Spherical Families of Polynomials: A Graphical Approach to Robust Stability Analysis

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1 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 Sphercal Famle of Polyomal: A Graphcal Approach to Robut Stablty Aaly Radek Matušů Abtract Th paper teded to preet ot o commo ad freuetly ued approach to the defto of ucertaty boudg et for ytem wth parametrc ucertaty ad related tool for robut tablty aaly. More pecfcally, the work deal wth phercal famle of polyomal. The et of llutratve example demotrate a eay-to-ue graphcal method of robut tablty tetg baed o the combato of the value et cocept ad the zero excluo codto by mea of the Polyomal Toolbox for Matlab. Keyword Sphercal Ucertaty, Weghted Eucldea Norm, Robut Stablty Aaly, Value Set Cocept, Zero Excluo Codto. R I. INTRODUCTION OBUSTNESS of cotrol ytem repreet attractve reearch topc wth a coutle umber of real lfe applcato [] [5]. Parametrc ucertaty commoly ued tool for the decrpto of real plat a t allow ug relatvely mple ad atural mathematcal model for procee whch behavor ca be much more complcated. The tructure (.e. order) of the model wth parametrc ucertaty codered to be fxed, but t parameter ca le wth gve boud. Wth th cotrbuto, thee boud are gog to be aumed a ot o freuetly appled way. The typcal, motly ued ad aturally compreheble approach aume the boud the hape of a box. Here, the alteratve approach, whch ue the boud the hape of a phere (ellpod), gog to be tuded. The cetfc lterature cota much more work related to the clacal box ucertate tha to the phercal oe. However, ome bac formato, a well a poble exteo ad varou applcato, ca be foud e.g. [6] []. Th paper focued o polyomal wth parametrc ucertaty ad phercal ucertaty boudg et. More pecfcally t deal wth the decrpto of a phercal polyomal famly ad wth tool for aaly of t robut tablty. Specal atteto pad to very uveral graphcal tool baed o the combato of the value et cocept ad the The work wa upported by the Mtry of Educato, Youth ad Sport of the Czech Republc wth the Natoal Sutaablty Programme project No. LO33 (MSMT-7778/4). Th atace gratefully ackowledged. Radek Matušů wth the Cetre for Securty, Iformato ad Advaced Techologe (CEBIA Tech), Faculty of Appled Iformatc, Toma Bata Uverty Zlí, ám. T. G. Maaryka 5555, 76 Zlí, Czech Republc (emal: rmatuu@fa.utb.cz). zero excluo codto [6]. The decrbed dea are followed by the et of llutratve example upported by plot from the Polyomal Toolbox for Matlab [], []. The paper the exteded vero of the prevouly publhed coferece cotrbuto [], [3]. The paper orgazed a follow. I Secto, bac otato ad theoretcal backgroud of ytem wth parametrc ucertaty ad ucertaty boudg et are provded. The Secto 3 the focued o the decrpto of phercal polyomal famle. Next, tool for robut tablty aaly wth epecal empha o the value et cocept ad the zero excluo codto are how Secto 4. The followg exteve Secto 5 preet three llutratve example of practcal robut tablty vetgato for elected phercal polyomal famle. Ad fally, Secto 6 offer ome cocluo remark. II. PARAMETRIC UNCERTAINTY AND UNCERTAINTY BOUNDING SET Geerally, the ytem wth parametrc ucertaty ca be decrbed through a vector of real ucerta parameter (ofte called jut ucertaty). The cotuou-tme ucerta polyomal, whch a typcal object of reearcher or egeer teret, ca be wrtte the form: p(, ) = ρ ( ) () = where ρ are coeffcet fucto. The, o-called famly of polyomal combe together the tructure of ucerta polyomal gve by () wth the ucertaty boudg et Q. Therefore, the famly of polyomal ca be deoted a: { (, ): } P = p Q () The ucertaty boudg et Q uually gve advace, typcally by uer reuremet. It uppoed a a ball a approprate orm. The mot freuetly ued cae utlze L orm: = max (3) ISSN:

2 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 whch mea that a ball th orm a box. Practcally, the box defed by the compoet,.e. by the real terval whch ca the ucerta parameter le wth. Aother approach employ L (Eucldea) orm: = (4) = or more geerally the weghted Eucldea orm: T = W (5) k where R ad W a potve defte ymmetrc matrx (weghtg matrx) of ze k k. Such defto of Q mea that a ball the orm ca be referred a a phere, or more geerally a a ellpod. Uder aumpto of r ad k R, the ellpod ( expreed by mea of: T ( ) ( ) W r k R ) whch cetered at ca be (6) or euvaletly by: (7) r,w The ellpod ca be ealy vualzed two-dmeoal pace ( k = ) for: may egeerg problem, the real ucerta phycal parameter are depedet of each other ad thu Q hould be a box aturally. However, accordg to [6], the ellpod could be ueful ad jutfable uder mprece decrpto of the ucertaty boud,.e. f actual Q located betwee ome mmum ad maxmum ad a utable ellpod ca terpolate them. The choce hould repect alo avalable tool for olvg the pecfc problem. Bede, the mathematcal model obtaed o the ba of phycal law uually have Q the hape of a box, but the detfcato method motly lead to the ellpod [4]. III. SPHERICAL POLYNOMIAL FAMILY The famly of polyomal gve by () called phercal oe [6] f p (, ) ha a depedet ucertaty tructure (all coeffcet of the polyomal are depedet o each other) ad Q a ellpod. I fact, oe ca work wth two bac repreetato of phercal polyomal famle. The frt type aume that polyomal cetered at zero: p(, ) = (9) = r where W a potve defte ymmetrc matrx, the omal ad r mea the radu of ucertaty. I the ecod repreetato, Q codered to be cetered at zero: W w = w (8) p(, ) = p ( ) + () r = a t how Fg. [6]. + r w r w r w Fg. A ellpod defed by weghted Eucldea orm + r w where moreover p () = p (, ). A a example, uppoe a phercal polyomal famly: ( ) ( ) ( ) p(, ) = W = 3 whch ca be cetered o the vector: () = (.5,.5,.5) () The, the reultg phercal polyomal famly, euvalet to (), ca be wrtte a: A deco o what type of orm hould be ued for ucertaty boudg et Q deped o everal factor. I ISSN:

3 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 p(, ) = + + W = 3 (3) Neverthele, the very uveral techue baed o the value et cocept ad the zero excluo codto, whch decrbed [6], applcable alo to the phercal polyomal famle. The value et at each freuecy ω for a phercal polyomal famly () uppoed the form: IV. ROBUST STABILITY ANALYSIS Obvouly, the mot mportat feature of all cotrol crcut ther tablty. Uder codto of parametrc ucertaty, th term ca be expaded to robut tablty, whch mea that the whole famly of cloed-loop cotrol ytem mut rema table for all poble allowed perturbato of ytem parameter. From the practcal tetg pot of vew, we are tereted the robut tablty of the famly of cloed-loop charactertc polyomal the form (). Th famly robutly table f ad oly f p (, ) table for all Q,.e. all root of p (, ) mut be located the trct left half of the complex plae for all Q. There are may reult for robut tablty aaly of ytem wth parametrc ucertaty for Q a hape of a box. Ther choce deped maly o the complexty of the tructure of vetgated polyomal (or ytem). Doubtle, the mot famou tool the Khartoov theorem [5] whch utable for vetgato of the robut tablty of terval polyomal (wth depedet ucertaty tructure). Moreover, everal modfcato ad geeralzato of clacal Khartoov theorem are alo avalable the lterature [6], [6]. Amog other kow tool t belog e.g. the edge theorem, the thrty-two edge theorem, the xtee plat theorem, the mappg theorem, etc. [6]. Furthermore, t ext a graphcal method whch applcable for wde rage of robut tablty aaly problem (from the mplet to the very complcated ucertaty tructure, for varou tablty rego, etc.). Th techue combe the value et cocept wth the zero excluo codto [6], [7]. Robut tablty aaly for ytem affected by parametrc ucertaty for the cae of Q a hape of a ellpod alo relatvely well developed ad there are everal method avalable. The Soh-Barger-Dabke theorem [8], [6] repreet the aalogcal tool to Khartoov theorem for phercal polyomal famle. Furthermore, exteo are provded by the theorem of Barmh ad Tempo [9], [6] baed o the dea of the pectral et ad the theorem of Berack, Hwag ad Bhattacharyya [], [6] whch olve the robut tablty for cloed-loop ytem wth affe lear ucertaty tructure (e.g. a phercal plat famly feedback coecto wth a fxed cotroller). Bede, well-kow Typk-Polyak fucto [] ca be ued for robut tablty tetg or actually for computato of robute marg uder phercal ucertaty. I fact, the phercal vero of Typk-Polyak crtero related to the reult gve by Soh-Berger-Dabke theorem [6]. p(, ) = p ( ) + ( ) (, ) deg p( ) = p = p = a r = (4) gve [6], [] by a ellpe cetered at omal p ( jω ), wth the major ax ( the real drecto) havg the legth: (5) eve R = r w ω ad wth mor ax ( the magary drecto) havg the legth: (6) odd I = r w ω where W a weghtg matrx: (,,, ) W = dag w w w (7) Moreover, for the pecal degeerate cae of ω =, the value et jut the real terval: p( j, Q) = a r, a + r (8) The practcal vualzato of the ellpodal value et ca be coveetly performed by mea of the Polyomal Toolbox.5 [], [], [] by ug the pherplot commad. The, the zero excluo codto ca be appled for tetg robut tablty the followg way: The phercal polyomal famly () wth varat degree ad at leat oe table member (e.g. omal polyomal) robutly table f ad oly f the complex plae org excluded from the value et p( jω, Q) at all freuece ω,.e. the phercal polyomal famly robutly table f ad oly f: p( jω, Q) ω (9) Geerally, the detaled decrpto, proof ad example of the zero excluo prcple applcato ca be foud [6] or for tace [4], [7]. ISSN:

4 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 A. Example V. ILLUSTRATIVE EXAMPLES Frt, uppoe the phercal polyomal famly defed by the ucerta polyomal: (, ) ( 4) ( 3.5 3) (.5 ) (.5 ) (.5 ) p 4 3 = ad by the ucertaty boudg et:.5 W.e.: = dag ( 5,4,3,,) 3 4 () () () The polyomal () ca be ealy expreed the form (4) a: p = (3) (, ) The omal polyomal table ad o the famly fulfll the codto of at leat oe table member. The value et for the rage of freuece from to 3 wth tep. wa obtaed wth the atace of the Polyomal Toolbox.5 for Matlab ad t route pherplot [], []. They are plotted Fg.. Imagary Ax Real Ax Fg. The value et for the famly (), () The zoomed vero of the ame value et, depcted Fg. 3, provde better vew of the eghborhood of the complex plae org whch crtcal area for deco o robut ()tablty. Imagary Ax Real Ax Fg. 3 The value et for the famly (), () a detaled vew ear the pot [, j] A ca be oberved, the zero pot cluded the value et whch mea that the phercal polyomal famly (), () ot robutly table. I other word, ot all member of the precrbed famly are table. The example of robutly table cae ca be llutrated e.g. jut by ug the arrower ucertaty boudg et:. W = dag ( 5,4,3,,) (4) The full overvew of the value et for the ame rage of freuece a the prevou plot ca be ee Fg. 4 ad the zoomed vero Fg. 5. Obvouly, the famly ha a table member ad the value et do ot clude the org of the complex plae ad coeuetly the famly (), (4) robutly table. Imagary Ax Real Ax Fg. 4 The value et for the famly (), (4) ISSN:

5 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 Imagary Ax Real Ax Fg. 5 The value et for the famly (), (4) a detaled vew ear the pot [, j] B. Example Now, aume aother phercal famly of polyomal gve by the ucerta polyomal of teth order: (, ) ( ) ( 8 9 ) ( 65 8 ) ( 5 7) ( 68 6) ( 5 5) 4 3 ( 68 4) ( 5 3) ( 65 ) ( 8 ) ( ) 9 8 p = ad by the correpodg ucertaty boudg et: (5) = ( ) (6).3; W dag,3,,,3,,,4,3,, The omal polyomal table whch mea that the codto of at leat oe table member fulflled. The value et for the rage of freuece from to wth tep. are plotted Fg. 6. Sce the polyomal (5) of the teth order, the value et from Fg. 6 uccevely go through te uadrat. Four varouly zoomed vero of the full Fg. 6 are how Fg. 7-. Imagary Ax -5 - x Real Ax x 6 Fg. 7 The value et for the famly (5), (6) a lttle zoomed vew Imagary Ax x Real Ax - x 4 Fg. 8 The value et for the famly (5), (6) a moderately zoomed vew x 9 5 Imagary Ax Imagary Ax Real Ax x 9 Fg. 6 The value et for the famly (5), (6) Real Ax Fg. 9 The value et for the famly (5), (6) a more zoomed vew ISSN:

6 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 Imagary Ax Real Ax Fg. The value et for the famly (5), (6) a detaled vew ear the pot [, j] The et of Fg. 6- reveal that the zero pot excluded from the value et. Sce the famly ha a table member ad the value et do ot clude the org, the famly (5), (6) robutly table. C. Example 3 The lat example teded to demotrate the mportace of at leat oe table member precodto fulfllmet, becaue f t gored t ca lead to the wrog reult. Aume the phercal polyomal famly: (, ) ( 4) ( 3) ( ) ( ) ( ) 4 3 = p W = dag ( 5,4,3,,) (7) The correpodg value et for the rage of freuece from to 3 wth tep. are depcted Fg. wth cloer look ear the org Fg.. Imagary Ax Real Ax Fg. The value et for the famly (7) Imagary Ax Real Ax Fg. The value et for the famly (7) a detaled vew ear the pot [, j] A the complex plae org obvouly excluded from the value et, t could (wrogly) dcate the robut tablty of the famly. However, the famly doe ot have ay table member ad thu the zero excluo codto ot fulflled actually. I fact, all member of the famly are utable whch the reao why the tablty border ot croed at all ad why the zero pot ot cluded. VI. CONCLUSION The paper ha bee amed to a alteratve boudg of ucerta parameter ytem wth parametrc ucertaty,.e. the ma object of teret ha bee the phercal polyomal famly ad t robut tablty aaly. The bac theoretcal decrpto have bee accompaed by the et of mple llutratve example upported by the Polyomal Toolbox for Matlab. REFERENCES [] A. Derrar, A. Nacer, D. Ghouraf, Robut off-le PSS automated cotrol deg baed H - loop hapg optmzato, Iteratoal Joural of Crcut, Sytem ad Sgal Proceg, vol. 9, 5, pp [] R. Matušů, Robut tablzato of terval plat by mea of two feedback cotroller, Iteratoal Joural of Crcut, Sytem ad Sgal Proceg, vol. 9, 5, pp [3] R. Matušů, Robut Stablty Aaly of Dcrete-Tme Sytem wth Parametrc Ucertaty: A Graphcal Approach, Iteratoal Joural of Crcut, Sytem ad Sgal Proceg, vol. 8, 4, pp [4] S. A. E. M. Ardjou, M. Abd, A. G. Aaou, ad A. Nacer, A robut fuzzy ldg mode cotrol appled to the double fed ducto mache, Iteratoal Joural of Crcut, Sytem ad Sgal Proceg, vol. 5, o. 4,, pp [5] J. Ezze, F. Tedeco, H Approach Cotrol for Regulato of Actve Car Supeo, Iteratoal Joural of Mathematcal Model ad Method Appled Scece, vol. 3, o. 3, 9, pp [6] B. R. Barmh, New Tool for Robute of Lear Sytem, Macmlla, New York, USA, 994. [7] A. Te, A. Vco, F. Vllore, Robut Stablty of Sphercal Plat wth Utructured Ucertaty, Proceedg of the Amerca Cotrol Coferece, Seattle, Wahgto, USA, 995. ISSN:

7 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume, 6 [8] B. T. Polyak, P. S. Shcherbakov, Radom Sphercal Ucertaty Etmato ad Robute, Proceedg of the 39th IEEE Coferece o Deco ad Cotrol, Sydey, Autrala,. [9] J. Che, S.-I. Nculecu, P. Fu, Robut Stablty of Qua-Polyomal: Freuecy-Sweepg Codto ad Vertex Tet, IEEE Traacto o Automatc Cotrol, vol. 53, o. 5, 8, pp [] Z. Hurák, M. Šebek, New Tool for Sphercal Ucerta Sytem Polyomal Toolbox for Matlab, Proceedg of the Techcal Computg Prague, Prague, Czech Republc,. [] PolyX: The Polyomal Toolbox, [ole], Avalable from URL: [] R. Matušů, Famle of phercal polyomal: Decrpto ad robut tablty aaly, Proceedg of the 8th Iteratoal Coferece o Sytem, Sator, Greece, 4, pp [3] R. Matušů, R. Prokop, Robut Stablty Aaly for Famle of Sphercal Polyomal, Advace Itellget Sytem ad Computg (Vol. 348) Proceedg of the 4th Computer Scece Ole Coferece 5 (CSOC5), Vol : Itellget Sytem Cyberetc ad Automato Theory, Sprger Iteratoal Publhg Swtzerlad, 5, pp [4] M. Šebek, Robutí řízeí, PDF lde for coure Robut Cotrol, ČVUT Prague,. (I Czech). [5] V. L. Khartoov, Aymptotc tablty of a eulbrum poto of a famly of ytem of lear dfferetal euato, Dfferetal'ye Uraveya, vol. 4, 978, pp [6] S. P. Bhattacharyya, H. Chapellat, L. H. Keel, Robut cotrol: The parametrc approach, Pretce Hall, Eglewood Clff, New Jerey, USA, 995. [7] R. Matušů, R. Prokop, Graphcal aaly of robut tablty for ytem wth parametrc ucertaty: a overvew, Traacto of the Ittute of Meauremet ad Cotrol, vol. 33, o.,, pp [8] C. B. Soh, C. S. Berger, K. P. Dabke, O the tablty properte of polyomal wth perturbed coeffcet, IEEE Traacto o Automatc Cotrol, vol. 3, o., 985, pp [9] B. R. Barmh, R. Tempo, O the pectral et for a famly of polyomal, IEEE Traacto o Automatc Cotrol, vol. 36, o., 99, pp. -5. [] R. M. Berack, H. Hwag, S. P. Bhattacharyya, Robut tablty wth tructured real parameter perturbato, IEEE Traacto o Automatc Cotrol, vol. 3, o. 6, 987, pp [] Y. Z. Typk, B. T. Polyak, Freuecy doma crtera for lp-robut tablty of cotuou lear ytem, IEEE Traacto o Automatc Cotrol, vol. 36, o., 99, pp [] PolyX: The Polyomal Toolbox for Matlab Upgrade Iformato for Vero.5, [ole],, Avalable from URL: Radek Matušů wa bor Zlí, Czech Republc 978. Curretly, he a Reearcher at Faculty of Appled Iformatc of Toma Bata Uverty Zlí, Czech Republc. He graduated from Faculty of Techology of the ame uverty Automato ad Cotrol Egeerg ad he receved a Ph.D. Techcal Cyberetc from Faculty of Appled Iformatc 7. The ma feld of h profeoal teret clude robut cotrol, ucerta ytem, PID cotrol, fractoal order ytem, ad algebrac method cotrol deg. ISSN: