A small-gain-theorem-like approach to nonlinear observability via finite capacity channels*

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A small-gain-theorem-like approach to nonlinear observability via finite capacity channels^*. / Pogromsky, A.; Matveev, A.; Proskurnikov, A.; Fridman, E.

в: IFAC-PapersOnLine, Том 50, № 1, 07.2017, стр. 15397-15402.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

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Pogromsky, A. ; Matveev, A. ; Proskurnikov, A. ; Fridman, E. / A small-gain-theorem-like approach to nonlinear observability via finite capacity channels^*. в: IFAC-PapersOnLine. 2017 ; Том 50, № 1. стр. 15397-15402.

BibTeX

@article{970633d9033d476f9924d20190c0aac4,

title = "A small-gain-theorem-like approach to nonlinear observability via finite capacity channels*",

abstract = "The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe chaotic systems via communication channels with finite data rates, with a focus on minimum data-rates needed for various types of observability. With the objective of developing tractable techniques to estimate these rates, the paper discloses benefits from regard to the operational structure of the system in the case where the system is representable as a feedback interconnection of two subsystems with inputs and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their topological entropy stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive finite upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.",

keywords = "Data rate estimates, Entropy, Nonlinear systems, Observability, Second Lyapunov method",

author = "A. Pogromsky and A. Matveev and A. Proskurnikov and E. Fridman",

note = "Funding Information: A.∗ Pogromsky∗∗,∗∗∗ A. Matveev∗ A. Proskurnikov∗∗∗∗ E. Fridman† ∗ Department of Mathematics and Mechanics, Saint Petersburg University, St. Petersburg, ∗∗ Department of Mathematics and Mechanics, Saint Petersburg University, St. Petersburg, Department of Mathematics and Mechanics, Saint Petersburg University, St. Petersburg, ∗∗ Department of MechanRiucassfiiEan(ge-inmeaeirfii:nagf,imEaitn1d7h1o2v@enyaUhnoiove.crosimty).of Technofiogy, Eindhoven, ∗∗DepartmentofMechanicafiRussiaEngineering(e-maifi: afimat1712@yahoo.com)., Eindhoven University of Technofiogy, Eindhoven, ∗∗ Department of MecThhaenNiceatfhi Eernfigainndese,r(ien-gm, aEiifin:dAh.oPvoegnroUmnsikvye@rsittuye.onffi)T.echnofiogy, Eindhoven, Department of Mechanicafi Engineering, Eindhoven University of Technofiogy, Eindhoven, ∗∗∗ Department of ContTrhoefi SNyesttheemrsfiaanndds,In(ed-umsatriifia:fiAR.Poobgortoicms,skSya@inttu-Pee.ntefi)r.sburg Nationafi Research ∗∗∗DepartmentofContTherofi SystemsNetherfiands,and Industriafi(e-maifi: ARobotics,.PogromskSy@tueaint-Pe.nfi).tersburg Nationafi Research ∗∗∗ DepaUrntmiveenrstiotyf CofoInntfroorfimSaystitoenmsTeacnhdnoInfiodguisetsriMafieRcohbaontiiccssa, nSdaiOntp-Ptiectser(IsTbMurOg )N,aRtuiosnsiaafi.Research Department of Controfi Systems and Industriafi Robotics, Saint-Petersburg Nationafi Research ∗∗∗∗UnDiveefirfstiCtyeonfteIrnffoorrmSaystitoenmTseacnhdnoCfioongtireosfiM, Deecfhifat nUincsivaenrsditOypotficTsec(IhTnMofiOog),y,RDusesfifita,.The University of Information Technofiogies Mechanics and Optics (ITMO), Russia. ∗∗∗∗ Defift Center fNoretShyesrtfeiamnsdsan(ed-mCaoinfit:raonfi,toDne.fpift.1U9n8i2v@eriseietye.oofrgT)e.chnofiogy, Defift, The Defift Center for Systems and Controfi, Defift University of Technofiogy, Defift, The DepartmentofEfiectricafiNetherfiandsEngineering(e-maifi:and Systems,anton.p.1982@ieeeTefi Aviv Univer.orgs)i.ty, Tefi Aviv 69978, Israefi, †† Department of EfiectricafiNetherfiandsEngineering(e-maifi:and Systems,anton.p.1982@ieeeTefi Aviv Univer.orgs)i.ty, Tefi Aviv 69978, Israefi, † Department of Efiectricafi Engin(ee-emrianigfi:aenmdifSiiyas@teemnsg,.tTaeufi.Aacv.iivfi)U. niversity, Tefi Aviv 69978, Israefi, Department of Efiectricafi Engineering and Systems, Tefi Aviv University, Tefi Aviv 69978, Israefi, (e-maifi: emifiia@eng.tau.ac.ifi). (e-maifi: emifiia@eng.tau.ac.ifi). Abstract: The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe Abstract: The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe Abstract: The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe rahtaeostniceesdyesdtefmosr vvairaiocuosmtympuensicoaftoiobnsecrhvaabninleitlys. wWiitthh ftihneitoebdjeactativreatoefs,dweviethlopainfogctursacotanblmeitneicmhunmiqudeasttao-chaotic systems via communication channels with finite data rates, with a focus on minimum data-eastteims nateeedtheedseforratveasr,itohuesptyappeersdoifscolbosseersvbabenileitfyit.sWfriotmh trheegaorbdjetcotitvheoofpedreavteiolonpailnsgtrturcatcutraebloefttehcehsnyisqtueems tion estimate these rates, the paper discloses benefits from regard to the operational structure of the system in estimate these rates, the paper discloses benefits from regard to the operational structure of the system in ahnedcoaustepuwths.eTreotthheisseynsdte,ma niosvreelperesstiemntaatbiolen maseathfoededisbaeclakbionrtaetrecdo,nwnhecictihoins oalfiktweoinsfulabvsoyrstteomthsewcietlhebinrpatuetds and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their oobpsoelrovgaibcialiltyenrtartoepsyasntdayaspbpoeunnddededbaysctohnesdtreulcatyivgerofiwnistewuitphpoeurt bliomuintsd.sTinhdisepisenedxetenntdoefdtohne tdheelasyt.udItieids topological entropy stays bounded as the delay grows without limits. This is extended on the studied shboswernvatbhialtittyhersaetebsouannddsaaprpeeansdyemdpbtoyticcoanllsytrtuigchtitvfeorfianittiemuep-dpeelrayboaunnaldosgionfdtehpeenbdoeunntcionfgtbhaellddeylanya.mIitciss. shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics. {\textcopyright} 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Observability, Nonlinear systems, Entropy, Second Lyapunov method, Data rate estimates Keywords: Observability, Nonlinear systems, Entropy, Second Lyapunov method, Data rate estimates Keywords: Observability, Nonlinear systems, Entropy, Second Lyapunov method, Data rate estimates 1. INTRODUCTION so that, e.g., the exact value of the topological entropy is still 1. INTRODUCTION so that, e.g., the exact value of the topological entropy is still 1. INTRODUCTION so that, e.g., the exact value of the topological entropy is still One of the fundamental issues in the rapidly emerging area uynsktneomws.nTehviesnisfcoornmsoannaynptrwoittohtyrpigicoaroluloswu-ndciommepnusitoabniallitcyhfaaocttisc; One of the fundamental issues in the rapidly emerging area systems. This is consonant with rigorous uncomputability facts; One of the fundamental issues in the rapidly emerging area systems. This is consonant with rigorous uncomputability facts; of control of networked systems is about constraints on come.g., even for piece-wise affine continuous maps and χ ≈ 0, no smuucnhiccaotniostnraainmtsonagretchoemnmetownolyrkmaogdeenltesd.bSaosmedeokneyacaospneccetpstof ppropgroraxmimcaatensgtheinseernatteroapyrawtiiothnaplrencuimsiobnerχi(nKaoifrianni,te20ti0m1e).that munication among the network agents. Some key aspects of program can generate a rational number in a finite time that csuocmhmcuonnisctraatiionntscahreancnoemlmwoitnhlyamlimoditeeldeddbaatasetdraonnsmaicsosinocneprattoef. approximatesthisentropywithprecisionχ(Koiran,2001). such constraints are commonly modeled based on a concept of approximates this entropy with precision χ (Koiran, 2001). communication channel with a limited data transmission rate. Nevertheless, it was recently shown that data rate thresholds Icnomthmisufnriacmateiwonorckh,aannperilmwairtyh ianqliumiriyteids adbaotauttrtahnesmmiisnsiimonalrraattee. Nevertheless, it was recently shown that data rate thresholds communication channel with a limited data transmission rate. Nevertheless, it was recently shown that data rate thresholds In this framework, a primary inquiry is about the minimal rate concerned with observability and introduced in (Matveev and nIneetdheisdftroamacehwieovrek,aadpersiimreadrycoinntqruoilryobijseacbtiovue.t Tthheismminoimmeanl troautes Nevertheless, it was recently shown that data rate thresholds In this framework, a primary inquiry is about the minimal rate concerned with observability and introduced in (Matveev and needed to achieve a desired control objective. This momentous Pogromsky, 2016) can yet be computed in closed form for some tnhereedsehdoltdohacahsiebveeenadaessuirbejdecctonotfrorlecoebnjetcetxivtee.nsTihviessmtuodmieesn;tosuees concernedwithobservabilityandintroducedin(Matveevand threshold has been a subject of recent extensive studies; see of the above nonlinear prototypical systems, e.g., the bounc-threshold has been a subject of recent extensive studies; see of the above nonlinear prototypical systems, e.g., the bounce.g., (Baillieul, 2004; De Persis and Isidori, 2004; Nair et al., ing ball system, H{\'e}non system and logistic and Lozy maps 2e.0g0.,4;(BLaibilelirezuoln,2a0n0d4H;eDsepaPnehras,is20an0d5;IDsiedoPrei,rs2is0,0240;0N5a;iSraevtkailn.,, oftheabovenonlinearprototypicalsystems,e.g.,thebounc-e.g., (Baillieul, 2004; De Persis and Isidori, 2004; Nair et al., ing ball system, He´non system and logistic and Lozy maps 2004; Liberzon and Hespanha, 2005; De Persis, 2005; Savkin, (Matveev and Pogromsky, 2016; Pogromsky and Matveev, 22000064;; LNiabierreztonala.,n2d0H07e;spManahtvae,e2v00an5d; DSeavPkeirns,is2,020090)5a;nSdavlikteinr-, ing ball system, H{\'e}non system and logistic and Lozy maps 2004; Liberzon and Hespanha, 2005; De Persis, 2005; Savkin, (Matveev and Pogromsky, 2016; Pogromsky and Matveev, 2006; Nair et al., 2007; Matveev and Savkin, 2009) and liter-2016a). This is thanks to the novel techniques for upper estima-a2t0u0r6e;thNearierine.t Tahl.i,s2d0a0t7a;-rMateattvhereevshaonldd hSaasvkaipnp,e2a0re0d9)toanbde aliltiekre-(Matveev and Pogromsky, 2016; Pogromsky and Matveev, ature therein. This data-rate threshold has appeared to be alike tion of these thresholds that are elaborated in (Pogromsky et al., aturetherein.Thisdata-ratethresholdhasappearedtobealike tion of these thresholds that are elaborated in (Pogromsky et al., in spirit to the topological entropy (Donarowicz, 2011) of the 2013; Pogromsky and Matveev, 2013, 2011, 2016b; Matveev sinysstpeimritattohtahnedsto,pboultoigsicnaolteanlwtroaypysi(dDeonntiacraolw;aicnzd,t2h0e1s1e)stoufdtihees tionofthesethresholdsthatareelaboratedin(Pogromskyetal., in spirit to the topological entropy (Donarowicz, 2011) of the 2013; Pogromsky and Matveev, 2013, 2011, 2016b; Matveev system at hands, but is not always identical; and these studies and Pogromsky, 2016) and turn off the classic road of the first hsyasvteemintartohdauncdeds, vbaurtioisusnoatnalwogasysoifdtehnitsiceanl;traonpdy t(hNesaeirsteutdaiel.s, 2013; Pogromsky and Matveev, 2013, 2011, 2016b; Matveev have introduced various analogs of this entropy (Nair et al., Lyapunov approach in study of topological entropy and the have introduced various analogs of this entropy (Nair et al., Lyapunov approach in study of topological entropy and the 2004; Savkin, 2006; Colonius and Kawan, 2009, 2011; Kawan, likes towards the second Lyapunov method. 2004; Savkin, 2006; Colonius and Kawan, 2009, 2011; Kawan, likes towards the second Lyapunov method. 2011;a2n0d11P;oHagiharaHgarogmihsakrya,anda2n0d1Nair,N6)a.ir, 2013;2013; ColoniusColonius etet al.,al., 2013;2013; MatvMatveeeevv Theesotbojwecardstivetheof thissecondpaperLyaisputonoaddvmethod.more functionality to thelikestowardsthesecondLyapunovmethod. andPogromsky,2016).andPogromskHagiharay,and2016).Nair,2013;Coloniusetal.,2013;Matveev The objective of this paper is to add more functionality to the andUnlikPogromske linear yplants,, 2016).computation or even fine estimation of The objective of this paper is to add more functionality to the just discussed approach of (Matveev and Pogromsky, 2016; Unlike linear plants, computation or even fine estimation of Pogromsky and Matveev, 2016a) via its further elaboration in a theseethresholdslinear plants,is ancomputationintricate matteror evforennonlinearfine estimationsystemsof Pogromsksituation wherey and Matvthe observeev, 2016a)ed dynamicsvia its furtherresult fromelaborationa feedbackin a ★these thresholds is an intricate matter for nonlinear systems Pogromsky and Matveev, 2016a) via its further elaboration in a ★ snittuearctioonnnwechtieorne tohfe towbosersvuebdsydsytenmams icwsitrhesiunlpt ufrtsomanadfeoeudtpbuactsk. ITMA.OPougnriovmersskityy abcyknGowovleedrngmesenhtisopfaRrtiuaslsisaunppFoerdtedruatriionng ghrisanstta(y07w4i-tUh0t1h)e, interconnectionwhere theof twobservo subsysted dynamicsems withresultinputsfromanda feedbackoutputs. ★A.PogromskyacknowledgeshispartialsupportduringhisstaywiththetheAs.ePotghrroemshskoyldascknisowalendgienstrhiicsapteartmialastuteprpofrotrdunrionnglhiniseastraysywsitthemthse interconnection of two subsystems with inputs and outputs. ITMO university by Government of Russian Federation grant (074-U01), interconnection of two subsystems with inputs and outputs. RTuMssOianuFneivdeerrsaittiyonbPyreGsiodveenrtnGmraenntt No1f4R.Yu3s1si.a1n6.9F2e8d1e-rHatIiIoIn, angdratnhte M(07in4i-sUtr0y1o)f, Such interconnection is very common in engineering practice ITMO university by Government of Russian Federation grant (074-U01), so that certain whole chapters of control theory assume that Russian Federation President Grant N14.Y31.16.9281-HIII, and the Ministry of so thatinterconnectioncertain whole ischaptersvery commonof controlin engineeringtheory assumepracticethat RussianFederationPresidentGrantN14.Y31.16.9281-HIII,andtheMinistryof thoethpalatncteirstaginivewnhoinlethcihsapfotermrs.oAfmcoonntgrotlhethme,ortyhearessuarmeesttuhda-t E,d3u,4c)a.tiAon. ManadtvSeceivenaccekonfowRluesdsgiaensFheisdesruaptipoonrt(pbryojRecStF141.4Z-5201.-3010.004013p1)a,n(Sdetchse. so that certain whole chapters of control theory assume that Education and Science of Russian Federation (project 14.Z50.31.0031), (Secs. the plant is given in this form. Among them, there are stud-S,a3i,n4t)P.eAte.rMsbautrvgeSevtaateckUnnoiwvelersdigtyes(Sheicsss.u2p,5p,o6r)t.bAy.PRrSoFsk1u4rn-2ik1o-v00a0c4k1npowanleddgthees the plant is given in this form. Among them, there are stud-SuapinptoPrteA.otefrMatvRsbFuBrgReeS,vtgaracknotaenUtsn1iwledgesv7e-0rs8i-ty00(7Shis1e5c,s1support.72-,058,6-)0.1byA72.8PRSFraonsdk14-21-00041pu1r7n-i0k8o-v01a2ck6n6o.wledgesandthe iesYeaskofoufboabsoluteavbiscohl,u2te00anda0n,d2robr0o0b2ustu;sMt ssettgaarbbeiitllsiittkyyi,,asee,snede,Re.g.ea.ngt.z((eWWr,ii1llll9ee9mm7ss),,,w1972;19h7e2re; ies of absolute and robust stability, see, e.g. (Willems, 1972; Saint Petersburg State University (Secs. 2,5,6). A. Proskurnikov acknowledges Yakubovich, 2000, 2002; Megretski and Rantzer, 1997), where support of RFBR, grants 17-08-00715,17-08-01728 and 17-08-01266. Yakubovich, 2000, 2002; Megretski and Rantzer, 1997), where support of RFBR, grants 17-08-00715,17-08-01728 and 17-08-01266. Publisher Copyright: {\textcopyright} 2017 Copyright: Copyright 2018 Elsevier B.V., All rights reserved.",

year = "2017",

month = jul,

doi = "10.1016/j.ifacol.2017.08.1864",

language = "English",

volume = "50",

pages = "15397--15402",

journal = "IFAC-PapersOnLine",

issn = "2405-8963",

publisher = "Elsevier",

number = "1",

}

RIS

TY - JOUR

T1 - A small-gain-theorem-like approach to nonlinear observability via finite capacity channels*

AU - Pogromsky, A.

AU - Matveev, A.

AU - Proskurnikov, A.

AU - Fridman, E.

N1 - Funding Information: A.∗ Pogromsky∗∗,∗∗∗ A. Matveev∗ A. Proskurnikov∗∗∗∗ E. Fridman† ∗ Department of Mathematics and Mechanics, Saint Petersburg University, St. Petersburg, ∗∗ Department of Mathematics and Mechanics, Saint Petersburg University, St. Petersburg, Department of Mathematics and Mechanics, Saint Petersburg University, St. Petersburg, ∗∗ Department of MechanRiucassfiiEan(ge-inmeaeirfii:nagf,imEaitn1d7h1o2v@enyaUhnoiove.crosimty).of Technofiogy, Eindhoven, ∗∗DepartmentofMechanicafiRussiaEngineering(e-maifi: afimat1712@yahoo.com)., Eindhoven University of Technofiogy, Eindhoven, ∗∗ Department of MecThhaenNiceatfhi Eernfigainndese,r(ien-gm, aEiifin:dAh.oPvoegnroUmnsikvye@rsittuye.onffi)T.echnofiogy, Eindhoven, Department of Mechanicafi Engineering, Eindhoven University of Technofiogy, Eindhoven, ∗∗∗ Department of ContTrhoefi SNyesttheemrsfiaanndds,In(ed-umsatriifia:fiAR.Poobgortoicms,skSya@inttu-Pee.ntefi)r.sburg Nationafi Research ∗∗∗DepartmentofContTherofi SystemsNetherfiands,and Industriafi(e-maifi: ARobotics,.PogromskSy@tueaint-Pe.nfi).tersburg Nationafi Research ∗∗∗ DepaUrntmiveenrstiotyf CofoInntfroorfimSaystitoenmsTeacnhdnoInfiodguisetsriMafieRcohbaontiiccssa, nSdaiOntp-Ptiectser(IsTbMurOg )N,aRtuiosnsiaafi.Research Department of Controfi Systems and Industriafi Robotics, Saint-Petersburg Nationafi Research ∗∗∗∗UnDiveefirfstiCtyeonfteIrnffoorrmSaystitoenmTseacnhdnoCfioongtireosfiM, Deecfhifat nUincsivaenrsditOypotficTsec(IhTnMofiOog),y,RDusesfifita,.The University of Information Technofiogies Mechanics and Optics (ITMO), Russia. ∗∗∗∗ Defift Center fNoretShyesrtfeiamnsdsan(ed-mCaoinfit:raonfi,toDne.fpift.1U9n8i2v@eriseietye.oofrgT)e.chnofiogy, Defift, The Defift Center for Systems and Controfi, Defift University of Technofiogy, Defift, The DepartmentofEfiectricafiNetherfiandsEngineering(e-maifi:and Systems,anton.p.1982@ieeeTefi Aviv Univer.orgs)i.ty, Tefi Aviv 69978, Israefi, †† Department of EfiectricafiNetherfiandsEngineering(e-maifi:and Systems,anton.p.1982@ieeeTefi Aviv Univer.orgs)i.ty, Tefi Aviv 69978, Israefi, † Department of Efiectricafi Engin(ee-emrianigfi:aenmdifSiiyas@teemnsg,.tTaeufi.Aacv.iivfi)U. niversity, Tefi Aviv 69978, Israefi, Department of Efiectricafi Engineering and Systems, Tefi Aviv University, Tefi Aviv 69978, Israefi, (e-maifi: emifiia@eng.tau.ac.ifi). (e-maifi: emifiia@eng.tau.ac.ifi). Abstract: The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe Abstract: The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe Abstract: The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe rahtaeostniceesdyesdtefmosr vvairaiocuosmtympuensicoaftoiobnsecrhvaabninleitlys. wWiitthh ftihneitoebdjeactativreatoefs,dweviethlopainfogctursacotanblmeitneicmhunmiqudeasttao-chaotic systems via communication channels with finite data rates, with a focus on minimum data-eastteims nateeedtheedseforratveasr,itohuesptyappeersdoifscolbosseersvbabenileitfyit.sWfriotmh trheegaorbdjetcotitvheoofpedreavteiolonpailnsgtrturcatcutraebloefttehcehsnyisqtueems tion estimate these rates, the paper discloses benefits from regard to the operational structure of the system in estimate these rates, the paper discloses benefits from regard to the operational structure of the system in ahnedcoaustepuwths.eTreotthheisseynsdte,ma niosvreelperesstiemntaatbiolen maseathfoededisbaeclakbionrtaetrecdo,nwnhecictihoins oalfiktweoinsfulabvsoyrstteomthsewcietlhebinrpatuetds and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their oobpsoelrovgaibcialiltyenrtartoepsyasntdayaspbpoeunnddededbaysctohnesdtreulcatyivgerofiwnistewuitphpoeurt bliomuintsd.sTinhdisepisenedxetenntdoefdtohne tdheelasyt.udItieids topological entropy stays bounded as the delay grows without limits. This is extended on the studied shboswernvatbhialtittyhersaetebsouannddsaaprpeeansdyemdpbtoyticcoanllsytrtuigchtitvfeorfianittiemuep-dpeelrayboaunnaldosgionfdtehpeenbdoeunntcionfgtbhaellddeylanya.mIitciss. shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Observability, Nonlinear systems, Entropy, Second Lyapunov method, Data rate estimates Keywords: Observability, Nonlinear systems, Entropy, Second Lyapunov method, Data rate estimates Keywords: Observability, Nonlinear systems, Entropy, Second Lyapunov method, Data rate estimates 1. INTRODUCTION so that, e.g., the exact value of the topological entropy is still 1. INTRODUCTION so that, e.g., the exact value of the topological entropy is still 1. INTRODUCTION so that, e.g., the exact value of the topological entropy is still One of the fundamental issues in the rapidly emerging area uynsktneomws.nTehviesnisfcoornmsoannaynptrwoittohtyrpigicoaroluloswu-ndciommepnusitoabniallitcyhfaaocttisc; One of the fundamental issues in the rapidly emerging area systems. This is consonant with rigorous uncomputability facts; One of the fundamental issues in the rapidly emerging area systems. This is consonant with rigorous uncomputability facts; of control of networked systems is about constraints on come.g., even for piece-wise affine continuous maps and χ ≈ 0, no smuucnhiccaotniostnraainmtsonagretchoemnmetownolyrkmaogdeenltesd.bSaosmedeokneyacaospneccetpstof ppropgroraxmimcaatensgtheinseernatteroapyrawtiiothnaplrencuimsiobnerχi(nKaoifrianni,te20ti0m1e).that munication among the network agents. Some key aspects of program can generate a rational number in a finite time that csuocmhmcuonnisctraatiionntscahreancnoemlmwoitnhlyamlimoditeeldeddbaatasetdraonnsmaicsosinocneprattoef. approximatesthisentropywithprecisionχ(Koiran,2001). such constraints are commonly modeled based on a concept of approximates this entropy with precision χ (Koiran, 2001). communication channel with a limited data transmission rate. Nevertheless, it was recently shown that data rate thresholds Icnomthmisufnriacmateiwonorckh,aannperilmwairtyh ianqliumiriyteids adbaotauttrtahnesmmiisnsiimonalrraattee. Nevertheless, it was recently shown that data rate thresholds communication channel with a limited data transmission rate. Nevertheless, it was recently shown that data rate thresholds In this framework, a primary inquiry is about the minimal rate concerned with observability and introduced in (Matveev and nIneetdheisdftroamacehwieovrek,aadpersiimreadrycoinntqruoilryobijseacbtiovue.t Tthheismminoimmeanl troautes Nevertheless, it was recently shown that data rate thresholds In this framework, a primary inquiry is about the minimal rate concerned with observability and introduced in (Matveev and needed to achieve a desired control objective. This momentous Pogromsky, 2016) can yet be computed in closed form for some tnhereedsehdoltdohacahsiebveeenadaessuirbejdecctonotfrorlecoebnjetcetxivtee.nsTihviessmtuodmieesn;tosuees concernedwithobservabilityandintroducedin(Matveevand threshold has been a subject of recent extensive studies; see of the above nonlinear prototypical systems, e.g., the bounc-threshold has been a subject of recent extensive studies; see of the above nonlinear prototypical systems, e.g., the bounce.g., (Baillieul, 2004; De Persis and Isidori, 2004; Nair et al., ing ball system, Hénon system and logistic and Lozy maps 2e.0g0.,4;(BLaibilelirezuoln,2a0n0d4H;eDsepaPnehras,is20an0d5;IDsiedoPrei,rs2is0,0240;0N5a;iSraevtkailn.,, oftheabovenonlinearprototypicalsystems,e.g.,thebounc-e.g., (Baillieul, 2004; De Persis and Isidori, 2004; Nair et al., ing ball system, He´non system and logistic and Lozy maps 2004; Liberzon and Hespanha, 2005; De Persis, 2005; Savkin, (Matveev and Pogromsky, 2016; Pogromsky and Matveev, 22000064;; LNiabierreztonala.,n2d0H07e;spManahtvae,e2v00an5d; DSeavPkeirns,is2,020090)5a;nSdavlikteinr-, ing ball system, Hénon system and logistic and Lozy maps 2004; Liberzon and Hespanha, 2005; De Persis, 2005; Savkin, (Matveev and Pogromsky, 2016; Pogromsky and Matveev, 2006; Nair et al., 2007; Matveev and Savkin, 2009) and liter-2016a). This is thanks to the novel techniques for upper estima-a2t0u0r6e;thNearierine.t Tahl.i,s2d0a0t7a;-rMateattvhereevshaonldd hSaasvkaipnp,e2a0re0d9)toanbde aliltiekre-(Matveev and Pogromsky, 2016; Pogromsky and Matveev, ature therein. This data-rate threshold has appeared to be alike tion of these thresholds that are elaborated in (Pogromsky et al., aturetherein.Thisdata-ratethresholdhasappearedtobealike tion of these thresholds that are elaborated in (Pogromsky et al., in spirit to the topological entropy (Donarowicz, 2011) of the 2013; Pogromsky and Matveev, 2013, 2011, 2016b; Matveev sinysstpeimritattohtahnedsto,pboultoigsicnaolteanlwtroaypysi(dDeonntiacraolw;aicnzd,t2h0e1s1e)stoufdtihees tionofthesethresholdsthatareelaboratedin(Pogromskyetal., in spirit to the topological entropy (Donarowicz, 2011) of the 2013; Pogromsky and Matveev, 2013, 2011, 2016b; Matveev system at hands, but is not always identical; and these studies and Pogromsky, 2016) and turn off the classic road of the first hsyasvteemintartohdauncdeds, vbaurtioisusnoatnalwogasysoifdtehnitsiceanl;traonpdy t(hNesaeirsteutdaiel.s, 2013; Pogromsky and Matveev, 2013, 2011, 2016b; Matveev have introduced various analogs of this entropy (Nair et al., Lyapunov approach in study of topological entropy and the have introduced various analogs of this entropy (Nair et al., Lyapunov approach in study of topological entropy and the 2004; Savkin, 2006; Colonius and Kawan, 2009, 2011; Kawan, likes towards the second Lyapunov method. 2004; Savkin, 2006; Colonius and Kawan, 2009, 2011; Kawan, likes towards the second Lyapunov method. 2011;a2n0d11P;oHagiharaHgarogmihsakrya,anda2n0d1Nair,N6)a.ir, 2013;2013; ColoniusColonius etet al.,al., 2013;2013; MatvMatveeeevv Theesotbojwecardstivetheof thissecondpaperLyaisputonoaddvmethod.more functionality to thelikestowardsthesecondLyapunovmethod. andPogromsky,2016).andPogromskHagiharay,and2016).Nair,2013;Coloniusetal.,2013;Matveev The objective of this paper is to add more functionality to the andUnlikPogromske linear yplants,, 2016).computation or even fine estimation of The objective of this paper is to add more functionality to the just discussed approach of (Matveev and Pogromsky, 2016; Unlike linear plants, computation or even fine estimation of Pogromsky and Matveev, 2016a) via its further elaboration in a theseethresholdslinear plants,is ancomputationintricate matteror evforennonlinearfine estimationsystemsof Pogromsksituation wherey and Matvthe observeev, 2016a)ed dynamicsvia its furtherresult fromelaborationa feedbackin a ★these thresholds is an intricate matter for nonlinear systems Pogromsky and Matveev, 2016a) via its further elaboration in a ★ snittuearctioonnnwechtieorne tohfe towbosersvuebdsydsytenmams icwsitrhesiunlpt ufrtsomanadfeoeudtpbuactsk. ITMA.OPougnriovmersskityy abcyknGowovleedrngmesenhtisopfaRrtiuaslsisaunppFoerdtedruatriionng ghrisanstta(y07w4i-tUh0t1h)e, interconnectionwhere theof twobservo subsysted dynamicsems withresultinputsfromanda feedbackoutputs. ★A.PogromskyacknowledgeshispartialsupportduringhisstaywiththetheAs.ePotghrroemshskoyldascknisowalendgienstrhiicsapteartmialastuteprpofrotrdunrionnglhiniseastraysywsitthemthse interconnection of two subsystems with inputs and outputs. ITMO university by Government of Russian Federation grant (074-U01), interconnection of two subsystems with inputs and outputs. RTuMssOianuFneivdeerrsaittiyonbPyreGsiodveenrtnGmraenntt No1f4R.Yu3s1si.a1n6.9F2e8d1e-rHatIiIoIn, angdratnhte M(07in4i-sUtr0y1o)f, Such interconnection is very common in engineering practice ITMO university by Government of Russian Federation grant (074-U01), so that certain whole chapters of control theory assume that Russian Federation President Grant N14.Y31.16.9281-HIII, and the Ministry of so thatinterconnectioncertain whole ischaptersvery commonof controlin engineeringtheory assumepracticethat RussianFederationPresidentGrantN14.Y31.16.9281-HIII,andtheMinistryof thoethpalatncteirstaginivewnhoinlethcihsapfotermrs.oAfmcoonntgrotlhethme,ortyhearessuarmeesttuhda-t E,d3u,4c)a.tiAon. ManadtvSeceivenaccekonfowRluesdsgiaensFheisdesruaptipoonrt(pbryojRecStF141.4Z-5201.-3010.004013p1)a,n(Sdetchse. so that certain whole chapters of control theory assume that Education and Science of Russian Federation (project 14.Z50.31.0031), (Secs. the plant is given in this form. Among them, there are stud-S,a3i,n4t)P.eAte.rMsbautrvgeSevtaateckUnnoiwvelersdigtyes(Sheicsss.u2p,5p,o6r)t.bAy.PRrSoFsk1u4rn-2ik1o-v00a0c4k1npowanleddgthees the plant is given in this form. Among them, there are stud-SuapinptoPrteA.otefrMatvRsbFuBrgReeS,vtgaracknotaenUtsn1iwledgesv7e-0rs8i-ty00(7Shis1e5c,s1support.72-,058,6-)0.1byA72.8PRSFraonsdk14-21-00041pu1r7n-i0k8o-v01a2ck6n6o.wledgesandthe iesYeaskofoufboabsoluteavbiscohl,u2te00anda0n,d2robr0o0b2ustu;sMt ssettgaarbbeiitllsiittkyyi,,asee,snede,Re.g.ea.ngt.z((eWWr,ii1llll9ee9mm7ss),,,w1972;19h7e2re; ies of absolute and robust stability, see, e.g. (Willems, 1972; Saint Petersburg State University (Secs. 2,5,6). A. Proskurnikov acknowledges Yakubovich, 2000, 2002; Megretski and Rantzer, 1997), where support of RFBR, grants 17-08-00715,17-08-01728 and 17-08-01266. Yakubovich, 2000, 2002; Megretski and Rantzer, 1997), where support of RFBR, grants 17-08-00715,17-08-01728 and 17-08-01266. Publisher Copyright: © 2017 Copyright: Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2017/7

Y1 - 2017/7

N2 - The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe chaotic systems via communication channels with finite data rates, with a focus on minimum data-rates needed for various types of observability. With the objective of developing tractable techniques to estimate these rates, the paper discloses benefits from regard to the operational structure of the system in the case where the system is representable as a feedback interconnection of two subsystems with inputs and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their topological entropy stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive finite upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.

AB - The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe chaotic systems via communication channels with finite data rates, with a focus on minimum data-rates needed for various types of observability. With the objective of developing tractable techniques to estimate these rates, the paper discloses benefits from regard to the operational structure of the system in the case where the system is representable as a feedback interconnection of two subsystems with inputs and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their topological entropy stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive finite upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.

KW - Data rate estimates

KW - Entropy

KW - Nonlinear systems

KW - Observability

KW - Second Lyapunov method

UR - http://www.scopus.com/inward/record.url?scp=85041514401&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2017.08.1864

DO - 10.1016/j.ifacol.2017.08.1864

M3 - Article

AN - SCOPUS:85041514401

VL - 50

SP - 15397

EP - 15402

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8963

IS - 1

ER -

ID: 36635201