Set point control in the state space setting

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1 Downloaded from orb.du.dk on: Apr 28, 28 Se pon conrol n he sae space seng Poulsen, Nels Kjølsad Publcaon dae: 29 Documen Verson Publsher's PDF, also known as Verson of record Lnk back o DU Orb Caon (APA): Poulsen, N. K. (29). Se pon conrol n he sae space seng. Kgs. Lyngby: echncal Unversy of Denmark, DU Informacs, Buldng 32. (D U Compue. echncal Repor; No. 29-4). General rghs Copyrgh and moral rghs for he publcaons made accessble n he publc poral are reaned by he auhors and/or oher copyrgh owners and s a condon of accessng publcaons ha users recognse and abde by he legal requremens assocaed wh hese rghs. Users may download and prn one copy of any publcaon from he publc poral for he purpose of prvae sudy or research. You may no furher dsrbue he maeral or use for any prof-makng acvy or commercal gan You may freely dsrbue he URL denfyng he publcaon n he publc poral If you beleve ha hs documen breaches copyrgh please conac us provdng deals, and we wll remove access o he work mmedaely and nvesgae your clam.

2 Se pon conrol n he sae space seng Nels Kjølsad Poulsen Deparmen of Informacs and Mahemacal Modellng he echncal Unversy of Denmark Absrac hs repor s nened as a supplemen or an eenson o he maeral used n connecon o or afer he courses Sochasc Adapve Conrol (242) and Sac and Dynamc Opmaon (27) gven a he Deparmen of Informacs and Mahemacal Modellng, he echncal Unversy of Denmark. he focus s n hs repor relaed o he problem of handlng a se pon or a consan reference n a sae space seng. In prncple jus abou any (sae space conrol) desgn mehodology may be appled. Here he presenaon s based on LQ desgn, bu oher ypes such as poleplacemen can be appled as well. hs s he Mone Perolo paper whch s a complaon of resuls gahered from he leraure. A major par of he resuls are colleced from he basc conrol leraure durng a sabacal year a Oford unversy and s furher compled and repored n Mone Perolo (Umbra, Ialy). Inroducon he focus n hs repor s se pon conrol or conrol of a dynamc sysem wh a pece wse consan reference. hs problem s sparsely handled n he leraure despe s praccal applcaon. hs s obvously due o s lack of heorecal conens and neres. We wll n hs repor ry o gve he smples presenaon and llusrae he eenon ncludng more general frameworks. For ha maer we wll n he frs par of he repor assume ha sysem s scalar (SISO or sngle npu sngle oupu).

3 2 2 he sandard regulaon problem he reference s n hs repor assumed o be consan or pece wse consan, whch off course nclude he sandard sep change n he reference. hs problem has wo major neres. he frs problem s he regulaon problem n whch he se pon s consan and he problem s o mach he sepon. he ne focus area s he closed loop properes n connecon o sepon changes. hs s he servo problem n connecon o a sep changes. Relaed problem whch are no addressed n hs repor are problems relaed o consan (or pece wse consan) dsurbances. Moreover, consrans n he conrol acons (or relaed sgnals) or saes are also omed n hs repor. Obvously, oher ypes of varaons (eg. harmonc) n he reference sgnal are also omed here. In hs repor we wll consder he problem of conrollng a sysem gven as: such ha oupu + A + Bu y C + Du mach a se pon (w). Ofen s no only he oupu ha eners no he objecve funcon. In general we can focus on C + D u he choce C I, D s que frequen. In he followng n wll denoe he sysem order and he number of sae, n u s he number of npus and n y s he number of oupus. 2 he sandard regulaon problem he objec n regulaon s o reduce he nfluence from dsurbances or smple o keep he sysem close o he orgn. he sandard regulaon problem has several formulaons. he mos common verson s an LQ formulaon n whch he cos funcon s quadrac n he saes and he conrol acons. In he H 2 formulaon he cos funcon s also a quadrac cos, bu n an augmened oupu vecor whch conans elemens of he errors (objecve) and he conrol acons (coss). 2. he LQ formulaon he sandard LQ problem s o fnd an npu sequence u ha ake he sysem + A + Bu from an nal sae along a rajecory such ha he cos nde J N Q + u Ru s mnmed. Evenually he horon, N, s nfne.e. N. he soluon o hs problem (see 2) can be formulaed as a sae feedback soluon u L ()

4 2.2 Oupu conrol 3 or as a me funcon u LΦ Φ A BL dependng only on he nal sae and age (.e. ). In a deermnsc seng he wo soluons are dencal, bu n an unceran envronmen he frs (he sae feed back verson) s more robus wh respec o uncerany and nose. he resuls can be found n Append D.. he LQ resuls are no resrced o he sandard presenaon gven above. I can, as ndcaed n Append D.2, also nclude cross erms n he cos funcon. J N Q + u Ru + 2 Su N u Q S S R u he resuls are que smlar o he ones n () ecep off course for he dependence on S. 2.2 Oupu conrol Ofen he problem s o conrol he oupu of a sysem y C wh due respec o he conrol effor. hs s ofen formulaed as a (LQ) problem n whch he cos funcons N J y 2 + ρ2 u 2 (2) has o be mnmed. Here ρ 2 s a desgn parameer. In he LQ framework hs s equvalen o Q C C R ρ 2 hs problem also emerge f we wan o mnme J N y 2 subjec o N u 2 C u C u s he desgn parameer. In ha case ρ 2 s a Lagrange mulpler. In pracce he wo approaches are much more relaed han epeced by a frs nspecon. he problem n (2) can also be formulaed as a mnmaon of J y u N 2 q q ρ 2 In general formulaon of he H 2 problem s a mnmaon of he cos funcon N J 2 q C + D u

5 4 2 he sandard regulaon problem here s a gh connecon beween C, D, q and Q, R, S. For eample, he cos funcon n (2) appears when C C D q ρ 2 In general he H 2 problem can be ransformed no he LQ problem hrough: Q S C S q C R D On he oher hand he H 2 problem can emerge from he LQ problem f Q S Q S R D s facored no C Q D q C D hs facoraon s by no means unque. A facor can for eample eher be n q or n C and D. In connuous me he problem s o conrol he sysem ẋ A + Bu such ha he cos funcon J 2 P + Z 2 Q + u Ru d s mnmed. he soluon o hs problem s n saonary ( ) gven by: u L : SA + A S + Q SBR B S and L R B S If he cos funcon has a cross erm Z J P + Q + Q + 2 Su d hen he soluon s : u R (B S + S ) S A + A S + Q (S B + S)R (B S + S ) he realon beween he LQ and H 2 problem s he same n connuous me as n dscree me.»» Q S C S R D W ˆ C D As n he dscree me case hs wll ypcally be he case f he problem arse from a mnmaon of Z J y 2 W d whch s a weghed (W) negral square of he oupu.e. he H 2 problem. y C + D u

6 5 3 Feed forward Le us now urn he he focus area of hs repor, namely conrol of a sysem such ha s oupu vecor mach he reference vecor. We assume ha a se pon w ess, s pecewse consan and he dmenson equals n. Probably he mos obvous way of nroducng he reference (see eg. ) s o nclude a feed forward erm (M, n u n y ) n he conrol. ha s o use conrol law In hs case he closed loop becomes u Mw L + Φ + BMw Φ A BL he feed forward erm, M, can be chosen such ha he oupu mach he reference n saonary,.e. y C(I Φ) B + D Mw Le us denoe he DC gan hrough he sysem as κ C(I Φ) B + D and assume ha s non ero (.e. nverble). In he normal case n u n y we can use: M κ If n u > n y we have an era flebly and can use M κ (κκ ) On he oher hand f n u < n u we can fulfll our objecve. If we wll mnme he dsance beween our objecve (w) and our possbly (KMw) hen we can use: M (κ κ) κ I s well known ha hs soluon s sensve o modellng errors. If he DC gan n he sysem model s wrong, hen he closed loop wll have a smlar error. In connuous me he dscrpon s ẋ A + Bu y C + Du and we have he same resul jus s he DC gan s gven by κ CA B + D

7 6 4 arge values 4 arge values Anoher very classcal way of handlng he se pon s o consder he sandard problem as a devaon away from a saonary pon. hs mehod has been eensvely been used n 3. We wll he denoe hs pon as a arge pon. ha s o consder he problem as a redefnon of orgn accordng o he reference. ha means he conrol s gven by: u u L( ) he arge pon, u has o be a saonary pon.e. o fulfll A + Bu he arge pon also have o mach he se pon,.e. o fulfll w C + Du In oal he saonary pon should sasfy he followng equaon A I B w C D u I hs ndcaes ha boh and u s proporonal o he se pon and Mw u + L M was dscussed n he prevous secon. hs soluon has, as he feed forward mehod, a hgh sensvy o modellng errors. he classcal way of handlng hs problem s use negral acon as we wll reurn o n Secon 9. Here we wll pursu anoher ype of approach. If we look a he problem of modellng errors, we can handle n saonary. One way o ackle he problem s o nclude a (consan) dsurbance n he npu or he oupu. + A + B(u + d) y C + D(u + d) he npu dsurbance has be esmaed wh an observer or a Kalman fler. In ha way, he dsurbance wll epress he modellng error a DC. Anoher, bu smlar, approach s o nclude en oupu dsurbance and operae wh a desgn model + A + Bu y C + Du + d Also here he dsurbance has o be esmaed. he wo approaces can be merged no he general form + A + Bu + Gd (3) y C + Du + Hd G and H are marces of approprae dmensons. In ha case we can fnd he arge values from: A I B G w C D H u + d I

8 7 d has o be esmaed. If we assume ha he dsurbance closely s consan we can model by he followng model d + d + ξ ξ N d (, R d ) and augmen he sochasc verson of (3). A G B v + u d I d + ξ + y C H + Du d + e hen d (and sae ) can be esmaed by means of an observer or a Kalman fler. (4) In connuous me he sysem s descrbed by d A + Bu + Gd d y C + Du + Hd and he arge values can be deermned by:»»» A B G + C D u H» d I w he sae varable whch s no neccesarely known can be esmaed from he descrpon» d d d» A G I d d d ξ» B +» v u + ξ 5 Opmal rackng In hs secon we wll eend he sandard dscree me LQ conrol problem o nclude a reference sgnal. hs approach s based on he presenaon and resuls n 2. In hs presenaon we wl resrc he approach and assume ha he reference sgnal s consan (e. s a se pon). Consder he problem of conrollng a dynamc sysem n dscree me such ha he cos funcon s mnmed. + A + Bu (5) N J y N r N 2 P + C r 2 Q + u 2 R In seady sae (N, r r) he soluon s gven by u L + Kv L R + B SB B SA K R + B SB B

9 8 6 Inernal model prncple (IMP) and S s he soluon o he algebrac Rca equaon; and v found from S C QC + A SA A SB R + B SB B SA v (I Φ ) C Qr Φ A BL Unless he sysem has (or has been mposed) a pure negraon, hs sraegy wll resul n a seady sae error (for R > ). In connuous me he problem s o conrol he sysem such ha he cos funcon ẋ A + Bu Z J y r 2 P + C r 2 Q + u 2 R d s mnmed. In seady sae (, r r) he soluon s gven by L R B S u L + Kv and S s he soluon o he algebrac Rca equaon; and v found from K R B SA + A S + C QC SBR B S v Φ C Qr Φ A BL Consder he problem of conrollng a dynamc sysem n connuous me gven by (2) such ha he cos funcon Z J C r 2 P + r 2 Q d C + Du s mnmed. In seady sae ( ) he soluon s gven by u L + Kv + Mr L ˆD QD (D QC + B S ) K ˆD QD B and S s he soluon o he algebrac Rca equaon; and v found from M ˆD QD D Q S A + A S + C QC (C QD + SB)(D QD) (D QC + B S ) v Φ (C DL) Qr 6 Inernal model prncple (IMP) Ye anoher way of ransformng he se pon problem (and oher ype reference problems) no a sandard regulaon problem s o use he socalled Inernal Model Prncple (IMP). I ha case a model of he se pon varance s creaed and buld no an augmened sysem descrpon.

10 9 he model r + r + ξ ξ s a whe nose sequence, s a suable model for se pons varaon (wh unknown changes and unknown nsans of changes). In ha case he oal descrpon becomes r + A r B + u + y C r One obvous problem n hs approach s ha he sysem descrpon s no conrolable. ha ofen resuls ofen resuls n problems when usng sandard sofware for solvng he Rca equaon. he soluon ess due o he fac ha he unconrollable par of he sae space s no vsble n he cos funcon. he soluon u L L r can be compared wh he soluon n secon 5. ξ In connuous me he sysem d d A + Bu has o be conrolled such he oupu s close o he reference. he model d r ξ d ξ s whe nose s a suable model for a sepon sgnal. he oal model can be gven as:»»»»» d A B + u d r r + ξ and he ask s o mnme he error y ˆ C» r (n some sense and wh respec o he conrol power). he soluon s u ˆ» L L r whch s a feedback from he sae and a feedforward from he se pon sgnal. 7 Conrol moves In hs secon we wll dscuss conrol formulaed n erms conrol moves. hs can be regarded as f he decson varable s he velocy of he conrol. I can be eended o nclude he dervave of he conrol acon o any order. If an opmal conrol sraegy from secon 5 or 6 s appled on a sysem (whou an negral acon) hen a non ero se pon wll resul n a seady sae error. hs s due a conflc beween seady sae error and seady sae conrol acon. One remedy s o dscharge he DC componen of he conrol acon n he cos funcon. he mos drec and smples way of don hs s o consder he conrol moves (conrol velocy) raher han he conrol acon self.

11 7 Conrol moves 7. Velocy conrol he problem of conrollng a dynamc sysem + A + Bu y C such ha he oupu s close o a ceran (consan) se pon, r, has some challenges. In order o avod he problem of a seady sae error we can formulae he objecve n erms of he conrol moves v. If he conrol moves are consan beween samples hen: u + s v + + s v and he descrpon of he sysem can be wren as: A B Bs + v (6) s + y C hs s more or less he same as nroducng (a dscree me) negraon n fron of he sysem. he problem s hen o conrol he sysem n (6) accordng o he mehod descrbed n secon 5 and 6. In ha case he cos s changed and he conrol relaed cos n he objecve funcon s shfed n frequency and he wegh s pu on he conrol velocy. he conrol acon can also be 7.2 Acceleraon conrol he mehod can be eended and he problem formulaed such ha he decson varable s he acceleraon, a, hen (f he acceleraon s consan beween samples): u + s v s a + + s v s a v + v + s a and he sysem can be wren as: A B B s s + v v + y C v 2 B 2 s 2 2 s s hs s he same as nroducng wo negraon n fron of he sysem. a (7)

12 7.3 General velocy conrol 7.3 General velocy conrol hese deas can be eended o jus abou any order. Le us llusrae hs n a forh order case. Le u be he conrol acon and le u 4 be he forh order dervave. hen (f u 4 s consan beween samples): u + s u s u s u s u 4 u u 2 u 3 or saed oherwse: u u 2 u s u s u s u s u 4 u + s u s u s u 4 u 2 + s u s u 4 u 3 + s u s s 2 s 2 s 2 s 2 s he oal sysem descrpon becomes: A B B s s u u 2 u B 2 s 2 2 s 6 B s 3 6 s 3 s 2 2 s s y C hs mehod can be eened o any order. u u 2 u 3 u u 2 u 3 u u 2 u s 6 3 s 2 2 s s + u 4 24 B 4 s 24 4 s 6 3 s 2 2 s s u Genereled velocy conrol In he prevuous secons we solved he problem of machng he se pon by means of usng a hgher order dervave as he decson varable. In hs secon we wll use a combnaons of hgher order dervaves as decson varables. ha means he conrol decson s a blend of dfferen frequences par Le us frs focus on he problem when we use he conrol move, v and he conrol level, ũ, as decson varable. In ha case: u ũ + + s v + + s v or smply: + A B y C B Bs ũ + s v

13 2 7 Conrol moves Par 2 If he decson varables are he acceleraon, he level of he conrol move (velocy) and he conrol level, hen we can nroduce v + v + ṽ + s a + + s (v + ṽ ) s a ũ and ṽ are perurbaons on he conrol and conrol move respecvely. he decson consss of he vecor ũ ṽ a In ha case: u ũ + + s (v + ṽ ) s a Consequenly, he acceleraon model n (7) can be ransformed no A B B s s + B B s 2 B s 2 s 2 2 ũ s ṽ v v s a + y C v (8) Par 3 he mehod can, as he smple mehod from he prevous secons, be eended o any fne order. For n 4 we have he resulng sysem descrpon: A B B s 2 B 2 s 6 B s 3 s 2 u 2 s 6 3 s s 2 u 2 s 2 u s u 2 u 3 u 3 + B B s 2 B 2 s 6 B 3 s 24 B s 4 s 2 + s 2 6 s 3 24 s 4 s 2 s 2 6 s 3 s 2 s 2 s y C u u 2 u 3 ũ ũ ũ 2 ũ 3 u 4 In connuous me he problem s o conrol he sysem d d A + Bu y C

14 7.4 Genereled velocy conrol 3 such ha he oupu mach he reference. Velocy conrol: If we wan o use he conrol velocy d u v d raher han he conrol sgnal self as he decson varable hen he sysem can be descrbed as:»»»» d A B + v d u u y ˆ C» u Acceleraon conrol: If we wan o use he conrol acceleraon d d u v d v a d as he decson varable hen he sysem can be descrbed as: 2 d u 5 4 A B u 5 d v v 2 3 y ˆ C General velocy conrol: As n descree me hese dears can be generaled o any order. Le he decson varable be v u (n) hen for n 4 we have he descrpon: d u d 6 u () 7 4 u (2) u (3) Generaled velocy conrol: u (n) dn d n u A B 4 u v y ˆ C u u () u (2) u (3) + u u () u (2) u (3) a If we wll use a hgh order dervave as well as perubaons of s negrals, we can use he followng agmenaon scheme. he for n 4 we have he descrpon: A B B ũ d u d 6 u () 7 4 u (2) 5 u u () 7 4 u (2) 5 + ũ () ũ (2) 7 4 ũ (3) 5 u (3) u (3) ũ (4) 2 y ˆ C 6 4 u u () u (2) u (3) v

15 4 8 Frequency weghng 8 Frequency weghng he reason for nroducng velocy conrol and s relaed s o le he conrol sgnal eners no he performance n such a way ha he resulng conroller can compensae for he sepon change n an approprae manner. Now assume he he sgnals n he performance nde are flered verson of he orgnal sgnals, such as he oupu, y, conrol u. J E { N ( y f y f H y (q)y ) ( u f O O 2 y f O2 O 2 u f H u (q)u hese frequency weghs or flers has a sae space represenaon such as: u f ) } y + Ay y + B y y u + Au u + Bu u If he sysem s gven by: y f C y y + D y y + A + Bu y C + Du u f C u u + D u u hen he sysem descrpon can be augmened o nclude he flered versons of he sgnals: y u + For he flhered oupu we have In a smlar way s If he augmened sae vecor A B y C A y A u y u y f D y C + C y y + D y Du ( D y C C y D y D ) u f ( C u D ) u y u y u u + y u u B B y D B u s nroduced he we can wre: y f ( ) D u f y C C y D y D C u D u u u

16 5 Insead of mnmng he orgnal cos we can mnme { N J E ( u ) ( Q Q )} 2 Q 2 Q 2 u C (D y ) (C y ) (C u ) D (D y ) (D y ) Q Q 2 Q 2 Q 2 O O 2 D y C C y D y D O2 O 2 C u D u Now he connuous me verson of he problem jus menoned. Assume he he sgnals n he performance nde are flered verson of he orgnal sgnals, such as he oupu, y, conrol u. ( Z»! ) J E y f u f O O 2 y f O2 O 2 u f d y f Hy(p)y uf Hu(p)u hese frequency weghs or flers has a sae space represenaon such as: d d y A y y + By y d d u A u u + B u u y f Cy y + Dy y u f Cu u + Du u If he sysem s gven by: d A + Bu d y C + Du hen he sysem descrpon can be augmened o nclude he flered versons of he sgnals: 2 d 4 y d u 3 5 For he flhered oupu we have In a smlar way s If he augmened sae vecor s nroduced he we can wre: " # y f u f 2 4 B y C A y A A u y 5 + u y f D y C + C y y + Dy Du ` D y C C y D y D u f ` C u D u y u y u u 2 4 B B y D B u y u u D y C C y D y D C u D u u 3 5 u Insead of mnmng he orgnal cos we can mnme j Z» J E u Q Q 2 Q 2 Q 2 «ff d u

17 6 9 Inegral acon C (D y ) (C y ) (C u ) D (D y ) (D y )» Q Q 2 Q 2 Q 2 3» 7 O O 2 5 O2 O 2» D y C C y D y D C u D u 9 Inegral acon Whn he conrol area here are a few mehods o avod a seady sae errors. One of he sandard rcks n conrol s o nroduce an negral acon. In he sae space seng nvolves a sae whch negrae he error, e. + + (r y ) ha resuls n an augmened sysem gven by A B + C + y C u + r If he conroller s gven by u u L hen he closed loop s gven by A BL BL C + B + B u + L + r hs can also be formulaed as: whch resuls n + u Mr L A BL BL C e C + r BM + he problem s o deermne M such ha he response s opmal. r Whn he conrol area here are a few mehods o avod a seady sae errors. One of he sandard rcks n conrol s o nroduce an negral acon. In he sae space seng nvolves a sae whch negrae he error, e. d r y d

18 7 ha resuls n an augmened sysem gven by»»»» d A B + d C y ˆ C»» u + r If he conroller s gven by» u u L hen he closed loop s gven by»»»» d A BL BL B + d C» B u +»» L + r hs can also be formulaed as: whch resuls n» d d» u Mr L» A BL BL C» e ˆ C» + r he problem s o deermne M such ha he response s opmal.» BM + r Concluson In hs paper we have revwed dfferen conrol mehods for handlng se pons n a sae space seng. he mehods covers area from feed forward and arge values o frequency weghs and negral acon. he paper s held n dscree me, bu he resuls s also gven n connuous me. Acknowledgemen hs research was graefully suppored by a gran from he Dansh Research Councl for echncal Scence, SVF (Saens eknsk-vdenskabelge Forsknngsråd) under lcense no References H. Kwakernaak and R. Svan. Lnear Opmal Conrol Sysems. Wleynerscence, F.L. Lews. Opmal Conrol. Wley, Gabrele Pannoccha, Nabl Laach, and James B. Rawlngs. A canddae o replace pd conrol: Sso-consraned lq conrol. AIChE Journal, 25.

19 8 B Quadrac forms II A Quadrac opmaon I Consder he problem of mnmng a quadrac cos funcon J u h h 2 h 2 h 22 u h + 2 h 2 u + u h 22 u I s que elemenary o fnd he dervave of he cos funcon and he saonary pon mus fulfll he saonary pon d d uj 2 h 2 + 2u h 22 (9) h 2 + h 22 u u h 22 h 2 s a mnmum o he cos funcon f h 22 s possve defne. Furhermore, he mnmum of he cos funcon s quadrac n. If we use (9) and complee he square hen: J h (u ) h 22 u ( h h 2 h 22 h 2) S S h h 2 h 22 h 2 B Quadrac forms II Le us now consder he more comple problem of mnmng + g gu or smply: J u h h 2 h 2 h 22 u u J h + 2 h 2 u + u h 22 u + g + g u u + σ A sandard resul gves he mnmum as or Applyng () gves he opmal cos: + σ 2 h 2 + 2u h 22 + g u () u h 22 h g u J h + g + σ (u ) h 22 u (h h 2 h 22 h 2 ) + (g g u h 22 h 2 ) + σ 4 g u h 22 g u h + g + σ

20 9 h h h 2 h 22 h 2 g g h 2 h 22 g u σ σ 4 g u h 22 g u C Feed forward hs append s a srah forward eenon of secon 3 and s vald boh n a descree and connuous me framework. In closed loop and n saonary we have ȳ Kū K R ny nu s he DC gan hrough he sysem. Our objecve s o fnd ū such ha In he feedforward sraegy we use M R nu ny. ȳ w ū Mw C. he balanced problem If n y n u and f K s nonsngular hen s rval ha ū K w or M K C.2 he overfleble problem Frsly, le n u > n y. ha means we have a surplus of flebly o acheve our objecve. We can hen choce o fnd ha saonary conrol whch has he lowes se (and sll acheve our objecve). hs can be formulaed as mnmng subjec o J 2ū ū w Kū he Lagrange funcon for hs problem s J L 2ū ū + λ (Kū w)

21 2 D he Dscree me LQ conrol problem whch s saonary wr. ū for ū + λ K or for ū K λ In order o acheve our objecve we mus have λ (KK ) w or hs resuls n ū K (KK ) w M K (KK ) C.3 he resrced problem Le us hen focus on he suaon n u < n y e. when we have less conrol flebly. In ha case we can acheve our objecve (ȳ w) bu have o fnd a ū (or a M) such ha he dsance beween he objecve and he possble s mnmed. In oher word we wll fnd an ū such ha s mnmed. hs s obaned for J 2 ε ε ε w Kū I ε K (w Kū) K or f hs resuls n: ū (K K) K w M (K K) K D he Dscree me LQ conrol problem In hs secon we wll revew he resuls relaed o conrol of a lnear me nvaran dynamc sysem + A + Bu () such ha a quadrac cos funcon s mnmed. D. he Sandard DLQ Conrol Problem Le us frs focus on he sandard problem. In hs cone we wll conrol he sysem n () such ha he (sandard LQ) cos funcon N J NP N + Q + u Ru (2) s mnmed. he Bellman equaon wll n hs case be V ( ) mn u Q + u Ru + V + ( + ) (3)

22 D.2 DLQ and cross erms 2 wh he end pon consrans If we es he canddae funcon V N ( N ) N P N V ( ) S hen he nner par of he mnmaon n (3) wll be I u Q + A S + A A S + B B S + A R + B S + B u he mnmum for he hs funcon s accordng o Append A gven by u L L R + B S + B B S + A and he canddae funcon s n fac a soluon o he Bellman equaon n (3) f S Q + A S + A A S + B R + B S + B B S + A S N P If he gan, L, s used n he recurson for S S Q + A S + (A BL ) S N P As a smple mplcaon from he proof we have ha V ( ) J S whch s usefull n connecon o an nerpreaon of S. D.2 DLQ and cross erms In order o connec he (very) relaed LQ formulaon and H 2 formulaon we wll augmen he sandard problem wh cross erms n he cos funcon. Assume ha a dscree me (LI) sysem s gven as n () and he cos funcon (nsead of (2)) s: N J N P N + Q + u Ru + 2 Su or J N P N + N u Q S S R u he suaon becomes a b more complcaed. he cross erms especally occurs f he conrol problem s formulaed as a problem n whch (he square of) an oupu sgnal y C + Du s mnmed,.e. In ha case Q S S R J N y 2 W C D W C D

23 22 D he Dscree me LQ conrol problem he Bellman equaon becomes n he specal case V ( ) mn u u Q S S R u + V + ( + ) V N ( N ) NP N and agan we wll ry he follwong canddae funcon V ( ) S hs can be solved head on or by ransformng he problem no he sandard one. D.2. Usng ransformaon echnque If R s nverble hen we can noduce a new decson varable, v, gven by: u v R S he nsanuous loss erm (frs erm n he Bellman equaon) can be epressed as: u Q S S R u Q + v Rv In smlar way we fnd for he dynamcs Q Q SR S + A + Bu (A BR S) + Bv Ā + Bv Ā A BR S For he fuure cos o go (he second erm n he Bellman equaon) we have: V + ( + ) + S + + (Ā + Bv ) S + (Ā + Bv ) We have now ransformed he problem o he sandard form and he nner mnmaon n he Bellman equaon V ( ) mn u Q + u Ru + V + ( + ) s hen smply: I wh he soluon v Q + Ā S + Ā Ā S + B B S + Ā R + B S + B v v L L R + B S + B B S + Ā he canddae funcon s a soluon o he Bellman equaon f S Q + Ā S + Ā Ā S + B R + B S + B B S + Ā (4) Ā S + (Ā B L ) + Q hs means ha u L + R S L R + B S + B B S + Ā

24 D.3 Opmal rackng 23 D.2.2 Drec mehod If R s no nverble hen we are forced o use a more drec approach whch resuls n he followng nner mnmaon (mnmaon of he nner par n he Bellman equaon): I wh he soluon and a Rcca equaon u Q + Ā S + Ā S + Ā S + B S + B S + Ā R + B S + B u u L L R + B S + B S + B S + A S Q + A S + A S + A S + B R + B S + B S + B S + A (5) Noce, ha (4) s he sandard Rcca equaon, as (5) conans (drecly) he cross erm S. he ransformaon mehod do requre ha R s nverble. D.3 Opmal rackng In hs secon we wll eend he sandard dscree me LQ conrol problem o nclude a reference sgnal. Laer on n hs presenaon, we wl resrc he approach and assume ha he reference sgnal s consan (e. s a se pon). Consder he problem of conrollng a dynamc sysem n dscree me gven by () such ha he cos funcon N J y N r N 2 P + C r 2 Q + u 2 R s mnmed. hs equvalen o he LQ cos funcon N J (C N r N ) P(C N r N ) + (C r ) Q(C r ) + u Ru (6) s o be mnmed. he Bellman equaon wll n case be V ( ) mn u (C r ) Q(C r ) + u Ru + V + ( + ) (7) wh he end pon consrans If we es he canddae funcon V N ( N ) (C N r N ) P(C N r N ) V ( ) S 2v + σ S N C PC v N C Pr N σ N r N Pr N hen he nner par of he mnmaon n (7) wll be I (C r ) Q(C r ) + u Ru +(A + Bu ) S + (A + Bu ) 2v +(A + Bu ) + σ + (C QC + A S + A) + u (R + B S + B)u + r Qr +2 A S + Bu 2(r QC + v + A) 2v + Bu + σ +

25 24 D he Dscree me LQ conrol problem he mnmum for he hs funcon s accordng o Append B gven by u R + B S + B B S + A B v + and he canddae funcon s n fac a soluon o he Bellman equaon n (7) f S C QC + A S + A A S + B R + B S + B B S + A v A v + + C Qr A S + B R + B S + B B v + σ σ + + r Qr v+b R + B S + B B v + If we nroduce he gans L R + B S + B B S + A K R + B S + B B hen he conrol law can be wren as u L + K v + and he Rca equaons becomes S C QC + A S + (A BL ) S N P v (A BL ) v + + C Qr v N C Pr N σ σ + + r Qr v + BK v + σ N r N Pr N In seady sae (N, r r) he soluon s gven by u L + Kv L R + B SB B SA K R + B SB B and S s he soluon o he algebrac Rca equaon; and v found from S C QC + A SA A SB R + B SB B SA v (I Φ ) C Qr Φ A BL Unless he sysem has (or has been mposed) a pure negraon, hs sraegy wll resul n a seady sae error (for R > ). D.4 Reference conrol wh cross erm In hs secon we wll eend he resul from he prevous secon a b furher. Consder he problem of conrollng a dynamc sysem n dscree me gven as n () such ha he cos funcon J C N N r N 2 P + r 2 Q

26 D.4 Reference conrol wh cross erm 25 C + Du s mnmed. hs equvalen o he (sandard LQ) cos funcon J ( C N r N ) P( C N r N ) (8) N + (C + Du r ) Q(C + Du r ) s o be mnmed. he Bellman equaon wll n hs case be V ( ) mn u (C + Du r ) Q(C + Du r ) + V + ( + ) (9) wh he end pon consrans If we es he canddae funcon V N ( N ) ( C N r N ) P( C N r N ) V ( ) S 2v + σ S N C P C v N C Pr N σ N r NPr N hen he nner par of he mnmaon n (9) wll be I (C + Du r ) Q(C + Du r ) +(A + Bu ) S + (A + Bu ) 2v +(A + Bu ) + σ (C QC + A SA) + r Qr + u (D QD + B SB)u + 2 (C QD + A SB)u 2(r QD + v +B)u 2v +A + σ + he mnmum for he hs funcon s accordng o Append B gven by u D QD + B SB ( (D QC + B SA) B v D Qr ) If we apply he resuls n Append B hen we can see ha he canddae funcon s n fac a soluon o he Bellman equaon n (25) f S A SA + C QC ( C SD + A SB ) D QD + B SB ( D QC + B SA ) v A v + ( C SD + A SB ) D QD + B SB ( B v + + D Qr ) σ σ + + r Qr ( r QD + v + B) D QD + B SB ( D Qr + B v + ) he nal (or raher ermnal) condons are: If we nroduce he gans S N C PC v N C Pr σ N r N Pr N L D QD + B SB ( D QC + B SA ) K D QD + B SB B M D QD + B SB D Q hen he conrol law can be wren as u L + K v + M r

27 26 E he Connuous me LQ Conrol and he Rca equaons becomes S v A S(A BL) + C QC C SDL (A BL ) v + L DQr σ σ + + r Qr ( r QD + v + B) D QD + B SB ( D Qr + B ) v + In seady sae (, r r) he soluon s gven by u L + Kv + Mr L D QD + B SB ( D QC + B SA ) K D QD + B SB B M D QD + B SB D Q and S s he soluon o he algebrac Rca equaon; S A SA + C QC ( C SD + A SB ) D QD + B SB ( D QC + B SA ) and v found from v I Φ L DQr Φ A BL E he Connuous me LQ Conrol In hs secon we wll revew he resuls gven n he prevous secons bu n connuous me. We wll sar wh he sandard LQ problem and hen n order o connec wh he H 2 formulaon revew he LQ problem wh a cross erm n he cos funcon. he problem s o conrol he LI sysem gven n connuous me such ha some objecves are mee. d d A + Bu (2) E. he Sandard CLQ Conrol problem Consder he problem of conrollng a connuous me LI sysem n (2) such ha he performance nde J P + Q + Q d s mnmed. he Bellman equaon s for hs suaon wh d d V ( ) mn u Q + u Ru + d d V ( ) (A + Bu ) V P as boundary condon. For he canddae funcon V ( ) S

28 E.2 CLQ and cross erms 27 hs (Bellman) equaon becomes hs s fulflled for Ṡ mn u Q + u Ru + 2 S A + 2 S Bu u R B S he canddae funcon s ndead a Bellman funcon f S s he soluon o he Rcca equaon Ṡ S A + A S + Q S BR B S S P In erms of he gan L R B S he Rcca equaon can also be epressed as Ṡ S A + A S + Q L RL Ṡ S (A BL ) + A S + Q (A BL ) S + S A + Q Ṡ S (A BL ) + (A BL ) S + Q + L RL I can be shown ha J S E.2 CLQ and cross erms Le us now focus on he problem performance nde has a cross erm,.e. J P + Q + Q + 2 Su d As n he dscree me case hs wll ypcally be he case f he problem arse from a mnmaon of J y 2 W d whch s a weghed (W) negral square of he oupu y C + Du.e. he H 2 problem. In ha case Q S C S R D W C D he Bellman equaon s now for hs suaon wh d d V ( ) mn u u Q S S R V P + d u d V ( ) (A + Bu ) as boundary condon. Agan we can go drecly for a soluon, bu f R s nverble, we can ranform he problem o he sandard form. (2)

29 28 E he Connuous me LQ Conrol E.2. Usng ransformaon echnque If we use he same mehod as n he dscree me and nroduce a new decson varable, v hrough u v R S hen nsananuous loss erm s rewren o u Q S S R u Q + v Rv Q Q SR S Furhermore he dynamcs s ransformed o A + Bu (A BR S ) + Bv Ā + Bv Ā (A BR S ) he Bellman equaon s now n he newly nroduced varable d d V ( ) mn Q + v Rv + dd V ( ) ( ) Ā + Bv v wh V P as boundary condon. For he canddae funcon hs Bellman equaons becomes V ( ) S Ṡ mn v Q + v Rv + 2 S Ā + 2 S Bv he soluon o hs problem s v L L R B S Ṡ S Ā + Ā S + Q S BR B S S P (22) he las equaon ensures ha he canddae funcon ndeed s a soluon. he oal soluon s consequenly gven as u ( L + R S ) L R B S or smpy as u R (B S + S ) Noce, ha (22) s he same Rcca equaon ha arse from he sandard problem ecep for he ransformaon of A and Q. Furhermore L s he same as arse from he sandard problem.

30 E.3 Reference conrol 29 E.2.2 Drec mehod If R s no nverble hen (2) mus be solved drecly. For he canddae funcon he Bellman equaon, (2), becomes V ( ) S Ṡ mn u Q + u Ru + 2 Su + 2 S A + 2 S Bu whch s mnmed for u R (B S + S ) Ṡ S A + A S + Q (S B + S)R (B S + S ) S P (23) I s que easy o check ha (for R beng nverble) he soluons o (23) and (22) are dencal. E.3 Reference conrol In hs secon we wll eend he sandard dscree me LQ conrol problem o nclude a reference sgnal. Consder he problem of conrollng a dynamc sysem n connuous me gven by (2) such ha he cos funcon J y r 2 P + C r 2 Q + u 2 R s mnmed. hs equvalen o he (sandard LQ) cos funcon J (C r ) P(C r ) + (C r ) Q(C r ) + u Ru d (24) s o be mnmed. he Bellman equaon wll n hs case be d d V ( ) mn u wh he end pon consrans If we es he canddae funcon (C r ) Q(C r ) + u Ru + d d V ( )(A + Bu ) V ( ) (C r ) P(C r ) V ( ) S 2v + σ S C PC v C Pr σ r Pr hen he nner par of he mnmaon n (25) wll be I (C r ) Q(C r ) + u Ru + 2 S v (A + Bu) (C QC + AS + SA ) + r Qr + u Ru 2 r QC + v A + 2( S v )Bu d (25)

31 3 E he Connuous me LQ Conrol he mnmum for he hs funcon s accordng o Append A gven by u R B S v he canddae funcon s n fac a soluon o he Bellman equaon n (25) f Ṡ S A + A S + C QC S BR B S S C PC v (A BR B S) v + C Qr v C Pr σ r Qr v BR B v σ r Pr If we nroduce he gans L R B S K R B hen he conrol law can be wren as and he Rca equaons becomes u L + K v Ṡ S A + A S + C QC L RL S C PC v (A BL ) v + C Qr v C Pr σ r Qr v K RK v σ r Pr In seady sae (, r r) he soluon s gven by u L + Kv L R B S K R B and S s he soluon o he algebrac Rca equaon; SA + A S + C QC SBR B S and v found from v Φ C Qr Φ A BL E.4 Reference conrol wh cross erm In hs secon we wll eend he resuls relaed o reference conrol and nclude a possble cross erm. Consder he problem of conrollng a dynamc sysem n connuous me gven by (2) such ha he cos funcon J C r 2 P + r 2 Q C + Du s mnmed. hs equvalen o he (sandard LQ) cos funcon J ( C r ) P( C r ) + d (C + Du r ) Q(C + Du r ) d (26)

32 E.4 Reference conrol wh cross erm 3 s o be mnmed. he Bellman equaon wll n case be d d V ( ) mn u wh he end pon consrans If we es he canddae funcon (C + Du r ) Q(C + Du r ) + d d V ( )(A + Bu ) V ( ) ( C r ) P( C r ) V ( ) S 2v + σ S C P C v C Pr σ r Pr hen he nner par of he mnmaon n (27) wll be I (C + Du r ) Q(C + Du r ) + 2 S v (A + Bu) (C QC + AS + SA ) + r Qr + u D QDu + 2 C QDu 2r QDu 2 r QC + v A + 2( S v )Bu (C QC + AS + SA ) + r Qr 2 r QC + v A +u D QDu + 2 C QDu 2r QDu + 2( S v )Bu he mnmum for he hs funcon s accordng o Append A gven by u D QD ( (D QC + B S ) B v D Qr ) he canddae funcon s n fac a soluon o he Bellman equaon n (25) f Ṡ S A + A S + C QC (C QD + SB)(D QD) (D QC + B S ) v (A B(D QD) (D QC + B S )) v + C Qr (C QD + SB)(D QD) D Qr σ r Qr (v B + r QD)(D QD) (B v + D Qr ) he ermnal condons are: If we nroduce he gans S C PC v C Pr σ r Pr L D QD (D QC + B S ) (27) K D QD B M D QD D Q hen he conrol law can be wren as and he Rca equaons becomes u L + K v + M r

33 32 E he Connuous me LQ Conrol Ṡ S A + A S + C QC L (D QD)L S C PC v (A BL ) v + (C DL) Qr v C Pr σ r Qr (v K + r M )(D QD)(K v + M r ) σ r Pr In seady sae (, r r) he soluon s gven by u L + Kv + Mr L D QD (D QC + B S ) K D QD B M D QD D Q and S s he soluon o he algebrac Rca equaon; S A + A S + C QC (C QD + SB)(D QD) (D QC + B S ) and v found from v Φ (C DL) Qr Φ A BL