Single integration optimization of linear time-varying switched systems

Transcription

1 Single inegraion opimizaion of linear ime-varying swiche sysems T. M. Calwell an T. D. Murphey Absrac This paper consiers he swiching ime opimizaion of ime-varying linear swiche sysems subjec o quaraic cos also poenially ime-varying. The problem is formulae so ha only a single se of ifferenial equaions nee o be solve prior o opimizaion. Once hese ifferenial equaions have been solve, he cos may be minimize over arbirary number of moes an moe sequences wihou requiring aiional simulaion. The number of marix muliplicaions neee o compue he graien grows linearly wih respec o he number of swiching imes, resuling in fas execuion even for high imensional opimizaions. Lasly, he ifferenial equaions ha nee o be simulae are as smooh as he sysem s vecor fiels, espie he fac ha he opimizaion iself is nonsmooh. Examples illusrae he echnique an is efficiency, incluing a comparison wih oher sanar echniques. I. INTRODUCTION This paper consiers he linear quaraic swiching ime opimizaion (LQSTO problem he problem of opimizing a quaraic cos subjec o he consrain of linear swiching ynamics where he moe orer is known ahea of ime an formulaes an efficien way of compuing opimal swiching imes. The primary resul of his paper is showing ha a single se of ifferenial equaions may be solve prior o opimizaion so ha uring opimizaion (e.g., uring compuaion of he graien no ifferenial equaions nee o be solve. This is in conras o he opimizaion approaches in [6, [7, [14, [15, which o no make full use of he lineariy of he swiche sysem an herefore solve a ifferenial equaion a every sep of he escen algorihm. Ohers have invesigae he specific case of linear swiching ime conrol. Giua e al. [6, [7 presen an opimal conrol law ha fins regions of he sae space where swiches shoul occur. Their approach assumes he linear moes are ime-invarian as well as sable, neiher of which are requiremens in he presen work. Xu an Ansaklis [14, [15 erive non-linear, swiching ime graien an Hessian calculaions of he cos for seepes escen an Newon s meho. The ieraive escen-base approach as inrouce in Xu an Ansaklis work form he basis for he work in This maerial is base upon work suppore by he Naional Science Founaion uner awar CCF as well as he Deparmen of Energy (DOE Office of Science Grauae Fellowship Program (DOE SCGF. The DOE SCGF Program was mae possible in par by he American Recovery an Reinvesmen Ac of The DOE SCGF program is aminisere by ORISE (which is manage by ORAU uner DOE conrac number DE-AC05-06OR Any opinions, finings, an conclusions or recommenaions expresse in his maerial are hose of he auhor(s an o no necessarily reflec he views of he NSF, DOE, ORAU, or ORISE. T. M. Calwell an T. D. Murphey are wih he Deparmen of Mechanical Engineering, Norhwesern Universiy, 2145 Sherian Roa Evanson, IL 60208, USA calwel@u.norhwesern.eu ; -murphey@norhwesern.eu his paper. Egerse e al. [5 propose a compuaionally more efficien ajoin-base graien calculaion, which only requires solving wo ifferenial equaions: he sae an cosae. In [3, [10, [11 he ajoin calculaion for he Hessian was ienifie, which also requires solving a consan number of ifferenial equaions. Xu an Ansaklis [14, [15 also consier he imeinvarian version of he LQSTO problem. Analogous o he linear quaraic regulaor problem of opimal conrols (see [1, Xu an Ansaklis foun ha he graien epens on he sae an a ifferenial equaion wih soluion specifying he linear relaionship beween he sae an co-sae. By only consiering ime-invarian swiche sysems, he evoluion of he sae is icae by he marix exponenial, which allows one o avoi solving he sae ifferenial equaion numerically. However, uner Xu an Ansaklis formulaion, he linear relaion of he sae an co-sae mus sill be numerically solve for a every sep of a escen algorihm. We presen a formulaion wih wo imporan ifferences. We refer o our formulaion as algebraic 1 linear quaraic swiching ime opimizaion (ALQSTO. Firs, ALQSTO makes no assumpion abou he ime-variance of he moes. Therefore, ALQSTO oes no exclue many imporan linear sysems. For example, power sysems are ofen linear imevarying [8, [12. Also, he linearizaion abou a rajecory of a non-linear sysem, in general, is ime-varying. Secon, ALQSTO generaes a single se of ifferenial equaions o be solve such ha hese ifferenial equaions are inepenen of he swiching imes as well as he assume moe sequence. Therefore, hese ifferenial equaions mus be solve only once an off-line from he opimizaion. By oing so, he sae an Riccai equaion may be compue algebraically from he soluions of hese ifferenial equaions an moreover, he opimaliy equaion (i.e., J(T = 0 oes no require any aiional ifferenial equaions o be solve. For his reason, he opimizaion complexiy oes no epen on he ifferenial equaions hey are only compue once. This paper is organize as follows: 2 Secion II reviews swiche sysems. Secion III, presens opimaliy coniions for linear swiche sysems wih quaraic cos. The algebraic linear quaraic swiching ime opimizaion (ALQSTO problem is propose in Secion IV. Differenial equaions are erive such ha he sae an a Riccai relaion may be compue from soluions o hese ifferenial 1 We will use he erm algebraic o refer o opimizaion proceures ha only nee aiion an muliplicaion o compue escen irecions. 2 This paper has been submie o he IEEE journal, Transacions on Auomaic Conrol. Tha version inclues a iscussion on reference racking.

2 equaions for any choice of swiching imes. Secion V shows how o apply he presene formulaion o solve he LQSTO problem using seepes-base mehos. II. REVIEW OF SWITCHED SYSTEMS Swiche sysems are a class of hybri sysems ha evolve over a sequence of coninuous moes of operaion, where he moes ransiion in a iscree manner. The imes for which he ransiions occur are referre o as swiching imes. We enoe he se of sricly monoonically increasing swiching imes as T = {T 1, T 2,..., T N 1 }, where N is he number of moes. In his paper, he moe sequence is assume o be known, an herefore, he opimizaion is only over he swiching imes. Mahemaically, a swiche sysem may be efine as [3, [5, [13, [15: f 1 (x(,, T 0 < T 1 ( f 2 (x(,, T 1 < T 2 x( = f x(, T, =... f N (x(,, T N 1 < T N subjec o: x(t 0 = x 0 (1 where T 0 is he iniial ime, T N is he final ime, x 0 is he iniial sae, an f i : R n R R n is he vecor fiel corresponing o he i h moe in he assume moe sequence. III. OPTIMALITY CONDITIONS FOR LINEAR SWITCHED SYSTEMS WITH QUADRATIC COST Accoring o [3, [5, he opimal swiching imes of he cos funcional, J(T = T N T 0 ( l, x( ( + m x(t N, mus saisfy he following equaions: 3 ( x( = f, x(, T, s..: x(t 0 = x 0 (2a ( T ( T ρ( = D 2 f, x(, T ρ( D2 l, x(, ( T (2b s..: ρ(t N = Dm x(t N ( ( 0 = ρ T (T i [f i T i, x(t i f i+1 T i, x(t i, i (1,..., N 1. (2c The evoluion of he sae is given by Eq (2a an he evoluion of he co-sae is given by Eq (2b. Then, given some se of opimal swiching imes, he corresponing opimal sae an co-sae mus saisfy Eq (2c. This se of equaions forms a necessary coniion for opimaliy. We refer o Eq (2c as he se of opimaliy equaions. Also, 3 The noaion D is he slo erivaive. For a funcion g, Dg( is he parial erivaive of g wih respec o is single argumen. Similarly, D i g(, is he parial of g wih respec o he i h argumen. noe ha he righ han sie of he opimaliy equaions is he graien of he cos, J(T. For his paper, we resric our focus o linear ime-varying swiche sysems, ( f, x(, T = A(, T x( where A(, T = A i ( for T i 1 < T i. Moreover, we choose a quaraic cos funcion such ha l(, x( := 1 2 x(t Q(x( an m(x(t N := 1 2 x(t N T P 1 x(t N. where Q an P 1 are symmeric posiive semi-efinie. Then, he necessary coniion, Eq (2, becomes (3 x( = A(, T x(, s..: x(t 0 = x 0 (4a ρ( = A(, T T ρ( Q(x(, s..: ρ(t N = P 1 x(t N. 0 = ρ T (T i [A i (T i A i+1 (T i x(t i, i (1,..., N 1. (4b (4c Noice ha Eqs (4a an (4b make up a wo-poin bounary value problem. As in he LQR from classical conrols, we will fin a linear mapping beween x an ρ in orer o ransform he problem ino an iniial value problem. A. The Linear Swiche Sysem Analog o he Riccai Equaion from LQR In he following heorem, he sae, x(, an co-sae, ρ(, are shown o be linearly epenen hrough a Riccai relaion 4. Theorem 1: For opimizing a quaraic cos funcion escribe by Eq (3 an subjec o a linear ime-varying swiche sysem of he form in Eq. (4a, he sae, x(, an co-sae, ρ(, are linearly epenen: ρ( = P (x(, where P ( is he soluion o he following linear ifferenial equaion: P ( = A(, T T P ( P (A(, T Q(, s..: P (T N = P 1. Proof: The proof follows from manipulaion of x( an ρ( using he sae-ransiion marix. Begin wih he soluion o x (see Eq (4a: x( = Φ(, T 0 x 0, where Φ(, T 0 is he sae-ransiion marix corresponing o A(, T (i.e. Φ(, T 0 = A(, T Φ(, T 0. Furhermore, he inegral form of he ajoin equaion, ρ(, is ρ( = Φ(T N, T P 1 x(t N + T N Φ(, T Q(x(. 4 We use he erm, Riccai relaion, as i is use in opimal conrol heory as he variable specifying he linear mapping beween he sae an he co-sae. Here, he Riccai relaion is he soluion o a linear ifferenial equaion an shoul no be confuse wih he 2 n -orer ifferenial equaions suie by he mahemaician Jacopo Riccai. (5

3 Plugging in for x(, = Φ(T N, T P 1 Φ(T N, T 0 x 0 + T N Noing Φ(T N, T 0 = Φ(T N, Φ(, T 0, Φ(, T Q(Φ(, T 0 x 0. = Φ(T N, T P 1 Φ(T N, Φ(, T 0 x 0 T N + Φ(, T Q(Φ(, Φ(, T 0 x 0. Pulling x( = Φ(, T 0 x 0 o he righ an enoing he marix inegral equaion as P ( [ T N = Φ(T N, T P 1 Φ(T N, + Φ(, T Q(Φ(, x(. } {{ } P ( The ifferenial form of P ( is Eq (5. Furhermore, x( an ρ( are linearly epenen hrough P (. B. Linear Quaraic Swiche Sysem Opimaliy Coniion Using he linear epenence of ρ( an x(, he opimaliy equaions become: 0 = x(t i T P (T i [A i (T i A i+1 (T i x(t i, i (1,..., N 1. The opimaliy equaions provies N 1 equaions an N 1 swiching ime unknowns such ha he problem of fining he opimal swiching imes may be formulae as a roofining problem where x(t i an P (T i may be calculae separaely (i.e. wo separae iniial value problems. Eq (6, along wih Eqs (4a an (5 ogeher make up he necessary coniion for opimaliy for he linear quaraic swiching ime opimizaion problem. IV. ALGEBRAIC LINEAR QUADRATIC SWITCHING TIME OPTIMIZATION (ALQSTO As seen in he previous secion, LQ swiching ime opimizaion is a roo-fining problem ha epens on he soluions o ifferenial equaions. Suppose he problem may be reformulae such ha all of he ifferenial equaions may be solve off-line from an opimizaion rouine. Consier he opimizaion problem characerize by he quaraic cos funcion escribe by Eq (3 an subjec o he linear swiche sysem of Eq (4a. We hypohesize ha an equivalen opimizaion problem exiss ha epens on he soluions o a se of ifferenial equaions, where once he soluions have been obaine, no aiional inegraion is necessary in orer o solve he opimizaion problem using sanar echniques (e.g. seepes escen, Newon s meho, ec. These ifferenial equaions are swiching ime an moe sequence invarian an because hey nee only be solve once, we shall refer o hem as he single inegraion ifferenial equaions (SIDE. Furhermore, we (6 shall refer o his equivalen opimizaion problem as he Algebraic Linear Quaraic Swiching Time Opimizaion (ALQSTO problem for i relies solely on aiion an muliplicaion once he SIDE have been solve for. In orer for he hypohesis o be rue, he sae, x(, an Riccai relaion, P (, (i.e. he soluions o Eq (4a an Eq (5 mus be compuable from he soluions o he SIDE for any arbirary, vali se of swiching imes. Furhermore, his compuaion mus no require solving any aiional ifferenial equaions. Then, he opimaliy equaions, Eq (6, may also be compue wihou solving aiional ifferenial equaions. The res of his secion is evoe o fining he SIDE as well as he associae compuaions of x( an P (. A. Compuing x( As sae, he SIDE for compuing x( are he saeransiion marices for each A i (. Using Φ i (, o enoe he STM corresponing o A i, he value of x a he i h swiching ime is [ 1 x(t i = Φ(T i, T 0 x 0 = Φ j (T j, T j 1 x 0. (7 j=i The prouc is from j = i own o j = 1 so ha he sequence of marix muliplicaions is correc. Now, suppose each Φ j (, T 0 has been solve for all imes [T 0, T N an for all j = 1,..., N an sore in memory. Then, Φ j (T j, T j 1 is Φ j (T j, T 0 Φ j (T j 1, T 0 1. I follows ha x( may be compue using only aiion an muliplicaion for any arbirary choice of swiching imes. B. Compuing P ( We wish o fin an analogous operaor o he STM o be he SIDE for compuing P (. We refer o such an operaor as he ajoin-ransiion marix, or ATM, an enoe Ψ i : R R R n n, i = 1,..., N as he ATM corresponing o each linear moe. We also consier he ATM ha correspons o A(, T, which we label as Ψ. We will fin ha Ψ may be calculae from he Ψ i for arbirary swiching imes using only marix muliplicaion an aiion. We will also fin ha he operaion, P ( = Ψ(, T N P 1, (8 gives he soluion o Eq (5. Firs, we efine he ATM an is properies an secon, use he properies o calculae P ( from Ψ i, i = 1,..., N an he erminal coniion P 1. Definiion 1: Consier he linear sysems efine by A(, associae STM, Φ(,, an cos funcion efine by Q(. Then, he ajoin-ransiion marix (ATM, Ψ(, : R R R n n, is efine as he soluion o Ψ(, = A(T Ψ(, Ψ(, A( Q( subjec o: Ψ(, = 0 n n (9 Noe ha he argumens of Ψ(, an Φ(, are swiche. This noaional choice is ue o how he sae an ajoin are solve in opposing irecions.

4 The following heorem efines he properies of he ATM ha will be use o compue P ( in a similar manner he properies of he STM are use o compue x(. Theorem 2: The ATM, Ψ, characerize by A( an associae STM, Φ(,, an cos funcion efine by Q( has he following properies: 1. Ψ(, = Φ(s, T Q(sΦ(s, s 2. Ψ(, = Φ(, T Ψ(, Φ(, P ( = Ψ(, P ( = Ψ(, + Φ(, T P (Φ(, (10 Ψ( 1, 3 = Ψ( 1, 2 Ψ( 2, 3 = Ψ( 1, 2 + Φ( 2, 1 T Ψ( 2, 3 Φ( 2, 1, where,, 1, 2, 3 R are arbirary imes. Proof: We prove each propery separaely. Propery 1: Propery 1 may be erive irecly from he efiniion of he ATM. Take he erivaive of Ψ(, wih respec o : Ψ(, = = = [ Φ(s, T Q(sΦ(s, s [ Φ(s, T Q(sΦ(s, s Φ(, T Q(Φ(, [ A( T Φ(s, T Q(sΦ(s, Φ(s, T Q(sΦ(s, A(s Q( = A( T Ψ(, Ψ(, A( Q(. Clearly, propery 1 is in he inegral form of he ATM. Furher, noice ha propery 1 is consisen wih he bounary coniion, Ψ(, = 0. Propery 2: Sar wih propery 1: Ψ(, = Φ(s, T Q(sΦ(s, s = Φ(, T = Φ(, T Ψ(, Φ(, Φ(s, T Q(sΦ(s, sφ(, Properies 3 an 4 are proven using he Funamenal Theorem of ODEs [9 by showing ha for each propery, he lef-han sie, L, an righ-han sie, R, share he same ifferenial equaion an iniial coniion. Propery 3: The lef an righ sies of propery 3 are L( = P ( an R ( = Ψ(, + Φ(, T P (Φ(,. The iniial coniions are L( = P ( an R ( = Ψ(, + Φ(, P (Φ(,. Simplifying R ( by noing ha Ψ(, = 0 an Φ(, = Φ(, = I, resuls in R ( = P (. Now, o check he ifferenial equaions: L( = P ( = A(T P ( P (A( Q(, (11 an [ R ( = Ψ(, + Φ(, T P (Φ(, = A( T Ψ(, Ψ(, A( Q( A( T Φ(, T P (Φ(, Φ(, T P (Φ(, A(. However, Ψ(, = P ( Φ(, T P (Φ(,, so R ( = A(T P ( P (A( Q(, hus saisfying he Funamenal Theorem of ODEs an proving propery 3. Propery 4: The lef an righ sies of propery 4 are L( 1 = Ψ( 1, 3 an R ( 1 = Ψ( 1, 2 + Φ( 2, 1 T Ψ( 2, 3 Φ( 2, 1. Furhermore, he iniial coniions are L( 3 = Ψ( 3, 3 = 0 an R ( 3 = Ψ( 3, 2 + Φ( 2, 3 T Ψ( 2, 3 Φ( 2, 3 = 2 3 Φ(s, 3 T Q(Φ(s, 3 s = +Φ( 2, 3 T = Φ(s, 2 T Q(Φ(s, 2 sφ( 2, 3 Φ(s, 3 T Q(Φ(s, 3 s 2 Φ(s, 3 T Q(Φ(s, 3 s = 0. Now, he equivalence of he ifferenial equaions of L an R are checke: 1 L( 1 = 1 Ψ( 1, 3 = A( 1 T Ψ( 1, 3 Ψ( 1, 3 A( 1 Q( 1, an 1 R ( 1 = 1 [Ψ( 1, 2 + Φ( 2, 1 T Ψ( 2, 3 Φ( 2, 1 = A( 1 T Ψ( 1, 2 Ψ( 1, 2 A( 1 Q( 1 A( 1 T Φ( 2, 1 T Ψ( 2, 3 Φ( 2, 1 Φ( 2, 1 T Ψ( 2, 3 Φ( 2, 1 A( 1. Recall, Ψ( 1, 2 = Ψ( 1, 3 Φ( 2, 1 T Ψ( 2, 3 Φ( 2, 1, so 1 R ( 1 = A( 1 T Ψ( 1, 3 Ψ( 1, 3 A( 1 Q( 1. The lef-han an righ-han sies of propery 4 share he same ifferenial equaion an iniial coniions, compleing he proof of he properies. The ATM may be solve from is ifferenial form, Eq (9. The ATM s inegral form, propery 1, is useful in proofs. The secon propery specifies he operaion of he ATM on he ajoin such ha he operaion ransiions he ajoin from one ime o anoher (i.e. P ( = Ψ(, P (. Propery 4 specifies how an ATM operaes on anoher ATM. Propery 4 will be useful for compuing Ψ from he se {Ψ i } N i.

5 Now o prove he ATM is he operaor ha saisfies Eq (8. Theorem 3: The operaor ha saisfies Eq (8 is Ψ(,, where Ψ(, is characerize by wo copies of he linear sysem efine by A(, T. Proof: Using propery 3 an recalling ha P (T N = P 1, Ψ(, T N P 1 = Ψ(, T N + Φ(T N, T P 1 Φ(T N,, which is he soluion o Eq (5. Using propery 4 of ATM an efining Ψ i := Ψ Ai,A i for i = 1,..., N, Ψ(, is compue as follows: Ψ(T j, T i = Ψ(T j, T j+1 Ψ(T i 1, T i = i k=j+1 Φ(T k 1, T j T Ψ k (T k 1, T k Φ(T k 1, T j. (12 Then, using propery 3, P (T i = Ψ(T i, T N P 1 = Ψ(T i, T N + Φ(T N, T i T P 1 Φ(T N, T i. (13 Now, we conuc a similar numerical exercise as one when compuing x(. Suppose, Ψ i (, T N has been solve for all i = 1,..., N an [T 0, T N. Consequenly, by noing ha Ψ k (T k 1, T k = Ψ k (T k 1, T N Ψ k (T N, T k =Ψ k (T k 1, T N +Φ k (T k, T k 1 T Ψ k (T N, T k Φ k (T k, T k 1 =Ψ k (T k 1, T N Φ k (T k, T k 1 T Ψ k (T k, T N Φ k (T k, T k 1, hen, Ψ(T j, T i, Eq (12, may be compue wihou solving any aiional ifferenial equaions for all i, j = 1,..., N. Aiionally, his resul implies he calculaion of P (T i (see Eq (13 also oes no require solving any exra ifferenial equaions. C. Example In he following example, we solve an LQSTO problem using he ALQSTO formulaion. The problem se up comes from he example in [4. Example 1: Suppose a swich sysem is escribe by he wo linear vecor fiels: Moe σ = 1 : f 1 (x( = A 1 x( an Moe σ = 2 : f 2 (x( = A 2 x(, where A 1 = ( an A 2 = ( Furher suppose ha a iniial ime T 0 = 0, he sysem is in moe σ = 1 wih iniial configuraion x 0 = (1, 1 T. I is also known ha he sysem ransiions beween moes 1 an 2 on 5 occasions (i.e. Σ = {1, 2, 1, 2, 1, 2} before he conclusion of he ime inerval a T N = 1. The goal is o fin he swiching imes ha opimize he cos funcion wih Lagrangian escribe by Q = iag(2, 2 an zero erminal cos. 5 In Mahemaica, we calculae he opimaliy coniions, Eq (6, wih variable swiching imes. The process o o 5 We use iag( as a compac represenaion of he iagonal marix. so begins by solving for {Φ i (, T 0 } an {Ψ i (, T N } [T 0, T N an soring in memory. To elaborae, we solve Φi (, T 0 = A i (Φ(, T 0, s.. Φ i (T 0, T 0 an Eq (9 for each A i respecively using Aam s meho for numerically solving ifferenial equaions an sore in memory he iscree aa poins chosen from he meho s aapive ime sepping. The value of {Φ i (, T 0 } an {Ψ i (, T N } is hen given by a hir-orer polynomial inerpolaion of he four aa poins surrouning he ime. In his example, a oal of 162 aa poins were save o fully consiue {Φ i (, T 0 } an {Ψ i (, T N } for all [T 0, T N. Following, we compue {x(t i } an {P (T i } for variable swiching imes accoring o Eqs (7 an (13, respecively. By obaining he opimaliy coniions in his way ransforms he LQSTO problem ino an ALQSTO problem. Uner his formulaion, he opimal swiching imes are given by fining he swiching imes ha are he roos o he opimaliy coniion. In Mahemaica, we conuc Newon s meho wih a rus region, saring wih an evenly isribue se of iniial swiching imes. Doing so resuls in he locally opimal swiching imes, T = (0.100, 0.297, 0.433, 0.642, T, which have an associae locally opimal cos of The algorihm converges afer six ieraions o wihin 10 8 of J(T = 0. V. IMPLEMENTATION AND COMPLEXITY For ALQSTO, we assume he STM, Φ i (, T 0, an ATM, Ψ i (T N,, corresponing o he N moes have been previously solve for an save in memory for all [T 0, T N. Then, following from he previous sub-secion, he compuaion of x( an P ( (an consequenly he opimaliy equaions relies simply on memory calls an marix algebra. For his reason, he number of ifferenial equaions one nees o solve a each ieraion of he escen irecion is 0. The complexiy of ALQSTO may insea be iscusse in erms of he number of marix muliplicaions. Theorem 4: Given Φ i (, T 0 an Ψ i (, T N for all i = 1,..., N an [T 0, T N, he number of marix muliplicaions neee o compue {x(t i } N 1 i=1, {P (T i} N 1 i=1 an he righ han sie of he opimaliy equaions, Eq (6 for ALQSTO, is 7(N 1. Proof: When compuing x(t i, begin wih x(t 0 = x 0. Then, recursively calculae where x(t i = Φ i (T i, T i 1 x(t i 1 (14 Φ i (T i, T i 1 = Φ i (T i, T 0 Φ i (T i 1, T 0 1. (15 Therefore, each aiional moe requires wo exra muliplicaions an a marix inverse. However, Φ i (, T 0 1 may oo be sore in memory for [T 0, T N if esire. Similarly, for P (T i, begin wih P (T N = P 1 an recursively calculae P (T i = Ψ i+1 (T i, T i+1 +Φ i+1 (T i+1, T i T P (T i+1 Φ i+1 (T i+1, T i.

6 Recall Ψ i+1 (T i, T i+1 = Ψ i+1 (T i, T N Φ i+1 (T i+1, T i T Ψ i+1 (T i+1, T N Φ i+1 (T i+1, T i an ha Φ i+1 (T i+1, T i has alreay been calculae using Eq (15 for {x(t i }. Therefore, P (T i = Ψ i+1 (T i, T N + Φ i+1 (T i+1, T i T [P (T i+1 Ψ i+1 (T i+1, T N Φ i+1 (T i+1, T i. (16 Thus, each aiional moe requires wo aiional muliplicaions o compue P (. Furhermore, each of he N 1 opimaliy equaions, Eq (6, require hree muliplicaions. In oal, hese compuaions require 7(N 1 marix muliplicaions. A. Implemenaion by Seepes Descen Eqs (14 an (16 in he proof of Theorem 4 show how x(t i an P (T i shoul be calculae for execuional efficiency. These equaions are easily implemene in a numerical opimizaion rouine. In Algorihm 1, we presen solving he ALQSTO problem using seepes escen. The seepes escen irecion is he negaive graien of he cos: J(T = {x(t i T P (T i [A i (T i A i+1 (T i x(t i } N 1 1, (17 which equals he 0 vecor when a a local opimum (i.e. saisfies Eq (6. Algorihm 1 ALQST O Seepes Descen: s( Argumens: Σ is he moe sequence T 0 is he se of iniial swiching imes Φ i (, T 0 are he STM i {1,..., N} an [T 0, T N Ψ i (, T N are he ATM i {1,..., N} an [T 0, T N T = s(σ,t 0,Φ i (, T 0,Ψ i (, T N : (1 k = 0; T k = T 0 ; x(t 0 = x 0 ; P (T N = P 1 while J(T k is no near 0 o (2 Recursively solve for {x(t i } N 1 i=1 wih swiching imes T k (see Eq (14 (3 Recursively solve for {P (T i } N 1 i=1 wih swiching imes T k (see Eq (16 (4 Calculae he graien, J(T k (see Eq (17 (5 Upae he swiching imes: T k+1 = T k γ J(T k, for some sep size γ (6 k = k + 1 en while If one nees o calculae he cos (e.g. for choosing he sep size from a line search as in [2, hen noice ha T N 1 J(T = 2 x(t Q(x( x(t N T P 1 x(t N T 0 [ T N Φ(, T 0 T Q(Φ(, T 0 = 1 2 xt 0 T 0 +Φ(T N, T 0 T P 1 Φ(T N, T 0 [ [ Ψ(T0, T N P 1 x0 Ψ(T0, T 1 P (T 1 x 0. = 1 2 xt 0 = 1 2 xt 0 x 0 The compuaion, Ψ(T 0, T 1 P (T 1 shoul be calculae from he alreay calculae P (T 1 (from sep 3 using Eq (16. Then, he number of aiional marix muliplicaions o calculae he cos is 4. B. Complexiy Comparison wih he Lieraure The lieraure has foun ha each sep of seepes escen for swiching ime opimizaion requires solving a consan number of ifferenial equaions. In [3, [5, he graien requires solving he sae, Eq (2a, forwar in ime, an he co-sae, Eq (2b, backwar in ime. This approach works for non-linear swiche sysems. In his iscussion, we refer o his seepes escen calculaion as he ifferenial graien swiching ime opimizaion, or DGSTO. Moreover, in [14, [15, Xu an Ansaklis presen an approach which solves for he Riccai relaion, Eq (5, an calculaes he sae using he marix exponenial. Their approach assumes he moes are linear ime-invarian. We shall refer o his echnique as Xu an Ansaklis swiching ime opimizaion, or XASTO. Each sep of he escen of DGSTO an XASTO requires solving an aiional consan number of ifferenial equaions. In comparison, each sep of escen for ALQSTO requires solving zero ifferenial equaions, which implies he ALQSTO escen irecion calculaion ime is invarian o any choice of ifferenial equaion solver. In comparison, he execuion ime of DGSTO an XASTO epens on he number of seps a variable (or fixe ODE inegraor requires o solve heir respecive ifferenial equaions. I is of noe, however, ha DGSTO generalizes o nonlinear swiche sysems. In fac, ALQSTO s resricion o quaraic coss ha are subjec o linear swiche sysems is he reason why ALQSTO oes no require solving any ifferenial equaions uring he opimizaion. Furhermore, he complexiy of ALQSTO comes a a price; ALQSTO raes compuaional complexiy for memory emans. Depening on he applicaion, soring he STM an ATM over he full ime inerval for each moe may be oo cosly, in which case, XASTO or DGSTO may be preferable. C. Example We compare he execuion imes of seepes escen for ALQSTO, XASTO an DGSTO in an example. Example 2: Consier he bi-moal sysem in Example 1. For his example, however, he number of moes, N, is varie an he execuion ime o calculae a single graien is compare beween ALQSTO an DGSTO. The moe sequence begins wih moe 1 an alernaes beween moes 1 an 2 over N 1 occasions. Fig.1 compares he calculaion imes of he graien for ALQSTO, XASTO an DGSTO for up o 100 moes. Fig.2 shows he graien calculaion ime for ALQSTO for up o 1000 moes. The iniial compuaion ime of secons for calculaing he SIDE is no facore ino he resuls for ALQSTO. The resuls were simulae in Mahemaica on a 2.26 GHz PC. Mahemaica s AbsolueTime[ funcion was use o ime he graien calculaions. The ifferenial equaions for DGSTO an XASTO were solve using

7 NDSolve[ wih meho ExpliciRungeKua an aapive ime sepping meho. inegraion is only one once an off-line from he opimizaion, he inegraion may be as slow as neee. CPU Times XASTO ALQSTO DGSTO of Moe Transiions Fig. 1. Comparison of he execuion imes for one calculaion of he graien for ALQSTO, XASTO an DGSTO. A 100 moes, one graien calculaion of ALQSTO execues in secons. CPU Times of Moe Transiions Fig. 2. The execuion imes for one calculaion of he seepes escen irecion for ALQSTO. The number of moes is varie beween 1 an Remark: A few remarks on he resuls 1 For his example, a 100 moes, Fig.1 shows ha XASTO is greaer han one orer of magniue slower han ALQSTO, while DGSTO is greaer han wo orers of magniue slower. 2 A ifferen choice of ODE inegraor may resul in a quicker execuion of XASTO an DGSTO. For insance, an inegraor wih a fixe sep size woul resul in a linear relaion beween he number of moes an execuion ime. However, such an inegraor woul lose accuracy, especially near he swiching imes ue o he sae an co-sae being non-iffereniable a hose imes. The choice of sep size woul likely epen on he number of moes an he sabiliy of he inegraion. This sor of iscussion is irrelevan for ALQSTO. The accuracy of ALQSTO epens on he numerical solver use o solve for {Φ i (, T 0 } an {Ψ i (, T N }. Since he numerical inegraion of hese STM an ATM are as smooh as each of he moes, ALQSTO has aiional numerical robusness. Furhermore, since he VI. CONCLUSION This paper consiers he swiching ime opimizaion of linear ime-varying swiche sysems subjec o quaraic cos. We formulae he problem such ha a se of ifferenial equaions, which we refer o as single inegraion ifferenial equaions (SIDE, may be solve once an off-line from he opimizaion rouine an ha once he SIDE have been solve, no more inegraion is necessary o solve he opimizaion problem using sanar echniques (e.g. seepes escen, Newon s meho, ec.. We refer o his formulaion as algebraic linear quaraic swiching ime opimizaion, or ALQSTO. Once he SIDE have been solve, he compuaion of he graien for an arbirary se of swiching imes requires an O(N number of marix muliplicaions, resuling in fas execuion. Furhermore, since he SIDE are as smooh as he swiche sysem s vecor fiels, he ALQSTO formulaion has aiional robusness o numerical errors compare wih solving he sae an co-sae irecly. REFERENCES [1 B. D. O. Anerson an J. B. Moore. Opimal Conrol: Linear Quaraic Mehos. Dover Publicaions, INC, [2 L. Armijo. Minimizaion of funcions having lipschiz coninuous firsparial erivaives. Pacific Journal of Mahemaics, 16:1 3, [3 T. M. Calwell an T. D. Murphey. Swiching moe generaion an opimal esimaion wih applicaion o ski-seering. Auomaica, 47:50 64, [4 M. Egerse, S.-I. Azuma, an H. Axelsson. Transiion-ime opimizaion for swiche-moe ynamical sysems. IEEE Transacions on Auomaic Conrol, 51: , [5 M. Egerse, Y. Wari, an F. Delmoe. Opimal conrol of swiching imes in swiche ynamical sysems. IEEE Conference on Decision an Conrol, 3: , [6 A. Giua, C. Seazu, an C. Van Der Mee. Opimal conrol of auonomous linear sysems swiche wih a pre-assigne finie sequence. In Proc. of he 2001 IEEE ISIC, pages , [7 A. Giua, C. Seazu, an C. Van Der Mee. Opimal conrol of swiche auonomous linear sysems. IEEE Conference on Decision an Conrol, 40: , [8 A. J. Golsmih an S.-G. Chua. Variable-rae variable-power mqam for faing channels. IEEE Transacions on Communicaions, 45(10: , [9 M. W. Hirsch an S. Smale. Differenial Equaions, Dynamical Sysems, an Linear Algebra. Acaemic Press, New York, [10 E. R. Johnson an T. D. Murphey. Secon orer swiching ime opimizaion for ime-varying nonlinear sysems. IEEE Conference on Decision an Conrol, 48: , [11 E. R. Johnson an T. D. Murphey. Secon-orer swiching ime opimizaion for non-linear ime-varying ynamic sysems. IEEE Transacions on Auomaic Conrol, (In Press. [12 M. J. Neely. Energy opimal conrol for ime varying wireless neworks. IEEE Transacions on Informaion Theory, 52(7: , [13 A. Schil, X. C. Ding, M. Egerse, an J. Lunze. Design of opimal swiching surfaces for swiche auonomous sysems. IEEE Conference on Decision an Conrol, 48: , [14 X. Xu an P. J. Ansaklis. An approach for solving general swiche linear quaraic opimal conrol problems. IEEE Conference on Decision an Conrol, 40: , [15 X. Xu an P. J. Ansaklis. Opimal conrol of swiche auonomous sysems. IEEE Conference on Decision an Conrol, 41: , 2002.