IET (2015) ISSN

Transcription

1 Grble Mchael and Majeck Pawel (5) on-lnear predctve generalsed n varance state-dependent control. IE Control heory and Applcatons. pp ISS hs verson s avalable at Strathprnts s desgned to allow sers to access the research otpt of the Unversty of Strathclyde. Unless otherwse explctly stated on the anscrpt Copyrght and Moral Rghts for the papers on ths ste are retaned by the ndvdal athors and/or other copyrght owners. Please check the anscrpt for detals of any other lcences that ay have been appled. Yo ay not engage n frther dstrbton of the ateral for any proftakng actvtes or any coercal gan. Yo ay freely dstrbte both the rl ( and the content of ths paper for research or prvate stdy edcatonal or not-for-proft prposes wthot pror persson or charge. Any correspondence concernng ths servce shold be sent to the Strathprnts adnstrator: strathprnts@strath.ac.k he Strathprnts nstttonal repostory ( s a dgtal archve of Unversty of Strathclyde research otpts. It has been developed to dssenate open access research otpts expose data abot those otpts and enable the anageent and persstent access to Strathclyde's ntellectal otpt.

2 onlnear Predctve GMV State-Dependent Control Mchael John Grble and Pawel Majeck Indstral Control Centre Dept. of Electronc and Electrcal Engneerng Unversty of Strathclyde 4 George Street Glasgow G XQ UK (phone: 44() ; e-al:.j.grble@strath.ac.k) Indstral Systes and Control Ltd Clzean Hose 36 Renfeld Street Glasgow G LU UK. Abstract (e-al: pawel.ajeck@gal.co) A onlnear Predctve Generalzed Mn Varance control algorth s ntrodced for the control of nonlnear dscrete-te state-dependent ltvarable systes. he process odel ncldes two dfferent types of sbsystes to provde a varety of eans of odellng the syste and nferental control of certan otpts s avalable. A state-dependent otpt odel s drven fro an nstrctred nonlnear npt sbsyste whch can nclde explct transport-delays. A lt-step predctve control cost-fncton s to be nsed nvolvng weghted error and ether absolte or ncreental control sgnal costng ters. Dfferent patterns of a redced nber of ftre controls can be sed to lt the coptatonal deands. Keywords: optal state-dependent predctve nonlnear n-varance. Acknowledgeents: We are gratefl for the dscssons and cooperaton wth Professor Yan Pang (Dalan Unversty of echnology) on the developent of the algorths.. Introdcton he objectve s to desgn an ndstral controller for nonlnear and state-dependent or lnear paraeter varyng systes whch has soe of the advantages of the poplar Generalsed Predctve Control (GPC) algorths. he control strategy blds pon prevos reslts on onlnear Generalzed Mn Varance (GMV) control []. he asspton was ade that the plant odel cold be decoposed nto a set of delay ters a very general nonlnear sbsyste that had to be stable and a lnear sbsyste. he plant descrpton sed here wll be assed

3 to be slar however the otpt sbsyste s assed to be represented n state-dependent possbly nstable for. he lt-step predctve control cost-fncton to be nsed nvolves both weghted error and control costng ters whch can be sed wth dfferent error and control horzons. wo alternatve types of control sgnal npt to the plant odel are consdered. he frst s the tradtonal control sgnal npt and t s ths sgnal whch s also penalzed n the predctve control crteron. However as s well know t s soetes desrable to agent the plant odel wth an ntegrator to provde a sple way of ntrodcng ntegral acton. In the agented syste the new syste npt s the change of control acton or ncreent and n ths case ths s the sgnal whch shold be penalzed n the crteron. he reslts wll apply to both cases and a paraeter change between = and = wll provde the necessary swtch. he cost ncldes dynac weghtngs on both error and control sgnals. here s a rch hstory of research on nonlnear predctve control ([] to [7]) bt the developent proposed s soewhat dfferent snce t s closer n sprt to that of a odel based fxed-strctre controller for a te-varyng syste. Part of the plant odel can be represented by a very general nonlnear operator and the plant can also nclde a state-dependent (or lnear paraeter varyng) otpt sb-syste odel rather than a LI odel as n prevos work. For eqvalent lnear systes stablty s ensred when the cobnaton of a control weghtng fncton and an error weghted plant odel s strctly n phase. For nonlnear systes t s shown that a related operator eqaton s reqred to have a stable nverse. he dynac cost-fncton weghtngs are chosen to satsfy perforance and stablty/robstness reqreents and a sple ethod s proposed for obtanng ntal vales for the weghtngs.. on-lnear Operator and State-Dependent Syste he plant odel can be nonlnear dynac and ay have a very general strctre. he otpt sbsyste and dstrbance odel s represented by a so-called state-dependent sb-syste n Fg.. he plant nvolves two nonlnear sbsystes and the frst s of a very general nonlnear operator for and wrtten as follows: k ( )( t) = z ( )( t) he second sbsyste s a state-dependent non-lnear for whch s slar to a te-varyng lnear syste. It s assed to be pont-wse stablzable and detectable and s represented by the operator wrtten as follows: k k ( )( t) = ( z )( t) k

4 r onlnear controller C onlnear operator sbsyste k Measreent or observatons sgnal z Dstrbances ζ d d State-dependent sbsyste d = d yd y y d = d yd Controlled Otpt ose v Fg. : Feedback Control wth Inferred or Controlled Otpts. Sgnal Defntons he otpt of the syste to be controlled y(t) ay be dfferent to that easred as shown n Fg. and ths otpt ncldes deternstc d(t) and stochastc yd () t coponents of the dstrbances. he easred otpt y () t also ncldes deternstc d () t and stochastc yd () t coponents of the dstrbances. he stochastc coponent s odelled by a dstrbance odel drven by zero ean whte nose{ ζ ( t)}. he easreent nose { v ( t)} s assed to be zero-ean whte nose wth covarance atrx R = R. here s no loss of generalty n assng that { t } whch s assed to be known. ζ () has an dentty covarance atrx. he controlled otpt st follow a reference rt (). State-Dependent Sb-Syste Models he second or otpt sbsyste s n a state-dependent/lpv for whch ncldes the plant and the error weghtng odels (see [8]). hs s assed to nclde a coon k-steps transport delay and has the state-eqaton: x ( t ) = ( x p) x ( t) ( x p) ( t k) ( x p) ζ ( t) ( x p) d ( t) () d f f where the vector p s a vector of known varables lke speed of an engne or alttde of an arcraft that change wth operatng condtons. he controlled otpt and easred otpts (wthot easreent nose): yt () = d() t ( x p) x () t ( x p) ( t k) () y () t = d () t ( x p) x () t ( x p) ( t k) (3)

5 where x t R. hs odel can be a fncton of the states npts and paraeters ( xt ( ) ( t k) pt ( )). he () n deternstc coponent of the npt dstrbance s d () t and the dstrbance on the otpt to be controlled d() t = dt () y() t ncldes a known deternstc coponent dt () and a stochastc coponent y () t. he d dstrbance on the easred otpt d () t = d () t y () t where d () t s deternstc and y () t s stochastc. d he plant ncldes a dstrbance odel on the otpt drven by zero ean whte nose (t): d nd x ( t ) = x () t ω() t x () t R (4) d d d d d y () t = x () t and y () t = x () t (5) d d d d d d he sgnals of nterest nclde the error on the otpt to be controlled and the easred otpt: Error sgnal: et ( ) = rt ( ) yt ( ) (6) Observatons sgnal: z ( t) y ( t) v ( t) = (7) he sgnal to be controlled wll nvolve the weghted trackng error n the syste: ( ) p x ( t ) = x () t r() t y() t x () t R (8) p p p p ( ) e() t = x() t rt () yt () (9) p p p p he tradtonal ethod of ntrodcng ntegral acton n predctve controls s to agent the syste npt by addng an ntegrator sng the npt sb-syste: p n d d n x ( t ) = β x ( t) ( t k) x () t R () ( t k) = β x ( t) ( tk) = ( z ) ( t k) () β he ( z ) for β = and the transfer () s an ntegrator wthot addtonal delay and f β = then ( t k) = ( t k). he reslts can therefore apply to systes sng control npt or rate of change of control..3 otal Agented Syste he state-space odel for the r ltvarable syste to be controlled s now defned n agented syste for. Cobnng the plant dstrbance ntegral and weghtng eqatons the agented state-vector becoes:

6 o splfy notaton wrte t t t and wth state () n t xt R t x() t x () t x () t x () t x () t = d p ( xt ( ) ( t k) pt ( )) = and slarly for the te-varyng atrces. he agented syste eqatons ay be wrtten as follows: xt ( ) = xt () ( t k) ξ () t d () t () t t t d yt () = dt () xt () ( tk) (3) t t y () t = d () t xt () ( tk) (4) t t z () t = v () t d () t xt () ( tk) t t e() t = d () t xt () ( tk) p p pt pt (5) (6) he agented syste has an npt () t and the change n actal control s denoted t () (theseare related as ( t) = k (..) t ( ))..4 Defnton of the Agented Syste Matrces he eqatons n. can be cobned wth a lttle anplaton to obtan the agented syste atrces. hat s the total state-eqaton odel ay be wrtten n ters of the agented syste atrces as follows: xt ( ) = xt () ( t k) ξ () t d () t (7) t t t d where the atrces n ths eqaton are defned fro the cobned odel eqatons: x( t ) β x( t) xd( t ) d xd( t) = x( t ) β I x( t) I xp( t ) p p d β p p xp( t) p ( t k) d ζ () t dd() t ω() t ( rt () dt ()) p (8)

7 he otpt to be controlled ay be wrtten n ters of agented syste odel n (3). hat s: yt () = dt () x () t x () t β x() t ( tk) d d where = [ ] = dt () xt () ( tk) (9) t t t d β and t = Slarly fro (3) and (5) the easred otpt ay be wrtten n the agented syste as follows: t t where = [ ] y () t = d () t xt () ( tk) () t d β and t = Also fro () and (9) the weghted trackng error to be nsed ay be wrtten as: e() t = d () t xt () ( tk) () p p pt pt where d() t = ( () ()) p p rt dt pt = p p d β p p and pt =. p he sbscrpt t on the state atrces here s sed for the agented syste and n a slght abse of notaton t also ndcates that these atrces are evalated at te t so that the syste atrx at t s wrtten as t. 3. State-Dependent Ftre State and Error Models A state-dependent odel predcton eqaton s reqred and later an estator for the state-dependent odels. he ftre vales of the states and otpts ay be obtaned by repeated se of () assng that the ftre vales of the dstrbance are known. Introdce the notaton:... t = t t t for > where for = t = I... t= t t t for > where = I for = () Ftre states: Generalsng ths reslt obtan for the state at any ftre te t ay be wrtten as: ( ξ ) j t t j t j t j dd j= xt ( ) = xt ( ) ( t j k) ( t j ) d ( t ) (3) t where j dd t j d j= d ( t ) = d ( t j) (4)

8 hese eqatons (3) and (4) are vald for f the saton ters are defned as nll for =. otng (6) the weghted error or otpt sgnal ep () t to be reglated at ftre tes (for ): e ( t ) = d ( t ) x( t ) ( t k) p p pt pt j pt t pt t j t j j= = d ( t ) xt () ( t j k) pd j pt t j t j ξ pt j= ( t j ) ( t k) (5) where d () t = ( rt () dt ()) and the deternstc sgnals: p p d ( t ) = d ( t ) d ( t ) (6) pd p p t dd 3. State Estates Usng State-Dependent Predcton Models he -steps predcton of the state for and the otpt sgnals ay be defned notng (3) as: j ˆ t t j t j dd j= xt ˆ( t) = xt ( t) ( t j k) d ( t ) (7) j where =... and t j t t t j j dd t j d j= d ( t ) = d ( t j) and for = the ddd ( t ) =. he predcted otpt: he weghted predcton error for : yˆ( t t) = d( t ) xˆ( t t) ( t k ) (8) t t eˆ ( t ) t = d ( t ) xˆ( t ) t ( t k) (9) p p pt pt he expresson for the ftre predcted states and error sgnals ay be obtaned by changng the predcton te n (7) t t k. hen for : ˆ j t k t k j t k j dd j= xˆ( t k t) = x( t k t) ( t j ) d ( t k ) (3) Predcted weghted otpt error: Sbstttng n (9) and splfyng for k and obtan:

9 eˆ ( t k ) t = d ( t k) ( t ) xˆ( t k ) t p pd p t k pt k t k j pt k t k j t k j j= ( t j) (3) j and eˆ ( ) ( ) ˆ p t t = d t ( ) ( ) pd pt t k pt t kx t t p t t j t j ( t j k) (3) he deternstc sgnals n ths eqaton: j= j d ( t k) = d ( t k) ( ) pd p pt k t k jd dt k j (33) j= and for = the ter d ( t k) = d ( t k). pd p 3. Vector Matrx For of Eqatons he predcted errors or otpts ay be copted for controls n a ftre nterval τε [ t t ] for. hese weghted error sgnals ay be collected n the followng vector for: eˆ p ( t k) d ( t k) pd pt ki pt k () t eˆ p ( t k) d ( t k ) pd pt k t k pt k ( t ) eˆ p ( t k) = d ( t k ) pd pt kt k xt ˆ( k ) t pt k ( t ) eˆ p ( t k) d ( t k ) pd pt k t k pt k ( t ) () t pt k t k ( t ) pt k t k t k pt k t k ( t ) pt k t k t k pt k t k t k pt k t k ( t ) (34) Ftre error and predcted error: Wth an obvos defnton of ters ths eqaton ay be wrtten as: Eˆ = D xt ˆ( k ) t ( ) U (35) Pt k Pt k Pt k t k Pt k t k Pt k t Defne the te-varyng atrx: Pt k = Pt k t k Pt k (36) so that Eˆ = D xt ˆ( k ) t U (37) Pt k Pt k Pt k t k Pt k t

10 Slarly the weghted ftre errors ay be wrtten ncldng Ξ t k as: E = D xt ( k) U Ξ (38) Pt k Pt k Pt k t k Pt k t Pt k t k t k Block atrces: otng (34) the vectors and block atrces for the general case of ay be defned as: = dag{... } Pt k pt k pt k pt k pt k = dag{... } (39) Pt k pt k pt k pt k tk I tk = tk tk tk tk = tk tk tk tk tk tk tk tk t k t k = t k t k t k t k t k t k t k t k ξ() t ξ( t ) Ξ = t ξ( t ) etk ˆ( ) p et ˆ( ) k p Eˆ = et ˆ( ) k Pt k p etk ˆ( ) p U () t ( t ) = ( t ) ( t ) t D Pt d () t pd d ( t ) pd = d ( t ) pd d ( t ) pd he sgnal denotes a block vector of ftre npt sgnals. ote that the block vector D Pt denotes a vector U t of ftre reference ns known dstrbance sgnal coponents. he above syste atrces t k t k t k are of corse all fnctons of ftre states and the asspton s ade that the state dependent sgnal xt () s calclable (f { () t } ξ s nll xt ˆ( t) = xt () can be calclated fro the odel). Fro (36) the atrx = ( ) can be assed to be fll-rank (deterned by the weghtngs). Pt k Pt k t k Pt k

11 3.3 Predcted rackng Error otng (38) the k-steps-ahead trackng error: E = D xt ( k) U Ξ (4) Pt k Pt k Pt k t k Pt k t Pt k t k t k he weghted nferred otpt s assed to have the sae denson as the control sgnal and Pt k sed n (4) and defned below for s sqare: Pt pt pt t pt = pt tt pt t pt pt t t pt t t pt t pt Pt = Pt t Pt (4) Based on (35) and (38) the predcton error ( E ˆ Pt k = EPt k EPt k ): E Pt k = DPt k Pt k t k xt ( k) Pt k Ut Pt k Ξt k t k ( D xt ˆ( k ) t U ) (4) Pt k Pt k t k Pt k t hence the nferred otpt estaton error: E Pt k = xt ( kt ) Ξ (43) Ptk tk Ptk tk tk where the state estaton error xt ( kt) = xt ( k) xt ˆ( kt ) s ndependent of the choce of control acton. Also recall xt ˆ( kt ) and xt ( kt ) are orthogonal and the expectaton of the prodct of the ftre vales of the control acton (assed known n dervng the predcton eqaton) and the zero-ean whte nose drvng sgnals s nll. It follows that E ˆPt k n (35) and the predcton error E are orthogonal. Pt k 3.4 e-varyng Kalan Estator n Predctor Corrector For he state estate xt ˆ( k ) t ay be obtaned k steps ahead fro a Kalan flter [9]. hese are well known bt the reslt below accoodates the delays on npt channels and throgh ters [9]. he estates can be copted sng:

12 xt ˆ( ) t = xt ˆ( ) t ( t k) d () t t t d ( ) xˆ( t t ) = xˆ( t t) z ( t ) zˆ ( t t) ft where zˆ ( ) ( ) ˆ t t = d t t x( t t) t ( t k) he state estate xt ˆ( k ) t ay be obtaned k steps-ahead n a coptatonally effcent for fro [9] where the nber of states n the flter s not ncreased by the nber of the delay eleents k. Fro (7) the k-steps predcton s gven as: xt ˆ k t = xt ˆ t kz t d t k (44) k ( ) t ( ) ( ) () dd ( ) he fnte plse response odel ter: k kj j k = t j t jz j= ( kz ) (45) where the saton ters n (45) are assed nll for k = so that ( z ) = d ( t ) = and dd k k j dd t j d j= d ( t k ) = d ( t j). 4. Generalzed Predctve Control for State-Dependent Systes A bref dervaton of a GPC controller s provded below for a state-dependent syste wth npt (t). hs s the frst step n the solton of the PGMV control solton derved sbseqently. he GPC perforance ndex: { } (46) p( ) p( ) λ j ( ( )) ( ) j== j J = E e t j k e t j k t j t j t where E{. t } denotes the condtonal expectaton condtoned on easreents p to te t and λ j denotes a scalar control sgnal weghtng factor. In ths defnton note that the error nzed s k-steps ahead of the control sgnal snce () t affects the error e ( t k) after k-steps. By stable defnton of the agented syste p the cost can nclde dynac error npt and state-costng ters. he ftre optal control sgnal s to be calclated for the nterval τ [ t t ] whch depends on the nber of steps ( ) n the control sgnal

13 costng ter n (46). If the states are not avalable for feedback then the Kalan estator st be ntrodced. Also recall fro (43) the weghted trackng error E ˆ = E E Assng the Kalan flter s ntrodced fro (47) Pt k Pt k Pt k { }. he lt-step cost-fncton: J= EJ { } = EE E Λ U U t (47) t Pt k Pt k t t J = E{ ( Eˆ E ) ( Eˆ E Λ) U U t} (48) Pt k Pt k Pt k Pt k t t Here the cost-fncton weghtngs on npts () t at ftre tes are wrtten as Λ = dag{ λ λ... λ }. he ters n the cost-ndex can then be splfed notng E ˆ s orthogonal to the estaton error E : Pt k Pt k J = Eˆ EˆΛ U U J (49) Pt k Pt k t t where J = EE { E } t s ndependent of control acton. Pt k Pt k 4. Connecton Matrx and Control Profle Instead of a sngle control horzon nber a control profle can be defned of the for: row{p }= [lengths of ntervals n saples nber of repettons] For exaple lettng P = [ 3; ; 3 ] represents 3 dfferent ntal controls for each saple then saples wth the sae control sed bt ths s repeated agan and fnally 3 saples wth the sae control sed. hs enables a control trajectory to be defned where ntally the control changes every saple nstant and then t only changes every two saple nstants and fnally t reans fxed for 3 saple ntervals. Based on a control profle t s easy to specfy the transforaton atrx relatng the control oves to be optzed (say vector V) to the fll control vector (U) that s U = V. For the above exaple the connecton atrx can be defned:

14 = and t ( ) t ( ) t ( ) v t ( 3) v t ( 4) = t ( 3) v 3 U = V = t ( 5) v4 t ( 6) = t ( 5) v 5 t ( 7) v 6 t ( 8) = t ( 7) t ( 9) = t ( 7) In the case of the ncreental control forlaton the connecton atrx: = and t ( ) t ( ) t ( ) v t ( 3) v t ( 4) = v 3 U = V = t ( 5) v4 t ( 6) = v 5 t ( 7) v6 t ( 8) = t ( 9) = Clearly ths represents a staton wth = 3 = 6 control oves and nvolves a total of = 3 3 saple ponts. here are 4 control oves that have not been calclated n ths exaple representng a sbstantal coptatonal savng. For splcty the sae sybol wll be sed to represent the connecton atrx for the control and ncreental control cases ( ) bt when sng t shold be recalled that dfferent defntons wll be needed. he control horzon ay be less than the error horzon and we ay defne the ftre control changes as U = U. U t t t 4. State Dependent GPC Solton o copte the vector of ftre weghted error sgnals note: U = U (5) Pt k t Pt k t

15 hen fro (37) and (5): Eˆ = D xt ˆ( k ) t U = D U (5) Pt k Pt k Pt k t k Pt k t Pt k Pt k t where D = D ˆ xt ( k ) t. otng (36) and sbstttng fro (35) for the vector of stateestates: Pt k Pt k Pt k t k J = ( D U ) ( D U ) U Λ U J Pt k Pt k t Pt k Pt k t t t = D D U D D U Pt k Pt k t Pt k Pt k Pt k Pt k t U t t k U t J (5) where =Λ. Fro a pertrbaton and gradent calclaton [9] notng that the J t k Pt k Pt k ter s ndependent of the control acton the vector of GPC ftre optal control sgnals: = t k Pt k ( Pt k Pt k t k ˆ( ) ) U = D t t k Pt k Pt k D xt k t (53) where () t ( t ) Ut = ( t ) ( t ) and D Pt d () t pd d ( t ) pd = d ( t ) pd. d ( t ) pd he GPC optal control sgnal at te t s defned fro ths vector based on the recedng horzon prncple [] and s taken as the frst eleent n the vector of ftre control ncreents. U t 4.3 Eqvalent Cost Optzaton Proble he above s eqvalent to a specal cost-nsaton control proble whch s needed to otvate the PGMV proble. Let =Λ that enters (53) be factorsed as: t k Pt k Pt k = = Λ (54) t k t k t k Pt k Pt k hen by copletng the sqares n (5) the cost becoes: ( Pt k Pt k t k t t k )( t k Pt k Pt k t k t ) J = D U D U D ( I ) D J (55) Pt k Pt k t k t k Pt k Pt k By coparson wth (55) the cost-fncton ay be wrtten as:

16 J = Ψˆ Ψ ˆ t k J() t tk (56) where the sqared ter n (55): Ψˆ = tk t k Pt k DPt k t k U t ( ˆ( ) ) = D x t k t U (57) t k Pt k Pt k Pt k t k t k t he cost-ters that are ndependent of the control acton J() t = J J() t where J () t = D ( I ) D (58) Pt k Pt k t k t k Pt k Pt k he optal control s fond by settng the frst ter to zero that s Ψˆ tk =. hs gves the sae optal control as (53). It follows that the GPC optal controller s the sae as the controller to nse the nor of the sgnal Ψˆ tk defned n (57). he vector of optal ftre controls: ( ) U = D D xt = ˆ( k ) t (59) t t k Pt k Pt k t k Pt k Pt k Pt k t k 4.4 Modfed Cost-Fncton Generatng GPC Controller he above dscsson otvates the defnton of a new lt-step n varance cost proble that s slar to the nsaton proble (56) bt where the lnk to GMV desgn can be establshed. he sgnal to be nsed n the GMV proble nvolves a weghted s of error and npt sgnals []. he vector of ftre vales for a lt-step crteron: Φ = P E F U (6) t k C t Pt k C t t where the cost-fncton weghtngs PC t = Pt k and F C t = Λ. hese are based on the GPC weghtngs n (47) and are jstfed later n heore below. ow defne a n-varance lt-step cost-fncton sng a vector of sgnals: J = EJ { } = E{ Φ Φ } t (6) t t k t k Predctng forward k-steps: Φ = P E F U (6) t k C t Pt k C t t

17 ow consder the sgnal Φ and sbsttte for E Pt k = EˆPt k E Pt k : t k t k Φ = P ( Eˆ E ) F U C C = ( ˆ ) t Pt k Pt k t t P E F U P E (63) C t Pt k C t t C t Pt k hs ay be wrtten as: Φ ˆ t k = Φ t k Φ t k (64) where the predcted sgnal Φ ˆ ( ˆ P E C t F U C t ) = and the predcton error = P E C t k Pt k t Φ t k t Pt k. he perforance ndex (6) ay therefore be splfed recallng E ˆPt k and E are orthogonal as follows: Pt k Jt () = EJ { } E{ Φ Φ= } t E {( Φˆ Φ )( Φˆ Φ ) } t t t k t k t k t k t k t k = Φˆ Φˆ E{ Φ Φ = } t Φˆ Φˆ J () t (65) t k t k t k t k t k t k where J () t == E{ Φ Φ } t EE { P P E } t. he predcton Φ ˆ C C t k ay be splfed as follows: t k t k Pt k t t Pt k = P Eˆ F U = P D U F U ˆ ( ) Φ t k C t Pt k C t t C t Pt k Pt k t C t t By sbstttng fro (54) (notng P F = ) C C t Pt k t t k ˆ = P D U (66) Φ t k C t Pt k t k t Recall the weghtngs are assed to be chosen so that t k s non-snglar. Fro a slar argent to that n the prevos secton the predctve control sets the frst sqared ter n (65) to zero Φ ˆ t k = and ths expresson s the sae as the vector of ftre GPC controls. heore : Eqvalent Mn Varance Cost Proble Consder the nsaton of the GPC cost ndex (46) for the syste and assptons ntrodced n where the nonlnear sbsyste k = I and the vector of optal GPC controls s gven by (53). Asse that the cost ndex s redefned to have a lt-step n varance for (6): Jt () = E{ Φ Φ } t where = P E F U C C t (67) t k t k Φ t k t Pt k t Let the cost-fncton weghtngs be defned relatve to the orgnal GPC cost-ndex as: P = and F = Λ C t Pt k C t he vector of ftre optal controls that nze (67) follows as:

18 ( ( ) ) U = D xt k t (68) ˆ t t k Pt k Pt k Pt k t k where =Λ. hs optal control (68) s dentcal to the vector of GPC controls. t k Pt k Pt k Solton: he proof follows by collectng the reslts above. 5. onlnear Predctve GMV Optal Control he a of the nonlnear control desgn approach s to ensre certan npt-otpt aps are fnte-gan stable and the cost-ndex s nzed. Recall that the npt to the syste s the control sgnal t () shown n Fg. rather than the npt to the state-dependent sb-syste () t. he cost-fncton for the nonlnear control proble st therefore nclde an addtonal control costng ter althogh the costng on the nteredate sgnal () t can be retaned. If the sallest delay n each otpt of the plant s of k-steps the control sgnal t affects the otpt k k-steps later. For GMV the sgnal costng ( )( t) = ( z )( t). ypcally ths weghtng on the nonlnear c sb-syste npt wll be a lnear dynac operator [] assed to be fll rank and nvertble. In analogy wth the GPC proble a lt-step cost ndex ay be defned that s an extenson of (6): ck J = E{ Φ Φ } t (69) p t k t k hs consder a sgnal whose varance s to be nsed nvolvng a weghted s of error npt and control sgnals ([] [3]): Φ t k = P C tept k F C t Ut k Ut (7) c he non-lnear fncton ck Ut wll norally be defned to have a sple block dagonal for: c ( )( ) ( )( ) ( )( ) ( U ) = dag{ t t... t } (7) k t ck ck ck ote the vector of changes at the npt of the state-dependent sb-syste: U = ( U ) (7) t k t hs s the otpt of the nonlnear npt-sbsyste k whch also has a block dagonal atrx for:

19 k Ut = dag k k k Ut ( ) {... } = [( )( t)...( )( t ) ] (73) k k 5. he PGMV Control Solton ote the state estaton error s ndependent of the choce of control acton. Also recall that the optal xt ˆ( k ) t and xt ( k ) t are orthogonal and the expectaton of the prodct of the ftre vales of the control acton (assed known n dervng the predcton eqaton) and the zero-ean whte nose drvng sgnals s nll. It follows that E ˆPt k and the predcton error E are orthogonal. he solton of the PGMV control Pt k proble follows fro slar steps to those n 3.3. Observe fro (6) that Φ = P E F U and t k C t Pt k C t t Φ =Φ ˆ Φ. It follows fro (7) that the predcted sgnal: t k t k t k Φ ˆ = Φ ˆ ( U ) t k t k c k t = P Eˆ F U ( U ) (74) C t Pt k C t t c k t and the estaton error: Φ = Φ = P E = E (75) C t k t k t Pt k Pt k Pt k he ftre predcted vales of the sgnal Φ ˆ nvolve the estated vector of weghted errors P ˆ C E Pt k whch are orthogonal to P E C t Pt k t k. he estaton error s zero-ean and the expected vale of the prodct wth any known sgnal s nll. he lt-step cost ndex ay therefore be wrtten as: Jt () = Φˆ Φˆ J () t (76) t k t k t he condton for optaltyφ ˆ = now becoes: t k P Eˆ F U U = (77) C t Pt k C t t c k t 5. PGMV Optal Control he vector of ftre optal control sgnals to nse (76) follows fro the condton for optalty n (77) P Eˆ Λ U U = C t Pt k k t ck t U = ( Λ ) ( P Eˆ ) (78) t ck k C Pt k An alternatve solton of (77) gves:

20 U = ( Λ Eˆ U ) (79) t ck Pt k Pt k k t Frther splfcaton by notng the condton for optalty Φ ˆ = ay be wrtten fro (5) (54) (7) and t k (74) as P Eˆ F U ( U ) = and becoes: C t Pt k C t t ck t ( ) P D U = (8) c C t Pt k t k k k t where D = D ˆ xt ( k ) t. he vector of ftre optal control becoes: Pt k Pt k Pt k t k ( c ) ( ˆ( ) C φ ) U = P D xt k t (8) t t k k k t Pt k where fro P = C t Pt k and φ t s defned as: t φ t == PC tpt k t k Pt k Pt k t k (8) An alternatve sefl solton follows fro (8) as: ( C ) U = P D U t ck t Pt k t k k t ( P D C φ t xt k t U ) = c ˆ( ) k t Pt k t k k t he control law s to be pleented sng a recedng horzon phlosophy. Let C = [ I...] and I C I [ I ] f = so that the crrent and ftre controls are t () = [ I...] Ut and Ut = CI Ut. heore : PGMV State-Dependent Optal Control Consder the lnear coponents of the plant dstrbance and otpt weghtng odels pt n agented state eqaton for () wth npt fro the nonlnear fnte gan stable plant dynacs k. Asse that the ltstep predctve controls cost-fncton to be nsed nvolves a s of ftre cost ters and s defned n vector for as: J = E{ Φ Φ } t (83) p t k t k where the sgnal Φ t k depends pon ftre error npt and nonlnear control sgnal costng ters: Φ t k = P C tept k FC t Ut c k U t (84)

21 Asse the error and npt cost-fncton weghtngs are ntrodced as n the GPC proble (46) and these are sed to defne the block atrx cost weghtngs P = C t Pt k and F = Λ. Also asse that the control C t sgnal cost weghtng s nonlnear and s of the for( )( t) = ( )( tk) where s fll rank and nvertble operator. hen the PGMV optal control law to nze the varance (83) s gven as: c ( C φ t ) Ut = c ˆ( ) k P tdpt k xt k t t k k Ut (85) where t k =Λ Pt kpt k and φ =. he crrent control can be copted ck t Pt k Pt k t k ck sng the recedng horzon prncple fro the frst coponent n the vector of ftre optal controls. Solton: he proof of the optal control was gven before the heore. he asspton to ensre closed-loop stablty s explaned n the stablty analyss that follows below. Rearks: he expressons for the PGMV control (8) and (85) lead to alternatve strctres for pleentaton bt the second n Fg. s ore stable for pleentaton. Inspecton of the cost ter (84) when the npt costng F s nll gves Φ = P E U and the ltng case of the PGMV controller s related C C t k t Pt k ck t to an GMV controller []. D Pt k P C t Plant Controlled Controller strctre otpt y - U t β - ck ( zc ) I y z xt ˆ( kt ) φ t tk k Dstrbances v d d d ( t k) U t Measred otpt or observatons sgnal Fg. : Ipleentaton For of PGMV State-Dependent Controller Strctre

22 6. Stablty of the Closed-Loop For lnear GMV desgns stablty s ensred when the cobnaton of a control weghtng and an error weghted plant odel transfer s strctly n-phase. For the nonlnear predctve control a nonlnear operator: ( I c k ( ) ) φ t t k t k C I t k k st have a stable nverse (shown below). It wll be assed that the stochastc external npts are nll and the only npts are those de to the deternstc sgnals. he state: ( ) xt z t k d t t k d t (86) () = (Iz t) ( t ( ) d()) = t t ( ) d() xt ( k) = ( ( t) d ( t k)) (87) t k t k d where = t ( I z t) z. he predcted state xt ˆ( k t) = xt ( k) = t k( t k( t) dd( t k)) and fro (85): ( C φ t ) U = c P D xt ˆ( k ) t U t k t Pt k t k k t ( P D C φt d t k φt t U ) = c ( ) () (88) k t Pt k t k d t k t k t k k t Assng the control costng s a lnear odel the condton for optalty (88): ( c k φtt kt kcik t k k ) Ut = ( P ( ) C tdpt k φtt kddt k ) he npt nonlnear sb-syste can be assed fnte-gan stable and k U t ay be wrtten as ( U ) = [( )( t)...( )( t ) ]. he vector of ftre optal controls becoes: k t k k ( ( ) ) c c φt ( ( ) C φt ) U = I C P D d t k (89) t k t k t k I t k k k t Pt k t k d he L Sbsyste ftre otpts follows as k Ut and the ftre plant otpts k necessary condton for stablty s that the operator that follows s fnte gan stable: ( I ( ) ) c φ t C t k k t k t k I t k k U t. It follows a = (9)

23 6. Sffcent Condton for Stablty and Robstness If the otpt sb-syste were lnear te-nvarant and not sbject to ncertanty a slar stablty argent to that n [4] cold be sed to arge fro (89) that no cancellaton of nstable odes cold occr f the controller s pleented n ts nal for. he robstness of the solton ay be consdered and a sffcent condton for stablty n the presence of ncertanty can be obtaned by frst notng the solton can be related to the wellknown Sth Predctor strctre. o establsh ths eqvalence consder the ore sal proble where syste otpts controlled are the sae as those easred and where absolte control s costed. he algebra s slar to the non-state-dependent probles consdered n [3]. he controller whch shold not be pleented n ths z for s shown n Fg. 3. he f ( ) ter n ths solton s obtaned by wrtng the Kalan flter loop n ters of the operator eqatons that follow: Estator: xt ˆ( t) = ( z )( zt ( ) dt ( )) ( z ) ( tk) f f he transfer operators here: f ( z ) = ( I z ( I ft t ) t) ft ( ) f ( z ) = ( I z ( I ft t ) t)) ( I ft t) tz ft t Unbased estates property: Observe that for the Kalan flter to be nbased: f( z )( tt( z ) t t) f ( z ) = t( z ) t he parallel paths n Fg. 3 fro control npt are sefl f the plant has an addtve ncertanty of the for =. he dagra n Fg. 3 ay then be redrawn as shown n Fg. 4. For the sffcent condton for optalty note that the operator t k actally represents the nternal feedback loop n Fg. 5. hs the operator S representng the path between and ncldes ths stable sb-syste and the Kalan flter sb-syste. he operator S and ncertanty odel S = can both therefore be assed stable. he sall gan theore [5] can now be nvoked to provde a sffcent condton for stablty. Recall ths can be sed to establsh npt-otpt stablty condtons for a feedback syste. It provdes a sffcent condton for fnte gan p stablty of the closed-loop syste. If two npt-otpt stable systes S and S are connected as shown n a feedback loop then the closed-loop s npt-otpt stable f the loop gan S. S < where the nor sed s any ndced nor. o deal wth nstable sgnals the space pe (see [6]) s sed where the pper lt of the nor saton s fnte. he sffcent condton for stablty reqres S < / so the gan of the nner feedback loop ter shold be sffcently sall when the ncertanty s large.

24 - L Plant - - Otpt Dstrbance Kalan predctor ose v Copensator - Observatons z Fg. 3: onlnear Sth Predctor Ipled by PGMV Copensator Strctre ( ( ) ) c c φt ( ( ) C φt ) U = I C P D d t k t k t k t k I t k k k t Pt k t k d D Ptk φ t P C t k t f Kalan predctor - - c k Copensator U t C t k φ t t k t k I k C I Uncertanty n nonlnear Fg. 4: Feedback Loop when Addtve Uncertanty Inclded 6. Cost Weghtngs and Relatonshp to Stablty Say there exsts a PID controller that wll stablze the nonlnear syste wthot transport delay then a set of cost weghtngs can be defned to garantee the exstence of ths nverse and hence ensre the stablty of the closed- loop. A stablsng control law can be fond fro cost-fncton weghtngs derved below. Asse Λ then fro (54) t k Pt k Pt k and fro (89):

25 ( c ( ) ) c φt ( ( ) C φt ) U I C P D d t k t k t k t k I Pt k Pt k k k t Pt k t k d In the case of a sngle-step cost wth a throgh ter the atrx Pt k = t k can be assed sqare and non- snglar. In the case = Pt = pt and P = C t Pt k = Pt k = pt k =. Hence φ t pt k pt k ( c ( ) ) k pt k pt k t kt k I pt k ( ) k ck pt k Pt k pt k t k d t () I C D d ( t k) Also asse the dynac weghtng s on the plant otpts yp () t = Pc ( z ) yt () then pt k pt kt kt k= P c k ( ) ck pt k c k k ck pt k( Pt k pt k t k d ) t () I P D d ( t k) (9) he ter ( k pt k k k ) I c P c ay be nterpreted as the retrn-dfference operator for a nonlnear syste wth delay-free plantk = k k. hs f the plant has a controller K that stablses ths odel the rato of PID weghtngs can be chosen as P K. c k pt k c = PID An extenson of ths dea s when a set of controllers say K (z - ) for = n k stablse the syste then a set of weghtngs can be defned to satsfy c Pc K. he best robst cost-weghtngs can then be chosen ( k pt k ) = sng a technqe lke Monte-Carlo slaton coverng a range of ncertanty [7]. 7. PGMV Specal Sple For In soe cases the nonlnear syste can be represented by the state-dependent odel only and the black-box odel can be set eqal to the dentty k = I (so that k k = I ). In ths case (t) = (t) and the control weghtng nvolves a cobnaton of the constant Λ and dynac ck weghtng ters. Fro (8): he vector of ftre controls: ( ) ( c ) P D xt ˆ( k ) t U = (9) C t Pt k Pt k t k t k k t ( ) ( ) Ut = c ˆ( ) t k k PC tdpt k φ xt k t (93) t where =Λ PC t = Pt k and =. t k Pt k Pt k φ t Pt k Pt k t k

26 7. Specal Weghtng Case Asse the dynac control weghtng ( z ) s lnear or alternatvely has a nonlnear decoposton nto a ck non-dynac or constant ter a b ck and an operator ter c ( ) k z ncldng at least a nt-delay a b ck ( z ) = ck k ( z ). In ths case frther splfcatons arse and there s no algebrac loop. ote the c block verson of these fnctons nvolves the decoposton of ck nto ters and a b the algorths ay be splfed by sbstttng ( z ) = ( z ). Fro (9) ( ) hence for a lnear control costng: c c c k k k c a k a b ( c ) c C t Pt k φ t t k k t k t b c ( z ) k P D xt ˆ( k ) t U ( z ) U = a b ( c ) ( ˆ( ) c ( ) C φ t ) t t k k t Pt k k t. Hence U = P D xt k t z U (94) where φ = and P = C. Slar reslts can be obtaned when ( ) can be t Pt k Pt k t k decoposed as ( )( t) = t ( t) () () k t Pt k. hs algorth s the splest PGMV solton shown n Fg. 5. k τ D Pt k P C t Controller Strctre y Plant U t - a ( t k ck ) C ( β z ) I - y Controlled otpt z t xt ˆ( k ) t φ b c ( z ) k Dstrbances v d d d t ( k) d d k z Measreents or observatons sgnal Fg. 5: Splfed PGMV Controller Strctre for Predcted State Feedback

27 8. Mltvarable Control of a wo-lnk Robotc Manplator One of the applcaton areas for nonlnear predctve control s n ndstral robotcs where the reference trajectory for the robot anplator s defned n advance (weldng or pant sprayng robots). Consder for exaple a planar anplator wth two rgd lnks. he objectve s to control the vector of jont anglar postons q wth the vector of torqes appled at the anplator jonts so that they follow a desred reference trajectory q d. hs proble was analysed n [8] and t was shown that a lt-loop PD controller cold be sed to control the lnks to desred fxed postons. Syste odel: he dynacs of the syste are hghly nonlnear and ay be descrbed by the followng contnos-te copled dfferental eqatons: H H q hq d h( q q ) q g H H = q hq d q g τ hs eqaton ay be wrtten n the followng ore concse dfferental eqaton atrx for: Hqq ( ) Cqqq ( ) gq ( ) = τ (95) he Hq ( ) s tered the nerta atrx Cqqq ( ) s a vector of Centrpetal and Corols torqes and gq ( ) s a vector of torqe coponents de to gravty. he paraeters d and d represent the syste dapng de to frcton (n the deal nonal case d = d = ). Asse the anplator s operatng n the horzontal plane so that gq ( ) =. he coponents of the atrx H are defned as: H = a a cos q a sn q H = H a a3 cos q a4 sn q 3 4 τ H = a he paraeters h= a3sn q a4cos q and a = I l I l l c c a = I l c a3= ll cos δ c e and a4= ll sn c δe. he followng nercal vales of paraeters were sed for the slaton trals = I =. l = l c =.5 = I =.5 l c =.6 δ e = 3 o (see [8])). he above syste has the statedependent eqaton for. hs s clear by rewrtng the prevos eqatons where the nvertablty of the atrx H s a physcal property of the syste as: x q q I q = = q H ( q) C( q) q H ( q) τ q y = q H ( q) = C( q ) H ( q) τ q (96)

28 8. wo Lnk Robot Ar State-Dependent Solton It was noted above that the two-lnk robot ar eqatons are n fact n a natral state-dependent for. In ths case the npt sb-syste can be replaced by the dentty and all the non-lnear odel can be absorbed n the statedependent otpt sb-syste. he control costng ter s lnear n ths case and hence the solton s gven by eqaton (94) and the controller can be pleented as n Fg. 5. he perforance of the nconstraned PGMV controller s shown n Fg. 6 for a changng reference and stochastc dstrbance npts. he nteracton s clearly evdent leadng to large torqe changes. he reslts for a well-tned PID controller (actally PD ters) are also shown n Fg. 6. ote that the PID controller dd not nclde any rate lts on plant npts as n the orgnal pblcaton bt the predctve control soltons both nclded sch lts (n the constraned case taken accont of drectly). he PID becoes nstable wth sch lts and the predctve control reslts are therefore pressve. o redce the apltde of control sgnals the constraned solton can be appled whch eans applyng a qadratc-prograng solton to nse (83) sng the sae atrces nvolved n (94). he area where the largest changes arse s llstrated n the expanded te-scale shown n Fg. 7. Ipleentng the constraned solton sng qadratc prograng s relatvely sple n ths PGMV case. It s not of corse very eanngfl to copare the actal vales of the dynacally weghted PGMV cost-fncton. hs only serves as a atheatcal eans to obtan desred syste propertes and by defnton the optal PGMV controller wll always provde the lowest cost for the PGMV cost-fncton. he able of varances below has therefore been copted for the ndvdal plant npts and otpts to enable a coparson of the dfferent controls. Clearly a dynacally weghted predctve controller does not nse the varances of these sgnals (ths wold reqre a n varance controller). he cost-fncton s sply a echans for controller desgn lke freqency response shapng of the senstvtes. hs s also a ltvarable proble and t s not therefore sply varances that are portant. Clearly cost weghtng gans can easly be odfed to change the portance of ltng partclar npts and otpts. Snce the plant rate lts were only appled to the predctve controls the reslts are good as entoned. able : Varances for PD and PGMV Unconstraned and Constraned Controllers RMS error (q ) RMS error (q ) St Dev (τ ) St Dev (τ ) PD PGMV Unconstraned PGMV Constraned

29 Fg. 6: PGMV and PID Control wth Increental Control Costng for Unconstraned Case State-Dependent Model and Free Weghtng Choce Fg. 7: PGMV Desgn for Increental Control Acton Cases and Free Error Weghtng Choce For Constraned and Unconstraned Cases

30 9. Concldng Rearks he PGMV control desgn proble for a state-dependent syste nvolves a lt-step predctve control costfncton and ftre set-pont nforaton. he trackng reslts are ore general than for GMV desgns becase of the ablty to dstngsh between sgnals that are to be penalzed and those whch are easred. he se of ether ncreental control or control costng ters over a control horzon and control profle deterned by the connecton atrx adds to the generalty of the reslts. he splfed control strctre has been shown to be partclarly valable for real applcatons and avods any algebrac-loop proble. he PGMV control has the property that f the syste s lnear then the controller redces to the Generalsed Predctve Controller for statedependent systes. he PGMV controller offers greater flexblty copared wth the GMV and GPC controllers at the expense of soe addtonal coplexty n the pleentaton ([9] []).. References. Grble M J 4 GMV control of nonlnear ltvarable systes UKACC Conference Control 4 Unversty of Bath 6-9 Septeber.. Kothare M. V. V. Balakrshnan and M. Morar 996 Robst constraned odel predctve control sng lnear atrx neqaltes Atoatca 3() pp Mchalska H. and D. Q. Mayne 993 Robst recedng horzon control of constraned non-lnear systes IEEE ransactons on Atoatc Control 38 pp Kovartaks B. M. Cannon and J.A. Rosster 999 onlnear odel based predctve control Int. J. Control 7() pp Mayne D.Q. J.B. Rawlngs C.V. Rao and P.O.M. Scokaert Constraned odel predctve control: stablty and optalty Atoatca 36(6) pp Allgower F. and R. Fndesen 998 on-lnear predctve control of a dstllaton coln Internatonal Sypos on on-lnear Model Predctve Control Ascona Swtzerland. 7. Caacho E.F. 993 Constraned generalzed predctve control IEEE ransactons on Atoatc Control 38 pp Haett K. D. 997 Control of on-lnear Systes va State-feedback State-dependent Rccat Eqaton echnqes Ph.D. Dssertaton Ar Force Insttte of echnology Dayton Oho. 9. Grble M.J. and Johnson M.A 988 Optal control and stochastc estaton Vols. I and II John Wley Chchester.. Kwon W.H. and Pearson A.E. 977 A odfed qadratc cost proble and feedback stablzaton of a lnear syste IEEE ransactons on Atoatc Control Vol. AC- o. 5 pp

31 . Grble M.J. and Majeck P. Polynoal Approach to onlnear Predctve Generalzed Mn Varance Control IE Control heory and Applcatons Vol.4 o.3 pp Grble M and Majeck P. 3 on-lnear generalsed n varance control sng nstable statedependent ltvarable odels IE Control heory and Applcatons Vole: 7 Isse: 4 pp DOI:.49/et-cta Grble M.J. and Majeck P. State-Space Approach to onlnear Predctve Generalzed Mn Varance Control Int. J. of Control Feb. Vole 83 Isse 8 pp ISS Grble M J 5 on-lnear generalsed n varance feedback feedforward and trackng control Atoatca Vol. 4 pp Zaes G. 966 On the npt-otpt stablty of te-varyng nonlnear feedback systes part : Condtons derved sng concepts of loop gan concty and passvty IEEE rans. on At. Control Vol. AC- o. pp Grble M. J. Jkes K A and D P Goodall onlnear flters and operators and the constant gan extended Kalan flter IMA Jornal of Matheatcal Control and Inf. Vol. pp Borhan H and E Hodzen 4 A Robst Desgn Optzaton Fraework for Systeatc Model-Based Calbraton of Engne Control Syste ASME Internal Cobston Engne Dvson Fall echncal Conference Keynote Paper Colbs Indana. 8. Slotne J. J. E and Wepng L 99 Appled nonlnear control Prentce-Hal Inc. Englewood Clff ew Jersey ISB: Grble M J 6 Robst ndstral control John Wley Chchester.. Grble M J and P Majeck 5 onlnear Generalsed Mn Varance Control Under Actator Satraton IFAC World Congress Prage Frday 8 Jly 5.