Abstract
This paper presents a proof of the Large Gain Theorem using the Nyquist Stability Criterion. The minimum gain constraint stipulated by the Large Gain Theorem guarantees the open-loop transfer function encircles the point (-1,0) exactly P times in the counterclockwise direction, where P is the number of open-loop open right-half plane poles. This guarantees asymptotic stability of the feedback system, even in the presence of an unstable open-loop transfer function. The Nyquist interpretation of the Large Gain Theorem is compared to Nyquist interpretations of the Large Gain and Passivity Theorems. Applications of the Large Gain Theorem are discussed and numerical examples illustrating the concept of minimum gain and the Large Gain Theorem are presented.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 3606-3611 |
| Number of pages | 6 |
| Journal | IFAC-PapersOnLine |
| Volume | 50 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jul 2017 |
| Externally published | Yes |
Bibliographical note
Funding Information:1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION The robust stabilization of open-loop unstable systems The robust stabilization of open-loop unstable systems The robust stabilization of open-loop unstable systems rneemerainings. aUncthilalrleencegnintlgy,ptrhoibslewmasinatmtroibduetrend ctoontarollacekngoi-f remains a challenging problem in modern control engi- nsteaebriinligty. rUesnutilltsrpeceerntatilny,intghtiso twhaesfeaetdtrbiabcukteindtetrocoannlaeccktionf stability results pertaining to the feedback interconnection stability results pertaining to the feedback interconnection ofifrsutndsetambolnesstyrastteemd st.hTathethwe oorpkeno-floGoeporggaiionuoeftaalf.ee(d1b9a9c7k) first demonstrated that the open-loop gain of a feedback first demonstrated that the open-loop gain of a feedback iinntpeurtc-oonuntepcuttiosntabmiluitsyt ibf aengyrseyastteermthwaitnhionntehetofeegdubaarcakntiene- input-output stability if any system within the feedback in- input-output stability if any system within the feedback in- theerdczoandneehcteiotnails. u(n2s0t0a8b)lep. rTovhiedLesarfgoermGaalinstTahbeiloitryemcroifteZraia- hedzadeh et al. (2008) provides formal stability criteria hedzadeh et al. (2008) provides formal stability criteria bmaosdedifieosn thheesreensuolttiotnos ianncdorBproirdagteemnaonnzaenrdo Fionritbieasl (c2o0n1d5i)- modifies the result to incorporate nonzero initial condi- modifies the result to incorporate nonzero initial condi- tTiohneos.reDme,spinitcelutdhiengapthpeeaalibniglitfyeattouraesseosfs tthhee Lcalorsged-Gloaoinp tions. Despite the appealing features of the Large Gain Tsthaeboilrietymo,fiancfleueddibnagckthinetearbciolintnyectotioanssoefsusntshteabclleosyedst-elomosp, Theorem, including the ability to assess the closed-loop sittarbeimlitayinosf a fereeldabtaivceklyintuenrkconnownenctsiotanboilfituynsrteasbullet.syTshteismsis, 2.1 Notation it remains a relatively unknown stability result. This is 2.1 Notation it remains a relatively unknown stability result. This is 2.1 Notation pitasrtpiraallcyticdaulietyt,owahicshoirstaegnetiroeflylitsepraantnuerde dbyemBornisdtgreamtianng In this paper, boldface letters represent matrices, script its practicality, which is entirely spanned by Bridgeman In this paper, boldface letters represent matrices, script its practicality, which is entirely spanned by Bridgeman leInttethisrs depanopteer,obpoeraldfatocrse,leandtterssimrepleprelettsenerst mdeatricnotees,scasclriptars. and Forbes (2015); Vasegh and Ghaderi (2012); Ghaderi letters denote operators, and simple letters denote scalars. and VFoarsbegehs (2015); VCaasveegrhlyaannddGFhorabdeesri(2(2001162).);PGrohoafdseorfi letters denote operators, and simple letters denote scalars. and Forbes (2015); Vasegh and Ghaderi (2012); Ghaderi letters denote operators, and simple letters denote sca2lars. athnedLVaarsgeeghGa(2in01T5h);eoCraevmerelympalnody Finoprbuet-so(u2t0p1u6t).thPerooroyfsZao-f Summation points within block diagrams are positive and Vasegh (2015); Caverly and Forbes (2016). Proofs of Sumesmsaottiohnerwpiosientnsotweidt.hiRnecbalollckthadtiagyra∈msL2arief p‖yo‖si2tiv=e hedzadehetal.(2008);BridgemanandForbes(2015),u∫0n∞letthehheedzLLaaadrrgegheeGGaetaiinanl.TT(hhe2e0oo0rre8e)mm; eeBmmrippdllgooeyymiinput-annpuat-nooduutputFpourttbttheehseoo(rry2y01ZZ5aa)--,∫∞ysTs(to)tyh(etr)wdtise<n∞ot,eda.ndRescimallilatrhlyatyy∈∈LL2e22ifif‖‖yy‖‖2T2222 = hwehdizchadedhoeestnaolt. a(2lw00a8y)s; oBffreirdgtehmeansamanedpFhoyrsbiceasl (i2n0si1g5h)t, ∫∞yT(t)y(t)dt < ∞, and similarly y ∈ L2e if ‖y‖2 = hedzadeh et al. (2008); Bridgeman and Forbes (2015), 0∞ yTT(t)y(t)dt < ∞, and simi+larly y ∈ L2e if ‖y‖2T = which does not always offer the same physical insight ∫0∞ y (t)y(Tt()td)tdt<<∞∞, ,anσd ∈simiR larl, why yere∈yLT(t)if=‖yy‖(t2)T for= washciclhassdicoaelsconnottroallwthaeyosryo.ffPeorptuhlaersianmpuet-pohuytpsiuctalstianbsiilgihtyt ∫00∞T + 2e 2T which does not always offer the same physical insight ∫∞yTT(t)yT(t)dt < ∞, σ ∈ R+, where yT(t) = y(t) for as classical control theory. Popular input-output stability 00≤ytTT≤(t)σyTa(ntd)dyt <(t)∞=,0σfor∈ tR>+,σwh. ThereeyHT(t)nor=my(oft)tfheor athsecolraesmsics,alsuccohntarsotlhteheSomrya.llPGoapiunlaarndinPpuasts-oivuitpyuTthsetoarbeimlitsy, 0 y (t)yT(t)dtT< ∞, σ ∈ R , where yT∞(t) = y(t) for as classical control theory. Popular input-output stability 00≤ tT≤ σ and yT (t) = 0 for t > σ . The H∞ norm of the theorems, such as the Small Gain and Passivity Theorems, 0tra≤nstfe≤r funcσ antiod nyTp(it()s)=is0‖pifor(st)‖> σ=. suThpe H∞|pin(orνωm)| of= γtheG. thhaevoereNmysq,usiustchinatsetrhpereStamtaiollnGs,awinhaicnhdmPoatsisvivaitteysTahNeoyrqeumisst, 0 ≤ t ≤ σ and yT(t) = 0 for t >∞σ. Theω∈HR∞ norm of the theorems, such as the Small Gain and Passivity Theorems, transfer function pi(s) is ‖pi(s)‖ = sup |pi(νω)| = γG. have Nyquist interpretations, which motivates a Nyquist transfer function pi(s) is ‖pi(s)‖∞ = supω∈R |pi(νω)| = γG. hinatveerpNreytqautiiostn ionftetrhpereLtaartgioenGs,aiwnhTichheomroemtiv.ates a Nyquist transfer function pi(s) is ‖pi(s)‖∞ = supω∈R |pi(νω)| = γG. have Nyquist interpretations, which motivates a Nyquist ∞ ω∈R interpretation of the Large Gain Theorem. 2.2 Problem Statement The novel contributions of this paper are a proof of the 2.2 Problem Statement The novel contributions of this paper are a proof of the 2.2 Problem Statement TLahregenoGvaeilncTonhteroibreumtiounssinogftthheisNpyqapuiesrt aSrteabailiptryoCofriotefrtiohne 2.2 Problem Statement Large Gain Theorem using the Nyquist Stability Criterion Consider the negative feedback interconnection of G : Large Gain Theorem using the Nyquist Stability Criterion Consider the negative feedback interconnection of G1 : ★ This work was supported in part by the Natural Sciences and CC2ooennss→iidederrLtt2hheeeannenedggaaGttii2vvee: ffeLeee2ddebbaa→cckkLiinn2ettee,rrcpcooicnnnetnuerccettiodionninooFffigGG.11::. ★★ This work was supported in part by the Natural Sciences and LL2e →→ LL2e anandd GG2 :: LL2e →→ LL2e,, picpictureturedd inin FigFig..1 11.. EnTghiniseewrinorgkRwesaesarscuhpCpoorutnedcilionf Cpaarntadbay’stPheosNtgartaudruaalteScSicehnocleasrsahnipd A2sesume G21eand G2 2are lin2ear time2-einvariant (LTI) single-EngineeringThis workRwesaesarsuppchCouncilorted iofn Canada’spart by thePosNtgaturalraduateSciencesScholarsandhip A2sesume G21eand G2 2are lin2ear time2-einvariant (LTI) single-prnoggirnaemer.ing Research Council of Canada’s Postgraduate Scholarship AinspsuutmseinGgl1e-aonudtpGu2t (aSrIeSlOin)esayrsttiemes-,inwviathriatrnatn(sLfeTrIf)usnicntgiloen-Engineering Research Council of Canada’s Postgraduate Scholarship Assume G1 and G2 are linear time-invariant (LTI) single-program. input single-output (SISO) systems, with transfer function program. input single-output (SISO) systems, with transfer function and a comparison to the Small Gain and Passivity The-and a comparison to the Small Gain and Passivity The-and a comparison to the Small Gain and Passivity Theorems. The goal of this work is to bring attention to the Lraermges.GTahine Tgohaeloroefmthaiss wa ourskefiusl tsotabbriilnitgy artetseunltt.ion to the orems. The goal of this work is to bring attention to the Large Gain Theorem as a useful stability result. This paper proceeds as follows. Preliminary definitions, This paper proceeds as follows. Preliminary definitions, This paper proceeds as follows. Preliminary definitions, perseusetnst,saNndyqtuhiesotrienmtesrparetiantciolundseodf itnheSeLcatriogne G2.aSine,ctSiomnal3l results, and theorems are included in Section 2. Section 3 presents Nyquist interpretations of the Large Gain, Small presents Nyquist interpretations of the Large Gain, Small Gain, and Passivity Theorems. Numerical examples of Gain, and Passivity Theorems. Numerical examples of LayrqgueisGtaainndThBeoodreempalortesinobSteacitnieodn 4w.hSeenctsioantis5fyiinncgludthees Nyquist and Bode plots obtained when satisfying the apaprgliecaGtiaoins TofhtehoereLmaragreeGianinSeTchtieoonre4m. . Seeccttiioonn 56 pinrcelsuednetss Large Gain Theorem are in Section 4. Section 5 includes applications of the Large Gain Theorem. Section 6 presents applications of the Large Gain Theorem. Section 6 presents concluding remarks. Appendix A and Appendix B include concluding remarks. Appendix A and Appendix B include proofs of the minimum gain of nonminimum phase and carsocoafds eodf styhsetemmisn, irmesupmectgiavienlyo. f nonminimum phase and proofs of the minimum gain of nonminimum phase and cascaded systems, respectively. cascaded systems, respectively. 2. PRELIMINARIES 2. PRELIMINARIES 2. PRELIMINARIES
Publisher Copyright:
© 2017
Keywords
- Large Gain Theorem
- Nyquist Stability Criterion
- input-output stability
- linear systems
- stability of feedback interconnections