Numerical Optimization

Transcription

1 Numerical Opimizaion A Workshop A Deparmen o Mahemaics Chiang Mai Universiy Augus 4-5, 9 Insrucor: Elecrical Engineering and Compuer Science Case Wesern Reserve Universiy Phone: , Fax: vira@case.edu Session: Opimizaion o Dynamic Sysems Case Wesern Reserve Universiy Elecrical Engineering and Compuer Science EECS.

2 Module Objecive o develop uncional knowledge and skills in modeling dynamic sysems and/or opimal conrol problems deermining a sae rajecory and/or a conrol o dynamic sysems ha opimizes a perormance uncional subjec o consrains on conrol and/or saes. 3 Module Highlighs wo pronged approach:. Focus on he wo radiional approaches or dealing wih dynamic opimizaion/opimal conrol problems he variaional approach based on calculus o variaions leading o he maximum principle he dynamic programming approach and is corresponding Hamilon-Jacobi-Bellman (HJB equaion. When applied o linear opimal conrol problem derivaion o similar resuls based on he concep o Lyapunov sabiliy will also be demonsraed.. Focus on numerical mehods or solving large real-world dynamic opimizaion and opimal conrol problems wih complex consrains. he wo numerical approaches are he indirec approach and he direc approach. MALAB will be used o implemen mehods discussed. Applicaions o engineering and economic problems will be illusraed hroughou 4

3 ex: Reerences Opimal Conrol heory, D.E. Kirk, Dover Publicaions, ISBN: , 4 Reerences:. Pracical Mehods or Opimal Conrol Using Nonlinear Programming, J.. Bes, SIAM, ISBN: ,. Applied Dynamic Programming or Opimizaion o Dynamical Sysems, R.D. Robine III, D.G. Wilson, G. Richard Eisler and J. E. Hurado, SIAM, ISBN: , 6 5 Dynamic Opimizaion ( n min J = gxx (,,.., x, d x: [, ] R ( n ( n subjec o, x( = x, ( =,.., x ( = x ( n ( n and possibly, x( = x, ( =,.., x ( = x Noe: he inal ime may or may no be speciied. I is speciied ixed-end-ime problem (or ixed-erminal-ime I is no speciied variable-end-ime problem ( or open-erminal-ime, open horizon I is speciied, and i x ( is also ixed ixed-end-poin problem x ( is consrained (i.e. x ( S consrained-erminal-poin problem x ( is no ixed or has no resricion ree-end-poin problem 6 3

4 Opimal Conrol Problems min J = h( x(, + g( x(, u(, d m u: [, ] R subjec o Sysem Dynamics: ( = a( x(, u(, ; where sae x: [, ] R Iniial condiions: x( = x erminal or inal condi ions: x( = x n Find a conrol u( o ake he sysem rom he iniial sae x( a he iniial ime o he inal sae x( a he inal ime. 7 Opimal Conrol Problems Possible addiional consrains: Conrains on conrol: u( U or all [, ] e.g. u u( u or all [, ] min max Conrains on sae: x( X or all [, ] e.g. xmin x( xmax or all [, ] 8 4

5 Noe: Opimal Conrol Problems As beore, may be speciied ( ixed-end-ime I is speciied, or may no speciied ( variable-end-ime or open erminalime x ( may be ixed ( ixed-end-poin or hard erminal consrain or consrained x ( S( so erminal consrain or no ixed or no resricion ( ree-end-poin 9 Various ypes o J : Opimal Conrol Problems g =, i.e. J = h( x(, Mayer Problem Special Mayer Problems: ( a J = cx( (Linear Mayer ( ( ( b J = x( - r( H x( - r( (erminal conrol problem h =, i.e. J = g( x(, u(, d lagrange Problem Special Lagrange Problems: (a g =, i.e. J = d = - (b g = u(, i.e. J = u( d (Minimum ime (Minimum uel (c g = u(, i.e. J = u( u( d (Minimum energy 5

6 (a racking problem: Opimal Conrol Problems J = h( x(, + g( x(, u(, d bolza Problems Special Bolza Problems: h = g = ( x( - r( H( x( - r( ( x(-( r Q( x(-( r + u( Ru( ( x r H( x r ( x r Q( J = ( -( ( - ( + (- ( x(-( r + u( Ru( d Noe: H, Q and R are w eighing marices (b Regulaor problem: h = x( Hx( g = x( Qx( + u( Ru( J = x( Hx( + x( Qx( + u( Ru( d Example : (Brachisochrome A bead o mass uni descends along a wire joining wo ixed poins (x,y and (x,y. We wish o ind he shape o he wire so ha he bead complees is slide in minimum ime. (x,y = (x,y 6

7 Example : (Brachisochrome Model: Dynamic Opimizaion Model: (x,y = y = y(x, y: [x,x ] R, y C ( x g y y( x (x,y min = subjec o yx ( = y yx ( x = y + y ( x dx Minimum ime Fixed erminal poin problem (Hard erminal consrain 3 (x,y = (, Example : (River-Crossing A boa ravels wih consan velociy V wih respec o he waer. In he region he velociy o he curren is parallel o x-axis bu varies wih y. Given he desinaion (x,y on he oher side o he river, ind he pah o be aken by he boa o minimize he ravel ime, y = V θ (x,y = v c (y Opimal Conrol Model: min = d + x Final condiions: x( = x ; y( = y Minimum ime Fixed erminal poin problem subjec o Dynamics (Equaion o Moion: = V cos θ + vc ( y y = V sinθ Iniial condiions: x( = x ; y( = y 4 7

8 (x,y = (, Example 3: (River-Crossing A boa ravels wih consan velociy V wih respec o he waer. In he region he velociy o he curren is parallel o x-axis bu varies wih y. Given he inal ime, ind he pah o be aken by he boa o maximize he landing disance on he oher side o he river, y = V θ (x,y = v c (y x Opimal Conrol Model: max J = x( subjec o Dynamics (Equaion o Moion: = V cos θ + vc ( y y = V sinθ Iniial condiions: x( = x ; y( = y Final condiions: y ( = y Linear Mayer problem wih semi-hard erminal consrain 5 Example 4: (Braking and Acceleraion: Wan o move a car o mass m rom o x in minimum ime. Minimum ime α ( +β ( Opimal Conrol: Hard erminal consrain x( x( = x min J = = d wih bound consrains EOM: x( = α( + β ( subjec o Sae: s x ( = x( Dynamics (Equaion o Moion: x( = ( = ( = x( Compac Dynamics: = α( + β( = Ax( + Bu( Iniial condiio ns: x( = ; x( = A = erminal condiions: x ( = x; x ( =, B = Conrains: For each [, ] : x( u( α( x( =, u( = = On conrol: u( M; u( M x( u( β ( On sae: x ( x ; x ( x( = 6 8

9 y v β( Example 5: (aking-o An aircra o mass m (assumed poin mass is o be lied by a consan rus o reach he cruising aliude H a ime, a maximum speed (along x- direcion. = H Opimal Conrol Model: max J = x ( Dynamics (Equaion o Moion: = = = 3 4 cos β m = 4 = sin β g x m Iniial condiions: x ( = x( = x3( = x4( = EOM: mx ( = cos β ( Final condiions: x3( = H; x4( = my ( = sin β ( mg Consrains: On conrol u ( π /, [, ] Saes: x( = x(; x( = ( = ( On sae: xi (, i =,..4; x3 ( H x3( = y(; x4( = 3( = y ( Conrol u ( = β ( x Linear Mayer problem wih semi-hard erminal consrain x 7 Example 6: (Rover Conrol Wan o conrol he speed o a Mariner Mars rover a abou 5 mph using as lile energy as possible. he conroller is he oupu volage o a baeryoperaed volage regulaing sysem. Baery Volage Regulaing Sysem i e( R L I a (consan R a L a θ( Viscous ricion coeicien B di ( Dynamics: Ri ( + L = e( d λ( = Ki( orque λ( = I θ( + B θ( + λl ( Saes: x( = i ( ; x( = θ ( Conrols: u ( = e(; u ( = λ ( L λ( λ L ( 8 9

10 Example 6: (Rover Conrol Min J = ( kx ( ( 5 + wx( u ( d i Dynamics : Baery R Volage R = x( + u( Regulaing e( L L Sysem L K B = x( - x( u( I I I I a (consan Iniial condi ions: x( = x( = R a Final con diions: None Viscous ricion Consrai ns : L coeicien B a θ( On conrol: u( emax, [, ] u( λmax, [, ] λ( λ L ( On sae: x( Imax, [, ] Minimum energy problem wih ree erminal x( Ωmax, [, ] condiions bu wih bound consrains 9 u( 5gal ank A 8lb sal a = Example 7: (Mixure Wan o ind inlow rae o resh waer (conrol o he wo-ank sysem so ha he sal concenraion in anks A and B are equal using minimum amoun o resh waer. u( 5gal ank B Fresh waer a = u( Opimal Conrol Model: min J = u( d Dynamics: u ( Assume: Well-mixed a all ime = x( 5 Incompressible luid u ( u ( Saes: x ( = lbs o sal in A a ime = x( x( 5 5 x ( = lbs o sal in B a ime Iniial condiions: x( = 8; x( = Co nrols: u ( = gal/m o resh waer erminal condiions: x( - x( = passing hru a ime Conrains: For each : Minimum conrol eor wih On conrol: u ( umax, [, ] So erminal consrain and bound consrains On sae: x ( 8; x (

11 Opimizing Yeas or Ehanol Producion in a Bioreacor Bioreacors are large vessels ha serve as an environmen or biochemical reacions o occur. ypical uses include he growh o microorganisms and he breakdown o producs. Source: D. Moore, MS hesis,, 7 he environmen wihin he vessel is conrolled o opimize perormance. ypical conrol variables include nurien eed rae, oxygen air low rae, and emperaure. here is large economic incenive o develop conrol sraegies o maximize he producion o baker s yeas and ehanol, wo imporan commercial producs produced in bioreacors. Yeas is ypically grown o a soluion conaining glucose and oher nuriens essenial or cellular growh. When glucose concenraion in he medium is high or when here is a limied supply o oxygen, he yeas microorganism excrees ehanol. Opimizing Yeas or Ehanol Producion in a Bioreacor We wish o maximize producion o yeas, or ehanol or boh by conrolling he subsrae eed rae, airlow (O and emperaure Model o Growh Dynamics X = yeas concenraion (g/l S = subsrae (glucose concenraion (g/l E = ehanol concenraion (g/l O = dissolved oxygen (O concenraion (g/l C = dissolved carbon dioxide (CO concenraion (g/l V = liquid volume (l F in = Subsrae eed rae (l/h S in = Inluen Subsrae Concenraion (g/l D = F in /V = Diluion rae (/h OR = k L a(o S O = O ranser rae (g/l h - CER = k V k L ac = CO evoluion rae (g/l h - m = Mainenance erm (g o S /g o X h - 7-Aug-9

12 wo Ways o Solve Dynamic Opimizaion/Opimal Conrol Problems. Indirec Mehod: Solve he necessary condiions or opimaliy derived hrough variaional principles rooed in Calculus o Variaions ha is: Solve wo-poin boundary value problems (PBVP 3 wo Ways o Solve Dynamic Opimizaion/Opimal Conrol Problems. Direc Mehod: Opimize he uncional direcly as consrained opimizaion Require conversion o nonlinear programs hrough ranscripion o he ODEs (dynamics o sysem Oen possesses high sparsiy and special srucure 4

13 Direc Mehod or Dynamic Opimizaion Conver o nonlinear programs hrough direc ranscripion o he ODEs (dynamics o sysem or he corresponding DAEs Use nonlinear opimizer such SQP o solve he resuling nonlinear programs (Soware such as SNOP by Boeing, ec. Fine-une he resul hrough meshreinemen echniques 5 Direc Mehod: ranscripion mehods Euler: x = x + h a( x, u, Classical Runge-Kaa: x x s k+ k k k k k k+ = k + hk k where sk = ( k + k + k3 + k4 6 k = hka( xk, uk, k hk k = hka( xk + k, uk+, k + hk k3 = hka( xk + k, uk+, k + k4 = hka( xk + k3, uk+, k+ rapezoidal: xk+ = xk + hk a( xk, uk, k + a( xk + hka( xk, uk, k, uk+, k+ Hermi-Simpson: ( hk xk+ = xk + ( a( xk, uk, k + a( xk + hka( xk, uk, k, uk+, k+ + ak+ 6 hk where xk+ = xk + xk + hka( xk, uk, k + a( xk, uk, k a( xk + hka( xk, uk, k, uk+, k+ 8 hk and ak+ = a( xk+, uk+, k + ( ( 6 3

14 Direc Mehod: ranscripion mehods Band, Sair-case Srucure and Sparsiy o resuling marix: Employ special numerical ricks o ake advanage o he special srucure and sparsiy o he resuling problem 7 Indirec Mehod: Derivaion o Opimaliy Condiions Euler-Lagrange Equaions and all Boundary Condiions Hamilon-Jacobi Jacobi-Bellman Condiions Ponryagin s Minimum Principle 8 4

15 Indirec Mehod: Fundamenals Variaions o a Funcional: Pah opimizaion: Funcional J( y( x: where x [ x, x] and y:[ x, x] R Δ J( y*, δy = δj( y*, δy + ο( δy incremen variaion error erm where ο( δy as δy Opimal conrol: Funcion J ( x( or J( x(, : where [, ] and x:[, ] R Δ J( δx = δj( δx + ο( δx incremen variaion error erm Δ J(*, x δx, δ = δj(*, x δx, δ + ο( δx, δ incremen variaion error erm n n 9 J Key variaion ormular: δj ( x, δx = δx x J J J = δx + δx δx x ( n ( x( = (,,..,, For example: J g x x x d J J J ( n hen δj( x, δx = δx+ δ δx ( n ( n =.. δx+ δ + + δx ( n d erm like δ xd are deal wih hru inegraion by par i.e. Indirec Mehod: Fundamenals δxd = δx δ xd n n 3 5

16 3 Fundamenal heorem o Calculus o Variaions: x* is an exremal o J( x only i δj( δx = or all admissible δx. 4 Fundamenal Lemmas o Calculus o Variaions: a Le α( x C[ a, b]. a b I α( xhxdx ( = or all hx ( Cab [, ] wih ha ( = hb ( = hen α( x = or all x [ a, b] b Le α( x C a b [, ]. b I α( xh ( xd x= or all h( x C [ a, b] wih h( a = h( b = a hen α( x = c or all x [ a, b] x Indirec Mehod: Fundamenals C a b c Le α( and β(x [, ]. ( α β b I ( xhx ( + ( xh ( x dx= or all hx ( C[ ab, ] wih ha ( = hb ( = a hen β ( x = α( x or all x [ ab, ] 3 Indirec Mehod: Necessary Condiions Now consider J( x = g( x,, d, x:[, ] R a Case: and x( are ixed (ixed end poin x* is an exremal o Jx ( only i δ Jx ( *, δ x = or all admissible δ x. J J = δ J( δ x = δ x+ δ or all admissible δx x = or all admissible δx+ δ d δx = δx + δ x δx d (inegraion by par x = or all admissible δxd δx (since δx ( = and δx ( = g = (by Fundamenal Lemmas o Calculus o Variaions 4a his is Euler-Lagrange Equaion 3 6

17 Indirec Mehod: Necessary Condiions For J( x = g( x,, d, x: [, ] R wih and x( ixed (ixed end poin Necessary Condiions: Euler-Lagrange Equaion x* is an exremal o Jx ( only i = (second-order ODE wih -boundary poin x ( = x and x ( = x Solved by shooing mehod (or example 33 Indirec Mehod: Necessary Condiions Example: J x = x x d x R g( x, x π ( ( ( (, :[, ] wih x( =, x( π = (ixed end poin Euler-Lagrange Equaion = x ( = x+ x = x ( = ccos+ c sin Wih x( = and x( π = x ( = cos+ sin = sin Solved numerically by he shooing mehod 34 7

18 Indirec Mehod: Necessary Condiions For J( x = g( x,, d, x:[, ] R b Case: is ixed and x( is ree (ree end poin x* is an exremal o J ( x only i δj( δx, δ x = or all admissible ( δx, δ x. J J = δj( δx = δx+ δ or all admissible δx = or all admissible δx+ δ d δx = δ x + δ x δ x d (inegraion by par x = x δ x ( + δxd or all admissible δx, and δx( (since δx ( = and δx( = ( Euler-Lagrange = x ( = 35 Indirec Mehod: Necessary Condiions Now consider J( x, = g( x,, d, x:[, ] R and is ree (variable end ime, ree erminal ime or horizon a Case: x ( is ree and is independen o (ree erminal condiions =δ J( δx, δx, δ or all admissible ( δx, δx, δ + δ = δ gxx (,, d+ (*, *, or all admissible x, x and gx d δ δ δ = δ x( + δ x d + g(*,*, x δ x = ( δx - * ( δ + ( *, *, δ xd + g x x δ x x (since δx ( = δx - *( δ g g = δ x + δ xd g( x* ( + δ or all admissible δx, δx and δ = (Euler-Lagrange =, gx (*,*, *( =, and x ( = 36 8

19 Indirec Mehod: Necessary Condiions Now consider J( x, = g( x,, d, x:[, ] R and is ree (variable end ime, ree erminal ime or horizon b Case: x ( = x (Hard erminal conrain = (Euler-Lagrange x ( = x, gx ( *, *, *( =, and x ( = c Case: x ( = θ ( (So erminal conrain -- δ x = θ( δ = (Euler-Lagrange θ ( + g( *, * ( =, x*, x*, and x ( =, x ( = θ ( 37 Example Derivaion o Case (c Now consider J( x, = g( x,, d, x:[, ] R and is ree (variable end ime, ree erminal ime or horizon c Case: x( = θ ( (So erminal conrain =δj( δx, δx, δ or all admissible δ x + δ = δ gx ( *, *, d + ( *, *, or all admissible gx d δ x = δx ( + δ xd g( δ + x = ( δx - *( δ + ( *, *, δ xd+ g x δ x (since δx ( = δx - *( δ = δ x + δ xd+ g( x *( δ x or all admissible δx, δx and δ = (Euler-Lagrange =, g(*,*, x *( =, and x( = 38 9

20 Indirec Mehod: Necessary Condiions Furher exensions: Piecewise Coninuous Soluion: Require addiional Weisrass Erdman Corner Condiions or ranversailiy Condiions Consrains on x( require he use o mulipliers Mos imporan cases are in opimal conrol problems: 39 Indirec Mehod: Necessary Condiions Opimal Conrol Problems: Opimize J ( xu,, = h( x(, + g( xu,, d s.. = a( x, u, Boundary condiions: x( = x Cas es: I is ixed and x( is ixed II is ixed and x( is ree III is ixed, and m( x( = IV is ree and x( is ree V is ree and x( is ixed VI is ree and x( = θ( VII is ree, and m( x( = VIII is ree, and m( x(, = 4

21 Indirec Mehod: Necessary Condiions Opimal Conrol Problems: Opimize J( xu,, = h( x(, + g( xu,, d s.. = a( x, u, Boundary condiions: x( = x Case IV: h( x(, = h ( x(, d + h( x(, ignored (consan H ( xup,,, Hamilonian h( x(, h( x(, = + d h( x(, h( x(, g a ( xxup,,,, = g( xu,, ++ p ( axu (,, + J ( xxup,,,, = g ( xxup,,,, d a a δja(*, x δx,*, u δu,*, p δp, δx, δ H * H * = d δu+ + p δx u h(*, x h( x *(, + p( δ x + g* + p a( u*, + = or all admissible δx, δu, δx, δ δ 4 Indirec Mehod: Necessary Condiions Necessary Condiions or Opimal Conrol Problems: Case IV: Free-end poin problems: ree and x( ree Hamilonian-Jacobi Condiions: ( u*is exremal, hen here exiss co-saes p*such ha: H * = ( u H * p = ( = a( x, u, (3 h(*, x h( x*(, p( δx + H* + δ = (4 and x( = x OHER CASES CAN BE SIMILARILY DERIVED 4