Projection-Based Optimal Mode Scheduling

Transcription

1 Projecion-Based Opimal Mode Scheduling T. M. Caldwell and T. D. Murphey Absrac This paper develops an ieraive opimizaion echnique ha can be applied o mode scheduling. The algorihm provides boh a mode schedule and iming of ha mode schedule wih convergence guaranees. Moreover, he algorihm aes advanage of a line search, and he number of ieraions in he line search is bounded. There are wo ey ingrediens in he algorihm. Firs, a projecion operaion is used ha aes arbirary curves and maps hem o feasible swiching conrols. Second, a descen direcion ha incorporaes he projecion is calculaed using he mode inserion gradien. Similar o derivaive-based finie dimensional opimizaion, he convergence guaranees and sufficien decrease crieria follow from a local approximaion of he cos in he direcion of he search direcion, bu his local approximaion is no he sandard quadraic approximaion. An example demonsraes he seps o implemen he opimizaion algorihm and illusraes convergence. I. INTRODUCTION This paper is concerned wih he problem of swiched sysem opimal conrol. Swiched sysems evolve over disinc dynamic modes, ransiioning beween he modes a discree imes. The problem is o schedule he modes i.e. calculae he sequence of modes and he ransiion imes ha opimize a performance index. As is common, we parameerize he mode schedule by a se of funcions of ime, u), wih values consrained o be eiher or [], [], [5]. While in general, opimizaion based on differeniabiliy is no applicable o ineger consrained problems, we use a projecion-based echnique so ha he mode scheduling problem shares underlying principles, paricularly absolue coninuiy of line search. Projecion operaors are commonly employed o solve consrained opimizaion problems. For example, in [4], he gradien projecion mehod is reviewed for finie dimensional inequaliy consrained opimizaion. Furhermore, in [9], a projecion operaor is used for opimal conrol of rajecory funcionals. Opimizaion echniques based on differeniabiliy locally approximae he cos funcion in order o calculae a new esimae of he opimum [4]. In finie dimensions, he descen direcion is calculaed from he gradien and Hessian which give he firs- and second-order approximaions of he This maerial is based upon wor suppored by he Naional Science Foundaion under award IIS-867 as well as he Deparmen of Energy Office of Science Graduae Fellowship Program DOE SCGF), made possible in par by he American Recovery and Reinvesmen Ac of 9, adminisered by ORISE-ORAU under conrac no. DE-AC5-6OR T. M. Caldwell and T. D. Murphey are wih he Deparmen of Mechanical Engineering, Norhwesern Universiy, 45 Sheridan Road Evanson, IL 68, USA caldwel@u.norhwesern.edu ; -murphey@norhwesern.edu cos [4]. Furhermore, ess such as descen direcion and sufficien descen depend on he gradien [], [4]. In his paper, we coninue our projecion-based swiched sysem wor in [4], [5]. In [4] we showed equivalency beween he projecion-based swiched sysem opimum wih a hybrid maximum principle. In [5] we showed ha he cos is absoluely coninuous wih respec o a search direcion. Wih such a propery one may expec a line search will resul in sufficien descen guaranees for convergence. Indeed, his paper finds his expecaion o be rue. For projecion-based opimal mode scheduling, he sae, x, and swiching conrol, u, are unconsrained. In oher words, x and u need no saisfy he dynamics and he value of u need no have ineger value or. However, unlie embedding mehods [], [5], [9] which embed u) in he inerval [, ], he cos J is calculaed on he projecion P of x, u) ono he se of non-chaering swiched sysem rajecories. In comparison o inserion mehods [7], [8], [8] since u is no consrained o he inegers he local variaions are curves in L [,T] as opposed o necessarily being needle variaions. While he underlying sraegy presened in his paper is fundamenally differen o inserion mehods, he high level algorihm is he same i.e. o ieraively aler he mode schedule so ha here are guaranees on convergence. Furhermore, boh sraegies base updae decisions using he mode inserion gradien, defined in he inserion lieraure. In [7], [8], he inserion ime and insered mode are calculaed direcly from he mode inserion gradien, while in [8] he inserion duraion is also calculaed using an Armijo-lie line search. In his paper, he negaive mode inserion gradien is an L [,T] variaion and is a search direcion similar o he negaive gradien in derivaive-based numerical opimizaion. In his paper, we propose an ieraive mode schedule opimizaion algorihm. The conribuions of his paper are: A) Approximaion of he cos funcion in he direcion of he negaive mode inserion gradien. B) Showing he negaive mode inserion gradien is a descen direcion. C) Tesing for sufficien descen. D) Showing ha bacracing will calculae a sep size ha saisfies sufficien descen in a finie number of ieraions. Similar o opimizaion echniques based on differeniabiliy, we will find ha Conribuions B, C, and D follow largely from Conribuion A. We show Conribuions C and D for he descen direcion calculaed from he mode inserion gradien. We leave he resuls for general descen direcions o fuure wor. This paper is organized as follows: Review of he projecion operaor, projecion-based opimal mode scheduling, and he mode inserion gradien is in Secion II. Secion III

2 reviews he ieraive opimizaion algorihm and discusses he challenges of calculaing a sep size for convergence guaranees. Secion IV examines he derivaive of he cos wih respec o he swiching imes. Conribuion A, he local approximaion of he cos, is in Secion V. Showing he negaive mode inserion gradien is a descen direcion, Conribuion B, is in Secion VI. Secion VII presens boh he sufficien descen and bacracing, Conribuions C and D. Finally, examples are in Secion VIII. II. REVIEW The following reviews swiching conrol of swiched sysems [4], [5], he swiching ime gradien [], [7], [], [], he max-projecion operaor for swiched sysems [4], [5], projecion-based opimal mode scheduling [4], [5], and he mode inserion gradien [7], [8], [8]. A. Swiched Sysems A swiched sysem evolves according o one of N modes f i : R n! R, i {,...,N} a any ime. The conrol problem is o deermine he schedule over he ime inerval [,T] where final ime T >. Noe we will alernaively label he iniial ime T := and final ime T M := T. We consider hree represenaions o parameerize a swiched sysem: mode schedule, swiching conrol, and acive mode funcion. Each represenaion is equivalen in ha a unique mapping exiss beween each. Depending on he maerial, one of he represenaions is ofen clearer for presenaion han he ohers. For his reason, hroughou he paper, we will swich beween he represenaions. The hree represenaions are: Definiion : The mode schedule is composed of he pair {, T} where = {,..., M } is he mode sequence and T = {T,...,T M } is he sricly monoonically increasing se of swiching imes. Here, each mode is i {,...,N}, each swiching ime is T i [,T], and he oal number of modes in he mode sequence is M N. Definiion : The curve u =[u,...,u N ] T composed of N piecewise consan funcions of ime is a swiching conrol if for almos each [,T], P N i= u i) =, and for each i {,...,N}, u i ) {, }, and for each i {,...,N}: u i does no chaer i.e. in he ime inerval [,T], he number of imes each u i swiches beween values and is finie. Denoe he se of all admissible swiching conrols as. Definiion : The piecewise consan funcion of ime : [,T]! {,...,N} is an acive mode funcion if does no chaer i.e. in he inerval [,T], he number of imes swiches beween values {,...,N} is finie. A unique mapping exiss beween each represenaion: mode schedule!swiching conrol) given a mode schedule, {, T}, he swiching conrol u is u) = e i for h [T i,t i ), i =,...,M where e i is he i vecor of he N dimensional ideniy marix; swiching We define he naurals N as he posiive inegers {,,...}. conrol!acive mode funcion) given a swiching conrol u, he acive mode funcion for each ime [,T] is he ) {,...,N} for which e ) = u); acive mode schedule!mode schedule) given an acive mode funcion ), he mode schedule is, T )={,..., M }, {T,...,T M }) where T = { [,T] + ) 6= )} and i = ) for [T i,t i ), i =,...,M. We will wrie u), T u)), when i is necessary o be explici he swiching conrol he mode schedule corresponds o. A swiched sysem is hen he sae and he swiching conrol, x, u) alernaively, x,, T )) or x, ) ha saisfies he sae equaions. Le X and U be ses of Lebesgue inegrable funcions from he ime inerval [,T] o, respecively, R n and R N. Consider a swiched sysem wih n saes x =[x,...,x n ] T X, and N swiching conrols u =[u,...,u N ] T U. The swiched sysem sae equaions are given by ẋ) =F x),u)) := NX u i )f i x)), x) = x. i= ) Formally, define a swiched sysem as: Definiion 4: The pair x, u) X U is a non-chaering swiched sysem if u and x) x) [,T]. R F x ),u ))d =for almos all Denoe he se of all such pairs of sae and swiching conrols by S. B. Swiching Time Gradien The problem of opimizing he swiching imes when he mode sequence is fixed is considered in [], [7], [], []. Consider he problem Z T min JT ):= `x ))d T consrained o he sae equaion Eq.) wih fixed. Supposing each mode, f i x)), and he running cos, `x)), is C, he i h swiching ime derivaive of he cos is [], [7], [], [], []) D Ti JT )= T T i )f i xt i )) f i+ xt i ))) ) where x is he soluion o he sae equaions, Eq.), and is he soluion o he following adjoin equaion where T )=. ) = Df i x)) T ) D`x)) T, T i <<T i for i {...,M} The inegral is he Lebesgue inegral. )

3 C. Projecion Operaor In [4], [5], we propose he max-projecion operaor. The projecion maps curves from he unconsrained se X U o he se of non-chaering swiched sysems, S. In order o define he max-projecion, we firs define he mapping Q : U!. Suppose µ U, hen Q i µ)) := NY µ i ) µ j )). 4) j6=i where :R! {, } is he sep funcion i.e. µ i ) µ j )) = if µ i ) µ j ) < and µ i ) µ j )) = if µ i ) µ j ). Noe Q is no well defined for all curves in U. For example, µ i and µ j may have equal greaes value for a conneced inerval of ime. For his reason, le us only consider a subse R U for which Q is well defined and maps o. We refer o his subse as he admissible subse of U. In [5], we give a sufficien condiion for a form of µ o be an elemen of R. Now, define he max-projecion as: Definiion 5: Tae µ R. The max-projecion, P : X R! S, a ime [,T] is ẋ) =F x),u)), x) = x P ),µ)) := u) =Qµ)). 5) Noice he max-projecion does no depend on. The unconsrained sae is included in he lef hand side of he definiion in order for P o be a projecion. Oher projecions proposed in [4] do depend on. D. Projecion-Based Opimal Mode Scheduling Define he usual cos funcion as Jx, u) = Z T `x ),u ))d where he running cos, ` : X U! R is coninuously differeniable wih respec o boh X and U. The problem of ineres is o minimize J wih respec o x and u under he consrain ha x and u consiue a feasible swiched sysem i.e. x, u) S. This paper furhers our wor in [4], [5], in which we consider an equivalen problem o he consrained problem where he design variables are elemens of an unconsrained se X, U) and he cos is evaluaed on he projecion of he design variables o he se of feasible swiched sysem rajecories: Problem : Suppose P : X U! S is a projecion i.e. PP,µ)) = P,µ). Solve arg min JP,µ)).,µ)X U Noice he cos is calculaed on admissible sae and swiching conrol rajecories. Furhermore, Problem is equivalen o he consrained problem arg min x,u)s Jx, u) [4], [5]. E. Mode Inserion Gradien For projecion-based swiched sysem opimizaion, he cos does no have a naural gradien. However, i does have a funcion wih a similar role in he opimizaion as he gradien does for finie dimensional opimizaion. This funcion is referred o as he mode inserion gradien [7], [8], [8]. The mode inserion gradien calculaes he change o he cos from insering a mode a some ime for an infiniesimal inerval. The mode inserion gradien a ime [,T] and mode a {,...,N} is d a ) := ) T f a x)) f ) x))) 6) where is he soluion o he adjoin equaion Eq.) and ) is he acive mode funcion [7], [8], [8]. Since he mode inserion gradien can be calculaed for each [,T] and mode a {,...,N}, define d : [,T]! R N o be he mode inserion gradien of u. I is he lis of he N mode inserion gradiens of each mode i.e. d) = {d ),...,d N )}. In Secion VII-A, he proof of sufficien descen relies on he assumpion ha d ab ) := d a ) db ) is Lipschiz coninuous. The following Lemma gives he condiions on f a and f b o ensure his assumpion is valid. Lemma Lipschiz condiion for d ab )): Suppose d is he mode inserion gradien for some u. If here exiss K > such ha for each [,T], x) R n and for each j {,...,N}, f j x)) is C and D f j x)) apple K hen here is an L > such ha for each a 6= b {,...,N} and, [,T], d ab ) dab ) apple L Proof: Firs, d ab = ) T f a x)) f b x))) where he sae and adjoin equaions are in Eqs.) and ). Consider each [,T], x) R n and j {,...,N}. Since D f j x)) <K, here is a K > and K > such ha f j x)) applek and Df j x)) applek. Therefore, for all u, ẋ) =F x),u)) apple K and ẍ) =D F x),u))f x),u)) apple K K. By he assumpions on F x),u)), for u, F x),u)) is piecewise coninuous wih respec o and Lipschiz wih respec o x). Therefore, x; u), defined as he soluion o Eq.) for u, is unique. Define g )) := D F x; u),u)) T ) D`x; u)). Since D F x; u),u)) applek and D F x; u),u)) is piecewise coninuous in for all u, g )) g )) appled F x; u),u)) T ) = K ) ). ) Therefore, ; u), defined as he soluion o Eq.) for u, is unique. Thus, here is a K > such ha for all u, ; u) apple K. Addiionally, here is a K > such ha for In his paper, he mode inserion gradien is defined as d, an n- dimensional lis of curves, while in [7], [8], [8], he mode inserion gradien is d a), he evaluaion of d for he a h mode a ime. 7)

4 all u, ; u) = g ; u)) applek. I follows ha for each, [,T], ; u) ; u) <K. From Eq.7), here is L such ha ; u) ; u) apple L. Noe, ; u) = D F x; u),u)) ; u),fx; u),u))) D F x; u),u)) T ; u) D `x; u))f x; u),u)). By he bounds on F, ), DF, ), and D F, ), and ha ; u) and ; u) are Lipschiz, here is L > such ha ; u) ; u) applel. Finally, by hese bounds and ha ; u), ; u) and ; u) are Lipschiz, dab is Lipschiz wih some consan L>. III. ITERATIVE OPTIMIZATION This paper pursues he problem of calculaing he swiching conrol u and swiched sysem sae x ha opimize he performance meric Jx, u) using projecion-based echniques. Similar o derivaive-based algorihms for opimizaion, an ieraive algorihm is proposed. Ieraive opimizaion mehods compue a new esimae of he opimum by aing a sep in a search direcion from he curren esimae of he opimum so a sufficien decrease in cos is achieved. The descen mus be sufficien so ha he sequence generaed by he ieraive opimizaion algorihm converges o a saionariy poin or a leas for an opimaliy funcion o go o zero. The problem of convergence of ieraive opimizaion algorihms is considered for boh smooh [], [4], [6] and non-smooh problems [], []. Pola and Wardi, in [7], consider he case where he cos minimizing sequence is no guaraneed or even liely o have an accumulaion poin. In he conex of his paper, he se of conrol inpus is infinie-dimensional and incomplee and herefore, he sequence of conrol inpus calculaed by he ieraive algorihm o minimize he cos migh no have an accumulaion poin. Indeed, in Wardi s recen wor wih Egersed, [8], on swiched sysem opimizaion, Wardi and Egersed argue his poin when comparing heir ieraive algorihm and convergence resuls o Gonzalez e al. s similar resuls [8]. Wardi and Egersed give heir convergence resul wih respec o an opimaliy funcion going o zero while Gonzalez e al. assume an accumulaion poin exiss. We provide a similar resul o Wardi and Egersed in Secion VII-C. The ieraive mehod follows. Noe, in he algorihm and for he res of he paper, a variable wih he superscrip implies ha he variable depends direcly on u. Algorihm : Choose u and se =. 4 ) Calculae d, Eq.6). ) Calculae sep size by bacracing, Secion VII-B. ) Updae: u + = Qu d ) Eq. 4). 4) If u + saisfies a erminaing condiion, hen exi, else, incremen and repea from sep. u ) d ) d ) u + = Qu d ) = u + ) u + ) u ) T Fig.. Example curves u =[u,u ]T and d =[ d, d ]T as well as he updaed curve u + = Qu d ) where =. The value is given in Eq.8) where is shown, he acive mode is )=and he insered mode is a =. Calculaing correcly is criical for he sequence of u generaed by execuing he algorihm o locally minimize he cos. Remars: ) The negaive mode inserion gradien, d mus be a descen direcion in order o guaranee here is a R + for which Ju + ) <Ju ). The definiion of and proof ha d is a descen direcion are given in Secion VI. ) The sep size mus be chosen so ha a sufficien descen is achieved i.e. so ha if he algorihm calculaes an infinie sequence {u } hen lim! Ju ) has locally minimal value. The sufficien descen condiion and proof of exisence of a sep size achieving sufficien descen is in Secion VII-A. Furhermore, Secion VII-B considers bacracing for calculaing such a sep size. An example of one ieraion of Algorihm is in Fig.. Noice in he example, he number of modes in he mode sequence of Qu d ) increases by 4 compared wih he mode sequence of u. Also noice if were much smaller han hen u + would equal u. In oher words, mus be large enough for Qu d ) 6= u. Noe for he res of he paper: Since he search direcion is he negaive mode inserion gradien, d, calculaed from u, we assume he condiions in Lemma are rue. In oher words, he sae x calculaed as he soluion o he sae equaions, Eq.), wih swiching conrol u is such ha x ) R n for each [,T], for each i {,...,N}, f i C, and here is K>such ha for each [,T] and i {,...,N}, D f i x )) applek.

5 A. Sufficienly Large Sep Size for Differing Mode Schedules As can be seen in Fig., if is small enough, hen Qu d ) equals u and he updaed mode schedule does no differ from he previous mode schedule. In oher words, here is > such ha for every [, ), u = Qu d ). We wish o calculae. Define ) {,...,N} as he acive mode of u a ime. By Eq.4), for Qu d ) o differ from u, here mus be a ime [,T] and mode a {,...,N}, a 6= ), for which u a ) ) d a ) ) >. Noe, u a ) ) :=u a) u ) ) = and d a ) ) = d a). Therefore, his mus be greaer han /d a). Consequenly, here mus be a a {,...,N} and [,T] for which d a) is negaive valued in order for he mode schedule of Qu d ) o change for any. The lower bound on R + for which u 6= Qu d ), labelled, is calculaed from he pair a, ): Define R: a, ) = arg min a{,...,n},[,t ] d a) This value is picured in Fig.. Finally, := d a ). 8) :=. 9) Noe is always non-posiive since d ) ) = for all [,T]. Since as!,!, we use as he opimaliy funcion for esing when o erminae Algorihm. Equivalen calculaions o are used in he mode inserion gradien lieraure, [7], [8], [8]. IV. DERIVATIVE OF THE COST WITH RESPECT TO THE STEP SIZE In he opimizaion procedure Algorihm, a new esimae of he opimum is obained by varying from he curren esimae and projecing he resul o he se of feasible swiched sysem rajecories. Fix u. We consider he cos as a funcion of only he sep size. Define J ):=JPu d )), which is only variable on R +. Recall Conribuion A of he paper. We wish o approximae J ) near. In order o do so we mus firs invesigae he derivaive of J ). Noe, when he mode sequence is consan, only he swiching imes of Qu d ) vary as varies. Define as he R + where he mode sequence changes. 5 := { R + 8 >, 9 B ) \ R +, where Qu d )) 6= Qu d ))}. The mode sequence is consan for all 6 u,v and only he swiching imes vary. For his reason, we use he mode schedule represenaion. Define ):= Qu d )) = 5 The ball B )=, + ). {,..., M } and T ):=TQu d )) = {T ),..., T M )}. The cos parameerized by he mode schedule is J ), T )) := J ) Assuming he cos is differeniable a cos wih respec o is, he derivaive of he DJ )=D J ), T )) DT ) ) where D J ), T )) is he swiching ime gradien, Eq.), and DT ) is he derivaive of he swiching imes wih respec o he sep size and is given in he following lemma. The proof is in [5]. Lemma Derivaive of swiching imes): If u and / u,v i.e. ) is consan, hen he i h elemen of he derivaive of T ), DT ) i = DT i ), is given for he following wo cases: ) If T i ) is no a criical ime of µ := u i i+ i i+ d, hen i i+ DT i )= u T i i+ i )) T i i+ i )), ) ) or if T i ) is a disconinuiy poin of µ and i i+ µ T i i+ i ) ),µ T i i+ i ) + )), hen DT i )=. As follows from Eq.), he derivaive of he cos, DJ ) is given by he do produc of he resul in Lemma wih he swiching ime gradien, Eq.), as long as he mode sequence is consan and each swiching ime saisfies he condiions for eiher case or case. In general, he derivaive will no exis everywhere. For example, DJ ) goes unbounded for where Qu d ) has a swiching ime T i ) ha approaches cri [,T] where d i i+ cri) = see Eq.). In fac, he derivaive of he cos will liely go unbounded a since d i i+ ) =a maximum and minimum poins. When his is he case, he cos can no be approximaed a direcly using a Taylor expansion. However, he cos may sill be approximaed, as we will see in he nex secion. V. APPROXIMATION OF THE COST AND SWITCHING TIMES Many of he algorihms and heory in opimizaion is designed from local approximaions of he cos funcion. Indeed, he gradien and Hessian are he soluions o local quadraic models []. Also, here are derivaive-free mehods ha mae descen decisions based on local approximaions [4]. For he projecion-based opimal mode scheduling problem, he design variable µ is infinie dimensional and U does no form a Hilber space. Therefore, he gradien of he cos is no expeced o exis. However, he cos may sill be approximaed Conribuion A of he paper). This approximaion will be useful for esing a candidae descen direcion Conribuion B), proving sufficien descen Conribuion C) and designing a bacracing algorihm Conribuion D) as we will see in Secions VI and VII.

6 u ) d ) u ) d ) d ) µ = u d d ) µ ) u ) u ) µ ) µ ) µ ) T i ) T i ) T i+ ) Fig.. Example curves of u, d and u d showing ype- lef) and ype- righ) swiching imes. The direcion in ime he swiching imes for > vary are also shown. A. Approximaion of he Swiching Times Recall Conribuion A in which we wish o locally approximae J ) in a neighborhood of for >. There exiss some > for which ) is consan for, + ), which follows from Lemma of [5]. Consequenly, only T ) varies for, + ) and he approximaion of J ) depends direcly on he approximaion of T ). For he mode schedule o vary for, + ), a leas one swiching ime of T + ) mus vary wih. Suppose T i ) is his swiching ime separaing modes i {,...,N} and i+ {,...,N} in he mode schedule. Ofen, a funcion approximaion is made from is Taylor expansion. For T i ), however, DT i + ) may be unbounded and in which case, T i ) would no have a firs-order Taylor expansion. Referring o Eq.), DT i + ) is unbounded when d T i i+ i + )) = and since d ) = a i i+ exremums, i is liely for DT i + ) o be unbounded. We will shorly presen an alernaive approximaion for when T i i+ i + )) =, bu since he approximaion depends on wheher T i i+ i + )) is zero or no, we label he swiching imes a wih a ype. Definiion 6: Suppose u, >, and T i ) T ) is he swiching ime beween modes i and i+ ) for, + ). The ype of swiching ime T i ) is m T i )) N where m T i )) = min{m N d m) T i i+ i )) 6= } For he purposes of his paper, we will only consider ype- and ype- swiching imes since we do no foresee a reason o expec ype- or greaer. In boh examples in Secion VIII, only ype- and ype- swiching imes were encounered. Fig. shows wo example ses of curves d for which ype- picured lef) and ype- picured righ) swiching imes occur. Before even considering an approximaion of T i ) for eiher ype, we firs show T i ) is coninuous in a neighborhood of T i + ). Lemma Coninuiy of swiching imes): Suppose u and here exiss > such ha for, + ), T i ) T ) is he swiching ime beween modes i and i+ ). If m T i )) = or, hen here is, ] such ha for all [, + ], T i ) is coninuous. Proof: The proof follows from he wo facs ha d i ) is Lipschiz and DT i+ i ) exiss when T i ) is no a criical poin of µ ) :=u i Lemmas and respecively. ) i+ d i ) see i+ If m T i )) =, hen d i i+ ) 6=. Since d ) is Lipschiz, ) is coninuous. Therefore, i i+ i i+ here is R such ha for all T i ) [T i ),T i )+ ], T i i+ i )) 6=. Since DT i ) exiss when T i i+ i )) 6=, here is, ] such ha for all [, + ], T i ) is coninuous. Finally, If m T i )) =, hen d i i+ ) = bu di i+ ) 6=. Furhermore, since d ) is i i+ Lipschiz, here is R such ha for all T i ) [T i ),T i )+ ], d T i i+ i )) and d i i+ i )) are sricly monoonic. Consequenly, T i i+ i )) 6= for T i ) T i ),T i )+ ] and hus by Lemma, DT i ) exiss. I follows ha here is, ] such ha for all, + ], T i ) is coninuous. All ha remains is o prove T i ) is coninuous from he righ. Recall u i T i+ i )) d i T i+ i )) = for T i ) o be a swiching ime. Rewriing, d i T i+ i )) = u i T i+ i ))/. Since d i ) is sricly monoonic for i+ [T i ),T i )+T i + )], d is bijecive in i i+ his domain and he inverse is coninuous. Thus, T i )= d i i+ i i+ i ))/ ) where d ) is he inverse i i+ funcion of d ). There is >, such ha T i i+ i ) T i ) <. Therefore, u i T! i+ i )) u i T i+ i ))! <. d ab d ab By he coninuiy of d ab ), here exiss some ) such ha u i i+ T i )) Following, < ) u i i+ T i )) = n < min < ). o ) + ), =: ), which proves coninuiy. The approximaion of T i ) for > near when m T i )) = or is given in he following lemma. Lemma 4 Approximaion of swiching imes): Consider u. Suppose here exiss > such ha for, + ), T i ) T ) is he swiching ime beween modes i and i+ ). There is, ]

7 such ha for all [, + ), m T i )) = implies T i )=T i + ) u i i+ + Ti ))d + Ti i i+ )) Ti + i i+ )) and m T i )) = implies T i )=T i + ± ) h u i i+ + Ti ))d + Ti i i+ )) d Ti + i i+ )) i )+o ) ) ) ) hspacep + o ) Proof: For T i ) o be a swiching ime, u T i i+ i )) d T i i+ i )) =. Define ) = T i ) T i ). Begin wih he case where m T i )) =. Taylor expand d ) around T i i+ i + ) for a neighborhood of T i + ): u i i+ T i + )) + d i i+ T i + )) + d i i+ T i + )) )+o )) =. Since d ) is Lipschiz, here is a R such ha i i+ his equaion is valid for each T i ) [T i ),T i )+ ). Furher, due o Lemma, T i ) is coninuous and herefore here exiss a, ] such ha his equaion is valid for [, + ). Noing d T i i+ i + )) = u T i i+ i + ))/ and reordering, )= u T i i+ i + )) T i i+ i + )) ) + o ) Taylor expanding around, concludes in Eq.). Now consider he case m T i )) = where T i i+ i + )) =. Taylor expand d ) i i+ around T i + ) in a neighborhood of T i ) and recall u T i i+ i )) d T i i+ i )) = : u i i+ T i + )) + d i i+ T i + )) + d i i+ T i + )) ) + o ) )=. Since d ) is Lipschiz, here is a R such ha i i+ his equaion is valid for each T i ) [T i ),T i )+ ). Furher, due o Lemma, T i ) is coninuous and herefore here is a, ] such ha his equaion is valid for [, + ). Noe d T i i+ i + )) = u T i i+ i + ))/. Reordering, i T i+ i + )) ) = u i T i+ i + )) +o ) ). Taylor expanding around, d T i i+ i + )) ) = u T i i+ i + )) +o )+o ) ), 4) By he Taylor expansion of d i i+ ) around T i + ), o ) ) is of lesser order han d i i+ T i + )) ). In order for he equaliy of Eq. 4) o be rue, o ) ) mus also be of lesser order han. Therefore, o ) )= o ). Recall = u i i+ T i + ))/d i i+ T i + )) and reorder: ) = u i i+ T i + )) d Ti + i i+ )) d Ti + i i+ )) ) +o ). 5) There is, ] such ha for each [, + ), d i i+ T i + )) d i i+ T i + )) ) >o ). Since m T i + )) =, u T i i+ i + )) and d T i i+ i + )) mus have opposie signs. Therefore, u T i i+ i + )) d T i i+ i + )) d T i i+ i + )) ) > As such, he righ side of Eq.5) has a single posiive real valued square roo and a single negaive real valued square roo for each, + ], compleing he proof. B. Approximaion of he Cos For smooh finie dimensional opimizaion, he firs order erm of he approximaion of he cos is he gradien composed wih he search direcion. We find ha similar o he finie dimensional gradien, he mode inserion gradien, Eq.6), has a similar role for approximaing he projecionbased swiched sysem cos. Le ) = {,..., M } and T ) = {T ),...,T M )} be he mode schedule for > near. Le J ) be he firs-order Taylor expansion of J ):=J ), T )), around T + ): J ) = J ) + D J + ), T + )) T ) T + )) The erm D J + ), T + )) is he swiching ime gradien Eq.) and hus J ) becomes J )=J ) + P M i= T i + )) T [f i xt i + ))) f i+ xt i + )))]T i ) T i + )). By he definiion of, here is a leas one T i ) T ) ha is no consan for > near. If increasing, noice he acive mode funcion a ime T i ) is T i )) = i+. Alernaively, if decreasing, noice T i )) = i. Thus, if T i ) is increasing, hen T i + )) T [f i xt i + ))) f i+ xt i + )))] = T i + )) T [f i xt i + ))) f T i + )) xt i + = d i T i + )) =, )))]

8 which is he mode inserion gradien of i jus afer T i ) and is also he opimaliy value, Eq.8), of u. Similarly, if decreasing, hen T i + )) T [f i xt i + ))) f i+ xt i + )))] = d T i+ i + )) =. Se! i =if T i ) is increasing or consan in value wih and! i = if decreasing i.e.! i = al.! i = ) implies here is > such ha for each, + ), T i ) T i ) al. T i ) <T i )). Then, J ) is J )=J ) + P M i= )!i T i ) T i )). 6) Approximaions of he swiching imes are given in Secion V-A. Recall he differen ypes of swiching imes. Pariion {,...,M } ino ses of equivalen ype of swiching ime. Define I as he se of indexes of he ype- swiching imes a and I as he se of indexes of ype- swiching imes a. In oher words, for j =,, I j = {i {,...,M } m T i + )) = j}. Furher, define m := max{m T i + ))} M i= ) 7) o have he value of greaes ype of swiching ime a. The approximaion of he swiching imes for m T i + )) = and is given in Lemma 4. We see ha he swiching imes wih he greaes ype will dominae he approximaion of he cos e.g. ype- swiching imes vary linearly wih while ype- swiching imes vary wih ). Label he approximaion of he cos wih he approximaion of he swiching imes as Ĵ m ; ). If m =, hen Ĵ ; )=J ) + X ) )!i ii T i + )) ), 8) while if m =, hen Ĵ ; )=J ) X ii p ) d T i + )) ), 9) The following lemma saes ha Ĵ m ; ) dominaes he remaining erms of J ) for > near. In oher words, Ĵ m ; ) is a valid approximaion of J ) near. Lemma 5 Approximaion of he Cos): Se J )=Ĵ m ; )+R ) where R ) is he remainder. If m =or, hen here exiss > such ha for all, + ), Ĵ m ; ) J ) R ). Proof: The firs order approximaion of J ) wih respec o ):=T ) T ) is J ), see Eq.6). Thus, J )= J )+o ) ). The approximaion Ĵ m ; ) is a furher approximaion from J ), which includes he approximaion of ) i := T i ) T i ) using Lemma 4. Consider m =firs. Se H =I ) c as he complemen of I. By he definiion of m, for each h H, ) h =. Therefore, using Eq.), ) varies linearly wih and hus, J )=Ĵ ; )+o ). Therefore, R ) = o ) and Ĵ ; ) J ) R ). Now for he case where m =. Firs, he approximaions of ) i = T i ) T i ) for i {,...,M } are a leas of order ) and hus o ) ) =o ) ). Second, se H =I ) c is he complemen of I. For each h H, ) h is a leas order ). Thus, for each h H, )! h ) h = o ). Finally, plugging Eq.) in for each i I, )!i ) i = = p ) d i +! Ti + i )) ) + o ) ). Therefore, he remainder erm is R )= X X o ) )+ o ii m ) hh +o ) )=o ) ). )) Since Ĵ ; ) J ) is no o ) ), he lemma is proven. As we show nex, he negaive mode inserion gradien is a descen direcion. VI. DESCENT DIRECTION In order o show sufficien descen Conribuion C) and for bacracing o be applicable Conribuion D), d mus be a descen direcion Conribuion B). In his secion we prove d is a descen direcion direcly from he approximaion of he cos Conribuion A). The search direcion d is a descen direcion if here is a > such ha for each, + ), J ) <J ). The following lemma saes ha d is a descen direcion. Lemma 6 Descen Direcion): If m =or and here exiss an a {,...,N} and a [,T] for which d a) <, hen here exiss > such ha for each, + ), J ) <J ). Proof: Firs, noe apple d a) <. The proof follows from showing Ĵ m ; ) J ) <, m =or, for > and invoing Lemma 5 o argue Ĵ m ; ) dominaes he remainder for near. Refer o Eqs.) and ) for Ĵ m ; ) and consider m =firs. Clearly Ĵ ; ) J ) < if for each i I, ) )!i <. ) T i + )) Recall for T i + ) o be a swiching ime, u i i+ T i + )) d T i i+ i + )) =. Using he! i noaion, u i+!i T i + )) T i + ))

9 d T i + )) = d T i + )) =. The derivaive wih respec o mus be zero: d T i + )) + d T i + )) T i + )= Rearranging and noing d T i + )) = <, d T i + )) = ii T i + ). Recall! i =implies T i + ) > and hus d T i i + )) >. Similarly,! i = implies T i + ) < and hus T i+ i + )) <. Therefore, Eq.) is rue. Now for m =. Since d T i + )) >, p X ) d T i + )) ) < and hus Ĵ ; ) J ) <. Since Ĵ m ; ) J ) <, m =or, for all > and by Lemma 5, Ĵ m ; ) J ) dominaes he remainder for near, he Lemma is proved. The following secion gives a condiion on he sep size for sufficien descen. VII. SUFFICIENT DESCENT Since d is a descen direcion, here is a > in he neighborhood of such ha J ) <J ). Therefore, by choosing such a, each execuion of he loop in Algorihm will resul in a cos decrease from he previous ieraion. Supposing J ) is bounded below by J R, he algorihm will converge o a cos H J. However, i is unclear wheher H is he cos a a local minimum unless each saisfies a sufficien descen condiion and is calculaed from bacracing. I can be unclear, hough, wha i means for H o be a local minimum. In finie dimensional derivaive-based opimizaion, he opimizaion algorihm converges o a saionariy poin where he gradien of he cos is zero. Since he se U is infinie dimensional and no a Hilber space, here is no reason o expec a gradien of JP )) o exis. Insead of he normed gradien, we choose a differen opimaliy funcion on U and give condiions for which i goes o zero. This opimaliy funcion is, which is calculaed from Eq.8). 6 If =, hen = / = which implies ha d has zero uiliy o reduce JPu )) furher. In ha respec, u is a saionariy poin for he descen direcion d. In his secion, we give he sufficien descen condiion Conribuion C), show ha a sep size ha saisfies he sufficien descen condiion can be calculaed in a finie number of bacracing ieraions Conribuion D) and finally ha execuing Algorihm for such a resuls in lim! =. Each of hese conribuions follows from he approximaion of he cos Conribuion A). 6 The opimaliy funcion has he same role in [6], [8], [8]. A. Type Sufficien Descen Condiion The sufficien descen condiion Conribuion C) follows direcly from he approximaion of he cos Ĵ m ; ), Eqs.8) and 9) Conribuion A). Se, ). The ypem sufficien descen condiion is J ) J ) < Ĵ m ; ) J )). We sudy he ype sufficien descen condiion since he greaes ype of swiching ime a is usually m =. In fac, in he example in Secion VIII, each of he 5 ieraions of Algorihm insered ype- swiching imes. Excep by design, m is rarely greaer han. However, m =is common. A, ype- swiching imes occur a swiching imes of u or a he boundary imes. Since he approximaion of ype- swiching imes is linear in ), for m =, sufficien descen and bacracing for projecion-based swiched sysem opimizaion and swiching ime opimizaion are equivalen see [], [7], [], [] for swiching ime opimizaion. For hese reasons, only he ype- sufficien descen is considered in his paper. Definiion 7: Se p ) s = X ii d T i + )) The ype sufficien descen condiion is ) J ) J ) < s ) ) The following Lemma finds ha here exiss a ˆ > for which each, ˆ] saisfies he ype- sufficien descen condiion. The sep size ˆ is he minimum of, and, each given in he lemma. The firs,, is he sep size where for each, ), J ) is differeniable. In oher words, is an upper bound on where he derivaivebased approximaion is valid. The second,, depends on he consan L ha saisfies he Lipschiz condiion on he second ime derivaive of d, which exiss based on he assumpions made in Secion III and due o Lemma. The hird,, is a consan scaling away from i.e. = apple where depending on, ), apple is beween p.577 and. In he following Lemma, se := min ii d T i + )). Lemma 7: Suppose m =and here exiss > such ha for each i I and, ), T i ) exiss. Se. and = 6L q p q A. Then, defining ˆ := min{,, }, he ype- sufficien descen condiion, Eq.), is rue for each, ˆ]. Proof: Recall from Eqs.8) and 9), = / = d T i + )) < for each i I. Also, since d ) is

10 Lipschiz and d T i )) =, for each i I, here is a neighborhood of for which d T i )) >, )!i d T i )) > and d T i )) >. Se H ):= p cardi ) ) ) Noice he righ hand side of he ype- sufficien descen condiion, Eq.), is greaer han H ) for all >. The proof follows by finding he, ) for which he derivaive of lef hand side of Eq.) is more negaive han he derivaive of he righ hand side. The derivaive of he lef hand side is DJ )= X ) d!i T i )) ii T i )) which is negaive valued. The derivaive of he righ hand side is bounded below by DH ): p DH ):= cardi ) ) ) ) The res of he proof shows DJ ) <DH ) for all, ˆ). Se i ) = T i ) T i ). Since d T i )) is Lipschiz, by he mean value heorem, )!i d T i )) apple d T i )) ) L ). Therefore, for i ) apple i,max := d i +! i Ti )) L )!i d T i )) apple d T i )) i ). 4) By Lipschiz, a lower bound of d T i )) for i ) apple i,max is d T i )) d T i + )) + L i ) d T i + )). By he Taylor expansion of d T i )) around T i ), wih remainder rt i )), + + rt i )) i ) =. For ) < i,max he lower bound of d T i )) is also he lower bound of he remainder erm. In oher words, rt i )) > d T i + )) and hus for i ) < i,max, i ) d T i + )) ). 5) Indeed, for each i I and, min{, }], he righ hand side of Eq.5) is less han or equal o i,max. Plugging ino he righ hand side of Eq.5) reduces o, L d T i + )) apple L apple i,max. Therefore, Eqs 4) and 5) are rue for every, min{, }]. For hese, an upper bound on )!i d T i )) is )!i d T i )) apple d T i + )) ). Le = max ii for each i I, d T i + )) and = / >. Thus, )!i d T i )) apple ) ). 6) To find a upper bound on d T i )), inegrae Eq.4) wih respec o i ). d T i )) < + R i ) d T i ))sds = + d 4 T i )) i ) Using he bound in Eq.5) and by seing ) = + ), d T i )) apple ) 7) Wih he bounds on d T i )), Eq.7), and )!i d T i )), Eq.6), DJ ) is bounded above by DJ ) apple cardi ) ) Comparing Eqs. ) and 8), ) p ) ) p, ). 8) implies DJ ) <DH ), which is valid for every min{,, } = ˆ. I follows ha each, ˆ] saisfies he sufficien descen condiion. B. Bacracing Calculaing ˆ =min{,, } direcly is compuaionally inefficien due o. Calculaing and is possible hough: is he neares > o for which J ) is no differeniable and herefore, can be calculaed from nowledge of he criical imes of u and d ; is a consan scaling from. Conversely, requires calculaing he Lipschiz consan L a priori. Similar o smooh finie dimensional opimizaion [], [], i is more efficien o calculae a sep size ha saisfies he sufficien descen crieria using a bacracing mehod han i is o calculae and hus ˆ direcly. Define j) as j) = Now, define j {,,...} for ) j +., ) as j := min{j = N J j)) J ) < s j) ) }. 9) Then, := j ) saisfies he sufficien descen condiion. Noe, if j =, hen =, which is a consan scaling from. Depending on, = apple where apple is a number beween approximaely.577 and. The following algorihm calculaes using bacracing. I should be implemened as an inner loop of Algorihm a sep. Algorihm : Se j = and calculae s from Eq.). ) If J j)) J ) < s j) ) hen reurn = j) and erminae. ) Incremen j and repea from Sep.

11 Lemma 8 Bacracing): If here exiss b > and b > such ha < b and for each of he i I, d T i )) >b, hen j is finie. Proof: The proof follows from Lemmas and 7. According o Lemma 7, ˆ = min{,, } saisfies he sufficien descen condiion. From Lemma, is bounded from. Furhermore, by he bounds on and v T i )), and are bounded from. Le b > bound ˆ from i.e. ˆ >b. Then, j = ceil log which is finie. 7 C. Locally Minimizing Sequence b For he ype- sufficien descen condiion, we have shown bacracing will find a for which he condiion is saisfied. In he following lemma, we find ha if {u } is he sequence calculaed using Algorihm from u where here is an infinie subsequence of {u } for which m =, hen he opimaliy funcion goes o zero. Lemma 9: Suppose u and S = {u } is an infinie sequence where ) Ju )=J<, ) Ju) is bounded below for all u, ) Ju + ) <Ju ), and 4) S S is an infinie subsequence where each u + S is calculaed from u + = Qu d ) and a) m =see Eq.7)), b) < or < see Lemma 7), c) here is K > such ha for each i I, d T i )) K, and d) = ) j + see Eq.9)). hen, lim! =. Proof: Since each Ju ) is sricly monoonically decreasing and bounded below, lim! Ju ) Ju + )=. Consider u + S which was calculaed from u using bacracing so ha u d saisfies he ype sufficien descen condiion, Eq.) and se := min ii and u is d T i + )). The cos difference beween u + Ju ) Ju + ) > p cardi ) ) ) ). ) Since S has infinie cardinaliy, i is he case ha as!, he righ hand side of Eq.) goes o zero. By Lemma 7 and he assumpion on,, and, any, min{, }], defined in Lemma 7, saisfies he ype- sufficien descen condiion. Le L be he Lipschiz consan of d a ) for each a {,...,N} and every u S. This consan exiss due o he assumpions made in Secion III. Recall = ) j + is calculaed by bacracing and herefore, if apple, hen j =and =. 7 The funcion ceil ) :R! Z rounds o he neares ineger of greaer value. Conversely, suppose <. Due o bacracing, i is possible for = ) j + <. If his is he case, hen ) j + >. Therefore, is in he inerval and hus [ ) +, = + ) ] ) 6L ) where [, ]. By assumpions, i mus be he case ha here are an infinie number of u + calculaed from u where eiher ) = or ) is given by Eq.). Since lim! Ju ) Ju + )=, he limi of he righ hand side of Eq.) goes o zero. If case ), hen lim p cardi q p A ) ) Since apple LT, lim! =. Now, if ), hen lim! p cardi ) =. 4L =. Since K and >, once again, lim! = and he proof is complee. The resricive assumpion in Lemma 9 is assumpion 4b. If he greaes for which he derivaive-based approximaion of he cos is valid goes o zero a a faser rae han goes o zero, hen he minimizing sequence is no be guaraneed o converge o =. VIII. EXAMPLE We consider wo examples. The firs is an insrucive example from he lieraure. The swiched sysem is composed of wo linear modes. The second example considers a simple aircraf model wih hree fligh modes. The airplane will eiher fly sraigh, ban righ, or ban lef, all a fixed velociies and a a fixed aliude. The goal is o schedule he fligh paern ha bes approximaes an infeasible desired urning maneuver. A. Scheduling a Linear Two-Mode Swiched Sysem Consider he linear ime-invarian swiched sysem example in [6] and [7]. Suppose x =, ) T and f x)) = A x) and f x)) = A x) where A = and A =. We wish o solve Problem i.e. o find he swiching conrol inpus ha minimize Jx, u) = R x )T Qx )d where Q is he ideniy marix. We execued Algorihm for 5 ieraions saring wih mode sequence = {} and swiching ime T = {}. This iniial mode schedule is he swiching conrol u =, ) T ). The procedure for one ieraion of Algorihm follows. The iniial sae, x, is calculaed from Eq.). Wih u

12 Ju ) Ju + ) 4 5 Ieraion # Fig.. Plo of Ju ) Ju + ) for 5 ieraions. Noice he difference beween successive coss decreases wih ieraion. and x, he adjoin for he firs ierae,, is calculaed from Eq.). The negaive mode inserion gradien, d is calculaed nex from Eq.6), which is guaraneed o be a descen direcion due o Lemma 6. Now, for any > he updae, Qu d ) is calculaed from Eq.4) and hus he projecion, x,u )=Pu d ), is calculaed from Eq.5). Noice he cos of he updae a, J ), is inegraed from `x,u ). The opimaliy funcion,, and he smalles sep size for which here is change o he cos,, are calculaed from Eqs.8) and 9) respecively. The greaes swiching ime ype a, m, is calculaed nex from Eq.7). If m =, hen bacracing using he ype- sufficien descen condiion may be done o calculae u. In his example, however, m =and hus ype- bacracing, Algorihm, is execued o calculae. We used = 4 and =. The new esimae of he opimum is found from u = Qu d ). We repeaed his process for 5 ieraions. In accordance wih Lemma 8, each j was finie and in fac he greaes number of bac sepping ieraions needed was. Noe we did no opimize he swiching imes beween each ieraion. Afer he 5 h ieraion, he cos reduced from.87 down o.. See Fig.) for he difference of curren and previous cos a each ieraion. The swiching conrol calculaed a he 5 h ieraions is given in Fig.4). Even hough u has zero mode ransiions, u has 6 mode ransiions and u 5 has 98. Finally, Fig.5) shows ha he opimaliy condiion,, rends oward where 5 = B. Scheduling Fligh Paern for Simple Aircraf Model For he second example, consider an aircraf wih hree modes of fligh a a fixed aliude. The plane s sae is given by is posiion in R, X, Y ) and is orienaion i.e. x) =X, Y, )). I flies sraigh, bans righ, or bans lef a fixed velociies. The hree modes are f i x)) = 4 cos )), 4 sin )),! i ) T where! =,! =, and! =. The goal is o schedule he fligh paern ha bes approximaes he urning maneuver u ) u 5 ) Fig. 4. Resuls of u and u Ieraion # Fig. 5. Plo of opimaliy condiion on a log scale for 5 ieraions. Noice he opimaliy condiion decreases wih ieraion. picured by he dashed lines in Fig.6). This desired rajecory is X des x des ) Y des A ) 8 p A < >< / = p. p + p A >: /6 The desired fligh rajecory is o fly along a sraigh pah for s, conduc an infeasible / radian poin urn, and fly sraigh once again. A bes, he aircraf can approximae he urn. Furhermore, he aircraf is no iniially posiioned or oriened wih he desired rajecory. The iniial condiion is x =5,, 5 /8) T. The objecive is o calculae he fligh schedule ha bes approximaes he desired rajecory. In oher words, he goal is o calculae he swiching conrol ha minimizes Ju) = Z 5 x ) x des )) T x ) x des ))d. Saring wih an iniial guess of u ) =,, ) T ), we execued Algorihm for 7 ieraions. We choose sufficien descen parameer =.4 and bacracing parameer =.4. Noe, swiching ime opimizaion is no conduced beween each ieraion. The swiching conrol and sae

13 Ju ) Ju + ) Fellowship Program DOE SCGF), made possible in par by he American Recovery and Reinvesmen Ac of 9, adminisered by ORISE-ORAU under conrac no. DE-AC5-6OR Ieraion # Fig. 7. Plo of Ju ) Ju + ) for 7 ieraions. -x Ieraion # Fig. 8. Plo of opimaliy funcion,, on a log scale for 7 ieraions. of ieraions 5, and 7 are picured in Fig.6). The desired rajecory is also picured for comparison. Noice, ha despie a poor iniial guess i.e. fly sraigh for he ime inerval by he 7 h execuion of he algorihm loop, he aircraf approximaes he desired rajecory. The difference of he cos beween successive ieraions is given in Fig.7). Furhermore, he value of he opimaliy funcion for each ieraion is given in Fig.8). IX. CONCLUSION An algorihm for opimal mode scheduling of swiched sysems is given. The algorihm has guaranees on convergence such as sufficien descen and bacracing for he search direcion given by calculaing he negaive mode inserion gradien. The mode inserion gradien is a single example of a descen direcion. In fuure wor, we will sudy sufficien descen and bacracing for general descen direcions. REFERENCES [] L. Armijo. Minimizaion of funcions having lipschiz coninuous firsparial derivaives. Pacific Journal of Mahemaics, 6:, 966. [] S. C. Bengea and R. A. DeCarlo. Opimal conrol of swiching sysems. Auomaica, 4: 7, 5. [] T. M. Caldwell and T. D. Murphey. Swiching mode generaion and opimal esimaion wih applicaion o sid-seering. Auomaica, 47:5 64,. [4] T. M. Caldwell and T. D. Murphey. Projecion-based swiched sysem opimizaion. American Conrol Conference,. [5] T. M. Caldwell and T. D. Murphey. Projecion-based swiched sysem opimizaion: Absolue coninuiy of he line search. IEEE Conference on Decision and Conrol,. [6] T. M. Caldwell and T. D. Murphey. Single inegraion opimizaion of linear ime-varying swiched sysems. IEEE Transacions on Auomaic Conrol, 57:59 597,. [7] M. Egersed, Y. Wardi, and H. Axelsson. Transiion-ime opimizaion for swiched-mode dynamical sysems. IEEE Transacions on Auomaic Conrol, 5: 5, 6. [8] H. Gonzalez, R. Vasudevan, M. Kamgarpour, S. S. Sasry, R. Bajcsy, and C. J. Tomlin. A descen algorihm for he opimal conrol of consrained nonlinear swiched dynamical sysems. Hybrid Sysems: Compuaion and Conrol, :5 6,. [9] J. Hauser. A projecion operaor approach o he opimizaion of rajecory funcionals. IFAC World Congress,. [] E. R. Johnson and T. D. Murphey. Second-order swiching ime opimizaion for non-linear ime-varying dynamic sysems. IEEE Transacions on Auomaic Conrol, 568):95 957,. [] K. Flaßamp, T. D. Murphey, and S. Ober-Blöbaum. Swiching ime opimizaion in discreized hybrid dynamical sysems. IEEE Conference on Decision and Conrol, pages 77 7,. [] C. Lemarechal. A view of line-searches. Opimizaion and Opimal Conrol, :59 78, 98. [] A. S. Lewis and M. L. Overon. Nonsmooh opimizaion via BFGS. SIAM Journal of Opimizaion, 9. [4] J. Nocedal and S. J. Wrigh. Numerical Opimizaion. Springer, 6. [5] J. Le Ny, E. Feron, and G. J. Pappas. Resource consrained LQR conrol under fas sampling. Hybrid Sysems: Compuaion and Conrol, 4:7 79,. [6] E. Pola. Opimizaion: Algorihms and Consisen Approximaions. Springer, 997. [7] E. Pola and Y. Wardi. A sudy of minimizing sequences. SIAM Journal on Conrol and Opimizaion, pages , 984. [8] Y. Wardi and M. Egersed. Algorihm for opimal mode scheduling in swiched sysems. American Conrol Conference, pages ,. [9] S. Wei, K. Uhaichana, M. Zefran, R. DeCarlo, and S. Bengea. Applicaions of numerical opimal conrol o nonlinear hybrid sysems. Nonlinear Analysis: Hybrid Sysems, :64 79, 7. [] X. Xu and P. J. Ansalis. Opimal conrol of swiched sysems via non-linear opimizaion based on direc differeniaions of value funcions. Inernaional Journal of Conrol, 75:46 46,. [] E. Zeidler. Applied Funcional Analysis: Applicaions o Mahemaical Physics. Springer, 995. X. ACKNOWLEDGMENTS This maerial is based upon wor suppored by he Naional Science Foundaion under award IIS-867 as well as he Deparmen of Energy Office of Science Graduae

14 ) T u ) X,Y ) X des ) Y ) ) T ) T ) T u 5 ) Y X des,y des ) X X ) des ) Y des ) ) ) T ) T ) T u ) X 5,Y 5 ) X,Y ) ) T ) T ) T u 7 ) ) T Y X 7,Y 7 ) X 7 ) ) T X Y 7 ) 7 ) Fig. 6. Resuls for ieraions, 5, and 7. The calculaed swiching conrol is shown lef, he X, Y ) rajecory is shown middle, and he sae x =X, Y, ) versus ime is shown righ. The desired rajecories are shown wih dashed lines.