1 Subdivide the optimization horizon [t 0,t f ] into n s 1 control stages,

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1 Opimal Conrol Formulaion Opimal Conrol Lecures 19-2: Direc Soluion Mehods Benoî Chachua Deparmen of Chemical Engineering Spring 29 We are concerned wih numerical soluion procedures for opimal conrol problems ha comply wih he following general formulaion: Deermine: u Ĉ[, f ] nu and v IR nv ha minimize: φ(x( f ),v) subjec o: ẋ() = f(,x(),u(),v); x( ) = h(v) ψ i (x( f ),v) ψ e (x( f ),v) = κ i (,x(),u(),v), f κ e (,x(),u(),v) =, f u() [u L,u U ], v [v L,v U ] Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 1 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 2 / 32 Direc Mehods of Opimal Conrol Discreize he conrol problem, hen apply NLP echniques o he resuling finie-dimensional opimizaion problem Sudied exensively over he las 3 years Proved successful for many complex applicaions Take advanage of he power of sae-of-he-ar NLP solvers Can be applied o ODE, DAE and even PDAE models Conrol Parameerizaion 1 Subdivide he opimizaion horizon [, f ] ino n s 1 conrol sages, < 1 < 2 < < ns = f 2 In each subinerval [ k 1, k ], approximae u() = U k (,ω k ) Lagrange Inerpolaing Polynomials (Degree M) u j () = U k j (,ω k ) = M i= ( ) ω k i,jφ (M) k 1 i k k 1, k 1 k u j () U k j (,ωk ) Three Main Varians 1 Direc simulaneous approach or orhogonal collocaion, or full discreizaion 2 Direc sequenial approach or single-shooing or conrol vecor parameerizaion (CVP) 3 Direc muliple-shooing approach φ (M) i (τ) = 1, if M = M τ τ q, if M 1 τ i τ q q= q i k 1 τ τ 1 τ M 1 wih collocaion poins τ τ 1 < < τ M 1 < τ M 1 ω k M,j ω1,j k ω,j k k τ Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 3 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 4 / 32

2 Conrol Parameerizaion piecewise consan piecewise linear wihou coninuiy u() piecewise linear wih coninuiy piecewise cubic wih coninuiy Direc Simulaneous Mehod Transcripion ino a finie-dimensional NLP hrough discreizaion of boh conrol and sae variables Sae Collocaion x() = X(,ξ k ), k 1 k, k = 1,,n s 1 k ns 1 ns = f Conrol bounds can be enforced by bounding he conrol coefficiens ω k in each subinerval Conrol coninuiy (or higher-order coninuiy) a sage imes k can be enforced via linear inequaliy consrains beween ω k and ω k+1 The choice of he collocaion poins τ i has no effec on he soluion Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 5 / 32 Lagrange Polynomial Represenaion (Degree N) X j (,ξ k j ) = N i= ξ k i,j φ(n) i ( ) k 1 k k 1, k 1 k wih: φ (N) i (τ q ) = δ i,q, q =,,N Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 6 / 32 Direc Simulaneous Mehod Transcripion ino a finie-dimensional NLP hrough discreizaion of boh conrol and sae variables Sae Collocaion x() = X(,ξ k ), k 1 k, k = 1,,n s Monomial Basis Represenaion (Degree N) X j (,ξ k j ) = ξ k,j + ( k k 1 ) N i=1 ( ) ξ k i,j Ω(N) k 1 i k k 1, k 1 k wih: Ω (N) i () = Ω (N) i (τ q ) = δ i,q, q = 1,,N Direc Simulaneous Mehod (con d) Original Opimal Conrol Problem Deermine: u Ĉ[, f ] nu and v IR nv ha minimize: φ(x( f ),v) subjec o: ẋ() = f(,x(),u(),v); x( ) = h(v) ψ i (x( f ),v) ψ e (x( f ),v) = κ i (,x(),u(),v), f κ e (,x(),u(),v) =, f u() [u L,u U ], v [v L,v U ] Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 6 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 7 / 32

3 Direc Simulaneous Mehod (con d) Fully-Discreized Opimizaion Problem Deermine: ω 1,,ω ns IR num,ξ 1,,ξ ns IR nxn and v IR nv ha minimize: φ(x ns ( ns,ξ ns ),v) subjec o: X k ( k,q,ξ k ) = f( k,q,x k ( k,q,ξ k ),U k ( k,q,ω k ),v), k,q X 1 (,ξ 1 ) = h(v); X k+1 ( k,ξ k+1 ) = X k ( k,ξ k ), k ψ i (X ns ( ns,ξ ns ),v) ψ e (X ns ( ns,ξ ns ),v) = κ i ( k,q,x k ( k,q,ξ k ),U k ( k,q,ω k ),v), k,q κ e ( k,q,x k ( k,q,ξ k ),U k ( k,q,ω k ),v) =, k,q ξ k [ξ L,ξ U ], ω [ω L,ω U ], v [v L,v U ] Direc Simulaneous Mehod (con d) Class Exercise: Discreize he following opimal conrol problem ino an NLP via he full discreizaion approach: min u() [u()]2 d s ẋ() = u() x(); x() = 1 x(1) = Consider a single conrol sage, n s = 1 Approximae he sae and conrol profiles using affine funcions, M = N = 1 wih k,q = k 1 + τ q ( k k 1 ), k = 1,,n s, q = 1,,N Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 8 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 9 / 32 Direc Simulaneous Mehod (con d) Pros and Cons very large-scale NLP problems in he variables p T = (ξ 1T,,ξ ns T,ω 1T,,ω ns T,v T ) The numbers of ime sages and collocaion poins as well as he posiion of he collocaion poins mus be chosen a priori The sage imes can be opimized as par of he decision vecor p oo Infeasible pah mehod: The ODEs are saisfied a he converged soluion of he NLP only Bu, his saves compuaional effor and allows unsable sysems Pah consrains are easily accommodaed by enforcing inequaliy consrain a he collocaion poins Class Exercise: Give sufficien condiions for exisence of an opimal soluion o he fully-discreized opimizaion problem Direc Sequenial Mehod Transcripion ino a finie-dimensional NLP hrough discreizaion of he conrol variables only, while he ODEs are embedded in he NLP problem Original Opimal Conrol Problem Deermine: u Ĉ[, f ] nu and v IR nv ha minimize: φ(x( f ),v) subjec o: ẋ() = f(,x(),u(),v); x( ) = h(v) ψ i (x( f ),v) ψ e (x( f ),v) = κ i (,x(),u(),v), f κ e (,x(),u(),v) =, f u() [u L,u U ], v [v L,v U ] Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 1 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 11 / 32

4 Direc Sequenial Mehod (con d) Parially-Discreized Opimizaion Problem Deermine: ω 1,,ω ns IR num and v IR nv ha minimize: φ(x( ns ),v) subjec o: ẋ() = f(,x(),u k (,ω k ),v); x( ) = h(v) ψ i (x( ns ),v) ψ e (x( ns ),v) = κ i (,x(),u k (,ω k ),v), k 1 k, k κ e (,x(),u k (,ω k ),v) =, k 1 k, k ω() [ω L,ω U ], v [v L,v U ] Direc Sequenial Mehod (con d) Class Exercise: Discreize he following opimal conrol problem ino an NLP via he sequenial approach: min u() [u()]2 d s ẋ() = u() x(); x() = 1 x(1) = Consider a single conrol sage, n s = 1 Approximae he conrol profile using affine funcions, M = 1 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 12 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 13 / 32 Direc Sequenial Mehod (con d) Direc Sequenial Mehod (con d) NLP Solver Conrol Variables U(, ω) Sensiiviy Variables x p () Gradiens OR Adjoin Variables λ() Numerical Inegraion Cos & Consrains Sae Variables x() Issue 1: How o calculae he cos and consrain values and derivaives? Issue 2: How o handle pah consrains? Reformulaion as inegral consrains: n s K e j (p) := n s K i j(p) := k=1 k=1 k k k 1 k 1 max ( κ i j (,x(),uk (,ω k ),v)) 2 d { ;κ i j(,x(),u k (,ω k ),v)} 2 d Bu, relaxaion needed o ensure regular consrains: K j (p) ǫ wih ǫ > a small nonnegaive consan Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 14 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 15 / 32

5 Direc Sequenial Mehod (con d) Direc Sequenial Mehod (con d) Issue 2: How o handle pah consrains? Discreizaion as inerior poin consrains: ( ) κ i j k,q,x( k,q ),U k ( k,q,ω k ),v ( ) k,q,x( k,q ),U k ( k,q,ω k ),v = κ e j a a given se of poins k,q [ k 1, k ] in each sage k = 1,,n s The combinaion of reformulaion and discreizaion approaches is recommended! Pros and Cons Relaively small-scale NLP problems in he variables p T = (ω 1T,,ω ns T,v T ) Sage imes can be opimized as par of he decision vecor p oo The accuracy of he sae variables is enforced via he error-conrol mechanism of sae-of-he-ar numerical solvers Feasible pah mehod: The ODEs are saisfied a each ieraion of he NLP algorihm Bu, his is compuaionally demanding and handles mildly unsable sysems only Class Exercise: Give sufficien condiions for exisence of an opimal soluion o he parially-discreized opimizaion problem Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 15 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 16 / 32 Scalar Dynamic Opimizaion Example Consider he dynamic opimizaion problem o: minimize: J(u) := 1 ( [x1 ()] 2 + [x 2 ()] 2) d subjec o: ẋ 1 () = x 2 (); x 1 () = ẋ 2 () = x 2 () + u(); x 2 () = 1 x 2 () + 5 8[ 5] 2, 2 u() 2, Soluion Approach Used Direc sequenial approach wih piecewise consan conrols Nonlinear program solved wih SQP mehods Inequaliy pah consrain reformulaed as: 1 max{;x 2 () + 5 8[ 5] 2 } 2 d ǫ, wih ǫ = 1 6 Scalar Dynamic Opimizaion Example (con d) u() Opimal Resuls for 1 Sages u() Opimal Cos x2() Opimal Resuls for 1 Sages x 2 () pah consrain n s J(u ) Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 17 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 18 / 32

6 Muliple-Shooing Mehod Transcripion ino a finie-dimensional NLP hrough discreizaion of he conrol variables only, wih sae disconinuiies allowed a sage imes Exra Decision Variables and Consrains New decision variables ξ k IR nx, k = 2,,n s, such ha ẋ k () = f(,x k (),U k (,ω k ),v), k 1 k { wih: x k h(v) if k = 1, ( k 1 ) = oherwise New (nonlinear) equaliy consrains: ξ k Muliple-Shooing Mehod (con d) Original Opimal Conrol Problem Deermine: u Ĉ[, f ] nu and v IR nv ha minimize: φ(x( ns ),v) subjec o: ẋ() = f(,x(),u(),v); x( ) = h(v) ψ i (x( ns ),v) ψ e (x( ns ),v) = κ i (,x(),u(),v), f κ e (,x(),u(),v) =, f u() [u L,u U ], v [v L,v U ] x k ( k ) ξ k+1 =, k = 1,,n s 1 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 19 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 2 / 32 Muliple-Shooing Mehod (con d) Muliple-Shooing Mehod (con d) Parially-Discreized Opimizaion Problem Gradiens Cos & Consrains Deermine: ω 1,,ω ns IR num,ξ 1,,ξns IRnx and v IR nv ha minimize: φ(x( f ),v) subjec o: ẋ k () = f(,x k (),U k (,ω k ),v), k 1 k, k NLP Solver x k Sensiiviy Variables p() OR λ k Adjoin Variables () Sae Variables x k () x 1 ( ) = h(v); x k ( k 1 ) = ξ k, k = 2,,n s ψ i (x( f ),v) ψ e (x( f ),v) = κ i (,x(),u k (,ω k ),v), k 1 k, k κ e (,x(),u k (,ω k ),v) =, k 1 k, k ξ k [ξl,ξ U ], ω() [ω L,ω U ], v [v L,v U ] Conrol Variables U(, ω) Numerical Inegraion Exac same procedure as he sequenial approach, excep ha muliple independen ses of differenial equaions Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 21 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 22 / 32

7 Muliple-Shooing Mehod Pros and Cons Lies beween he sequenial and simulaneous approaches: sequenial infeasible-pah mehod Shares many aracive feaures of he sequenial and simulaneous approaches: accurae soluion of he differenial equaions abiliy o deal wih unsable sysems safely SQP solvers exploiing he special srucure of he resuling NLP problem can/mus be devised Muliple shooing lends iself naurally o parallelizaion Class Exercise: Give sufficien condiions for exisence of an opimal soluion o he parially-discreized opimizaion problem Handling Funcionals wih Sae Variables Paricipaing Consider he funcional in Mayer form: F(p) = φ(x( f ),p) subjec o he parameric iniial value problem: ẋ() = f(,x(),p), f ; x( ) = h(p) Key Issues in Sequenial Approaches: Given ha a unique soluion x(; p) o he ODEs exiss for given p P, 1 Calculae he funcional value, F( p) Use sae-of-he-ar mehods for differenial equaions 2 Calculae he funcional derivaives, F pj ( p), wr p 1,,p np 3 possible procedures: finie differences, forward sensiiviy analysis, adjoin sensiiviy analysis Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 23 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 24 / 32 Funcional Derivaives Finie Differences Approach: F pi (p) φ(x( f),p 1,,p i + δp i,,p np ) φ(x( f ),p) δp i Procedure for Finie Differences Approach Iniial Sep: Solve ODEs for he acual parameer values p; Calculae he value of F(p) Loop: j = 1,,n p Se p i := p i, i j; p j := p j + δp j ; Solve ODEs for he perurbed parameer values p F( p) F(p) Calculae he value of F( p), and F pj (p) δp j End Loop Trade-off: Choose δp j small enough, bu no oo small! Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 25 / 32 Funcional Derivaives (con d) Sae Sensiiviies: Under which condiions is x(; ) coninuously differeniable wr p 1,,p np, a a poin p P, for f? 1 A unique soluion x(; p) o he ODEs exiss on [, f ] 2 f is coninuously differeniable wr x and p, and piecewise coninuous wr 3 h is coninuously differeniable wr p Sensiiviy Equaions: The sae sensiiviies, x pj () = x() p j, saisfy he ODEs: ẋ pj () = f x (,x(),p) x pj () + f pj (,x(),p) wih he iniial condiions x pj ( ) = h pj (p) One sensiiviy equaion for each parameer p 1,,p np! Funcional derivaive: F pj (p) = φ x (x( f ),p)x pj ( f ) + φ pj (x( f ),p) Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 26 / 32

8 Funcional Derivaives (con d) Procedure for Forward Sensiiviy Approach Sae and Sensiiviy Numerical Inegraion: f ẋ() = f(,x(),p); x( ) = h(p) ẋ p1 () = f x (,x(),p) x p1 () + f p1 (,x(),p); x p1 ( ) = h p1 (p) ẋ pnp () = f x (,x(),p) x pnp () + f pnp (,x(),p); x pnp ( ) = h pnp (p) Funcion and Gradien Evaluaion: F(p) = φ(x( f ),p) F p1 (p) = φ x (x( f ),p)x p1 ( f ) + φ p1 (x( f ),p) F pnp (p) = φ x (x( f ),p)x pnp ( f ) + φ pnp (x( f ),p) Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 27 / 32 Funcional Derivaives (con d) Forward Sensiiviy Approach: F pi (p) = φ x (x( f ),p) x pi ( f ) + φ pi (x( f ),p) wih: ẋ pi () = f x (,x(),p) x pi () + f pi (,x(),p); x pi ( ) = h pi (p) Number of sae & sae sensiiviy equaions: (n x + 1) n p Bu, independen of he number of funcionals, n F Adjoin (Reverse) Sensiiviy Approach: f F pi (p) = φ pi (x( f ),p) + λ( ) T h pi (p) + f pi (,x(),p) T λ()d wih: λ() = f x (,x(),p) T λ(); λ( f ) = φ x (x( f ),p) Number of sae & adjoin equaions: 2n x n F Bu, independen of he number of parameers Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 28 / 32 Funcional Derivaives (con d) Procedure for Adjoin Sensiiviy Approach Sae Numerical Inegraion: f ẋ() = f(,x(),p); x( ) = h(p); Sore sae values x() a mesh poins, < 1 < < M = f Adjoin Numerical Inegraion: f λ() = f x (,x(),p) T λ(); λ( f ) = φ x (x( f ),p) T q 1 () = f p1 (,x(),p) T λ(); q 1 ( f ) = Funcional Derivaives (con d) Procedure for Adjoin Sensiiviy Approach (con d) Funcion and Gradien Evaluaion: F(p) = φ(x( f ),p) F p1 (p) = φ p1 (x( f ),p) + λ( ) T h p1 (p) + q 1 ( ) F pnp (p) = φ pnp (x( f ),p) + λ( ) T h pnp (p) + q np ( ) q np () = f pnp (,x(),p) T λ(); q np ( f ) = ; Need o inerpolae sae values x() beween mesh poins Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 29 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 3 / 32

9 Funcional Derivaives (con d) Class Exercise: Consider he funcional F(p 1,p 2 ) = x(1) wih: ẋ() = x() + p 1 ; x() = p 2 1 Calculae he value of F by firs solving he ODE analyically; hen, calculae he derivaives F p1 and F p2 2 Derive he sensiiviy equaions and heir respecive iniial condiions, hen find a soluion o hese equaions; apply he forward sensiiviy formula o calculae F p1 and F p2 3 Derive he adjoin equaion and is erminal condiion, hen find a soluion o his equaion; apply he adjoin sensiiviy formula o calculae F p1 and F p2 Exisence of an Opimal Soluion (Finie Dimension) Recalls: Weiersrass Theorem Le P IR n be a nonempy, compac se, and le φ : P IR be coninuous on P Then, he problem min{φ(p) : p P} aains is minimum, ha is, here exiss a minimizing soluion o his problem Why closedness of P? Why coninuiy of φ? Why boundedness of P? φ φ(c) φ φ a b a c b a (a) (b) (c) + Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 31 / 32 Benoî Chachua (McMaser Universiy) Direc Mehods Opimal Conrol 32 / 32