4. Advanced Stability Theory

Transcription

1 Applied Nonlinear Conrol Nguyen an ien Advanced Sabiliy heory he objecive of his chaper is o presen sabiliy analysis for non-auonomous sysems 41 Conceps of Sabiliy for Non-Auonomous Sysems Equilibrium poins and invarian ses For non-auonomous sysems, of he form & f (, (41) equilibrium poins * are defined by f ( *, (4) Noe ha his equaion mus be saisfied, implying ha he sysem should be able o say a he poin * all he ime For insance, we can easily see ha he linear ime-varying sysem & A( (43) has a unique equilibrium poin a he origin unless A( is always singular Eample 41 he sysem a( & (44) has an equilibrium poin a However, he sysem a( & + b( (45) wih b (, does no have an equilibrium poin I can be regarded as a sysem under eernal inpu or disurbance b ( Since Lyapunov heory is mainly developed for he sabiliy of nonlinear sysems wih respec o iniial condiions, such problem of forced moion analysis are more appropriaely reaed by oher mehods, such as hose in secion 49 Eensions of he previous sabiliy conceps Definiion 41 he equilibrium poin is sable a if for any R >, here eiss a posiive scalar r ( R, ) such ha ( ) < r ( ) < R (46) Oherwise, he equilibrium poin is unsable he definiion means ha we can keep he sae in ball of arbirarily small radius R by saring he sae rajecory in a ball of sufficienly small radius r Definiion 4 he equilibrium poin is asympoically sable a if i is sable r( ) > such ha ( ) < r( ) ( ) as he asympoic sabiliy requires ha here eiss an aracive region for every iniial ime Definiion 43 he equilibrium poin is eponenially sable if here eis wo posiive numbers, α and λ, such ha for sufficienly small ( ), λ( ) ( α e Definiion 44 he equilibrium poin is global sable ( ), ( ) as Eample 4 A firs-order linear ime-varying sysem Consider he firs-order sysem & ( a( ( Is soluion is a( r) dr ( ( e ) hus sysem is sable if a(, I is asympoically sable if a ( r) dr + I is eponenially sable if here eiss a sricly posiive number such ha, + a( r) dr γ, wih γ being a posiive consan For insance he sysem & /( is sable (bu no asympoically sable) he sysem & /( is asympoically sable he sysem & is eponenially sable Anoher ineresing eample is he sysem & ( sin dr + r he soluion can be epressed as ( ) 1 sin ( ) ( ) e Since dr, he sysem is eponenially sin ( r) convergen wih rae1 / Uniformiy in sabiliy conceps he previous conceps of Lyapunov sabiliy and asympoic sabiliy for non-auonomous sysems boh indicae he imporance effec of iniial ime In pracice, i is usually desirable for he sysems o have a cerain uniformiy in is behavior regardless of when he operaion sars his moivaes us o consider he definiions of uniform sabiliy and uniform asympoic sabiliy Non-auonomous sysems wih uniform properies have some desirable abiliy o Chaper 4 Advanced Sabiliy heory 19

2 Applied Nonlinear Conrol Nguyen an ien - 4 wihsand disurbances he behavior of auonomous sysems is dependen of he iniial ime, all he sabiliy properies of an auonomous sysem are uniform Definiion 45 he equilibrium poin is locally uniformly sable if he scalar r in Definiion 41 can be chosen independen of, ie, if r r(r) Definiion 46 he equilibrium poin a he origin is locally uniformly asympoically sable if i is uniformly sable here eis a ball of aracion B R, whose radius is independen of, such ha any rajecory wih iniial saes in B R converges o uniformly in By uniform convergence in erms of, we mean ha for all R1 and R saisfying < R < R1 R, ( R1, R ) > such ha, ( ) < R1 ( ) < R + ( R1, R ) ie, he rajecory, saring from wihin a ball B R 1, will converges ino a smaller ball B R afer a ime period which is independen of By definiion, uniform asympoic sabiliy always implies asympoic sabiliy he converse (ñaû o ñeà ) is generally no rue, as illusraed by he following eample Eample 43 Consider he firs-order sysem & his sysem has 1 + general soluion ( ( ) he soluion asympoically converges o zero Bu he convergence is no uniform Inuiively, his is because a larger requires a longer ime o ge close o he origin he concep of globally uniformly asympoic sabiliy can be defined be replacing he ball of aracion B R by he whole sae space 4 Lyapunov Analysis of Non-Auonomous Sysems In his secion, we eend he Lyapunov analysis resuls of chaper 3 o he sabiliy of non-auonomous sysems 41 Lyapunov sdirec mehod for non-auonomous sysems he basic idea of he direc mehod, ie, concluding he sabiliy of nonlinear sysems using scalar Lyapunov funcions, can be similarly applied o non-auonomous sysems Besides more mahemaical compleiy, a major difference in nonauonomous sysems is ha he powerful La Salle s heorems do no apply his drawback will parially be compensaes by a simple resul in secion 45 called Barbala s lemma Definiion 47 A scalar ime-varying funcion V (, is locally posiive definie if V (, and here eis a ime-varian posiive definie funcion V ( ) such ha, V (, V ( ) (47) hus, a ime-varian funcion is locally posiive definie if i dominaes a ime-varian locally posiive definie funcion Globally posiive definie funcions can be defined similarly Definiion 48 A scalar ime-varying funcion V (, is said o be decrescen if V (,, and if here eis a ime-varian posiive definie funcion V1 ( ) such ha, V (, V1 ( ) (47) In oher word, a scalar funcion V (, is decrescen if i is dominaed by a ime-invarian pd funcion Eample 44 Consider ime-varying posiive definie funcions as follows V (, (1 + sin ( 1 + ) V ( ) 1 + V 1( ) ( 1 + ) V (, dominaes V ( ) and is dominaed by V 1( ) because V ( ) V (, V1 ( ) Given a ime-varying scalar funcion V (,, is derivaive along a sysem rajecory is + & + f (, (48) ime-varying posiive definie funcions and decrescen srenghened by requiring hav & be negaive definie, hen funcions he equilibrium poin is asympoically sable Chaper 4 Advanced Sabiliy heory d V d Lyapunov heorem for non-auonomous sysem sabiliy he main Lyapunov sabiliy resuls for non-auonomous sysems can be summarized by he following heorem heorem 41 (Lyapunov heorem for non-auonomous sysems) Sabiliy: If, in a ball BR around he equilibrium poin, here eis a scalar funcion V (, wih coninuous parial derivaives such ha 1 V is posiive definie V & is negaive semi-definie hen he equilibrium poin is sable in he sense of Lyapunov Uniform sabiliy and uniform asympoic sabiliy: If, furhermore 3 V is decrescen hen he origin is uniformly sable If he condiion is

3 Applied Nonlinear Conrol Nguyen an ien - 4 Global uniform asympoic sabiliy: If, he ball B R is replaced by he whole sae space, and condiion 1, he srenghened condiion, condiion 3, and he condiion 4 V (, is radially unbounded are all saisfied, hen he equilibrium poin a is globally uniformly asympoically sable Similarly o he case of auonomous sysems, if in a cerain neighborhood of he equilibrium poin, V is posiive definie andv &, is derivaive along he sysem rajecories, is negaive semi-definie, henv is called Lyapunov funcion for he nonauonomous sysem Eample 45 Global asympoic sabiliy Consider he sysem defined by & 1( 1 ( e ( & ( 1( ( Chose he Lyapunov funcion candidae V (, 1 + (1 + e ) his funcion is pd, because i dominaes he ime-invarian pd funcion 1 + I is also decrescen, because i is dominaed by he ime-invarian pd funcion 1 + Furhermore, V& (, [ (1 + e )] his shows ha V& (, ( ) ( 1 ) 1 hus, V& (, is negaive definie, and herefore, he poin is globally asympoically sable 4 Lyapunov analysis of linear ime-varying sysems Consider linear ime-varying sysems of he form & A( (417) Since LI sysems are asympoically sable if heir eigenvalues all have negaive real pars Will he sysem (417) be sable if any ime, he eigenvalues of A ( all have negaive pars? Consider he sysem A simple resul, however, is ha he ime-varying sysem (417) is asympoically sable if he eigenvalues of he symmeric mari A ( + A ( (all of which are real) remain sricly in he lef-half comple plane λ >, i,, λ ( A ( + A ( ) λ (419) i his can be readily shown using he Lyapunov funcion V, since V & & + & ( A( + A ( )) λ & λv λ so ha, V ( V () e and herefore ends o zero eponenially I is imporan o noice ha he resul provides a sufficien condiion for any asympoic sabiliy Perurbed linear sysems Consider a linear ime-varying sysem of he form & [ A1 + A ( ] (4) where A1 is consan and Hurwiz and he ime-varying mari A ( is such ha A ( as and A ( d < (ie, he inegral eiss and is finie) hen he sysem (4) is globally sable eponenially sable Eample 48 Consider he sysem defined by 5 8 & 1 ( ) 1 & + 43 & 3 ( + sin 3 Since 3 ends o zero eponenially, so does 3, and herefore, so does Applying he above resul o he firs equaion, we conclude ha he sysem is globally eponenially sable Sufficien smoohness condiions on he A( mari Consider he linear sysem (417), and assume ha a any ime, he eigenvalues of A( all have negaive real pars α >, i,, λi[ A ( ] α (41) & 1 1 & e 1 1 (418) Boh eigenvalues of A( equal o -1 a all imes he soluion If, in addiion, he mari A( remains bounded, and A ( A( d < (ie, he inegral eiss and is finie) of (418) can be rewrien as () e, & () e Hence, he sysem is unsable hen he sysem is globally eponenially sable Chaper 4 Advanced Sabiliy heory 1

4 Applied Nonlinear Conrol Nguyen an ien he linearizaion mehod for non-auonomous sysems Lyapunov s linearizaion mehod can also be developed for non-auonomous sysems Le a non-auonomous sysem be described by (41) and be an equilibrium poin Assume ha f is coninuously differeniable wih respec o Le us denoe f A( ) (4) he for any fied ime (ie, regarding as a parameer), a aylor epansion of f leads o herem 43 If he Jacobian mari A ( is consan, A ( ) A, and if (43) is saisfied, hen he insabiliy of he linearized sysem implies ha of he original non-auonomous nonlinear sysem, ie, (41) is unsable if one or more of he eigenvalues of A has a posiive real par 43 Insabiliy heorems 44 Eisence of Lyapunov Funcions 45 Lyapunov-Like Analysis Using Barbala s Lemma & A( + f h o (, If f can be well approimaed by fh o (, lim sup hen he sysem A ( for any ime, ie, (43) & A( (44) is said o be he linearizaion (or linear approimaion) of he nonlinear non-auonomous sysem (41) around equilibrium poin Noe ha: - he Jacobian mari A hus obained from a nonauonomous nonlinear sysem is generally ime-varying, conrary o wha happens for auonomous nonlinear sysems Bu in some cases A is consan For eample, he nonlinear sysem & + / leads o he linearized sysem & - Our lae resuls require ha he uniform convergence condiion (43) be saisfied Some non-auonomous sysems may no saisfy his condiion, and Lyapunov s linearizaion mehod canno be used for such sysems For eample, (43) is no saisfied for he sysem & + Given a non-auonomous sysem saisfying condiion (43), we can asser is (local) sabiliy if is linear approimaion is uniformly asympoically sable, as saed in he following heorem: herem 4 If he linearized sysem (wih condiion (43) saisfied) is uniformly asympoically sable, hen he equilibrium poin of he original non-auonomous sysem is also uniformly asympoically sable Noe ha: - he linearized ime-varying sysem mus be uniformly asympoically sable in order o use his heorem If he linearized sysem is only asympoically sable, no conclusion can be draw abou he sabiliy of he original nonlinear sysem - Unlike Lyapunov s linearizaion mehod for auonomous sysem, he above heorem does no relae he insabiliy of he linearized ime-varying sysem o ha of he nonlinear sysem Asympoic sabiliy analysis of non-auonomous sysems is generally much harder han ha of auonomous sysems, since i is usually very difficul o find Lyapunov funcions wih a negaive definie derivaive An imporan and simple resul which parially remedies (khaé c phuï c) his siuaion is Barbala s lemma When properly used for dynamic sysems, paricularly for non-auonomous sysems, i may lead o he saisfacory soluion of many asympoic sabiliy problem 451 Asympoic properies of funcions and heir derivaives Before discussing Barbala s lemma iself, le us clarify a few poins concerning he asympoic properies of funcions and heir derivaives Given a differeniable funcion f of ime, he following hree facs are imporan o keep in mind f& > f converges he fac ha f & does no imply ha f ( has a limi as f converges > f & he fac ha f ( has a limi as does no imply ha f & If f is lower bounded and decreasing ( f & ), hen i converges o a limi 45 Barbala s lemma Lemma 4 (Barbala If he differeniable funcion f( has a finie limi as, and if f & is uniformly coninuous, hen f & ( as Noe ha: - A funcion g( is coninuous on [, ) if 1, R >, η( R, 1) >,, 1 < η g( g( 1) < R - A funcion g( is said o be uniformly coninuous on [, ) if R >, η( R) >, 1,, 1 < η g( g( 1) < R or in oher words, g( is uniformly coninuous if we can always find anη which does no depend on he specific poin 1 - and in paricular, such haη does no shrink as and 1 play a symmeric role in he definiion of uniform coninuiy Chaper 4 Advanced Sabiliy heory

5 Applied Nonlinear Conrol Nguyen an ien A simple sufficien condiion for a differeniable funcion o be uniformly coninuous is ha is derivaive be bound his can be seen from he finie differen heorem:, 1, ( 1) such ha g( g( 1) g& ( )( 1) And herefore, if R1 > is an upper bound on he funcion g&, we can always useη R / R1 independenly of 1 o verify he definiion of uniform coninuiy Eample 41 Consider a sricly sable linear sysem whose inpu is bounded hen he sysem oupu is uniformly coninuous Indeed, wrie he sysem in he sandard form & A + Bu y C Since u is bounded and he linear sysem is sricly sable, hus he sae is bounded his in urn implies from he firs equaion ha & is bounded, and herefore from he second equaion ha y & C& is bounded hus he sysem oupu y is uniformly coninuous Using Barbala s lemma for sabiliy analysis o apply Barbala s lemma o he analysis of dynamic sysems, one ypically uses he following immediae corollary, which looks very much like an invarian se heorem in Lyapunov analysis: Lemma 43 (Lyapunov-Like Lemma) If a scalar funcion V (, saisfies he following condiions V (, is lower bounded V& (, is negaive semi-definie V& (, is uniformly coninuous in ime hen V& (, as Indeed, V he approaches a finie limiing valuev, such ha V V ((),) (his does no require uniform coninuiy) he above lemma hen follows from Barbala s lemma Eample 413 Consider he closed-loop error dynamics of an adapive conrol sysem for a firs-order plan wih unknown parameer e& e + θ w( & θ e w( where e andθ are he wo saes of he closed-loop dynamics, represening racking error and parameer error, and w( is a bounded coninuous funcion Consider he lower bounded funcion V e +θ Is derivaive is V& e[ e + θ w( ] + θ [ e w( ] e his implies ha V ( V (), and herefore, ha e andθ are bounded Bu he invarian se canno be used o conclude he convergence of e, because he dynamics is non-auonomous o use Barbala s lemma, le us check he uniform coninuiy of V & he derivaive of V & is V& 4e( e + θ w) his shows ha V & is bounded, since w is bounded by hypohesis, and e and θ were shown above o be bounded Hence, V & is uniformly coninuous Applicaion of Babarla s lemma hen indicaes ha e as Noe ha, alhough e converges o zero, he sysem is no asympoically sable, because θ is only guaraneed o be bounded Noe ha: Such above analysis based on Barbala s lemma shall be called a Lyapunov-like analysis here are wo imporan differences wih Lyapunov analysis: - he funcion V can simply be a lower bounded funcion of and insead of a posiive definie funcion - he derivaive V & mus be shown o be uniformly coninuous, in addiion o being negaive or zero his is ypically done by proving ha V & is bounded 46 Posiive Linear Sysems In he analysis and design of nonlinear sysems, i is ofen possible and useful o decompose he sysem ino a linear subsysem and a nonlinear subsysem If he ransfer funcion of he linear subsysem is so-called posiive real, hen i has imporan properies which may lead o he generaion of a Lyapunov funcion for he whole sysem In his secion, we sudy linear sysems wih posiive real ransfer funcion and heir properies 461 PR and SPR ransfer funcion Consider raional ransfer funcion of n h -order SISO linear sysems, represened in he form m m 1 bm p + bm 1 p + K+ b h( p) n n 1 p + an 1 p + K+ a he coefficiens of he numeraor and denominaor polynomials are assumed o be real numbers and n m he difference n m beween he order of he denominaor and ha of he numeraor is called he relaive degree of he sysem Definiion 41 A ransfer funcion h(p) is posiive real if Re[ h ( p)] for all Re[ p ] (433) I is sricly posiive real if h( p ε) is posiive real for some ε > Condiion (433) is called he posiive real condiion, means ha h( p) always has a posiive (or zero) real par when p has posiive (or zero) real par Geomerically, i means ha he raional funcion h( p) maps every poin in he closed RHP (ie, including he imaginary ais) ino he closed RHP of h ( p) Chaper 4 Advanced Sabiliy heory 3

6 Applied Nonlinear Conrol Nguyen an ien - 4 Eample 414 A sricly posiive real funcion 1 Consider he raional funcion h( p), which is he p + λ ransfer funcion of a firs-order sysem, wih λ > Corresponding o he comple variable p σ + jω, 1 σ + λ jω h ( p) ( σ + jω) + λ ( σ + λ) + ω Obviously, Re[ h( p)] if σ hus, h( p) is a posiive real funcion In fac, one can easily see ha h( p) is sricly posiive real, for eample by choosing ε λ / in Definiion 49 heorem 41 A ransfer funcion h( p) is sricly posiive real (SPR) if and only if i h( p) is a sricly sable ransfer funcion ii he real par of h( p) is sricly posiive along he jω ais, ie, ω Re[ h ( jω)] > (434) he above heorem implies necessary condiions for assering wheher a given ransfer funcion h( p) is SPR: is sricly sable he Nyquis plo of h( jω) lies enirely in he RHP Equivalenly, he phase shif of he sysem in response o sinusoidal inpus is always less han 9 has relaive degree of or 1 is sricly minimum-phase (ie, all is zeros are in he LHP) Eample 415 SPR and non-spr funcions Consider he following sysems p 1 h1 ( p) p + ap + b 1 h3 ( p) p + ap + b p 1 h ( p) p p + 1 p + 1 h 4 ( p) p + p + 1 which shows ha h4 is SPR (since i is also sricly sable) Of course, condiion (434) can also be checked direcly on a compuer he basic difference beween PR and SPR ransfer funcions is ha PR ransfer funcions may olerae poles on he jω ais, while SPR funcions canno Eample Consider he ransfer funcion of an inegraor h ( p) Is p σ jω value corresponding o p σ + jω is h ( p) We σ + ω can easily see from Definiion 49 ha h( p) is PR bu no SPR heorem 411 A ransfer funcion h( p) is posiive real if, and only if, is a sable ransfer funcion he poles of h( p) on he jω ais are simple (ie, disinc and he associaed residues are real and non-negaive Re[ h( jω)] for anyω such ha jω is no a pole of h ( p) he Kalman-Yakubovich lemma If a ransfer funcion of a sysem is SPR, here is an imporan mahemaical propery associaed wih is sae-space represenaion, which is summarized in he celebraed Kalman-Yakubovich (KY) lemma Lemma 44 (Kalman-Yakubovich) Consider a conrollable linear ime-invarian sysem & A + bu y c he ransfer funcion 1 h( p) c [ pi A] b (435) is sricly posiive real if, and only if, here eis posiive marices P and Q such ha he ransfer funcion h1,h and h 3 are no SPR, because h 1 is non-minimum phase, h is unsable, and h 3 has relaive degree larger han 1 Is he (sricly sable, minimum-phase, and of relaive degree 1) funcion h 4 acually SPR? We have jω + 1 ( jω + 1)( ω jω + 1) h 4 ( jω) ω + jω + 1 (1 ω ) + ω (where he second equaliy is obained by muliplying numeraor and denominaor by he comple conjugae of he denominaor) and hus ω + ω 1 Re[ h 4 ( jω)] (1 ω ) + ω (1 ω ) + ω A P + PA -Q (436a) P b c (436b) In he KY lemma, he involved sysem is required o be asympoically conrollable A modified version of he KY lemma, relaing he conrollabiliy condiion, can be saed as follows Lemma 45 (Meyer-Kalman-Yakubovich) Given a scalar γ, vecor b and c, any asympoically sable mari A, and a symmeric posiive definie mari L, if he ransfer funcion γ 1 H ( p) + c [ pi A] b Chaper 4 Advanced Sabiliy heory 4

7 Applied Nonlinear Conrol Nguyen an ien - 4 is SPR, hen here eis a scalar ε >, a vecor q, and a symmeric posiive definie mari P such ha A P + P A -qq ε L Pb c + γ q his lemma is differen from Lemma 44 in wo aspecs he involved sysem now has he oupu equaion γ y c + u he sysem is only required o be sabilizable (bu no necessary conrollable) 463 Posiive real ransfer marices he concep of posiive real ransfer funcion can be generalized o raional posiive real marices Such generaion is useful for he analysis and design of MIMO sysems Definiion 411 An m m mari H( p) is call PR if H( p) has elemens which are analyic for Re( p ) > H ( p) + H ( p*) is posiive semi-definie for Re( p ) > where he aserisk * denoe he comple conjugae ranspose H( p) is SPR if H( p ε ) is PR for some ε > 47 he Passiviy Formalism 48 Absolue Sabiliy 49 Esablishing Boundedness of Signal 41 Eisence and Uniciy of Soluions Chaper 4 Advanced Sabiliy heory 5