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A ROBUS REPEIIVE CONROL SCHEME WIH RELAXED MINIMUM IME CRIERION Andrzej rna Macej Szymkat Adam Korytowsk and Krzyszto Kołek AGH Unversty o Scence and echnology al Mckewcza Kraków Poland emal:

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A ROBUS REPEIIVE CONROL SCHEME WIH RELAXED MINIMUM IME CRIERION Andrzej rna Macej Szymkat Adam Korytowsk and Krzyszto Kołek AGH Unversty o Scence and echnology al Mckewcza Kraków Poland emal: Abstract: A repettve control scheme s proposed or constraned nonlnear optmal control problems he lower level algorthm adjsts swtchng tmes or bang control arcs and parameters o nterval polynomal approxmatons or nteror control arcs It s based on a lnearzaton o optmal controller and perorms redced optmzaton wth changes o control strctre he pper level nds the optmal control and recalclates the lnearzaton each tme the devaton rom the optmal solton becomes too large he lnearzed controller s analytcally derved he pper level ses the MSE method to determne the reerence optmal control strctre Smlaton and expermental tests show that the proposed approach yelds an optmzng nonlnear controller able both to ensre close to mnmm-tme pont-to-pont transton as well as to stablze the state Copyrght 5 IFAC Keywords: mnmm tme control nonlnear systems robst control INRODUCION he prpose o ths paper s to mprove the adaptve control algorthm rst presented n (Korytowsk et al ) It s an extenson o the neghborng optmm eedback (Bryson Jr 999; Pesch 989a b) he controller reacts to state dstrbances on two levels he lower level adjsts control parameters basng on a lnearzaton o optmal controller and perorms redced optmzaton whereas the pper level detects changes n the optmal control strctre and recalclates the lnearzaton each tme the devaton o trajectory rom the optmal one becomes too large Both the redced and ll optmzaton rely on orced changes o control strctre whch take place stable tests are satsed he lnearzed controller s descrbed by an explct lnear relatonshp between the measred or compted devatons o state and the correctons o control parameters It s analytcally derved wth the se o dscontnos matrx soltons o the canoncal varatonal system he modcatons o the reslts o (Korytowsk et al ) nclde: - ntrodcton o the horzon as a decson varable - lmtaton o ampltde o control varatons n the vcnty o the target state - mproved convergence de to stronger stablzaton reqrements he seqence o bondary/nteror arcs determnes the strctre o optmal control Pesch s approach (Pesch 989b) known as the repeated correcton method reles on the assmpton o xed control strctre hs lmtaton s overcome sng precompted neghborng extremals (Pesch 989a) at a hgh comptatonal cost he approach proposed below apples also to varyng control strctre hs s acheved by combnng the lnearzed eedback scheme wth the monotonos strctre evolton (MSE) method (Szymkat et al 3) whch makes t possble to generate or redce arcs wthot consderable comptatonal eort Comparsons wth the repeated correcton scheme (Pesch 989b) and ll repettve optmzaton show that the new method gves good dstrbance rejecton at a low comptatonal cost OPIMAL CONROL PROBLEM Consder the mnmm tme problem o steerng the state o the system x &( = ) ) + t t ( t x x ) = to an ε -neghborhood o a gven state n ) x( R ( R () x ( ε ) ) x ) ) x ) ε () s the control horzon he nctons and are twce contnosly derentable he set o admssble controls U conssts o all rght-contnos nctons :[ t [ [ ] Dene the Hamltonan H ( ψ = ψ ( where the adjont var- n able ψ ( R satses Let ψ& = H x ( ψ t [ ] (3) ψ ( ) = x x( ) (4) x g( ψ ) = H ( ψ = ψ ( ) (5) he respectve swtchng ncton s dened as φ ( = g( x( ψ ( ) (6) Its projecton onto U at an admssble pont s gven by U sgnφ( = φ ( = (7) φ ( otherwse By the Maxmm Prncple φ U s dentcally zero on an optmal control he optmal control satses = v( ψ ) = sgn g( ψ ) g( ψ ) (8) he swtchng ncton φ s contnosly derentable (Korytowsk et al ) We assme that φ takes zero vale at most at a nte nmber o ponts (swtchng tmes) τ τ m t τ K τ m and ts dervatve s derent rom zero at every pont τ = m he canoncal system o eqatons s obtaned by sbstttng the control (8) n () and (3) X & = F( X v( X )) t [ ] X = col( ψ ) F( X = col( ( H x ( ψ ) wth the bondary condtons as n () (4) he varatonal eqaton or the canoncal system wth jmp condtons at swtchng moments was gven n (Lastman 978) δ X & ( = J ( δ X ( (9) A( J ( = X F( X ( ) = () B( A( where t t [\ { τ τ K τ } ] m A( = ) x B( = xxh ψ ( ) () he termnal condton reslts rom (4) [ I ] X ( ) = I δ () δ X s n general dscontnos at τ τ K τ m δ X ( τ ± ) = Z δx ( τ m) = m (3) Z m F m F g( X ( τ )) = I ± ψ ( τ ) [ ] = F( X ( τ ) τ )) F( X ( τ ) τ + )) (4) where [ ] s the Le bracket Obvosly Z = ( Z+ ) he dependence o the swtchng tme varaton δτ on δ X s determned rom the dentty g ( X ( )) see (Korytowsk et al ) τ g( X ( τ)) δ X ( τ ± ) δτ = (5) ψ ( τ ) [ ] Note that the ncton t a g( X ( ) δ X ( s contnos at every swtchng tme Dene a n n matrx solton V o the varatonal eqaton satsyng V & ( = J ( V ( t ] [\ { τ τ K τ m } (6) V ( ) = col( I I) (7) he jmps at the swtchng moments are gven by V ( τ ) = Z V ( τ + ) = m (8) + Let V = col( V V ) wth sqare matrces V and V hs or any solton o (9) () (3) and every t = V ( ) δ ψ ( = V ( ) (9) 3 LINEARIZED CONROLLER he constrcton o the lnearzed controller s based on (5) and the relatonshp between the varatons o state and adjont trajectores t δψ ( = V ( V ( ) () From now on t s assmed that V ( s nonsnglar or every t hs crcal assmpton s genercally satsed n practcal control problems he coecent matrx K ( = V ( V ( s symmetrc or every t and derentable everywhere except or the swtchng tmes It satses a lnear Rccat eqaton (Korytowsk et al ) Accordng to ormla (9) the matrx ( ) W ( = V t () represents the senstvty o the termnal state o the optmal solton wth respect to the state at t Formlas (5) and () yeld a relatonshp between the varaton o the state trajectory and the varatons o the swtchng tmes δ τ Λ τ ±) = m () = ± ψ ( τ ) τ )) + τ )) K( τ ± ) Λ ± = ψ ( τ ) [ ] Sppose that the state ncrement δ x satses the varatonal eqaton n the nterval [ t ] or some t [ t [ and s the reslt o a pertrbaton o the state at t by a known vale he vales o τ ±) can be compted n advance by solvng eqatons (9) () (3) and the respectve correctons δτ o the swtchng tmes τ t can be appled drng the control process provded τ δτ τ δτ τ + δτ t (3) + or { m} By denton τ m + + δτ m+ = τ + δτ = t I τ t and τ + δτ t the control at t shold change sgn o avod too reqent swtchngs the ntervals between the tme moments t at whch the control n [t ] s corrected have a xed length t From (9) τ ± ) = V ( τ ± ) V ( Pttng ths nto () and sng () we obtan the general orm o the lnearzed swtchng controller δτ = ΠW ( δx( = m (4) Π g( X ( τ )) V ( τ ± ) = ψ ( τ ) [ ] g( X ( τ )) V ( τ ± ) = ψ ( τ ) τ )) V ( τ ± ) + τ )) V ( τ ± ) he vale o Π does not depend on whch lmt s taken n the rght-hand sde o (4) he varaton o the horzon de to a varaton o state at the tme t s obtaned n a smlar way ) x ) V ( δ = ) x ) ) )) I s known at t ] τ + δτ τ + δτ [ or some m the correcton δτ can be compted n another way Dene the contnos n n matrx solton o the eqaton Φ( t s)/ t = A( Φ( t s) Φ ( s s) = I or every ts n [ ] he eqalty δ τ = Λ ± Φ( τ s then eqvalent to (4) where Λ s taken or t τ and Λ + or t τ he decson abot the vale o the correcton δ τ shold be taken as late as possble hs crtcal last moment t lls t τ = Λ ± Φ( τ = Π W ( 4 BASIC REPEIIVE SCHEME he overall repettve comptatonal scheme has two levels On the lower level the lnearzed controller s appled combned wth redced optmzaton In each tme step t addtonally calclates two qanttes or the remanng part o the control tme nterval hese are the norm o the projecton φ n the control space see (7) and the expected vale o the axlary cost nctonal x( ) x ) ) x ) When both o them ( exceed some predetermned thresholds the pper level algorthm s actvated On the pper level the MSE method (Szymkat et al 3) s adopted hs dynamc optmzaton algorthm n the varant applcable here ses swtchng tmes as decson varables It atomatcally adjsts the control strctre by generatng and redcng swtchngs he pper level algorthm s contned ntl the projecton norm decreases below another threshold I the axlary cost s not below ts threshold vale at that tme the MSE algorthm has to be rentalzed Ater a sccessl completon o the pper level comptatons the lnearzed controller s recalclated and the lower level algorthm restarted he dstnctve eatre o ths adaptve scheme s that the control strctre s adapted n the corse o the control process hs dstngshes the approach descrbed here rom the repeated correcton method o (Pesch 989a b) Recall that the dervatve o the perormance ndex wth respect to a swtchng tme τ s eqal to (Szymkat et al 3) S φ( τ ) ( τ ) (5) τ = he repettve control scheme conssts o the ollowng steps Set t : = Fnd optmal solton sng MSE started wth crrent control approxmaton e determne horzon reerence control and swtchng tmes τ = m ; calclate matrx V and vectors Π or all swtchng tmes 3 Choose tme step t t (pdate nterval) execte control n t t + ] and sbsttte t : = t + t [ t 4 Determne state devaton x( t ) Calclate and τ = ΠV ( t) x( t) or all τ t Set corrected vales τ : = τ + τ : = + and ths determne new control 5 Update ntal state Compte the norm o φ U n control space (7) and the expected vale o the axlary cost nctonal ) x ) ) x ) I thresholds or both are exceeded retrn to Otherwse go to 6 6 I φ ( t ) t ) e the dervatve (5) wold be negatve add a control swtchng at t and perorm redced (xed strctre) optmzaton Stop when a swtchng tme hts the bondary o the admssble set or the gradent norm termnaton condtons are met Retrn to 3 U he ntrodcton o an addtonal swtchng tme at t (step 6 ) does not ntally change the control bt creates the possblty o mprovng the vale o perormance ndex by movng ths swtchng tme to the rght For a more detaled treatment o MSE ncldng control swtchng generatons and redctons see (Szymkat et al 3) he restranng o optmzaton n step 6 to xed strctre (wth the addtonal ntal swtchng) yelds relatvely good reslts at low comptatonal cost he area o applcaton o the algorthm can be extended to cases where condtons (3) are not llled n [ t ] (step 4 ) Denote = mn{ : τ t} For s ncreasng rom to sccessvely remove every swtchng or whch the respectve vale o τ (s) = τ + s τ m ( s) = + s hts the bondary o the admssble set Every tme the constrant τ ( s) t s ht the control ntal vale t ) changes ts sgn ( 5 EXAMPLE We show the applcaton o the repettve optmzng scheme to the well-known benchmark problem o steerng a pendlm hnged on a cart whch s a strongly nonlnear orth order system Denote the cart poston by x ts velocty by x 3 the angle between the pward drecton and the pendlm by x and the anglar velocty o the pendlm by x 4 Pt x = col( x x4) = col( 4) s = sn x c = cos x S = sn x C = cos x θ = tanhx3 w = kx3 + k k3θ l x4s w = k4s + k5x4 D = ( ec ) hen ( = x3 ( = x4 = D( w + lw ) = D( aw c + ) 3( c 4( w wth k = 785 k = 6646 k 3 = k 4 = 43 k 5 = l = 4375 e = 9996 a = 866 Consder the optmzaton problem o secton wth x = col( π) and open-loop nstable target state 6 x = Assme ε = 5 By (8) the optmal control satses ( = sgnφ ( where φ = k D( ψ + ac 4) he adjont eqaton has the orm 3 ψ ψ& = A ψ where A dened n () has the ollowng nonzero elements A = A 3 4 = 3 = ld ( x4c + k4c k5x4s a 3 S A 33 θ A = D ( kk3( A = ld x s + k ) A 34 ( 4 5c 4 = D( ex4c aws + k4c e4s A 43 = ac A 33 A44 = D( k5 alx4s) )) ) ) he perormance o the repettve control scheme s evalated n a seres o smlaton experments wth modelng o tme dscretzaton o control (wth constant hold ntervals) and stochastc state dstrbances he length o the pdate nterval n step 3 o the algorthm s constant eqal to Gassan state dstrbances wth zero mean generated by the MALAB expresson p*[44]*randn(4) are added to the crrent state at every tme moment o pdate ypcal trajectores and controls are shown n Fgres and For p 9 the perods o eectve stablzaton are long (Fg shows the reslts or p = 9 ) and or greater p the probablty o alre rapdly ncreases (see Fg where p = ) Observe that the process o Fg can be dvded nto two stages: swngng the pendlm p and stablzng t n the pper poston Althogh the se o bang-bang controls s lly jsted n the rst stage and garantees a close to mnmm tme transton to the neghborhood o the nstable pper poston t may seem prposel to seek a more relaxed control behavor n the stablzaton stage resltng n smaller oscllatons hs wll be realzed by addng an ntegral o sqared control to the axlary cost nctonal wth a weght actor growng as the dstance to the target state decreases x x / Fg Example o eectve stablzaton x / Fg Example o alre x In most analyzed smlaton rns and real-tme experments the above proposed control scheme ensres stablzaton o the system n a vcnty o the open-loop nstable eqlbrm However n some cases we observe large oscllatons that cannot be solely explaned by the dstrbances A more detaled analyss leads to the dsclosre o a trap phenomenon It conssts n the alre o the MSE algorthm employed n step to properly denty the tre mnmzer o the axlary cost nctonal de to the presence o a competng solton wth a low vale o the crteron or the gven horzon Independently o control ths solton departs rom the vcnty o target state shortly ater In order to avod sch canddate soltons n the corse o monotonos MSE search a tal term n the orm o an ntegral over an addtonal nterval [ + ] wll be ntrodced o gve some hnts how to handle the trap phenomenon consder the staton when a new startng pont or the MSE algorthm has to be generated hs occrs eg ater a step wth the optmal horzon smaller than the pdate nterval he proposed new vale o the horzon has to be scently large to avod local mnma at whch the target s mssed An approprate ncrease o the penalty coecent also proves helpl However sch measres are problem specc and not always ecent A more general solton s sggested below 6 ROBUSIFIED REPEIIVE SCHEME he axlary optmal control problem conssts now n the mnmzaton o the ollowng crteron nctonal on the trajectores o () d Sρ ( ) = + α( x) t ρ x( ) x + x( x dt (6) he decson varables that s the control and horzon are sbject to constrants: t ( or t ( = or t (7) he constant s nonnegatve and ρ s postve he weght actor α x ) monotonosly decreases as ( x departs rom the target state startng rom a postve vale For x x greater than a certan threshold vale t s dentcally zero Sch a constrcton garantees approprate reglarty o the strctre evolton or the transton rom the pontto-pont to the stablzng eedback type control he hamltonan or the basc optmal control problem s as ollows H = ψ ( ) α ρ σ ( x( x (8) where σ ( = or t and σ ( = or t he adjont varable ψ satses the adjont eqaton ψ& = ( ψ + ρ σ ( x x ) (9) x wth a termnal condton ψ ( + ) = and a jmp ψ ( ) ψ ( ) + ρ( x( ) x ) = + I α ( x ) = the extremal control e the control that maxmzes the hamltonan (8) sbject to (7) s gven by = sgn( ψ ( ) t (3) I α ( x ) the extremal control satses = sat( α ψ ( ) t (3) he sat ncton s dened by ξ ξ sat( ξ ) = (3) sgn ξ otherwse he dea o the robsted repettve scheme o secton 4 remans generally nchanged wth the ollowng modcatons In step each tme the MSE procedre s recalled the vale o α ( x ) s pdated I the crrent vale s zero the rest o the algorthm s exected wthot any essental change or bang-bang type controls I α ( x ) s nonzero the control s contnos or t and may have both bondary and nteror (non-satrated) arcs Its rst dervatve may be dscontnos only at the ends o the bondary arcs We assme that approxmatons o optmal control also have these propertes Let t = σ σ K σ N = be end ponts o sbseqent control approxmaton arcs Some σ concde wth τ k dvdng bondary and nteror arcs Let n every nteror arc = p w( t σ ) (33) ( σ where p s a vector o parameters and w a vector o Hermte cbc polynomals w w ( t σ σ ) = w3 ( t σ σ ) = 3 ( tσ ) (t+ σ 3σ ) /( σ σ ) t σ σ = w t σ σ = ( ) 4( ) ( t ) ( tσ ) /( σ σ ) σ hs ( σ ) = p & ( σ + ) = p ( σ ) = p 3 & ( σ ) = p 4 o ensre contnty at dvson ponts and smoothness between neghborng nteror arcs some parameters p are xed or made dentcal k Let Σ denote the perormance ndex as a ncton o the parameters dvson ponts and horzon Its dervatve wrt p reads pk k Σ = Ωk (ψ dt (34) H pk where the dervatves o are determned by (33) and Ω k s the non o [ σ σ ] and possbly one o ts neghborng nteror ntervals he dervatve wrt σ σ N beng the rght-hand end o an nteror nterval s gven by σ Σ = & ( σ ) p3 Σ && ( σ ) p 4 Σ & ( σ + ) + p 4 where p Σ and + Σ 4 p are compted accordng to 4 (34) bt wth Ω k eqal to [ σ σ ] and [ σ σ + ] respectvely For the let-hand end ponts we have σ Σ = + p+ p+ p+ & ( σ + ) Σ && ( σ ) Σ & ( σ + ) Σ I σ s an end pont o a bondary arc the terms wth vanshng control dervatves are dropped he dervatve wrt horzon Σ = + α( x) ) + α( x) σ N σ N + ρ x( ) x ) ( x( + ) x x( ) x ) + ρ )) dt For step 4 o the scheme n the case o α ( x ) we consder controls parameterzed wth dvson ponts and vectors p (33) or segments wthn the nteror arcs A varatonal approach analogos to that descrbed n secton 3 can be employed to get lnearzed parametrc controllers An example o solton optmal accordng to the perormance ndex (6) s gven n Fg 3 For comparson the mnmm tme solton or the same ntal condton s shown n Fg 4 7 CONCLUSIONS he constrcton o optmal closed-loop controller or systems wth non-lnear state eqatons s a complex comptatonal task he adaptve optmzng controller s a practcal solton whch can be appled n real tme n a vcnty o a reerence trajectory compted beorehand A combnaton wth repettve optmzaton sng the MSE method enlarges the area o applcaton at the cost o more on-lne comptatons An mportant observaton s that the mplementaton o redced optmzaton largely decreases the comptatonal cost o the algorthm wth nsgncant deteroraton o ts perormance he nclson o the horzon nto the optmzaton process mproves the overall ecency o the repettve control scheme he se o relaxed mnmm tme crteron (6) orces the comptatonal procedre to reject certan nsae controls and assres reqred robstness by restranng the control ampltde n the neghborhood o the target state Σ x x Fg 3 Robsted optmal solton Fg 4 Mnmm tme solton REFERENCES Bryson Jr AE (999) Dynamc Optmzaton Addson Wesley Longman Menlo Park Korytowsk A M Szymkat and A rna () Adaptve lnearzed swtchng controller or constraned optmal control problems Proc 7th IEEE Int Con MMAR Mędzyzdroje Poland 87-9 Lastman GJ (978) A shootng method or solvng two-pont bondary-vale problems arsng rom non-snglar bang-bang optmal control problems Int J Control 7 (4) Pesch HJ (989a) Real-tme comptaton o eedback controls or constraned optmal control problems Part : Neghborng extremals Optmal Control Applcatons and Methods () 9-45 Pesch HJ (989b) Real-tme comptaton o eedback controls or constraned optmal control problems Part : A correcton method based on mltple shootng Optmal Control Applcatons and Methods () 47-7 Szymkat M and A Korytowsk (3) Method o monotone strctral evolton or control and state constraned optmal control problems Eropean Control Conerence Unversty o Cambrdge UK x x

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