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1 Control Lyapnov Fnctons: New Ideas From an Old Sorce Randy A. Freeman Department of Electrcal and Compter Engneerng Nortwestern Unversty, Tecnologcal Insttte 5 Serdan Road, Evanston, IL freeman@ece.nw.ed James A. Prmbs Control and Dynamcal Systems Calforna Insttte of Tecnology Pasadena, Calforna 95 jprmbs@ot.caltec.ed Abstract A control desgn metod for nonlnear systems based on control Lyapnov fnctons and nverse optmalty s analyzed. Ts metod s sown to recover te LQ optmal control wen appled to lnear systems. More generally, t s sown to recover te optmal control wenever te level sets of te control Lyapnov fncton matc tose of te optmal vale fncton. Te metod can be readly appled to feedback lnearzable systems, and te resltng nverse optmal control law s generally mc derent from te lnearzng control law. Examples n two dmensons are gven to llstrate bot te strengts and te weaknesses of te metod. Control Lyapnov fnctons We consder sngle-npt, control-ane nonlnear systems of te form _x = f(x) + g(x) () were x IR n s te state vector, IR s te control npt, and f and g are known contnos fnctons. Or goal s to constrct a contnos state feedback law = k(x) sc tat x = s a globally asymptotcally stable eqlbrm pont of te resltng closedloop system. Or control desgn wll be based on knowledge of a control Lyapnov fncton (clf), tat s, a C, proper, postve dente fncton V : IR n! IR + sc tat nf L f V (x) + L g V (x) < () for all x 6= [, ]. Te exstence of a clf for te system () s eqvalent to te exstence of a globally asymptotcally stablzng control law = k(x) wc s smoot everywere except possbly at x = []. Moreover, one can calclate sc a control law k explctly from f, g, and V [3]. We wll say tat V as above s a weak clf wen te neqalty () s non-strct, namely, wen nf L f V (x) + L g V (x) (3) for all x. Te exstence of a weak clf does not garantee global stablzablty as does te exstence of a clf. Neverteless, n many cases a weak clf can ndeed be sed to desgn a globally stablzng control law as we wll see n Secton below. Gven a general system of te form (), t may be df- clt to nd a clf or even to determne weter or not one exsts. Fortnately, tere are sgncant classes of systems for wc te systematc constrcton of a clf s possble. As we wll see n Secton 3, tese nclde te class of (globally) feedback lnearzable systems. Inverse optmalty Once we ave fond a clf V, we can constrct a control law = k(x) sc tat te Lyapnov dervatve V _ s negatve at every pont n te state space. To prevent orselves from makng absrd coces n ts constrcton, we wll nsst tat te control law k be optmal wt respect to some meanngfl cost fnctonal. A meanngfl cost fnctonal s one tat places stable penalty on bot te state varable x and te control varable so tat seless conclsons lke \every stablzng control law s optmal" are avoded. Te problem of assocatng some cost fnctonal wt a control law k or a clf V s known as an nverse optmal control problem [, 5, 6, 7, 8, 9,,,, 3,, 5]. As sown n [5], every clf V s te vale fncton for some meanngfl cost fnctonal. In oter words, every clf solves te Hamlton-Jacob-Bellman (HJB) eqaton assocated wt a meanngfl cost. Moreover, one can compte te resltng (nverse) optmal control law k drectly from V, f, and g wtot recorse to te HJB eqaton. Unfortnately, te constrcton of an nverse optmal control law k s not nqe becase V may be te vale fncton for many derent cost fnctonals, eac of wc may ave a derent optmal control. Neverteless, te nverse optmal constrcton can be sed to narrow down te coces for k to tose wc satsfy an optmalty crteron. One metod for generatng an nverse optmal control law k gven a clf V s trog te pontwse mnmzaton of control eort. It s sown n [5] tat control

2 laws of te form n o k(x) = arg mn jj : L f V(x)+L g V(x)?(x) () are nverse optmal, were (x) s cosen to be contnos, postve dente, and sc tat L f V (x)?(x) wenever L g V (x) =. Ts desgn \parameter" represents te desred amont of negatvty of te closedloop Lyapnov dervatve _ V, and derent coces for reslt n derent nverse optmal control laws k. Te contnty of te control law gven by te formla () depends on te coce for (x). If s cosen so tat te strct neqalty L f V (x) <?(x) olds wenever L g V (x) = and x 6=, ten k wll be contnos except possbly at x = [5]. A coce for resltng n a non-strct neqalty cold lead to a dscontnos control law k, so sc fnctons sold be cosen wt care. Also, n some cases we may want to consder a postve semdente, especally wen V s only a weak clf. We wll now specfy a partclarly nterestng coce for (x). Sppose tat we ws to mnmze a cost fnctonal of te form Z J = q(x) + dt (5) were q s a contnos, postve semdente fncton. Let s coose as follows: q Lf (x) = V (x) + q(x) L g V (x) (6) Wen sbsttted nto te formla (), ts yelds te control law 8 q? L f V + [L f V ] + q [L g V ] >< L g V k(x) = (7) wen L g V (x) 6= >: wen L g V (x) = wc was orgnally proposed n [3]. Ts nverse optmal control law s contnos everywere except possbly at x =. More mportantly, t redces to te optmal control for te cost (5) wenever te clf V as te same level sets as te vale fncton. To see ts, let V? (x) be te vale fncton assocated wt (5), and assme tat t satses te HJB eqaton = mn q(x) + + L f V? (x) + L g V? (x) = q(x) + L f V? (x)? Lg V? (x) (8) Sppose tat V = (V? ) for some smoot class K fncton (n oter words, sppose tat V and V? ave te same level sets). Becase te dervatve s always postve, from (7) and (8) we ave k(x) =? L f V? + p [L f V? ] + q [L g V? ] L g V? =? L f V? + q q + [L gv? ] L g V? =? L f V? + q + [L gv? ] L g V? =? L gv? (x) (9) wen L g V (x) 6=, wc s exactly te optmal control for te cost (5). Wen L g V (x) s zero, ten k(x) = wc stll matces te optmal control becase L g V? (x) s also zero. To smmarze, f te level sets of te clf matc tose of te vale fncton, and f s cosen as n (6), ten te nverse optmal control s n fact te optmal control. As a specal case, sppose tat te system s lnear and te cost s qadratc: _x = Ax + B () Z J = x T Qx + dt () If standard stablzablty and detectablty assmptons are satsed, ten tere exsts a nqe symmetrc postve dente solton P to te Rccat eqaton A T P + P A? P BB T P + Q = () One can verfy tat V (x) = x T P x s a clf for ts lnear system. If we coose as n (6), namely, q (x) = [x T (A T P + P A)x] + [x T Qx][x T P BB T P x] q = [x T (Q?P BB T P )x] + [x T Qx][x T P BB T P x] = x T Q + P BB T P x (3) ten te formla (7) generates te standard LQ lnear optmal feedback law = k(x) =?B T P x. 3 Feedback lnearzable systems Let s llstrate an nverse optmal desgn for te class of (globally) feedback lnearzable systems (see [6]). Sppose tere exsts a deomorpsm = (x) wt () = wc transforms or system nto _ = A + B b() + a() () were te matrx par (A; B) s stablzable and te smoot fnctons a and b are sc tat b() = b () =, a() =, and a() 6= for all (we ave normalzed a and b so tat (A; B) represents te Jacoban lnearzaton of te system). Let Q be sc tat T Q approxmates q(x) n te cost fnctonal (5) arond x =, and let P be te symmetrc postve dente solton to te Rccat eqaton (). Ten te fncton V (x) = (x) T P (x) = T P s a clf for ts system, and te nverse optmal control law () s 8 () ><? k(x) = T P B a() wen () > (5) >: wen ()

3 were te fncton s gven by () = T A T P + P A + T P Bb() + () (6) If s cosen as n (6), ten te control law (5) wll locally approxmate te LQ optmal control?b T P for te lnearzed system. We can compare te nverse optmal control law (5) wt te feedback lnearzng control law gven by k(x) =? b() + BT P a() (7) x_ 5 3 Altog bot control laws (5) and (7) globally asymptotcally stablze te system () and locally mnmze te cost (5), te nverse optmal control law (5) s (globally) optmal wt respect to a meanngfl cost fnctonal, wereas te feedback lnearzng control law (7) s not (n general). For example, te feedback lnearzng control law for te system _x =?x 3 + (8) wold cancel te stablzng nonlnearty?x 3, bt te nverse optmal control law wold not becase sc a cancellaton s contrary to meanngfl optmalty. Unfortnately, one-dmensonal examples are not rc enog to llstrate potental ptfalls of te clf desgn metod, prmarly becase all clf's for a scalar system possess essentally te same level sets. Examples n two dmensons We wll rst consder te feedback lnearzable system _x = x (9) _x =?x + x sn(x + x ) + () One can verfy tat te control law? =?x e x +x () mnmzes te cost fnctonal Z J = x + dt () and tat te assocated vale fncton s V? (x) = e x +x? (3) Te Rccat eqaton () yelds P = I, wc means te feedback lnearzng control law (7) s FL =?x + sn(x + x ) () Altog ts control law s not te same as te optmal control law (), t as te same qaltatve sape. Let s now try an nverse optmal desgn sng te clf V (x) = x T P x = x + x. Ts s actally a weak clf for x_ Fgre : Te vale fncton V? (x) from (9). ts system becase te neqalty n () s non-strct wen x =. Neverteless, we can proceed wt te nverse optmal desgn provded we do te followng: rst, we mst coose to be postve semdente and ceck tat te resltng control law s contnos; second, we mst make sre tat LaSalle's teorem apples so tat we can conclde global asymptotc stablty. Keepng tese catons n mnd, we see tat n (6) s (x) = x sn(x + x ) + (x) (5) One can verfy tat te coce = x recovers te feedback lnearzng control law (). Also, te coce = x cos(x + x ) from (6) recovers te optmal control law (); ts was to be expected becase te vale fncton V? and te clf V bot ave crcles as level sets. We next consder an example for wc te level sets of te vale fncton are far from beng ellpsod: _x = x (6) _x =?e x (x + x ) + x e x+3x + e x+x (7) One can verfy tat te control law? =?x e x+x (8) mnmzes te cost fnctonal () and tat te assocated vale fncton s V? (x) = x +? e?x ( + x ) (9) Ts vale fncton s smoot and postve dente, bt t s not proper as can be seen from te noncompactness of some of ts level sets (Fgre ). Ts no clf wll ave te same level sets as V?, and t remans to be seen weter or not some clf desgn can recover te optmal control.

4 x x tme tme x x tme tme tme Fgre : Soltons to (6){(7) from ntal condton (?; ) wt optmal control (8) (sold), nverse optmal control (3) (dased), and feedback lnearzng control (7) (dotted). We wll now try te clf desgn otlned n Secton 3. We let (A; B) be te lnearzaton of te system (6){(7) abot zero: A = ; B = (3)? Wt Q cosen accordng to te cost (), te solton to te Rccat eqaton () s P = I. Ts we wll se te weak clf V (x) = x T P x = x + x. We wll coose as n (6) so tat or control law s gven by te formla (7). For ts example we ave L f V (x) = x x p(x ) + e x+3x? e x (3) L g V (x) = x e x+x (3) were p(x ) represents te smoot fncton p(x ) =? ex x (33) (te apparent snglarty at x = s removable). Upon sbstttng tese expressons nto (7) sng q(x) = x, we obtan te followng control law: k(x) =? x e x?x?x p + e x+3x? e x (3) + qx p + x p e x+3x? e x + e x +3x + e x Ts control law s contnos even tog we sed only a weak clf n te formla (7). Moreover, k(x) = wen x = wc means LaSalle's teorem garantees te global asymptotc stablty of te closed-loop system. Note tat ts nverse optmal control (3) concdes wt te optmal control (8) at ponts were eter x = or x = tme Fgre 3: Soltons to (6){(7) from ntal condton (; ) wt optmal control (8) (sold), nverse optmal control (3) (dased), and feedback lnearzng control (7) (dotted). Fgre sows smlaton reslts for te system (6){ (7) from te ntal condton (?; ). Not srprsngly, te optmal control (8) (sold lne), wc generates a cost () of J = from ts ntal condton, yelds better reslts tan eter te nverse optmal control (3) (dased lne, J = 39) or te feedback lnearzng control (7) (dotted lne, J = 38). In fact, te nverse optmal control generates te gest cost from ts ntal condton. Ts does not contradct nverse optmalty becase ts control (3) optmzes a derent, nspeced cost fnctonal. Fgre 3 sows smlaton reslts from te ntal condton (; ); ere te costs are J = (optmal), J = : (nverse optmal), and J = :5 (feedback lnearzaton). Note tat all tree controllers provde nearly te same performance for small ntal condtons. We conclde ts secton by sowng ow an alternatve clf desgn can recover te optmal control (8) for ts system (6){(7). Recall tat we cannot se te vale fncton (9) as or clf becase t s not a proper fncton (some of ts level sets are not compact). However, we sold be able to nd a vald clf wose level sets look more lke te ones n Fgre tan lke te crclar level sets of te clf sed above. Rater tan coose or clf trog feedback lnearzaton as above, we wll take te system nonlneartes nto accont drng or constrcton of te clf. We rst rewrte te system (6){(7) as follows: _x = x (35) _x =?e x (x + x ) + e x+x ( + x e x+x ) (36) We next observe tat te system obtaned by droppng te second term n (36), namely, _x = x (37)

5 x_ x_ Fgre : Te clf V (x) from () wt c =. _x =?e x (x + x ) (38) s already globally asymptotcally stable. Any Lyapnov fncton for ts trncated system (37){(38) wll be a clf for te complete system (35){(36); or strategy s to nd sc a fncton. Let te fncton W (x) be gven by W (x) = x (x + x ) (39) We dene a C fncton : IR! IR + wt dervatve as follows: (s) = (s) = ( s +s wen s wen s < ( s(+s) wen s (+s) wen s < () () Te dervatve satses (s) = for s and < (s) < for s >. Frtermore, we ave (s)! as s!. It follows tat te fncton V dened by V (x) = (c + )V? (x) + (W (x)) () s C, postve dente, and radally nbonded, were c > s a desgn parameter. One can verfy tat V s n fact a Lyapnov fncton for te trncated system (37){(38), wc means we may se V as a clf n or control desgn. Te level sets of V, sown n Fgre, are smlar n sape to tose of te vale fncton V?. Te rst step n te constrcton of te control law from te clf V s to compte te dervatve of V along soltons to te orgnal system (35){(36): _V = L f V (x) + L g V (x) =?x [c +? (W )]? (W )e x (x + x )? (W )e x W + e x+x ( + x e x+x ) (c + )x e?x + (W )(x + x ) (3) From ts expresson we can conclde tat V s a weak clf: te control =? x e x+x renders V _ negatve semdente. We ave left to coose te fncton n te formla () for te nverse optmal control law. Coosng as n (6) sold prodce a near-optmal control law becase te level sets of or new clf () closely matc tose of te vale fncton (9). Indeed, smlatons from te ntal condton (?; ) sow tat te cost () generated by ts new nverse optmal control s J = : wc s only slgtly ger tan te optmal cost J =. A derent coce for, namely, (x) =?L f V (x) + x e x+x L g V (x) = x [c +? (W )] + (W )e x (x + x ) + (W )e x W + x e x+x L g V (x) () wll exactly recover te optmal control (8). One can verfy tat ts coce () s postve semdente and s terefore a vald coce n te nverse optmal desgn. Te conclson drawn from tese examples s tat te qalty of te clf desgn wt nverse optmalty can depend eavly on te coce for te clf V and te desred sze of ts dervatve. Fndng te best clf for a gven cost wold reqre solvng an HJB eqaton, a task wc te nverse optmal desgn s meant to avod. Wat we need, terefore, are metods for mprovng te coce for te clf wen te controller desgns t generates are nsatsfactory. 5 Dscsson Te Lyapnov desgn dscssed n ts paper conssts of two basc steps: Step #: Step #: nd a clf V se te clf to constrct an nverse optmal control law k Wle a clf wll always exst wenever te system s stablzable, te task of constrctng a clf n Step # may not be feasble for a general nonlnear system. For ts reason, Lyapnov desgn s best sted to systems avng specal strctres wc we know ow to explot (e.g. te feedback lnearzable systems of Secton 3). If a clf cannot be fond for te complete system, t may be possble to \decentralze" te problem, nd clf's for te separate sbsystems, and se varos nterconnecton/small-gan teorems [7, 8, 9, ] to prove te stablty of te complete system. For tose classes of systems for wc metods for constrctng clf's are avalable, great care sold be taken to explot desgn exbltes n Step # so tat good coces for te control law k are possble n Step #. We ave sown tat f te level sets of te clf matc tose of te vale fncton, ten te nverse optmal desgn, wt cosen as n (6), s n fact optmal. Moreover, for feedback lnearzable systems we can always

6 nd a clf wose level sets matc tose of te vale fncton n a negborood of te eqlbrm, and te resltng nverse optmal desgn s locally optmal. In most cases, owever, te level sets wll not matc globally, and te smplest coces n te clf desgn metod may generate control laws wc are far from optmal for large devatons from te eqlbrm. In sc cases, non-obvos coces n te clf desgn may be reqred to approac global optmalty. Peraps te greatest advantage of te Lyapnov desgn metod s ts ablty to accont for ncertanty n te fnctons f and g n te system descrpton (). Indeed, te nverse optmal desgn dscssed n ts paper was orgnally developed for ncertan nonlnear systems admttng \robst" clf's (rclf's) [5]. Altog feedback lnearzaton metods do not easly yeld rclf's for ncertan systems, more powerfl backsteppng metods can often be sccessflly appled []. However, backsteppng desgn metods are extremely exble, and agan te smplest desgn coces n Step # may exclde all good control laws n Step # []. Mc as yet to be dscovered abot ow to make smart coces n Lyapnov desgn. Please see ttp://ot.caltec.ed/ doyle for frter detals and a comparson of reslts on several examples. Acknowledgments: Te ators wold lke to tank Jon Doyle for s contrbtons to dscssons wc revealed tat (6) s a partclarly nterestng coce n te proposed desgn metod. References [] Z. Artsten, \Stablzaton wt relaxed controls," Nonlnear Anal., vol. 7, no., pp. 63{73, 983. [] E. D. Sontag, \A Lyapnov-lke caracterzaton of asymptotc controllablty," SIAM J. Contr. Optmz., vol., no. 3, pp. 6{7, 983. [3] E. D. Sontag, \A `nversal' constrcton of Artsten's teorem on nonlnear stablzaton," Syst. Contr. Lett., vol. 3, no., pp. 7{3, 989. [] R. E. Kalman, \Wen s a lnear control system optmal?," Trans. ASME Ser. D: J. Basc Eng., vol. 86, pp. {, 96. [5] B. D. O. Anderson and J. B. Moore, Lnear Optmal Control. Englewood Cls, New Jersey: Prentce- Hall, 97. [6] B. P. Molnar, \Te stable reglator problem and ts nverse," IEEE Trans. Atomat. Contr., vol. 8, pp. 5{59, Oct [7] P. J. Moylan and B. D. O. Anderson, \Nonlnear reglator teory and an nverse optmal control problem," IEEE Trans. Atomat. Contr., vol. 8, pp. 6{ 6, Oct [8] P. J. Moylan, \Implcatons of passvty n a class of nonlnear systems," IEEE Trans. Atomat. Contr., vol. 9, pp. 373{38, Ag. 97. [9] D. H. Jacobson, Extensons of Lnear-Qadratc Contol, Optmzaton and Matrx Teory. London: Academc Press, 977. [] C. A. Harvey and G. Sten, \Qadratc wegts for asymptotc reglator propertes," IEEE Trans. Atomat. Contr., vol. 3, pp. 378{387, Jne 978. [] D. H. Jacobson, D. H. Martn, M. Pacter, and T. Gevec, Extensons of Lnear-Qadratc Contol Teory, vol. 7 of Lectre Notes n Control and Informaton Scences. Berln: Sprnger-Verlag, 98. [] S. T. Glad, \Robstness of nonlnear state feedback A srvey," Atomatca, vol. 3, no., pp. 5{35, 987. [3] K. E. Lenz, P. P. Kargonekar, and J. C. Doyle, \Wen s a controller H -optmal?," Mat. Control Sgnals Systems, vol., pp. 7{, 988. [] J. Feng and M. C. Smt, \Wen s a controller optmal n te sense of H loop-sapng?," IEEE Trans. Atomat. Contr., vol., pp. 6{39, Dec [5] R. A. Freeman and P. V. Kokotovc, \Inverse optmalty n robst stablzaton," SIAM J. Contr. Optmz., vol. 3, pp. 365{39, Jly 996. [6] R. A. Freeman and P. V. Kokotovc, \Optmal nonlnear controllers for feedback lnearzable systems," n Proceedngs of te 995 Amercan Control Conference, (Seattle, Wasngton), pp. 7{76, Jne 995. [7] Z.-P. Jang, A. R. Teel, and L. Praly, \Smallgan teorem for ISS systems and applcatons," Mat. Control Sgnals Systems, vol. 7, pp. 95{, 99. [8] I. M. Y. Mareels and D. J. Hll, \Monotone stablty of nonlnear feedback systems," J. Mat. Syst. Estm. Contr., vol., pp. 75{9, 99. [9] A. R. Teel, \A nonlnear small gan teorem for te analyss of control systems wt satraton," IEEE Trans. Atomat. Contr.. To appear. [] E. D. Sontag, \Frter facts abot npt to state stablzaton," IEEE Trans. Atomat. Contr., vol. 35, pp. 73{76, Apr. 99. [] R. A. Freeman and P. V. Kokotovc, Robst Nonlnear Control Desgn. Boston: Brkaser, 996. [] R. A. Freeman and P. V. Kokotovc, \Desgn of `softer' robst nonlnear control laws," Atomatca, vol. 9, no. 6, pp. 5{37, 993.