Fig. 1. Block diagram of he dynamic sysem Any working device may be parly or even oally subjeced o differen ype defecs. Here one can menion dangerous asymmery in elecronic devices such as operaional and acousic amplifiers as well as generaors [3]. In roaing mechanical sysems very undesirable are non-uniformly disribued resisances [3]. Also in physics pleny of symmerical oscillaors are analysed [3]. Every linear ime-invarian dynamical sysem has a symmery propery. Dynamical sysem failures menioned above can be modeled by a saic sauraion elemen characerized by a non-symmerical characerisic. A block diagram of a defeced dynamical sysem is presened in Figure 2. u() f(x, u, ) x() g(x, u, ) v() y() Fig. 2. Block diagram of he dynamic sysem wih non-symmeric sauraion elemen 1.2. Non-symmeric sauraion elemen Now i is assumed ha he dynamical sysem impairmen (failure, lesion) sympoms are modeled by a non-symmeric sauraion elemen conneced o he oupu signal. The elemen considered in his paper is described by a following saic equaion y for v v A = As fm v f A f e m + for v < A s m s where: A s, A s he upper and lower limis of a linear behavior, f m he asymmeric sauraion elemen parameer, v() he inpu signal, y() he oupu signal. The asymmeric sauraion elemen saic characerisics for differen lineariy limis is ploed in Figure 3. I should be noed ha he proposed non-linear elemen possess a coni- s (3)

3 Simple Mehod of a Dynamical Sysem Asymmeric Transiens Deecion 457 nuiy propery which is ypical in pracice. Realise ha commonly considered sauraion elemens wih sharp lineariy/non-lineariy cross pracically doesn exis. I means ha for he considered sauraion elemen is firs order derivaive of y() wih respec o v() 0 for v As dy = As fm v dv f f m me for v < As (4) is a coninuous funcion. Plo of (4) for differen sauraion levels is given in Figure 4. Fig. 3. Saic characerisics of he non-symmeric sauraion elemen dy() dv() Fig. 4. Characerisic (4) of he non-symmeric sauraion elemen

4 458 Pior Osalczyk, Kaarzyna Maciaszczyk, Anna Pajor 2. Asymmery marker To deec dynamical asymmery behavior of he sysem one inroduces is addiional sae by connecing an ideal inegraor. The inegraion can be also performed by numerical inegraion of he oupu signal. The shape of his new addiional sae will indicae a sysem non symmery level. Now one assumes ha he sysem descripion (1) is ransformed o he form where he higher sae is a firs-order derivaive wih respec o ime of he previous one. Hence a sae equaion akes a form L n+ L m L L x& 1 f1 x1, x2,, x 1, u1, u2,, u, x& 2 f2 x1, x2,, xn+ 1, u1, u2,, um, M = M x& n fn x1, x2, L, xn+ 1, u1, u2, L, um, x n+ 1 & fn+ 1 x1, x2, L, xn+ 1, u1, u2, L, um, x2 x3 = M xn fn+ 1 x1, x2,, xn+ 1, u1, u2,, um, L L = (5) The oupu equaion is This means ha = [ ] y = τ τ+ ( 0) x x d x x1 x2 L M (6) xn x n+ 1 and he shape of his funcion can be reaed as a dynamic sysem asymmery marker. One should menion ha x 3 dx2 = d (7) (8)

5 Simple Mehod of a Dynamical Sysem Asymmeric Transiens Deecion 459 The saes defined above are elemens of he so called phase space [5, 6]. The firs hree may be ploed is 3D subspace which reveals he dynamical sysem asymmery propery. 3. Linear oscillaor analysis Now one considers a linear, ime invarian second order sysem wih addiional inegraor. I is described as follows x& 1 x2 x& = x 2 3 x& 3 a0x3 + a0u x1 y = [ 0 1 0] x2 x 3 (9) (10) where a 0 he sysem parameer. The plos of x 1 (), x 2 (), x 3 () in a seady-sae evaluaed numerically for u() = 0 and non-zero iniial condiions are presened in Figure 5. Fig. 5. Transien characerisics of he second order sysem (9) (10) Appropriae 3D plo is given in Figure 6.

6 460 Pior Osalczyk, Kaarzyna Maciaszczyk, Anna Pajor Fig. 6. 3D plo of x 1 (), x 2 (), x 3 () Now a sysem faul characerised by a saic asymmery modelled by he sauraion elemen is considered. Analogous ransiens are presened in Figures 7 and 8a, b, respecively. Fig. 7. Transien characerisics of he second order sysem (9) (10) wih asymmery

7 Simple Mehod of a Dynamical Sysem Asymmeric Transiens Deecion 461 a) b) x 2 () Fig. 8. 3D plo of x 1 (), x 2 (), x 3 () wih asymmery (a); anoher poin of view (b) Figure 9 shows he same ransiens as hose which were presened in Figure 7 bu in wider ime range. Here wo black lines define an angle represening an asymmery level.

8 462 Pior Osalczyk, Kaarzyna Maciaszczyk, Anna Pajor Fig. 9. Non-symmeric ransien characerisics of he second order sysem (9) (10) wih wo black lines revealing an angle beween x 2 () and x 1 () 4. Signal asymmery measure Now one defines wo funcions relaed o he saes x 2 () and x 1 () 1 x1 = x1 d τ τ 0 1 x1 x2 = x2 d τ τ= 0 (11) Definiion 1 An angle describing a dynamical sysem asymmery level is an angle beween x1 and x2 for > NT where T denoes an oscillaion period and N is a number of periods afer which sysem oscillaions achieves a seady-sae α= { x, x } 1 2 > NT (12) I is well known ha sauraion inroduces higher frequencies. Is differeniaion leads o a periodic funcion wih high ampliudes. Hence one can define anoher asymmery measure. I is formulaed in a Definiion 2.

9 Simple Mehod of a Dynamical Sysem Asymmeric Transiens Deecion 463 Definiion 2 A measure of high frequency componens in a non-symmerical periodic signal is a raio of maximal ampliudes of x 2 () and x 3 () max x > NT β= max x > NT 3 2 (13) 5. Voice signal analysis Voice, as he carrier of informaion, consiues an imporan elemen of social communicaion. I is an acousic phenomenon produced in larynx due o synergy of aerodynamic and psychoneural mechanisms. Three anaomical and physiological srucures play a significan role in voice producion: subgloal air reservoir wihin he lower respiraory rac (lungs, bronchi, rachea), which issues a high-pressure airsream; acousic energy generaor in larynx (glois and vocal folds), producing laryngeal one (fundamenal frequency); resonanse chambers (laryngeal vesibule, pharynx, nasal caviy, oral caviy) which form vocal imbre and audiory elemens of speech ha is phonemes [9, 11]. Glois is a par of inermediae caviy of larynx and is buil of wo vocal folds. A vocal fold consiss of mucous membrane, vocalis muscle (cover-body complex) and has a complex, mulilayer srucure. According o he myoelasic-aerodynamic heory vibraions of vocal folds depend on compeiion of he forces which clench (increasing subgloal pressure) and close he glois (conracions of he adducor muscles of vocal folds, elasic forces and viscosiy of issues, he Bernoulli effec). Vocal folds movemens during phonaion occur in hree planes: verical (opening of glois upwards), horizonal (backwards) and sagial (wave moions of vocal fold mucosa along is medial surface) [8, 11, 12]. Voice disurbances (dysphonia) occur as a resul of moor discoordinaion in he vocal organ. I leads o irregulariies of phonaory vibraions wih he lack of he phase of full gloal closure. Proper, laminar flow of he air urns ino he urbulen one hus causing murmurs perceived as hoarseness. We disinguish organic dysphonia resuling from primary organic lesion of larynx and funcional dysphonia conneced wih funcional disorders wihou angible organ lesion (hiper- and hipofuncional dysphonia). The mos frequen causes of organic dysphonia are acue and chronic laryngiis, cys, polyp of vocal fold, neoplasm of larynx, neurological, hormonal or posraumaic movemen disorders of vocal fold and laryngeal disurbances in he course of gasroesophageal reflux. Funcional dysphonia develops as a resul of incorrec voice producion (phonoponosis), psychogenic disorders (phononeurosis) or phono-respiraory-ariculaive discoordinaion (phonasenia). I ofen occurs in professional voice users like eachers, lawyers or acors. Funcional dysphonia, due o vocal abuse, can induce secondary organic lesions (vocal nodules) [9, 11].

10 464 Pior Osalczyk, Kaarzyna Maciaszczyk, Anna Pajor Complex voice evaluaion, according o suggesions of he Commiee on Phoniarics of he European Laryngological Sociey, includes audiory percepual esimaion, videolaryngosroboscopy, acousic analysis, aerodynamic assessmen (maximum phonaion ime) and subjecive voice evaluaion done by paiens hemselves (e.g. according o a quesionnaire deermining voice handicap, Voice Handicap Index VHI) [2]. In an experimen a long-lasing vowel aaa a (in Polish) was digially regisered and numerically processed. In Figures 10, 11 and 12 a voice signal, is firs order derivaive and an inegral are presened [2, 4, 7, 10]. One can easily find signal asymmery and calculae an angle (13). Finally in Figure 13 he 3D plo of x 1 (), x 2 (), x 3 () is presened. Rys. 10. Plo of a digially regisered long-lasing vowel aaa a (in Polish) x 3 () x 2 () Rys. 11. Plo of a numerically evaluaed firs-order derivaive of a digially regisered vowel aaa a (in Polish)

11 Simple Mehod of a Dynamical Sysem Asymmeric Transiens Deecion 465 Rys. 12. Plo of a numerically evaluaed inegral of a digially regisered vowel aaa a (in Polish) x 3 () x 1 () x 2 () x 1 () Fig D plo of x 1 (), x 2 (), x 3 () of a vowel aaa a 6. Final conclusions The voice analysis show applicabiliy of he proposed mehod o signal asymmery deecion. The shape of 3D orbi may serve as an indicaor of sysem asymmery and higher harmonics conens. The angle defined by Formula (12) indicaes an asymmery level.

12 466 Pior Osalczyk, Kaarzyna Maciaszczyk, Anna Pajor In fuure works a relaion beween he angle (12) and he healh sae will be esablished. One may expec ha he angle may vary due o a herapy effecs. The las saemen should be proved by experimens on numerous samples. References [1] Box G.E.P., Jenkins G.M., Time Series Analysis forecasing and conrol. Holden Day, San Francisco [2] Dejonckere P.H., Bradley P., Clemene P., Cornu G., Crevier-Buchman L., Friedrich G. van De Heyning P., Remacle M., Woisard V., A. basic proocol for funcional assessmen of voice pahology, especially for invesigaing he efficacy of (phonosurgical) reamens and evaluaing new assessmen echniques. Guideline elaboraed by he Commiee on Phoniarics of he European Laryngological Sociey (ELS). Eur. Arch. Oorhinolaryngol., 2001, 258(2), [3] Fossard A.J., Modeling and esimaion. Chapman & Hall, London [4] Ifeachor E.C., Digial Signal Processing. A Pracical Approach. Addison Wesley, Edinburgh Gae, [5] Kaczorek T., Teoria serowania i sysemów. Wydawnicwo Naukowe PWN, Warszawa [6] Kailah T., Linear Sysems. Prenice-Hall, Englewood Cliffs, New York [7] Owen M., Pracical signal processing. Cambridge Universiy Press, Cambridge [8] K³aczyñski M., Zjawiska wibroakusyczne w kanale g³osowym cz³owieka. Disseraion, AGH, Kraków [9] Obrêbowski A. (red.), Narz¹d g³osu i jego znaczenie w komunikacji spo³ecznej, wyd. 1. Wydawnicwo Naukowe Uniwersyeu Medycznego im. Karola Marcinkowskiego w Poznaniu, Poznañ 2008 (in Polish). [10] Proakis J.G., Monolakis D.G., Digial signal processing. Principles, Algorihms, and Applicaions, Prenica Hall, Upper Saddle River, [11] Rubin J.S., Saaloff R.T., Korovin G.S. (red.), Diagnosis and reamen of voice disorders, 3rd ed. Plural Publishing Inc., San Diego Oxford [12] Œwidziñski P., Przydanoœæ analizy akusycznej w diagnosyce zaburzeñ g³osu. Usefulness of acousic analysis in diagnosics of voice disorders. Habiliaion hesis, Poznañ 1998 (in Polish).