Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 C. Bag-bag Cotrol. Itroucto hs chapter eals wth the cotrol wth restrctos: s boue a mght well be possble to have scotutes. o llustrate some of the basc cocepts volve whe cotrols are boue a allowe to have scotutes we start wth a smple physcal problem: Derve a cotroller such that a car move a stace a wth mmum tme. he moto equato of the car x u where (.) u u( t), α u β (.) represets the apple accelerato or ecelerato (brakg) a x the stace travele. he problem ca be state as mmze (.3) o subect to (.) a (.) a bouary cotos x ( ), (), x( ) a, ( ) (.4) he methos we evelope the last chapter woul be approprate for ths problem except that they caot cope wth equalty costrats of the form (.). We ca chage ths costrat to a equalty costrat by troucg aother cotrol varable, v, where v ( u + α )( β u) (.5) Sce v s real, u must satsfy (.). We trouce th usual state varable otato x x so that x u v u& v& p (.) p µ ( β α u) (.) v µ (.) (.) v or µ. We wll coser these two cases. () µ p be mpossble. p () v (.5): v ( u + α )( β u) u α or u β Hece β t τ α τ < t the swtch takg place at tme τ. Itegratg usg bouary cotos o x β t t τ x (.3) α ( t ) τ < t Itegratg usg bouary cotos o x β t x α ( t ) + a t τ τ < t (.4) Both stace, x, a velocty, x, are cotuous at t τ, we must have x x x ), x ( ) a (.6) u x ( ), x ( ) (.7) We ow form the augmete fuctoal + + + * { p ( x ) p ( u ) µ [ v ( u + α )( β u)]} (.8) (.4) βτ α ( τ ) (.5) βτ a α ( τ ) Elmatg gves the swtchg tme as aα τ β ( α + β ) (.5) where p, p, η are Lagrage multplers assocate wth the costrats (.6), (.7) a (.5) respectvely. he Euler equatos for the state varables x, x a cotrol varables u a v are x (.9) a the fal tme s a(α + β ) (.6) αβ he problem ow s completely solve a the optmal cotrol s specfe by Chapter Bag-bag Cotrol 53
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 β t τ α τ < t (.7) (,, ) (.3) x ths s llustrate Fg.. β α cotrol u τ tme t Fg.. Optmal Cotrol hs fgure shows that the cotrol: - has a swtch (scotuty) at tme t τ - oly take ts maxmum a mmum values hs type of cotrol s calle bag-bag cotrol.. Potryag s Prcple (early 96s) Problem: We are seekg extremum values of the fuctoal J f t ( x, ) (.8) subect to state equatos f ( x, t) (,, ) (.9) tal cotos x x a fal cotos o x, x, L, xq ( q ) a subect to u U, the amssble cotrol rego. For example, the prevous problem, the amssble cotrol rego s efe by U { u : α u β} As secto.4, we form the augmete fuctoal J f p f x * + ( & ) (.) a efe the Hamtoa H f + p f (.) For smplcty, we coser that the Hamtoa s a fucto of the state vector x, cotrol vector a aot vector p, that s, H H ( x,. We ca express J * as J* H p x & (.) a evaluatg the Euler equatos for x, we obta as secto.4 the aot equatos he Euler equatos for the cotrol varables, u, o ot follow as secto.4 as t s possble that there are scotutes u, a so we caot assume that the partal ervates H / u exst. O the other ha, we ca apply the free e pot coto (9.3): to obta p k ( ) k q +, (.4) that s, the aot varable s zero at every e pot where the correspog state varable s ot specfe. As before, we refer to (.4) as traversalty cotos. Our ffculty ow les obtag the aalogous equato to / u for cotuous cotrols. For the momet, let us assume that we ca fferetate H wth respect to a coser a small varato δu the cotrol u such that u + δu stll belog to U, the amssble cotrol rego. Correspog to the small chage u, there wll be small chage x, say δ x, a p, say δ p. he chage the value of J * wll be δ J *, where δ J* δ { H p x & } o he small chage operator, δ, obeys the same sort of propertes as the fferetal operator / x. Assumg we ca terchage the small chage operator, δ, a tegral sg, we obta δj* [ δ ( H p )] [ δh δ ( p )] [ δh δ p pδ ] Usg cha rule for partal fferetato m δ H δu + δx + δp u x p so that δj* m δu u + p p x x p p x δ & δ δ δ & p sce / x. Also, from (.) f p Chapter Bag-bag Cotrol 54
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 usg (.9): f ( x, t),(,, ). hus m δ J* δu u m δu u ( δ x + p δ ) ( p δ x ) We ca ow tegrate the seco part of the tegra to yel ( ) m δ J* pδ x + δu (.5) u At t : x (, ) are specfe δ x ( ) At t : x (, q) are fxe δ x ( ) For q +,, from the trasversalty cotos, (,4) p ( ) δ x ( ) for,, L, q, q +,. We ow have m δ J* δu u where δu s the small varato the th compoet of the cotrol vector u. Sce all these varatos are epeet, a we requre δj* for a turg pot whe the cotrols are cotuous, we coclue that u (,, m) (.6) But ths s oly val whe the cotrols are cotuous a ot costrae. I our preset case whe u U, the amssble cotrol rego a scotutes u are allowe. he argumets presete above follow through the same way, except that ( / u ) u must be replace by H ( x; u + δ u, um; H ( x, We thus obta m δ J* [ H ( x; u, u + δu, um; H ( x, ] I orer for u to be a mmzg cotrol, we must have δj* for all amssble cotrols u + δu. hs mples that H ( x; u + δ u, um; H ( x, (.7) for all amssble δu a for, m. So we have establshe that o the optmal cotrol H s mmze wth respect to the cotrol varables, u, L, um. hs s kow as Potryag s mmum prcple. We frst llustrate ts use by examg a smple problem. We requre to mmze J subect to x & x, u where α u β a x ( ) a, x ( ), x ( ) x (). Itroucg aot varables p a p, the Hamltoa s gve by H + px + pu We must mmze H wth respect to a where u U [ α, β ], the amssble cotrol rego.. Sce H s lear t clearly attas ts mmum o the bouary of the cotrol rego, that s, ether at u α or u β. hs llustrate Fg..3. I fact we ca wrte the optmal cotrol as α β f p > f p < α H β Fg..3 he case p > cotrol u But p wll vary tme, a satsfes the aot equatos, x p x hus p A, a costat, a p At + B, where B s costat. Sce p s a lear fucto of t, there wll at most be oe swtch the cotrol, sce p has at most oe zero, a from the physcal stuato there must be at least oe swtch. So we coclue that () the cotrol u α or u β, that s, bag-bag cotrol; () there s oe a oly oe swtch the cotrol. Aga, t s clear from the basc problem that tally u β, followe by u α at the approprate tme..3 Swtchg Curves I the last secto we met the ea of a swtch a cotrol. he tme (a posto) of swtchg from oe extremum value of the cotrol to aother oes of course epe o the tal startg pot the phase plae. Bycoserg a specfc example we shall show how these swtchg postos efe a swtchg curve. Chapter Bag-bag Cotrol 55
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 Suppose a system s escrbe by the state varables x, x where x & x + u (.8) x & u (.9) Here u s the cotrol varable whch s subect to the costrats u. Gve that at t, x a, x b, we wsh to f the optmal cotrol whch takes the system to x, x mmum tme; that s, we wsh to mmze J (.3) whle movg from ( a, to (,) the x x phase plae a subect to (.8), (.9) a x B k log ( x k) / A Now f u, that s, k, the the traectores are of the form x B log ( x ) / A that s x log x + C (.3) where C s costat. he curves for fferet values of C are llustrate Fg..4 u x u (.3) Followg the proceure outle secto., we trouce the aot varables p a p a the Hamltoa H + p( x + u) + pu u( p + p ) + px x Sce H s lear a u, H s mmze wth respect to u by takg + f f p + p < p + p > A So the cotrol s bag-bag a the umber of swtches wll epe o the sg chages p + p. As the aot equatos, (.3), are p x t p Ae, A a B are costat p B x a p p Ae t + + B, a ths fucto has at most oe sg chage. Fg..4 raectores for u Follow the same proceure for u, gvg x log x + C (.33) a the curves are llustrate Fg..5. x B So we kow that from ay tal pot ( a,, the optmal cotrol wll be bag-bag, that s, u ±, wth at most oe swtch the cotrol. Now suppose u k, whe k ±, the the state equatos for the system are x + k k We ca tegrate each equato to gve u x t x Ae + k, A a B are costats x kt + B he x x plae traectores are fou by elmatg t, gvg Fg..4 raectores for u Chapter Bag-bag Cotrol 56
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 he basc problem s to reach the org from a arbtrary tal pot. All the possble traectores are llustrate Fgs..4 a.5, a we ca see that these traectores are oly two possble paths whch reach the org, amely AO Fg..4 a BO Fg..5. u x B Followg the usual proceure, we form the Hamltoa H + px + pu (.39) We mmze H wth respect to where u, whch gves + f p < f p > he aot varables satsfy x x p x p A p At + B A Fg..4 raectores for u u Combg the two agrams we evelop the Fg..6. he curve AOB s calle swtchg curve. For tal pots below AOB, we take u + utl the swtchg curve s reache, followe by u utl the org s reache. Smlarly for the pots above AOB, u utl the swtchg curve s reache, followe by u + utl the org s reache. So we have solve the problem of fg the optmal traectory from a arbtrary startg pot. Sce x ( ) s ot specfe, the trasversalty coto becomes p ( ). Hece At + B a p A( t). For < t <, there s o chage the sg of p, a hece o swtch u. hus ether u + or u, but wth o swtches. We have x ( x B) whe u (.4) x ( x B) whe u (.4) hese traectores are llustrate Fg..7, the recto of the arrows beg eterme from x & u u + x x u hus the swtchg curve has equato log( + x ) for x > x (.34) log( + x) for x < x x.4 rasversarlty cotos o llustrate how the trasversalty cotos ( p ( ) f x s ot specfe) are use, we coser the problem of fg the optmum cotrol u whe u for the system escrbe by x & x (.35) x & u (.36) whch takes the system from a arbtrary tal pot x ( ) a, x () b to ay pot o the x axs, that s, x ( ) but x ( ) s ot gve, a mmze J. (.37) Fg..7 Possble Optmal raectores x + { { ( a, ( a, A x subect to (.35), (.36), the above bouary cotos o x a x, a such that u (.38) Fg..8 Ital pot below OA We frst coser tal pots ( a, for whch a >. For pots above the curve OA, there s oly oe traectory, Chapter Bag-bag Cotrol 57
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 u whch reaches the x -axs, a ths must be the optmal curve. For pots below the curve OA, there are two possble curves, as show Fg..8. From (.36): x & u, that s ±, a the tegratg betwee a gves x ( ) x () ± that s x ( ) x () (.4) Hece the moulus of the fferece fal a tal values of x gve the tme take. hs s show the agram as + for u + a for u. he complete set of optmal traectores s llustrate Fg..9. u + x u x Fg..9 Optmal raectores to reach x -axs mmum tme.5 Exteso to the Boltza problem Chapter Bag-bag Cotrol 58