CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XIII - Lyapunov Design - Shuzhi Ge

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1 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge LYAPUNOV DESIGN Shuzhi Ge Depatet of Electical ad Copute Egieeig, he Natioal Uivesity of Sigapoe, Sigapoe Keywods: Cotol Lyapuov Fuctio, Lyapuov Desig, Model Refeece Adaptive Cotol, Adaptive Cotol, Backsteppig Desig. Cotets 1. Itoductio 2. Cotol Lyapuov Fuctio 3. Lyapuov Desig via Lyapuov Equatio 3.1. Lyapuov Equatio 3.2. MRAC fo Liea ie Ivaiat Systes 3.3. MRAC fo Noliea Systes 4. Lyapuov Desig fo Matched ad Uatched Ucetaities 4.1. Lyapuov Desig fo Systes with Matched Ucetaities Lyapuov Redesig Adaptive Lyapuov Redesig Robust Lyapuov Redesig 4.2. Backsteppig Desig fo Systes with Uatched Ucetaities Backsteppig fo Kow Paaete Case Adaptive Backsteppig fo Ukow Paaete Case Adaptive Backsteppig with uig Fuctio 5. Popety-based Lyapuov Desig 5.1. Physically Motivated Lyapuov Desig 5.2. Itegal Lyapuov Fuctio fo Noliea Paaeteizatios 6. Desig Flexibilities ad Cosideatios 7. Coclusios Ackowledgeets Glossay Bibliogaphy Biogaphical Sketch Suay SAMPLE CHAPERS his chapte gives a oveview o soe state-of-the-at appoaches of Lyapuov desig by dividig systes ito seveal distict classes, though i geeal thee is o systeatic pocedue i choosig a suitable Lyapuov fuctio cadidate fo cotolle desig to guaatee the closed-loop stability fo a give oliea syste. Afte a bief itoductio ad histoic eview, this chapte sequetially pesets (i) the basic cocepts of Lyapuov stability ad cotol Lyapuov fuctios, (ii) Lyapuov equatios ad odel efeece adaptive cotol based o Lyapuov desig fo atched systes, (iii) Lyapuov edesig, adaptive edesig ad obust desig fo atched systes, (iv) adaptive backsteppig desig fo uatched oliea systes, (v) Lyapuov desig by exploitig physical popeties fo special classes of systes, ad (vi) desig flexibilities Ecyclopedia of Life Suppot Systes (EOLSS)

2 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge ad cosideatios i actual desig. 1. Itoductio Lyapuov desig has bee a piay tool fo oliea cotol syste desig, stability ad pefoace aalysis sice its itoductio i he basic idea is to desig a feedback cotol law that edes the deivative of a specified Lyapuov fuctio cadidate egative defiite o egative sei-defiite. Lyapuov s diect ethod is a atheatical itepetatio of the physical popety that if a syste s total eegy is dissipatig, the the states of the syste will ultiately each a equilibiu poit. he basic idea behid the ethod is that, if thee exists a kid of cotiuous scala eegy fuctio such that this eegy diiishes alog the syste s tajectoy, the the syste is said to be asyptotically stable. Sice thee is o eed to solve the solutio of the diffeetial equatios goveig the syste i deteiig its stability, it is usually efeed to as the diect ethod (see Lyapuov Stability). Although Lyapuov s diect ethod is efficiet fo stability aalysis, it is of esticted applicability due to the difficulty i selectig a Lyapuov fuctio. he situatio is diffeet whe facig the cotolle desig poble, whee the cotol has ot bee specified, ad the syste ude cosideatio is udeteied. Lyapuov fuctios have bee effectively utilized i the sythesis of cotol systes. he basic idea is that, by fist choosig a Lyapuov fuctio cadidate, a feedback cotol law ca be specified such that it edes the deivative of the specified Lyapuov fuctio cadidate egative defiite, o egative sei-defiite whe ivaiace piciple ca be used to pove asyptotic stability. his way of desigig cotol is called Lyapuov desig. Lyapuov desig depeds o the selectio of Lyapuov fuctio cadidate. hough the esult is sufficiet, it is a difficult poble to fid a Lyapuov fuctio (LF) satisfyig the equieets of Lyapuov desig. Fotuately, duig the past seveal decades, ay effective cotol desig appoaches have bee developed fo diffeet classes of liea ad oliea systes based o the basic ideas of Lyapuov desig. Lyapuov fuctios ae additive, like eegy, i.e., Lyapuov fuctios fo cobiatios of subsystes ay be deived by addig the Lyapuov fuctios of the subsystes. his poit ca be see clealy i the adaptive cotol desig ad backsteppig desig i this chapte. SAMPLE CHAPERS hough Lyapuov desig is a vey poweful tool fo cotol syste desig, stability ad pefoace aalysis, the costuctio of a Lyapuov fuctio is ot easy fo geeal oliea systes, ad it is usually a tial-ad-eo pocess owig to the lack of systeatic ethods. Diffeet choices of Lyapuov fuctios ay esult i diffeet cotol stuctues ad cotol pefoace. Past expeiece shows that a good desig of Lyapuov fuctio should fully utilize the popety of the studied systes. Lyapuov desig is used i ay cotexts, such as dyaic feedback, output-feedback, estiatio of egio of attactio, ad adaptive cotol, aog othes. he chapte is ot eat to be copehesive but to seve as a itoductio to the state-of-the-at of full-state feedback desig based o Lyapuov techiques fo seveal typical classes of autooous systes. Ecyclopedia of Life Suppot Systes (EOLSS)

3 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge Itesive eseach i adaptive cotol was fist otivated by the desig of autopilots fo high pefoace aicaft i the ealy 1950s. Because thei dyaics chage dastically whe they fly fo oe opeatig poit to aothe, costat gai feedback cotol caot hadle it effectively. he lack of stability theoy ad oe disastous flight test led to the diiished iteest i adaptive cotol i the late 1950s. he 1960s saw ay advaces i cotol theoy ad adaptive cotol i paticula. Siultaeous developet i coputes ad electoics ade the ipleetatio of coplex cotolles possible, ad iteest i adaptive cotol ad its applicatios was eewed i the 1970s with seveal beakthough esults ade. he studies of o-obust behavio of adaptive cotol subject to sall distubace ad uodeled dyaics i 1979 ad the ealy 1980s, led to bette udestadig of the istability echaiss ad the desig of obust adaptive cotol i the late 1980s though it was vey cotovesial iitially. hey wee all systes satisfyig the atchig coditio. I the late 1980s ad ealy 1990s, the atchig coditio was elaxed to the exteded atchig coditio, which fo oe peiod was egaded as the fotie that could ot be cossed by Lyapuov desig, ad the futhe elaxed to the stict-feedback systes with geeal uatched ucetaities though backsteppig desig, which is the state-of-the-at of adaptive cotol. his chapte gives a oveview of the state-of-the-at appoaches of Lyapuov desig ad ways of choosig Lyapuov fuctios i the aea of full-state adaptive cotol. Sectio 2 pesets the cocepts of Lyapuov stability aalysis ad cotol Lyapuov fuctios. I Sectio 3, Lyapuov fuctios fo liea tie ivaiat systes ae peseted fist, the the esults ae utilized to solve Model Refeece Adaptive Cotol (MRAC) pobles fo classes of liea ad oliea systes which ca be tasfoed to systes havig stable liea potio. Fo this class of pobles, the choice of Lyapuov fuctios is systeatic ad cotolle desig is stadad. I Sectio 4, afte the pesetatio of Lyapuov Redesig, Adaptive Lyapuov Redesig, Robust Lyapuov Redesig fo a class of atched systes, backsteppig cotolle desig is discussed fo uatched oliea systes. By exploitig the physical popeties of the systes ude study, Sectio 5 shows that diffeet choices of Lyapuov fuctios ad bette cotolles ae possible. Sectio 6 discusses the desig flexibilities ad cosideatios i actual applicatios of Lyapuov desig, ad futhe eseach wok. 2. Cotol Lyapuov Fuctio SAMPLE CHAPERS hough Lyapuov s ethod applies to oautooous systes x = f( x,t), fo claity ad siplicity, we shall estai ou discussio to tie-ivaiat oliea systes of the fo x = f( x), (1) whee x R, ad fx ( ): R R is cotiuous. he basic idea of Lyapuov diect ethod cosists of (i) choosig a adially ubouded positive defiite Lyapuov Ecyclopedia of Life Suppot Systes (EOLSS)

4 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge fuctio cadidate V ( x ), ad (ii) evaluatig its deivative V ( x) alog syste dyaics (1) ad checkig its egativeess fo stability aalysis. Lyapuov desig efes to the sythesis of cotol laws fo soe desied closed-loop stability popeties usig Lyapuov fuctios fo oliea cotol systes x = f( x, u), (2) whee x R is the state, o ( xu, ), ad f (0, 0) = 0. u R is the cotol iput, ad fxu (, ) is locally Lipschitz he usefuless of Lyapuov diect ethod fo feedback cotol desig ux ( ) ca be see as follows: Substitutig u= u( x ) ito (2), we have the autooous closed-loop dyaics x = f( x, u( x)) ad Lyapuov diect ethod ca the be used fo stability aalysis. I actual applicatios, Lyapuov desig ca be coceptually divided ito two steps: (a) choose a cadidate Lyapuov fuctio V fo the syste, ad (b) desig a cotolle which edes its deivative V egative. Soeties, it ay be oe advatageous to evese the ode of opeatio, i.e., desig a cotolle that is ost likely to be able to stabilize the closed-loop syste fist by exaiig the popeties of the syste, ad the choose a Lyapuov fuctio cadidate V fo the closed-loop syste to show that it is ideed a Lyapuov fuctio. Lyapuov desig is sufficiet. Stabilizig cotolles ae obtaied if the pocesses succeed. If the attepts fail, o coclusio ca be daw o the existece of a stabilizig cotolle. Let fuctio V ( x ) be a Lyapuov fuctio cadidate. hus, the task is to seach fo ux ( ) to guaatee that, fo all satisfy x R, the tie deivative of V ( x ) alog syste (2) SAMPLE CHAPERS V ( ) V x ( x) = f( x, u( x)) W( x) x, (3) whee W ( x ) is a positive defiite fuctio. Fo affie oliea systes of the fo x = f( x) + g( xu ), f( 0) = 0 (4) the iequality (3) becoes V V ( ) + ( ) ( ) W ( ) x fx x g xux x. (5) I geeal, this is a difficult task. A syste fo which a good choice of V ( x ) ad W ( x ) Ecyclopedia of Life Suppot Systes (EOLSS)

5 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge exists is said to possess a cotol Lyapuov fuctio. A sooth positive defiite ad adially ubouded fuctio V( x ) : R R + is called a Cotol Lyapuov Fuctio (CLF) fo (2) if V ( x) if { fxu (, )} < 0, x 0. (6) x u R If V ( x ) is a CLF fo affie oliea syste (4), the a paticula stabilizig cotol law, ux ( ), sooth fo all x 0, is give by the Atstei ad Sotag s uivesal cotolle V 2 4 fx ( ) + ( V fx ( )) + ( V gx ( )) V x x x ( ) 0 V, gx ux ( ) = gx ( ) x x V 0, gx ( ) = 0 x he steps of Lyapuov desig ad cocept of Cotol Lyapuov Fuctio ae used fo systes with cotols to diffeetiate the classical te Lyapuov fuctio fo systes without cotols. As a desig tool fo geeal oliea systes, the ai deficiecy of the CLF cocept is that a CLF is ukow. he task of fidig a appopiate CLF ay be as coplex as that of desigig a stabilizig feedback law. Howeve, fo seveal ipotat classes of oliea systes, these two tasks ca be solved siultaeously. Whe V ( x) is oly egative seidefiite, asyptotic stability caot be cocluded fo Lyapuov fuctio ethod diectly. Howeve, if x= 0 is show to be the oly solutio fo V ( x) = 0, the asyptotic stability ca still be daw by evokig LaSalle s Ivaiace Piciple, Ivaiat Set heoe, which basically states that, if V ( x) 0 of a chose Lyapuov fuctio cadidate V ( x ), the all solutios asyptotically covege to the lagest ivaiat set i the set { x V ( x) = 0} as t. I fact, this appoach has bee fequetly used i the poof of asyptotic stability of a closed-loop syste. SAMPLE CHAPERS Lea 2.1 [Babalat] Coside the fuctio () t : R R. If φ () t is uifoly cotiuous ad t 0 φ + li φ ( τ ) dτ exists ad is fiite, the li t φ( t) = 0. t (7) heoe 2.1 [LaSalle] Let (a) Ω be a positively ivaiat set of x = f( x), (b) V( x) :Ω R + be a cotiuously diffeetiable fuctio such that V ( x) 0, x Ω, ad (c) E = { x Ω V ( x) = 0}, ad M be the lagest ivaiat set cotaied i E. he, evey bouded solutio x () t statig i Ω coveges to M as t. o show that a vaiable ideed coveges to zeo, Babalat s Lea is fequetly used. Ecyclopedia of Life Suppot Systes (EOLSS)

6 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge If Ω is the whole space R, the the above local Ivaiat Set heoe becoes the global oe. o pove the asyptotic stability of the syste, we oly eed to show that o solutio othe tha x () t 0ca stay foeve i E. It should be oted that thee ay exist ay Lyapuov fuctios fo a give oliea syste. Specific choices of Lyapuov fuctios ay yield bette, cleae cotolles tha othes. Usually, Lyapuov fuctios ae chose as quadatic fo due to its elegacy of atheatical teatet. Howeve, it is ot exclusive. Othe fos have also bee used i the liteatue, such as eegy-based Lyapuov fuctios, itegal-type Lyapuov fuctios, which have bee applied i the desig of cotolles fo classes of ucetai oliea systes. 3. Lyapuov Desig via Lyapuov Equatio Model Refeece Adaptive Cotol (MRAC) was oigially poposed to solve the poble i which the desig specificatios ae give by a efeece odel, ad the paaetes of the cotolle ae adjusted by a adaptatio echais/law such that the closed-loop dyaics of the syste ae the sae as the efeece odel which gives the desied espose to a coad sigal. I solvig this class of pobles, the Lyapuov equatio plays a vey ipotat ole i choosig the Lyapuov fuctio ad deivig the feedback cotol ad adaptatio echais. I fact, the costuctio of Lyapuov fuctios is systeatic ad staightfowad fo the class of systes which ca be tasfoed ito systes with two potios: (i) a stable liea potio so that liea stability esults ca be diectly applied, ad (ii) atched oliea potio which ca be hadled usig diffeet techiques such as adaptive o obust cotol techiques i diffeet situatios. hus, MRAC ca also be viewed as Lyapuov desig based o Lyapuov equatios. o explai the cocepts clealy, Lyapuov equatio ad Lyapuov stability aalysis ae fistly peseted fo liea tie-ivaiat systes, the adaptive cotol desig fo classes of ukow liea tie ivaiat systes ad ukow oliea systes is peseted by utilizig Lyapuov equatio Lyapuov Equatio SAMPLE CHAPERS hough liea systes ae well udestood, it is iteestig to look at the i the Lyapuov laguage, ad povide a basis of Lyapuov desig fo systes havig liea potios. Fo siplicity, coside the followig siple cotollable Liea ie Ivaiat (LI) systes descibed by x = Ax+ bu, (8) whee x R, ad u R ae the states, ad cotol vaiable, espectively, A R ad b R. It is well kow that thee is always a global quadatic LF, ad the stabilizig cotolle ca be obtaied costuctively. Let the state feedback cotol be Ecyclopedia of Life Suppot Systes (EOLSS)

7 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge u = kx (9) the esultig closed-loop syste will be of the fo x = A x, A = A bk. (10) Fo the liea syste theoy, thee ae ay ways to desig k fo a desiable stable closed-loop syste. he ost ituitive ad diect oe ight be the pole-placeet ethod. I the cotext of this chapte, we shall look at the poble i the sese of Lyapuov desig. Not supisigly, Lyapuov fuctios ca be systeatically foud to descibe stable liea systes owig to the followig theoe. heoe 3.1 he LI syste x = A x is asyptotically stable if ad oly if, give ay syetic positive-defiite atix Q, thee exists a syetic positive-defiite atix P, which is the uique solutio of the so-called Lyapuov equatio PA + A P = Q. (11) Fo such a solutio, the positive defiite quadatic fuctio of the fo V ( x) = x Px (12) is a LF fo the closed-loop syste (10), sice V ( x) = x Qx< 0, x 0. (13) Aothe ethod to desig k is the well kow optial liea quadatic (LQ) desig ethod. o ivestigate the poble i the cotext of CLF, coside the followig Lyapuov fuctio cadidate fo (8) V = x Px, (14) SAMPLE CHAPERS whee P= P > 0. Fo P to defie a CLF (6), the followig iequality should hold if{ x A Px + x PAx + u b Px + x Pbu} < 0 x 0, (15) u R which iplies that u( x ) should take the fo that u( x) = γb Px with γ > 0 (the coespodig liea feedback gai k = γ b P). hus, P defies a CLF if if { A P + PA 2 γ Pbb P } < 0. (16) γ R Such a P ca always be foud though the solutio of algebaic Riccati equatio Ecyclopedia of Life Suppot Systes (EOLSS)

8 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge A P+ PA+ Q 2γ Pbb P = 0, (17) which is kow to solve the optial liea quadatic (LQ) state desig which iiizes 1 2 xqx ad subject to the dyaic costaits 2γ the cost fuctio J = [ + u ] dt 0 iposed by (8) (see Optial Liea Quadatic Cotol (LQ)). Equatio (17) guaatees that V = xqx, ad i tu asyptotic stability of the closed-loop systes. he esults ae eadily available fo ulti-iput-ulti-output (MIMO) systes. echiques i dealig with liea systes i state space ae well established. (see Classical Desig Methods fo Cotiuous LI-Systes, Desig of State Space Cotolles (Pole Placeet) fo SISO Systes, Pole Placeet Cotol, Optial Liea Quadatic Cotol (LQ)) MRAC fo Liea ie Ivaiat Systes o illustate the basic steps i solvig MRAC fo liea tie ivaiat systes, coside the followig LI plat descibed by the state-space odel x = Ax+gbu, (18) whee x R ; u R ae the states ad iput espectively, the cotolle caoical fo as A= 0 0 1, b= 0 a a a A R ad SAMPLE CHAPERS b R ae i with ukow costats ai, i = 1,,, ad cotol iput gai g > 0 is a ukow costat. he objective is to dive x to follow soe desied efeece tajectoy x R ad guaatee closed-loop stability. Let the efeece tajectoy x be geeated fo a efeece odel specified by the LI syste (19) x = A x +g b, (20) whee R is a bouded efeece iput, R A is a stable atix give by A = a a a 1 2 (21) Ecyclopedia of Life Suppot Systes (EOLSS)

9 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge i 1 with a, i = 1,,, chose such that s + a s + + a is a Huwitz polyoial. he efeece odel ad iput ae chose such that x ( ) epesets a desied tajectoy that x has to follow, i.e., x x as t. Coside a geeal liea cotol law of the fo 1 u = k() t x + k () t, (22) whee k ad k ay be chose feely. he closed-loop syste the becoes t x = ( A+ gbk) x+ gkb. (23) It is clea that thee exist costat paaetes k ad k coditios i i i such that the atchig a + gk = a, gk = g (24) hold, i.e., equatios (20) ad (23) ae equivalet. Sice a i ad g ae ukow, so ae k ad k, which eas that cotolle (22) with k = k ad k = k is ot feasible. his poble ca be easily solved usig o-lie adaptive cotol techiques. Fo ease of discussio, let θ = [ k, k ], ˆ θ = [ k( t), k ( )] t, defie paaete estiatio eos ˆ θ = θ θ = [ θx, θ ] with θ x = k k( t), θ = k k( t) ad deote φ = [ x, ]. Accodigly, equatio (23) ca be witte as x = ( A+ gbk ) x+ gbk gb θ xx gb θ A x g b gbφ = + Defie the tackig eo closed-loop eo equatio e= Ae gbφ θ. (25) SAMPLE CHAPERS e= x x. Copaig equatios (20) ad (25) give the θ, (26) which has a stable liea potio ad a ukow paaetic ucetaity iput, which tus out to be easily solvable usig the facts that (i) give ay stable kow atix A, fo ay syetic positive-defiite atix Q, thee exists a uique syetic positive-defiite atix P satisfyig PA + A P = Q (27) Ecyclopedia of Life Suppot Systes (EOLSS)

10 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge as detailed i Subsectio 3.1, ad (ii) the liea-i-the-paaete ucetaity bφ θ ca be dealt with usig adaptive techiques. Owig to the above obsevatios, choose the Lyapuov fuctio cadidate by augetig the Lyapuov fuctio i (11) with a quadatic paaete estiatio eo te as follows 1 V( e, θ) = e Pe+ g θ Γ θ, Γ = Γ > 0. (28) Noticig that θ = ˆ θ, the tie deivative of V is give by 1 V = eqe 2gePbφ θ + 2g θ Γ θ 1 2 = eqe g ( φepb ˆ θ Γ Γ θ ). (29) Appaetly, choosig the paaete adaptatio law as ˆ θ = Γ φe Pb (30) leads to V = eqe 0. Accodigly, the followig coclusios ae i ode: (i) the boudedess of e ad θ, (ii) the boudedess of x ad ˆ( θ t) (i.e., k ( t) ad k ( t ) ) by otig the boudedess of x ad θ, ad the boudedess of the cotol sigal u, ad (iii) the tackig eo li t e 0 usig Babalat Lea 1 because (a) ee < c 0 with costat c > 0 obtaiable fo (30), ad (b) e is uifoly cotiuous sice e is bouded as ca be see fo equatio (26). he basic ideas ae ot oly eadily applicable to MIMO LI systes, they ca also be exteded fo a class of oliea systes as will be detailed ext SAMPLE CHAPERS O ACCESS ALL HE 40 PAGES OF HIS CHAPER, Click hee Bibliogaphy Asto K. J. ad Witteak B. (1995). Adaptive Cotol, 2d Editio, Addiso-Wesley Publishig Copay. [his is a widely used textbook o adaptive cotol, which gives the histoy developet ad all the essetials of adaptive cotol] Ge S.S., Lee.H. ad Hais C.J. (1998). Adaptive Neual Netwok Cotol of Robotic Maipulatos, Ecyclopedia of Life Suppot Systes (EOLSS)

11 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge Wold Scietific, Lodo. [his books teats adaptive eual etwok cotol of obots systeatically ad igoously usig Lyapuov desig] Ge S. S., Hag C.C., Lee.H. ad Zhag. (2001). Stable adaptive Neual Netwok Cotol, Kluwe Acadeic Publishes, Nowell, USA. [his book deals with eual etwok cotol fo soe geeal classes of oliea systes ad poves closed-loop stability igoously] Levie W.S. (Ed.) (1996). he Cotol hadbook, CRC Pess, Boca Rato, FL, [his book is a vey coplete itoductoy efeece] Ioaou P. A. ad Su J. (1996). Robust Adaptive Cotol, Petice Hall, New Jesey. [his is a selfcotaied tutoial book that uifies, siplifies, ad pesets ost of the existig techiques i desigig ad aalyzig odel efeece adaptive cotol systes] Khalil H. K. (1996). Noliea Systes, 2d Editio, Petice Hall, New Jesey. [his efeece book coves all the essetial theoies o stability, Lyapuov desig, oliea cotol, ad backsteppig desig] Kstic M., Kaellakopoulos I. ad Kokotovic P. (1995), Noliea ad Adaptive Cotol Desig, Joh Wiley & Sos, Ic., New Yok. [his book gives a systeatic ad copehesive teatets of backsteppig desig ad its applicatios] Lewis F.L., Abdallah C.. ad Dawso D.M. (1993). Cotol of Robot Maipulatos, Maxwell Macilla Iteatioal, [his book gives a coehesive teatet of dyaics ad thei popeties, ad diffeet cotol techiques] Maio R. ad oei P. (1995). Noliea Cotol Desig: Geoetic, Adaptive ad Robust, Petice Hall, Eglewood Cliffs, New Jesey. [his book gives a systeatic teatet of cotol syste desig based o diffeetial geoety] Naeda K.S. ad Aasway A.M. (1989). Stable Adaptive Cotol, Petice Hall, Eglewood Cliffs, New Jesey. [his book povides a systeatic teatet of odel efeece adaptive cotol] Qu Z. (1998). Robust Cotol of Noliea Ucetai Systes, Joh Wiley & Sos, New Yok. [his book povides a copehesive teatet of obust cotol based o Lyapuov desig] Slotie J. J. E. ad Li W. (1991). Applied Noliea Cotol, Petice Hall, New Jesey. [his book is a widely used text book o applied oliea cotol, the essetial cocepts of stability ad cotolle desig ae peseted elegatly ad easy to udestad] Spog M.W. ad Vidyasaga M. (1989). Robot Dyaics ad Cotol, Joh Wiley & Sos, New Yok, [his book povides the essetial cocepts of obot dyaics ad fudetal tools ad theoies o obot cotol] Biogaphical Sketch SAMPLE CHAPERS Shuzhi Sa Ge, IEEE Fellow, P.Eg, is the Diecto of Social Robotics Lab, Iteactive Digital Media Istitute, ad Supeviso of Edutaiet Robotics Lab, Depatet of Electical ad Copute Egieeig, he Natioal Uivesity of Sigapoe. He eceived his PhD degee ad DIC fo the Ipeial College, Lodo, ad BSc degee fo Beijig Uivesity of Aeoautics & Astoautics. He has (co)-authoed thee books: Adaptive Neual Netwok Cotol of Robotic Maipulatos (Wold Scietific, 1998), Stable Adaptive Neual Netwok Cotol (Kluwe, 2001) ad Switched Liea Systes: Cotol ad Desig (Spige-Velag, 2005), edited a book: Autooous Mobile Robots: Sesig, Cotol, Decisio Makig ad Applicatios (aylo ad Facis, 2006), ad ove 300 iteatioal joual ad cofeece papes. He seves as Vice Pesidet of echical Activities, , ad Mebe of Boad of Goveos, , ad Chai of echical Coittee o Itelliget Cotol, , of IEEE Cotol Systes Society. He seved as Geeal Chai ad Poga Chai fo a ube of IEEE iteatioal cofeeces. He is the Edito-i-Chief, Iteatioal Joual of Social Robotics, Spige. He has seved/bee sevig as a Associate Edito fo a ube of flagship jouals icludig IEEE asactios o Autoatic Cotol, IEEE asactios o Cotol Systes echology, IEEE asactios o Neual Netwoks, ad Autoatica, ad Book Edito fo aylo & Facis Autoatio ad Cotol Egieeig Seies. He Ecyclopedia of Life Suppot Systes (EOLSS)

12 CONROL SYSEMS, ROBOICS AND AUOMAION - Vol. XIII - Lyapuov Desig - Shuzhi Ge was the ecipiet of Chagjiag Guest Pofesso, Miisty of Educatio, Chia, 2008; Fellow of IEEE, USA, 2006; Outstadig Oveseas Youg Reseach Awad, NSF, Chia, 2004; Iaugual easek Youg Ivestigato Awad, Sigapoe, 2002; Outstadig Youg Reseache Awad, Natioal Uivesity of Sigapoe, He is the Chaia ad foudig Diecto of Pesoal E-Motio (PEM) Pte Ltd specialized i iteactive digital ultiedia authoig platfo fo educatio ad e-publishig of e-books. Its poduct, koobits, was the wie of the pestigious IfoCo Sigapoe Awad, Septebe 2008, ad Asia Pacific IC Awad of the E-Leaig Categoy, Jakata, Idoesia, Novebe SAMPLE CHAPERS Ecyclopedia of Life Suppot Systes (EOLSS)