Explicit Stochastic Nonlinear Predictive Control Based on Gaussian Process Models

Transcription

1 Explct Stochatc Nolear Predctve Cotrol Baed o Gaua Proce Model Alexadra Gracharova, Juš Kocja, ad Tor A. Johae Abtract Nolear Model Predctve Cotrol (NMPC) algorthm are baed o varou olear model. Recetly, a o-le optmzato approach for tochatc NMPC baed o a Gaua proce model wa propoed. A gfcat advatage of the Gaua proce model that they provde formato about predcto ucertate, whch would be of help NMPC deg. O the other had, a explct oluto to the tochatc NMPC problem baed o Gaua proce model would allow effcet o-le computato a well a verfablty of the mplemetato. Th paper ugget a approxmate mult-parametrc Nolear Programmg approach to explct oluto of tochatc NMPC problem for cotraed olear ytem baed o Gaua proce model. I partcular, the referece trackg problem codered. The approach buld a orthogoal earch tree tructure of the tate pace partto ad cot cotructg a feable PWL approxmato to the optmal cotrol equece. N I. INTRODUCTION ONLINEAR Model Predctve Cotrol (NMPC) volve the oluto at each amplg tat of a fte horzo optmal cotrol problem ubject to olear ytem dyamc ad tate ad put cotrat [], [], [3], [4]. A recet urvey of the ma o-le optmzato tratege of NMPC gve [5]. Mathematcal model of egeerg ytem uually cota ome amout of ucertaty (ukow addtve dturbace ad/or ucerta model parameter). Therefore, the robut MPC problem formulato requre the model ucertaty to be take to accout. I may applcato, the ytem to be cotrolled decrbed by a tochatc model where the probabltc dtrbuto of the ucertaty aumed to be kow. Several approache for cotraed MPC baed o tochatc model (tochatc MPC) are propoed [6] []. The approache [6], [7], [8] are baed o lear tate pace Th work wa poored by the Ad-Futura cece ad educato foudato of Republc of Slovea ad by the Reearch Coucl of Norway through the Strategc Uverty Programme o Computatoal Method Nolear Moto Cotrol. A. Gracharova wth the Ittute of Cotrol ad Sytem Reearch, Bulgara Academy of Scece, Acad. G. Bochev tr., Bl., P.O.Box 79, Sofa 3, Bulgara (phoe: ; fax: ; e-mal: alexadra.gracharova@abv.bg). J. Kocja wth the Departmet of Sytem ad Cotrol, Jozef Stefa Ittute, Jamova 39, Ljubljaa, Slovea ad wth Uverty of Nova Gorca, School of Egeerg ad Maagemet, Vpavka 3, 5 Nova Gorca, Slovea (e-mal: ju.kocja@j.). T. A. Johae wth the Departmet of Egeerg Cyberetc, Norwega Uverty of Scece ad Techology, 749 Trodhem, Norway (e-mal: Tor.Are.Johae@tk.tu.o). model wth tochatc parameter ad/or addtve oe ad they optmze the expected value of the cot fucto ubject to hard put cotrat [6] or probabltc cotrat [7], [8]. I [9], [], [], [], tochatc MPC approache corporatg a probabltc cot ad probabltc cotrat are developed. The method uggeted [9] baed o a movg average (MA) model wth radom coeffcet. It wa further exteded to lear tme-varyg MA model [] ad to tate pace model wth tochatc ucertaty the output or the put map [], []. It hould be oted that the tochatc MPC approache [6] [] are baed o parametrc probabltc model. Alteratvely, the tochatc ytem ca be modeled wth o-parametrc model whch ca offer a gfcat advatage compared to the parametrc model. Th related to the fact that the o-parametrc probabltc model provde formato about predcto ucertate whch are dffcult to evaluate approprately wth the parametrc model. The Gaua proce model a example of a o-parametrc probabltc black-box model ad up to ow t ha bee appled to model maly tatc olearte. The ue of Gaua procee the modellg of dyamc ytem a recet developmet e.g. [3], [4], [5]. I [6], [7], [8], a o-le optmzato approach for tochatc NMPC baed o Gaua proce model propoed. It ha recetly bee how that the feedback oluto to lear ad quadratc cotraed MPC problem ha a explct repreetato a a pece-we lear (PWL) tate feedback defed o a polyhedral partto of the tate pace [9]. The beeft of a explct oluto, addto to the effcet o-le computato, clude alo verfablty of the mplemetato, whch a eetal ue afetycrtcal applcato. For olear ad tochatc MPC the propect of explct oluto are eve hgher tha for lear MPC, ce the beeft of computatoal effcecy ad verfablty are eve more mportat. A approach for effcet o-le computato of NMPC for cotraed put-affe olear ytem ha bee uggeted []. I [], [], [3], approache for off-le computato of explct ub-optmal PWL predctve cotroller for geeral olear ytem wth tate ad put cotrat have bee developed, baed o the mult-parametrc Nolear Programmg (mp-nlp) dea [4]. The metoed method for explct NMPC are baed o determtc frt prcple model of the ytem.

2 Th paper ugget a approxmate mp-nlp approach to explct oluto of tochatc NMPC problem for cotraed olear ytem baed o a Gaua proce model (referred to a GP-NMPC problem). I partcular, the referece trackg problem codered. The cotrbuto of the preet work the formulato of a more geeral GP-NMPC problem compared to [6], [7], [8], ad repreetg t a a mp-nlp problem. Further, the approxmate mp-nlp approach [3] appled to buld a orthogoal earch tree tructure of the tate pace partto ad cotruct a feable PWL approxmato to the optmal cotrol equece. Thu, the approach propoed th paper ca be codered a a applcato of the approxmate method [], [3], [5] for explct oluto of MPC problem to the cae where the ytem dyamc decrbed by a probabltc (Gaua proce) model. The followg otato wll be ued the paper. A mea that the quare matrx A potve defte. For T x, the Eucldea orm x = x x ad the weghted orm defed for ome ymmetrc matrx A a x A T = x Ax. For a radom varable y wth Gaua dtrbuto, N ( µ ( y), σ ( y)) deote t probablty dtrbuto, ad µ ( y) ad σ ( y) are repectvely t mea ad varace. II. MODELLING OF DYNAMIC SYSTEMS WITH GAUSSIAN PROCESSES A Gaua proce a example of the ue of a flexble, probabltc, oparametrc model whch drectly provde u wth ucertaty predcto. It ue ad properte for modellg are revewed [6]. A Gaua proce a collecto of radom varable whch have a jot multvarate Gaua dtrbuto. Aumg a relatohp of the form y = f( z) betwee a D put z ad output y, we have y(), y(),..., y( M) N (, Σ), where Σ pq = Cov( y( p), y( q)) = C( z( p), z( q)) gve the covarace betwee the output pot y( p ) ad yq ( ) correpodg to the put pot z( p ) ad zq ( ). Thu, the mea µ ( z) (uually aumed to be zero) ad the covarace fucto Czp ( ( ), zq ( )) fully pecfy the Gaua proce. Note that the covarace fucto Czp ( ( ), zq ( )) ca be ay fucto wth the property that t geerate a potve defte covarace matrx. A commo choce : D Czp ( ( ), zq ( )) = vexp w( z( p) z( q)) + vα pq () = where Θ= [ w,..., wd, v, v] are the hyperparameter of the covarace fucto, z deote the -th compoet of the D -dmeoal put vector z, ad α pq the Kroecker operator. Other form of covarace fucto utable for dfferet applcato ca be foud [7]. For a gve problem, the hyperparameter are leared (detfed) ug the data at had. After the learg, oe ca ue the w parameter a dcator of how mportat the correpodg put compoet (dmeo) are: f w zero or ear zero t mea that the put dmeo cota lttle formato ad could pobly be removed. Coder a et of M D-dmeoal put vector Z = [ z(), z(),..., z( M)] T ad a vector of output data Y = [ y(), y(),..., y( M)] T. Baed o the data ( Z, Y ), ad gve a ew put vector z, we wh to etmate the probablty dtrbuto of the correpodg output y. Ulke other model, there o model parameter determato a uch, wth a fxed model tructure. Wth th model, mot of the effort cot tug the parameter of the covarace fucto. Th doe by maxmzg the log-lkelhood of the parameter, whch computatoally relatvely demadg ce the vere of the data covarace matrx (M M) ha to be calculated at every terato. The decrbed approach ca be ealy utlzed for regreo calculato. Baed o a trag et Z, a covarace matrx K of ze M M determed. A already metoed before, the am to etmate the probablty dtrbuto of the correpodg output y at ome ew put vector z. For a ew tet put z, the predctve dtrbuto of the correpodg output y z,( Z, Y) ad Gaua, wth mea ad varace: T µ ( z ) = k( z ) K Y () T σ ( z ) = k( z ) k( z ) K k( z ) + v where kz ( ) = [ Cz ( (), z),..., CzM ( ( ), z)] T the M vector of covarace betwee the tet ad trag cae ad k ( z ) = C( z, z ) the covarace betwee the tet put ad telf. Gaua procee ca be ued to model tatc olearte ad ca therefore be ued for modellg of dyamc ytem f delayed put ad output gal are ued a regreor [3]. I uch cae a autoregreve model codered, uch that the curret predcted output deped o prevou etmated output, a well a o prevou cotrol put: zt ( ) = [ yt ˆ( ), yt ˆ( ),..., yt ˆ( L), ut ( ), T ut ( ),..., ut ( L)] (3) yt ˆ( ) = f( zt ()) + η() t where t deote coecutve umber of data ample, L a gve lag, ad η () t the predcto error. The qualty of the predcto wth a Gaua proce model aeed by

3 computg the average quared error (ASE): M ASE= [ µ ( y ˆ( )) y ( )] (4) M = ad by the log dety error (LD) [3]: M µ y ˆ y log( π) log[ σ ( yˆ ( ))] (5) = σ ( y ( )) [ ( ( )) ( )] LD = + + M ˆ I (4), (5), µ ( y ˆ( )) ad σ ( yˆ ( )) are the predcto mea ad varace, y () the ytem output ad M the umber of the trag pot. The teratve mult-tep ahead predcto ca be doe the followg way, a decrbed [8]: ) by feedg back at each tme tep the predctve mea oly; ) by feedg back at each tme tep both the predctve mea ad the predctve varace; 3) by Mote Carlo mulato. Thu, the ucertaty attached to each termedate predcto take to accout. The Gaua proce model ow ot oly decrbe the dyamc charactertc of the o-lear ytem, but at the ame tme provde formato about the cofdece the predcto. The Gaua proce ca hghlght area of the put pace where predcto qualty poor, due to the lack of data, by dcatg the hgher varace aroud the predcted mea. It worthwhle otg that the dervatve of mea ad varace wth repect to put data ca be calculated traghtforward maer. For more detal ee [8]. III. FORMULATION OF THE GP-NMPC PROBLEM AS AN MP- NLP PROBLEM Coder a tochatc ytem decrbed by a ucerta olear dcrete-tme model: x( t+ ) = f( x(), t u()) t + ξ () t (6) m where xt () ad ut () are the tate ad put varable, ξ () t are Gaua dturbace, ad m f : a olear cotuou fucto. The ucertaty cot that the aalytcal expreo of f( x, u ) ot kow ad ether are the mea value ad the covarace of the dturbace ξ () t. The relatohp (6) repreeted the form: yt () = fg ( zt ()) + ξ () t (7) + m where yt () = xt ( + ) ad zt () = [ xt (), ut ()]. Suppoe that we have a output data et Y = [ y(), y(),..., y( M )], =,,..., correpodg to a put data et Z = [ z(), z(),..., z( M )]. Aume that the relatohp (7) approxmated wth Gaua procee wth dtrbuto: Y N (, Σ), Y N (, Σ),, Y N (, Σ) (8) where the covarace fucto Σ, pq = Cov ( y ( p), y ( q)) = C ( z( p), z( q)),, Σ, pq= Cov ( y( p), y( q)) = C( z( p), z( q)) wth p =,,..., M, q =,,..., M, deped o the gve put ad output data et. Havg obtaed the Gaua proce model (8), the probablty dtrbuto of the output ym ( ) correpodg to a ew put zm ( ) ca be determed a decrbed the prevou ecto: Z N µ σ y ( M) z( M ),(,Y ) ( ( y ( M)), ( y ( M))) Z N µ σ y ( M) z( M ),(,Y ) ( ( y ( M)), ( y ( M))) I (9), µ ( y ( M)) ad σ ( y ( M)) deote repectvely the mea ad the varace of the output varable y ( M ), =,,...,. We troduce the vector µ ( M) = [ µ ( y ( M)),..., µ ( y ( M))] ad y y = [ Y, Y,..., Y ] σ ( M) = [ σ ( y ( M)),..., σ ( y ( M))] ad the matrx Y. The, the relato (9) repreeted: y (9) ym ( ) zm ( ),( ZY, ) N ( µ ( M), σ ( M)) () A how [8], t poble to obta a mult-tep ahead predcto: ym ( + k) zm ( + k ),( ZY, ) N ( µ y( M+ k), σy( M+ k)) () k =,,..., N Suppoe the tal tate x() t = xtt ad the cotrol put ut ( + k) = ut+ k, k=,,..., N are gve. The, by takg to accout that yt () = xt ( + ) ad zt () = [ xt (), ut ()], from () we obta the probablty dtrbuto of the predcted tate xt+ k+ t, k =,,..., N whch correpod to the gve tal tate x tt ad cotrol put ut+ k, k =,,..., N : xt+ k+ t xt+ k t, ut+ k N ( µ ( xt+ k+ t), σ ( xt+ k+ t)) () k =,,..., N The 95% cofdece terval of the radom varable x t + k + t [ µ ( x ) σ ( x ); µ ( x ) + σ ( x )], where t+ k+ t t+ k+ t t+ k+ t t+ k+ t σ ( + + ) the tadard devato. x t k t Here, we coder a referece trackg problem where the goal to have the tate vector x() t track the referece gal rt (). I the problem formulato, the type of the cot fucto lke the oe ued [9]. Suppoe that a full meauremet of the tate x() t avalable at the curret tme t. For the curret x() t, the referece trackg GP-NMPC olve the followg optmzato problem: Problem P: V ( x( t), r( t), u( t )) = m J( U, x( t), r( t), u( t )) (3) ubject to x = xt () ad: tt U y

4 µ ( x ) σ ( x ) x, k =,..., N (4) t+ k t t+ k t m µ ( x ) + σ ( x ) x, k =,..., N (5) t+ k t t+ k t max u u u, k =,,..., N (6) m t+ k max um ut+ k umax, k =,,..., N (7) max{ µ ( xt+ t) σ( xt+ t) r( t), µ ( xt+ t) + σ( xt+ t) r( t)} δ (8) ut+ k = ut+ k ut+ k, k =,,..., N (9) xt+ k+ t xt+ k t, ut+ k N ( µ ( xt+ k+ t), σ ( xt+ k+ t)) k =,,..., N () wth U = [ ut, ut+,..., ut+ N ] ad the cot fucto gve by: JU (, xt ( ), rt ( ), ut ( )) = µ ( x ) rt ( ) + t+ t N µ ( xt k t) r( t) u + + Q t+ k R k = P () Here, N a fte horzo. From a tablty pot of vew t derable to chooe δ (8) a mall a poble []. However, due to the ucertaty of the x t + t predcto, characterzed by the varace σ ( x t + N t ), the feablty of problem P wll rely o δ beg uffcetly large. A part of the GP-NMPC deg wll be to addre th tradeoff. If the ytem aymptotcally table (or pre-tablzed), N large, ad the Gaua proce model ha a mall predcto ucertaty, the t more lkely that the choce of a mall δ wll be poble. A more geeral tochatc MPC problem formulated [9], [], [], [], where a probabltc formulato of the cot troduced that clude the probabltc boud of the predcted varable. Alo thee referece, a probabltc formulato of the cotrat ued,.e. the radom varable hould ot exceed a certa boud wth a gve probablty. The tochatc MPC problem codered th paper (problem P) of a more pecal form compared to the geeral problem formulated [9] []. Here, the cot fucto () clude the mea value of the radom varable ad the cotrat (4), (5) ad (8) are equvalet to the followg probabltc cotrat: Pr( xt+ k t xm) = p, k =,..., N () Pr( xt+ k t xma = p, k =,..., N (3) Pr( xt+ t rt ( ) δ ) = p (4) where the probablty p.95 (the cofdece terval ued (4), (5) ad (8) aocated wth th level of probablty). The followg aumpto are made: A. PQR,,. A. x m < xmax <, u m < < umax ad um < < umax. We troduce a exteded tate vector: xt () = [ xt (), rt (), ut ( )], = + m (5) Let x be the value of the exteded tate at the curret tme t. The, the optmzato problem P ca be formulated a compact form a follow: Problem P: V ( = m J ( U, (6) ubject to: G ( U, (7) The GP-NMPC problem defe a mp-nlp, ce t NLP U parameterzed by x. A optmal oluto to th problem deoted U = [ ut, ut+,..., ut+ N ] ad the cotrol put choe accordg to the recedg horzo polcy ut () = ut. Defe the et of N-tep feable tal tate a follow: Nm X f = { x G( U, x ) for ome U } (8) If δ (8) choe uch that the problem P feable, the X f a o-empty et ad due to aumpto A, the org a teror pot X f. I parametrc programmg problem oe eek the oluto U ( a a explct fucto of the parameter x ome et X X f [4]. The explct oluto allow u to replace the computatoally expeve real-tme optmzato wth a mple fucto evaluato. I th paper we ugget a computatoal method for cotructg a explct PWL approxmate oluto of the referece trackg GP-NMPC problem. IV. APPROXIMATE MP-NLP APPROACH TO EXPLICIT GP- NMPC Here, the computatoal ue related to the ocovexty of the optmzato problem are treated a way mlar to that [3]. A. Cloe-to-global oluto of mp-nlp I geeral, problem P ca be o-covex wth multple local mma. Therefore, t would be eceary to apply a effcet talzato of problem P o to fd a cloe-toglobal oluto. Oe poble way to obta th to fd a cloe-to-global oluto at a pot v X by comparg the local mma correpodg to everal tal guee ad the to ue th oluto a a tal gue at the eghbourg pot v X, =,,..., N,.e. to propagate the oluto. The followg procedure ued to V = v, v, v,..., vn, where geerate a et of pot { } v X, =,,,..., N. Procedure (geerato of et of pot): Coder ay hyper-rectagle X X f wth vertce { λ, λ,..., λ } N λ Λ = ad ceter pot v. Coder alo U

5 j the hyper-rectagle X X, j =,,..., N j wth vertce j j j j repectvely { λ λ λ } Λ =,,..., N, j =,,..., N j. Suppoe X X... X. For each of the hyperrectagle X ad X X, j =,,..., N j, determe a j et of pot that belog to t facet ad deote th et j j j j { φ φ φ } φ Φ =,,..., N, j =,,,..., N j. Defe the et of all pot V { v, v, v,..., vn } N j λ =, where Nj Nj j j v Λ Φ, =,,..., N. j= j= The followg procedure appled to fd a cloe-toglobal oluto at the pot v V, =,,,..., N: Procedure (cloe-to-global oluto of problem P): Coder ay hyper-rectagle X X f wth a et of pot {,,,..., N } V = v v v v determed by applyg Procedure. The: a). Determe a cloe-to-global oluto of problem P at the ceter pot v through the followg mmzato: local local local local U { U,..., UN U } local U U ( v ) = arg m J( U, v ), (9) where U, =,,..., N correpod to local mma of the cot fucto J ( Uv, ) obtaed for a umber of tal guee U, =,,..., NU. b). Determe a cloe-to-global oluto of problem P at the pot v V, =,,..., N the followg way:. Determe a cloe-to-global oluto of P at the ceter pot v by olvg problem (9). Let =.. Let V = { v, v, v,..., vn } V be the ubet of pot at whch a feable oluto of P ha bee already determed. 3. Fd the pot v V that mot cloe to the pot v,.e. v = arg m v v. Let the oluto at v be U ( v ). v V 4. Solve P at the pot v wth tal gue for the optmzato varable et to U ( v ). 5. If a oluto of P at the pot v ha bee foud, mark v a feable ad add t to the et V. Otherwe, mark v a feable. 6. Let = +. If N, go to tep. Otherwe, termate. B. Computato of feable PWL oluto Defto (Feablty o a dcrete et): X = v, v,..., v Q be a dcrete et. A fucto Let { } (x ) U feable o X f GU ( ( v), v), {,,..., Q}. We retrct our atteto to a hyper-rectagle X where we eek to approxmate the optmal oluto U ( to problem P. We requre that the tate pace partto orthogoal ad ca be repreeted a a k d tree. The ma dea of the approxmate mp-nlp approach to cotruct a feable pecewe lear (PWL) approxmato U ˆ ( x ) to U ( o X, where the cottuet affe fucto are defed o hyper-rectagle coverg X. I cae of covexty, t uffce to compute the oluto of problem P at the vertce of a codered hyper-rectagle X by olvg up to NLP. I cae of o-covexty, t would ot be uffcet to mpoe the cotrat oly at the vertce of the hyper-rectagle X. Oe approach to reolve th problem to clude ome teror pot addto to the et of vertce of X [3]. Thee addtoal pot ca repreet the vertce ad the facet ceter of oe or more hyper-rectagle cotaed the teror of X. Baed o the oluto at all pot, a feable local lear approxmato U ˆ ( = Kx + g to the optmal oluto U (, vald the whole hyper-rectagle X, determed by applyg the followg procedure: Procedure 3 (computato of explct approxmate oluto): Suppoe A ad A hold. Coder ay hyper-rectagle X X f wth a et of pot V = { v, v, v,..., vn } determed by applyg Procedure. Compute K ad g by olvg the followg NLP: Problem P3: N K, g = + m ( J( K v g, v ) V ( v ) + β Kv+ g U ( v) ) (3) ubject to: G( Kv + g, v), v V (3) I (3), the parameter β > a weghtg coeffcet. C. Etmato of error boud Suppoe that a tate feedback U ( that feable o V X ha bee determed by applyg Procedure 3. The, for the cot fucto approxmato error X we have: ε( x ) = Vˆ ( x ) V ( x ) ε, x X (3) where V ˆ( = J ( Uˆ ( ), x the ub-optmal cot ad V ( deote the cot correpodg to the cloe-to-global oluto ( U,.e. V ( = J ( U (,. The followg ˆ

6 procedure ca be ued to obta a etmate ˆε of the maxmal approxmato error ε X. Procedure 4 (computato of the error boud): Coder ay hyper-rectagle X X f wth a et of pot {,,,..., N } V = v v v v determed by applyg Procedure. Compute a etmate ˆε of the error boud ε through the followg maxmzato: ˆ ε = max ( Vˆ ( v) V ( v)) (33) {,,,..., N } D. Approxmate mp-nlp algorthm for explct GP- NMPC Aume the tolerace ε > of the cot fucto approxmato error gve. The followg algorthm propoed to deg explct referece trackg GP-NMPC: Algorthm (explct referece trackg GP-NMPC). Italze the partto to the whole hyper-rectagle,.e. Π= { X}. Mark the hyper-rectagle X a uexplored.. Select ay uexplored hyper-rectagle X Π. If o uch hyper-rectagle ext, termate. 3. Compute a oluto to problem P at the ceter pot v of X by applyg Procedure a. If P ha a feable oluto, go to tep 4. Otherwe, plt X to two hyperrectagle X ad X by applyg the heurtc rule from [3]. Mark X ad X uexplored, remove X from Π, add X ad X to Π, ad go to tep. 4. Defe a et of pot V = { v, v, v,..., vn } by applyg Procedure. Compute a oluto to problem P for x fxed to each of the pot v, =,,..., N by applyg Procedure b. If P have a feable oluto at all thee pot, go to tep 6. Otherwe, go to tep Compute the ze of X ug ome metrc. If t maller tha ome gve tolerace, mark X feable ad explored ad go to tep. Otherwe, plt X to hyperrectagle X, X,, X N by applyg the heurtc rule from [3]. Mark X, X,, X from Π, add X, X,, X N uexplored, remove X N to Π, ad go to tep. 6. Compute a affe tate feedback U ( ug Procedure 3, a a approxmato to be ued X. If o feable oluto wa foud, plt X to two hyperrectagle X ad X by applyg the heurtc rule 3 from [3]. Mark X ad X uexplored, remove X from Π, add X ad X to Π, ad go to tep. 7. Compute a etmate ˆε of the error boud ε X by applyg Procedure 4. If ˆε ε, mark X a explored ad feable ad go to tep. Otherwe, plt X to two ˆ hyper-rectagle X ad X by applyg Procedure 4 from [3]. Mark X ad X uexplored, remove X from Π, add X ad X to Π, ad go to tep. V. SIMULATION EXAMPLE A. The olear ytem Coder the tochatc ytem decrbed by the followg olear tate pace model: 3 x = tah( x+ u ) + ξ (34) where ξ whte oe wth varace.5 ad zero mea. The amplg tme, determed accordg to ytem dyamc, wa elected to be T =.5. The Euler approxmato of ytem (34) : 3 x( t+ ) = x() t T tah( x() t + u()) t + ξ () t (35) where ξ() t = T ξ() t. B. Gaua proce model detfcato The cotrol gal u wa geerated by a radom umber geerator wth ormal dtrbuto. The cotrol gal blockg wa Tu = 6T,.e. t kept cotat for 6 tme tat. The umber M of the put gal ample ued for the detfcato determe the dmeo of the covarace matrx. I our cae, M =. Let x m be the mea value of the tate of ytem (35) obtaed for the M geerated cotrol gal,.e. xm = xt (). By M troducg the varable yt () = xt () xm, we would lke to obta a Gaua proce model for the followg dcretetme tochatc ytem: 3 yt ( + ) = yt () T tah( yt () + ut () ) + ξ () t (36) Baed o the geerated data et, the dcrete-tme ytem (36) approxmated wth Gaua proce wth zero mea ad covarace fucto of the form (). The maxmum lkelhood framework wa ued to determe the hyperparameter. The optmzato method appled for detfcato of the Gaua proce model wa the cojugate gradet method wth le earche [8]. The followg et of hyperparameter wa foud: Θ = [ w, w, v, v] = [.395,.9754,.333,.354] (37) A valdato cotrol put gal wa geerated by radom umber geerator wth ormal dtrbuto ad rate of chage that dfferet from the oe ued for the detfcato gal. The repoe of the Gaua proce model to the valdato gal how Fg.. The aocated average quared error ad log dety error are repectvely ASE =.7 ad LD =.476.

7 mea value ad 95% cofdece terval of y(t) tme tat valdato data model mea 95% cofdece terval Fg.. Repoe of the Gaua proce model to the exctato gal ued for valdato. C. Deg of explct referece trackg GP-NMPC cotroller The mp-nlp approach decrbed ecto IV appled to deg a explct referece trackg GP-NMPC cotroller for the ytem (35) baed o the obtaed Gaua proce model. I the GP-NMPC problem formulato (problem P), the predcted tate x t + k + t of ytem (35) ued. Th predcto obtaed the followg way. Frt, we obta the predcto of y t + k + t from the Gaua proce model of ytem (36): yt+ k+ t yt+ k t, ut+ k N ( µ ( yt+ k+ t), σ ( yt+ k+ t)) (38) k =,,..., N The, the predcted x + + : t k t x = y + x (39) t+ k+ t t+ k+ t m The teratve mult-tep ahead predcto wa doe by feedg back at each tme tep the predctve mea oly. The followg cotrol put ad rate cotrat are mpoed o the ytem: u ;.5 u.5 (4) The predcto horzo N = 8 ad the termal cotrat : max{ µ ( xt+ t) σ( xt+ t) r( t), (4) µ ( xt+ t) + σ( xt+ t) r( t) } δ where δ =.5. The weghtg matrce the cot fucto () are Q =, R =, P =. The GP-NMPC mmze the cot fucto () ubject to the Gaua proce model (38) (39) ad the cotrat (4), (4). The formulated GP-NMPC problem reult optmzato problem P wth 8 optmzato varable ad 33 cotrat. Oe teral rego X X ued Procedure,, 3 ad 4. Th reult problem P3 whch ha 3 optmzato varable ad 85 cotrat. I (3), t choe β =. The approxmato tolerace choe the followg way: ε( X ) = max( ε, ε m V ( x )), (4) a r x X where ε a =.5 ad ε r =.5 are the abolute ad the relatve tolerace, repectvely. The exteded tate vector 3 xt () = [ xt (), rt (), ut ( )], whch lead to a 3- dmeoal tate pace to be parttoed. The latter defed by X = [.,.] [.7,.7] [, ]. The partto ha 49 rego ad 8 level of earch. Totally, 4 arthmetc operato are eeded real-tme to compute the cotrol put (8 comparo, 3 multplcato ad 3 addto). The performace of the cloed-loop ytem wa mulated for the followg et pot chage: rt ( ) =.5, t [;5] ; rt ( ) =., t [5;] (43) rt ( ) =., t [;5] ; rt ( ) =.5, t [5;] ad tal codto for the tate ad cotrol varable x () = ad u () =, repectvely. The reultg cloedloop repoe depcted Fg. to Fg u(t) tme tat Fg.. The cotrol put. The dahed curve wth the approxmate explct GP-NMPC ad the dotted curve wth the exact GP-NMPC mea value of x(t) ad et pot tme tat Fg. 3. The mea value of the tate varable predcted wth the Gaua proce model. The dahed curve wth the approxmate explct GP- NMPC, the dotted curve wth the exact GP-NMPC, ad the old curve the et pot.

8 % cofdece terval of x(t) ad et pot tme tat Fg. 4. The 95% cofdece terval of the tate varable predcted wth the Gaua proce model. The dahed curve wth the approxmate explct GP-NMPC, the dotted curve wth the exact GP-NMPC, ad the old curve the et pot. The reult how that the exact ad the approxmate oluto are almot dtguhable. VI. CONCLUSIONS I th paper, a approxmate mp-nlp approach to explct oluto of referece trackg NMPC problem baed o Gaua proce model developed. The approach buld a orthogoal earch tree tructure of the tate pace partto ad cot cotructg a feable PWL approxmato to the optmal cotrol equece. REFERENCES [] S. S. Keerth ad E. G. Glbert, Optmal fte horzo feedback law for a geeral cla of cotraed dcrete-tme ytem: Stablty ad movg horzo approxmato, Joural of Optmzato Theory ad Applcato, vol. 57, pp , 988. [] D. Q. Maye, J. B. Rawlg, C. V. Rao, ad P. O. M. Scokaert, Cotraed model predctve cotrol: Stablty ad optmalty, Automatca, vol. 36, pp ,. [3] F. Allgöwer ad A. 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