Robust cross-directional control of large scale sheet and lm processes

Transcription

1 Journal of Proce Control (00) 49±77 Robut cro-drectonal control of large cale heet and lm procee Jeremy G. VanAntwerp, Andrew P. Feathertone, Rchard D. Braatz * Large Scale Sytem Reearch Laboratory, Department of Chemcal Engneerng, Unverty of Illno at Urbana-Champagn, 600 South Mathew Avenue, Box C-3, Urbana, IL , USA Abtract Sheet and lm procee, whch nclude papermakng, polymer lm extruon, and adheve coatng, are of ubtantal ndutral mportance. The procee are poorly condtoned and truly large cale, wth up to hundred of manpulated varable and thouand of enor locaton. The uncertante n heet and lm proce model requre that they be explctly taken nto account durng the control degn procedure. Numercally e cent algorthm are developed that provde robut optmal controller for a wde varety of uncertanty decrpton. The robut optmalty of the controller can be relaxed to provde low order controller utable for real tme mplementaton. Robut controller are degned for a mulated paper machne, baed on a realtc decrpton of the nteracton acro the machne, and the level of model naccurace. # 00 Elever Scence Ltd. All rght reerved. Keyword: Cro-drectonal control; Sheet and lm procee; Papermakng; Robut control; Large cale ytem; Polymer lm extruon. Introducton Sheet and lm procee have two man control objectve (ee Fg. ). One the mantenance of the average heet property pro le, whch referred to a the machne-drecton (MD) control problem. The other the mantenance of at pro le acro the machne web, referred to a the cro-drectonal (CD) control problem. Snce the MD problem [±8] ha been extenvely tuded and much le d cult than the CD problem [9], only the CD problem condered here. Pro le properte are controlled by actuator whch are almot alway located at evenly paced pont along the cro-drecton [0]. The number of actuator can be 00 or more. Pro le properte that have been controlled nclude ba weght (weght per unt area), moture content, calper, and opacty. Senor meaurement are taken after proceng (e.g. preng, dryng, tretchng) and are located ome dtance down the machne. In the pat, due to ther hgh cot, a mall number of cannng enor were ued. Each enor * Correpondng author. Tel.: ; fax: E-mal addre: braatz@uuc.edu (R.D. Braatz). meaured a zgzag porton of the heet/ lm, a llutrated n Fg.. From th lmted number of meaurement, the pro le properte were etmated at each amplng tme for ue by the control algorthm (for ntance, by ung a tme-varyng Kalman lter, a ha been decrbed by numerou author [±3]. Recently enor have become avalable whch multaneouly meaure the CD pro le a nely a every mllmeter at rate of up to 0,000 tme per mnute [4]. Th could reult n a many a 0,000 enor acro the machne. The control problem to calculate the 00+control move baed on the meaured or etmated pro le of 500±0,000 enor poton at each amplng tme. The large dmenonalty and the poor condtonng of heet and lm plant matrce make thee procee challengng to control. Further, t mpoble to generate a hghly accurate heet/ lm proce model, ether phenomenologcally or va nput-output dent caton, becaue of unknown dturbance, naccurate value for the phycal parameter, cro-drectonal movement of the web [5,6], lack of complete undertandng of the underlyng phycal phenomena (for example, durng dryng) [0], tatc frcton, and equpment wear [6±9]. The large cale nature make accountng for model uncertanty more /0/$ - ee front matter # 00 Elever Scence Ltd. All rght reerved. PII: S (00)

2 50 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 Fg.. Generc heet and lm proce wth cannng gauge (not drawn to cale). mportant and more challengng [0±]. The popular robut controller ynthe and model reducton oftware package have demontrated numercal naccurace for procee wth large number of nput and output [3]. Th paper preent computatonally e cent algorthm for degnng robut optmal CD controller for heet and lm procee. Further, t hown how the optmalty of the controller can be relaxed to gve low order controller that are eaer to mplement... Relatonhp to prevou work There an extenve lterature on non-robut optmal control tratege for heet and lm procee. Th lterature nclude paper on controllng procee wth ymmetre, one of the early paper beng by Roger Brockett and Jan Wllem [4], and everal recent paper whch have propoed model predctve control (MPC) tratege [5±30]. The fat MPC algorthm of VanAntwerp and Braatz [3±33] wa degned to avod exctng uncontrollable plant drecton, but doe not actually guarantee robutne to all common type of model uncertante. The reader referred to [0,34] for a detaled revew of optmal control degn algorthm for heet and lm procee that do not explctly addre model uncertanty. Laughln, Morar and Braatz (LMB) [35] ued crculant matrx theory to develop method for degnng conervatve robut multvarable controller baed on the degn of only one ngle loop controller. The LMB reult appled to heet and lm procee wth very hghly tructured nteracton. Crculant ymmetrc, Toepltz ymmetrc, and centroymmetrc ymmetrc model were all covered by the theory. The controller were retrcted to be ether decentralzed or decentralzed controller n ere wth a contant decoupler matrx. Forcng the controller to have thee partcular tructure retrct the performance that can be acheved wth thee algorthm. There are ubtantal d erence between the reult of LMB and the reult preented here. Frt, LMB treated only retrctve type of nteracton matrce, whle our approach handle arbtrary nteracton matrce. Second, LMB condered only parametrc uncertante n the nteracton matrce, wherea here we treat nonparametrc uncertante. Thrd, the robut controller ynthe and analy theorem preented here are much le conervatve. Fourth, applcaton of the LMB approach to a proce wth a d erent number of enor than actuator would requre quarng-up to gve a quare tranfer functon matrx. Although quarng-up procedure have been appled ndutrally for at leat the lat 5 year [36], they ntroduce an unneceary approxmaton and can reult n a lo of performance []. Duncan [9] developed a robut controller degn algorthm for heet and lm procee wth arbtrary nteracton acro the machne. Su cent condton for robut performance wth multplcatve nput and output uncertante were derved n term of atfyng robut performance for ngle-nput ngle-output (SISO) ubytem mlar to thoe treated here. The robut controller ynthe and analy theorem preented here are potentally le conervatve, and treat much broader type of uncertanty. Hovd, Braatz and Skogetad (HBS) [3,37,38] preented everal robut control reult that are applcable to CD procee. Stewart et al. [39] propoed a varaton on the robut CD control algorthm of HBS, but wth more trngent aumpton on the model uncertante, performance objectve, and nteracton acro the machne. A varaton of the control algorthm ha been mplemented by Honeywell-Meaurex on ndutral CD control hardware workng wth a hardware-n-the-loop paper machne mulator [39]. The algorthm preented here are extenon and re nement of reult by HBS. There are x gn cant new contrbuton. Frt, the HBS reult are pecalzed for applcaton to heet and lm procee, leadng to ubtantally mpl ed tatement of both the theory and the reultng algorthm. Second, the theorem here provde explct expreon for lower dmenon robut control problem whoe oluton can be ued to contruct the robut controller for the orgnal large cale control problem, wherea HBS provded only condton for the extence of the lower dmenon problem. Thrd, for many uncertanty type, we provde a much more complete model reducton. For example, where HBS may reduce the multvarable robut control problem to a large number of ngle-nput ngle-output (SISO) robut control problem, n many cae our reult can reduce the multvarable problem to a ngle SISO robut control problem. Fourth, algorthm for the degn of low order robut controller are nvetgated n detal. Ffth, nonlnear a well a lnear perturbaton

3 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 5 are addreed. Sxth, the algorthm are appled to a mulated paper machne, baed on a realtc decrpton of the nteracton acro the machne, and the level of model naccurace. Th mulated example of ubtantally hgher dmenonalty than that of any robut control problem ever condered.. Background on the robut control formulaton for heet and lm procee Here we decrbe heet and lm proce model, the performance objectve, approprate uncertanty decrpton, and provde ome background on robutne analy and ynthe... Sheet and lm proce model The proce model relate the manpulated varable move to the pro le properte meaured downtream. All reported heet and lm proce model have the form y ˆP u ; P ˆp P CD ; where y a vector of meaurement, u a vector of actuator poton, p repreent calar dynamc, and P CD a contant matrx repreentng nteracton between nput and output. Takng the ngular value decompoton (SVD) [40] of the nteracton matrx P CD allow the proce tranfer functon to be decompoed nto the peudo-svd form P ˆp P CD ˆ p UV T ˆ Up V T ˆ U P V T : where U and V are real orthogonal matrce. The element of the dagonal matrx P are tranfer functon and are not ordered n any partcular manner. Thee dagonal element P ; are referred to a peudo-ngular value [4]. The peudo-svd form u cently general to allow for non-quare P CD wth arbtrary nteracton. For non-quare P CD, rt augment P wth row or column of zero to make a quare matrx. Then compute the SVD of the quare matrx to reult n quare U and V. The peudo-ngular value correpondng to the addtonal row or column wll be equal to zero. Although there are more compact way to de ne the peudo-svd for a non-quare nteracton matrx, th de nton lead to the mplet notaton throughout the manucrpt. For ymmetrc P CD P CD ˆ P T CD, an orthogonal decompoton of P CD (e.g. Theorem 3 of [4]) allow U to be choen equal to V. In th cae, U T ˆ U and the dagonal element of P can be nterpreted a peudoegenvalue. Whle many modern polymer lm extruder have quare nteracton matrce, modern paper machne have many more enor than actuator, reultng n a non-quare nteracton matrx [3]. Although almot all of the reult n th manucrpt wll apply to the general model (), omewhat tronger reult wll be reported for ymmetrc model... Performance objectve A block dagram of the cloed loop ytem hown n Fg.. The objectve of the controller K to mnmze the e ect of dturbance d on the pro le properte y. Snce the entvty functon I PK the tranfer functon between d and y, th objectve can be quant ed by W P I P K up ˆj! W P I P K ; 3 where A refer to the maxmum ngular value of A, and the weght W P elected to de ne the dered performance (e.g. bandwdth). The weght alo ued to normalze the dered performance objectve: W P I P K 4: 4 The goal of the CD control problem to mantan at pro le acro the entre wdth of the machne, mplyng that the performance weght W P hould be elected a a calar weght w P multpled by the dentty matrx. The mot commonly ued weght ha the form w P ˆb a ; 5 a where a and b are contant real calar [35]. Wth th performance weght, the maxmum dturbance ampl caton wll be le than =b at all frequence, and the cloed loop ytem wll have a bandwdth of at leat =a. Fg.. Standard feedback control ytem.

4 5 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 A phycally meanngful performance weght mut atfy 0 < b < and a > 0. The performance objectve n (3) elected for everal reaon. Frt, the objectve allow the drect pec caton of the cloed loop bandwdth, or equvalently, the cloed loop peed of repone [th equal to a n (5)]. Second, the objectve (3) the wort-cae gan for nuodal nput at any frequency, whch allow the engneer to drectly pecfy a bound on the e ect of ocllatory dturbance on the cloed loop ytem. Thrd, the objectve (3) bound the ntegral of the quared pro le devaton acro the machne ubject to dturbance bounded by the ntegral-quared-error norm (th can be nterpreted for determntc or tochatc dturbance, n whch cae the expected value ued [43]). Mnmzng the quared pro le devaton the tated goal of mot CD control ytem [0,35,44]. Fourth, t mpler mathematcally to develop robut control algorthm baed on the performance objectve (3) than for mot other performance objectve. Detaled dcuon on electng performance objectve and performance weght are avalable [43,45]..3. Uncertanty decrpton Due to ther poor condtonng and the lmted nputoutput data avalable, a heet/ lm proce model only an approxmaton of the true proce. The naccuracy repreented by decrbng the proce model a a et of plant P^, gven by a nomnal model P and a et of norm bounded perturbaton. The x major type of multvarable uncertanty decrpton are lted n Table [43,46]. Through weght each perturbaton normalzed to be of ze one 4; 6 where a table tranfer functon repreentng unmodeled dynamc. In the more general cae where not treated a beng lnear tme nvarant, other norm on are ued [47±49]. Uncertante whch have been carefully characterzed nclude nonlnear tme nvarant Table Sx major type of multvarable uncertanty decrpton (dependence on uppreed for brevty) Uncertanty type Addtve Multplcatve nput Multplcatve output Invere addtve Invere multplcatve nput Invere multplcatve output Mathematcal repreentaton P^ ˆ P w A A P^ ˆ PI w I I P^ ˆ I w O O P P^ ˆ I w IA P IA P P^ ˆ PI w II II P^ ˆ I w IO IO P (NLTI), nonlnear tme varyng (NLTV), lnear tme varyng (LTV) [49], and arbtrarly-low tme varyng (SLTV) [48]. Multplcatve nput uncertanty repreent naccurace aocated wth the actuator, wherea multplcatve output uncertanty repreent naccurace aocated wth the meaurement. Addtve and multplcatve output uncertante are the mot commonly ued to repreent unmodeled proce dynamc. The ``nvere'' uncertante allow for procee n whch t not known wth certanty whether pole near the magnary ax are untable or table. Invere multplcatve output uncertanty provde a convenent mathematcal mean to addre performance pec caton wthn the context of robut tablty (th explaned n Secton.4). Each uncertanty block of dmenon compatble wth the nomnal model P. Th mple that A ha the ame dmenon a P, I and II are quare matrce of dmenon equal to the number of actuator, and O and IO are quare matrce of dmenon equal to the number of enng locaton. Each uncertanty block can have tructure. In the lterature, addtve uncertanty (typcally repreentng unmodeled proce dynamc) normally repreented a a full matrx, wherea multplcatve uncertante are treated a beng ether full or dagonal. Further, dagonal uncertanty block can be repreented a havng dagonal element that are ndependent calar, ˆ dag j, or repeated calar, ˆ I. A repeated dagonal uncertanty decrpton may be approprate for modelng naccurace n the enor model, nce the enor uually of the trackng type, wth the ame enor beng ued to take all meaurement. An ndependent dagonal uncertanty decrpton would be more approprate for repreentng naccurace n the actuator model [9] nce each actuator expected to have omewhat d erent dynamc repone. The uncertanty weght n Table aume that component of the ame type (for example, lce lp crew) have the ame level of uncertanty aocated wth ther repectve model. Th a good aumpton for heet/ lm machne component, nce each component of a partcular type almot alway manufactured by the ame company to provde the ame level of reproducblty. Note that th aumpton doe not necearly requre that the model for each component of a partcular type are precely equal for all plant wthn the uncertanty decrpton, only that the level of naccuracy of each component the ame. The electon of uncertanty weght decrbed n everal reference [43,45,46]. Procedure have been developed for heet and lm procee for dentfyng both nomnal model and uncertanty weght baed on proce data [4,50]. Now we de ne a non-tradtonal addtve uncertanty decrpton for a heet and lm proce, n whch the

5 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 53 peudo-ngular value n () are uncertan. The uncertanty decrpton repreented a ndependent dagonal addtve uncertanty (that, DA ˆ dagf k g, k ˆ ;...; n) P^ ˆ U DA DA V T ˆ P U DA DA V T ; 7 where DA a dagonal weghtng matrx, not necearly equal to a calar multpled by the dentty matrx. Theoretcal jut caton of th uncertanty decrpton, ncludng method to compute DA from expermental data, are provded elewhere [4,50]..4. Robut tablty and performance Algebrac manpulaton performed ether by hand [43,46,5] or wth program [5±55] can be ued to collect the uncertante aocated wth varou component n the ytem nto the block-dagonal hown n Fg. 3. The generalzed plant G de ned by the nomnal model P, the performance pec caton, and the magntude and locaton of the uncertante. The generalzed plant G and the controller K can be combned to produce the nomnal cloed loop ytem matrx M.IfG parttoned to be compatble wth K, then M decrbed by the lnear fractonal tranformaton (LFT), where ha been uppreed for brevty, M ˆ F l G; K ˆ G G KI G K G : 8 The LFT F l G; K de ned for any well-poed ytem [th equvalent to the extence of the nvere of (I G K)]. Eq. (6) mple that each block-dagonal matrx wthn the uncertanty decrpton n the et D, where D dag k j k 9 4; k table; k ˆ ;...; u where each k ha the ame dmenon a P, and u the number of uncertanty type. The tructure of each k can be repeated dagonal, ndependent dagonal, or full block. Fg. 3. Equvalent ytem repreentaton (dependence on uppreed by brevty). The cloed loop ytem ad to atfy robut tablty f t table for all table norm-bounded perturbaton D. The cloed loop ytem ad to be atfy robut performance f the performance pec caton (3) hold for all D. The cloed loop ytem robutly table to lnear tme nvarant (LTI) perturbaton f and only f the nomnal cloed loop ytem table (that, the pole of M are n the open left half plane) and the tructured ngular value D Mj! le than for all frequence (ee [43,46,56,57] for more detal). The value of the matrx functon D Mj! at each frequency depend on both the element of the matrx M and the tructure of D. The correpondng tet for robut performance exactly a for the robut tablty tet, except wth the performance pec caton treated a though t were an addtonal nvere multplcatve output uncertanty (that, w IO et equal to w P, wth full block IO repreentng the performance pec caton). It a key dea that provde a general analy tool for determnng robut tablty and performance wth repect to LTI uncertanty [58±60]. Any ytem wth uncertanty adequately modeled a n (6) can be put nto M form, wth robut tablty and robut performance wrtten a a -tet. Although exact computaton of the matrx functon can be computatonally expenve [6,6], upper and lower bound for can be computed n polynomal tme (M can alway be augmented wth zero to a quare matrx wth the ame value of, o wthout lo of generalty M wll be taken to be quare n what follow): max MU ˆ D M 4 DMD ; 0 UU DD where U the et of untary matrce wth the ame block dagonal tructure a D, A the pectral radu of A, and D the et of all matrce that commute wth every D, that, D ˆfDjDˆD for all D g [56,63]. Th de nton mple that each D D a block-dagonal matrx wth u block, the tructure of each block de ned by the correpondng block of D. In partcular, D k full block for repeated calar k, D k repeated calar for full block k, and D k ndependent calar for ndependent calar k. The maxmzaton n (0) not convex, and extng algorthm ether provde only a local maxmum or are computatonally expenve [64,65], hence the reference to the maxmzaton a beng a ``lower bound,'' although the equalty n (0) hold [5]. The upper bound can be formulated a a lnear matrx nequalty and olvable n polynomal tme ung ether ellpod or nteror pont algorthm [66,67]. The computed lower and upper bound are uually tght. However, computatonal experence ndcate that the bound become more conervatve a the ytem dmenon ncreae [64,65]. Robut uboptmal controller are almot alway computed ung the upper bound.

6 54 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 The H -optmal control problem to compute a tablzng K that mnmze F l G; K (ee Fg. 3). The tate-pace approach for olvng the H control problem mplemented n o -the-helf oftware [5,53]. The DK-teraton method (often called -ynthe) an ad hoc method that attempt to mnmze the upper bound of, that, t attempt to olve [5,53] up D F l G ; K D ; D D nu ˆj! K K n where K n the et of all nternally tablzng controller of dmenon n n, and D nu the et of all nu nu table mnmum phae tranfer functon that atfy D ˆ D at each frequency. The approach n DKteraton to alternatvely mnmze D M D up ˆj! ˆ up ˆj! D F l G ; K D for ether K or D whle holdng the other contant. For xed D, the controller ynthe olved va H - optmzaton. For xed K, the quantty () mnmzed for each D ung lnear matrx nequalte [66,67] or ome other approach [5,53]. The reultng nvertble table mnmum-phae tranfer functon D wrapped back nto the nomnal nterconnecton tructure G. Th ncreae the number of tate of the caled G, whch caue the econd H -ynthe tep to produce a hgher order controller. The teraton between D and K top after the quantty () le than or no longer dmnhed. The reultng hghorder controller typcally reduced ung Hankel model reducton [68]. Although the DK-teraton method not guaranteed to converge to a global mnmum, t ha been ued to degn robut controller for many mechancal ytem, e.g. exble pace tructure [69], mle autoplot [70,7], and rocket [7]. Bede beng an approxmaton to the orgnal condton for LTI perturbaton, () alo nteretng n t own rght, a t objectve le than one a neceary and u cent condton for robutne to arbtrarly low lnear tme varyng (SLTV) perturbaton [48] when all the perturbaton are full block. Alo, the objectve n () le than one a neceary and u cent condton for robutne to fat lnear tme varyng (FLTV), nonlnear tme nvarant (NLTI), or nonlnear tme varyng (NLTV) perturbaton when the matrce n D are retrcted to be contant matrce [49]. 3. Optmal robut controller degn To tate the reult, t ueful to recall that G an open loop tranfer functon matrx de ned by the uncertanty weght w j, the uncertanty locaton (n Table ), and the open loop nomnal model P. For the uncertanty type n Table and n (7), G can be wrtten n term of ubmatrce that nclude only the followng term (ncludng multplcaton of the term): P, w j I n, I n, 0 n, U, and V, where I n the n n dentty matrx, and 0 n the n n matrx of zero. De ne the n lower dmenon tranfer functon G~, whch are contructed from G by the followng ubttuton [ee (45) and (49) for example]: P! P; w j I n! w j I n! 0 n! 0 U! V! 3 Each of the G~ correpond to a peudo-ngular value P; of the plant P. To mplfy the tatement of the reult, P wll be treated a beng quare. A dcued earler, th wthout lo n generalty. The reult of th ecton are of two type. Frt, t hown that for varou uncertanty type the robut controller of the form K ˆV K U T 4 optmal. Second, t hown how controller of th form mplfy robutne analy and ynthe by ether partally or completely decouplng the MIMO controller degn problem nto SISO control problem, or a ngle SISO control problem. The robutne analy and ynthe reult are rt preented for heet and lm procee wth general nteracton. Then omewhat tronger reult are tated for ymmetrc nomnal model. 3.. Procee wth general nteracton matrx For the cae where all the uncertanty block are full and arbtrarly-low lnear tme varyng, the followng theorem provde condton for whch the robut optmal controller ha the form K ˆV K U T, and decrbe how th mpl e the computaton of the robut optmal controller. Proof of all reult are n the appendx. Theorem (Robut optmalty wth SLTV ). Conder a nomnal model P ˆU P V T, where U and V are real orthogonal matrce and P a dagonal tranfer functon matrx. Suppoe there are multple full block uncertante of the form lted n Table and a dagonal addtve uncertanty of the form (7). Then a controller of the form K ˆV K U T mnmze D F l G ; K D 5 D D nu

7 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 55 where u the number of uncertante and dagfd k g; k ˆ ;...; u; D ˆ dag d ; ˆ ;...; n; D k ˆ dag d k I n ; k ˆ ;...; u: 6 D nu The generalzed plant G contructed from the nomnal model P, the type of uncertante, and the uncertanty weght, wth the row and column of G arranged uch that the ndependent dagonal addtve uncertanty the upper block of D. Furthermore, K K n ˆ D D nu max d k ˆ;...;n kˆ;...;u D^ D F l G ; K D ( K; K ) ; d D^ F l G~ ; K; 7 where G^ contructed from G a de ned n (3), D^ ={dag d ; d ;...; d u dk table and mnmum phae; k ˆ ; ;...; u}, and K; are the dagonal element of K. For the cae wth no ndependent dagonal addtve uncertanty, the d and the correpondng n mum n (7) are dropped. Theorem (Robut optmalty wth NLTV, NLTI, and LTV ). Conder the aumpton of Theorem, except wth the SLTV perturbaton replaced by NLTV, NLTI, or LTV perturbaton. All reult of Theorem hold, wth the calng matrce D nu retrcted to be contant matrce. For SLTV, NLTV, NLTI, and LTV full block uncertante, Theorem and ndcate that the robut controller ynthe problem for K can be reduced to n mldly coupled SISO robut controller ynthe problem for the K;. If DK-teraton ued to degn to a robut uboptmal controller, then the K tep cont of n ndependent SISO H -optmal control problem, one for each of the SISO ubplant P; of P. The D tep coupled, nce many of the element of D enter n more than one of the SISO H -optmal control problem. After the DK teraton have converged to reult n the nal K;, they are collected nto a dagonal matrx K, and the nal controller computed from (4). The next reult for the cae where the uncertante are lnear tme nvarant. Theorem 3 (Robut optmalty wth LTI ). Conder a nomnal model P ˆU P V T, where U and V are real orthogonal matrce and P a dagonal tranfer functon matrx. Suppoe there any combnaton of uncertante of the followng form: () one full block uncertanty of any type, () any number of repeated dagonal multplcatve and nvere multplcatve uncertante of the form lted n Table, () an ndependent dagonal addtve uncertanty of the form (7). Then a controller of the form K ˆV K U T mnmze up ˆj! D F l G ; K 8 where the generalzed plant G contructed from the nomnal model P, the type of uncertante, and the uncertanty weght. Furthermore, up D F l G ; K K K n ˆj! ( ) ˆ max up ˆ;...;n K; K ˆj! D~ F l G~ ; K; ; 9 where D ~ ˆ dagf k g j k j4; k C; k ˆ ;...; ug and G~ contructed from G a de ned n (3). For ome heet and lm procee, Theorem 3 ndcate that the robut controller ynthe problem for K can be reduced to n completely ndependent SISO robut controller ynthe problem for K;, one for each of the SISO ubplant P; of P. To make the comparon wth Theorem clearer, conder the Corollary. Corollary (Robut optmalty wth LTI ). Conder the condton n Theorem 3, wth the addtonal condton that equal to t upper bound. Then up D F l G ; K K K n ˆj! ( K; K D^ D^ u ˆ max ˆ;...;n D^ F l G~ ; K; D^ ) 0 n u where D^ ˆ D^ D^ ˆdag d k ; dk table and mnmum phae; k ˆ ;...; ug: It much mpler to olve for the controller n (0) than n (7), although (7) ha fewer varable to optmze over. The SISO problem n (7) are coupled whle thoe n (0) are completely decoupled. If DK-teraton were appled n both cae, the computaton for D n (7) coupled, whle the computaton for each D n (0) not. In both cae, the K tep decoupled. A dcued n the Background ecton, t common for to be equal to or nearly equal to t upper bound.

8 56 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 The next reult aume th to generalze Theorem to addre a wder range of uncertanty tructure. Theorem 4 (Robut optmalty wth SLTV or LTI ). Conder a nomnal model P ˆU P V T, where U and V are real orthogonal matrce and P a dagonal tranfer functon matrx. Suppoe there any combnaton of uncertante of the followng form: () multple full block uncertante and repeated dagonal multplcatve and nvere multplcatve uncertante of the form lted n Table, () an ndependent dagonal addtve uncertanty of the form (7). Aume that equal to t upper bound. Then a controller of the form K ˆV K U T mnmze up ˆj! D F l G ; K ˆ D D nu D F l G ; K D where the generalzed plant G contructed from the nomnal model P, the type of uncertante, and the uncertanty weght. Let f refer to the number of full block, and d refer to the number of repeated and ndependent calar dagonal block, and let the row and column of G be arranged uch that all the full block appear a the lower block n D. Then D F l G ; K D K K n ˆ D D nu D^ f D^ f 6 4 D^ d D^ d 6 4 max ˆ;...;n 8 >< >: K; K D^ D^ d d 3 7 5F l G~ ; K; D^ f D^ f >= >; where f D^ ˆ dag d f;k jdf;k table and mnmum phae; k ˆ ;...; f ; 3 d D^ ˆ dag d d;k dd;k table and mnmum phae; k ˆ ;...; d ; 4 G~ contructed from G a de ned n (3), and K; are the dagonal element of K. The upper bound not exactly equal to for many problem, at whch cae the aumpton of Theorem 4 wll be an approxmaton. However, th approxmaton a wdely accepted one, and ued n all extng o -the-helf oftware for robut controller ynthe [5,53]. The next reult how that, under ncreaed retrcton on the uncertante, t poble to contruct the multvarable robut optmal controller by olvng a ngle SISO robut ynthe problem. Theorem 5 (Robut optmalty wth multplcatve LTI uncertante). Conder the condton of Theorem 3 wth the addtonal condton that: () all the uncertante are multplcatve or nvere multplcatve (the full block uncertanty mut correpond to a multplcatve or nvere multplcatve uncertanty), () the P; 6ˆ08, and () the P; have ame rght half plane (RHP) pole and zero, 8. De ne K;;opt a the optmal controller for any of the SISO robut ynthe problem n the rght hand de of (9). Then the other n SISO robut optmal controller can be computed by K;;opt ˆK;;opt P; P; 5 Theorem 6 (Robut optmalty wth addtve LTI uncertante). Conder the condton of Theorem 3 wth the addtonal condton that: () there one addtve, nvere addtve, or dagonal addtve uncertanty, () the P; 6ˆ08, and () the P; have ame RHP pole and zero, 8. De ne K;;opt a the optmal controller for any of the SISO robut ynthe problem n the rght hand de of (9). Then the other n SISO robut optmal controller can be computed by K;;opt ˆK;;opt P; P; 6 Aumpton () of Theorem (5) and (6) not retrctve, a heet and lm procee have the ame dynamc for each peudo-ngular value, and o hare the ame pole and zero. Aumpton () only requre that a peudo-ngular value not precely equal to zero o that the rato n (5) and (6) are well-de ned [note that th aumpton doe allow P; to have zero]. When a peudo-ngular value exactly zero, whch occur for ome quare and all non-quare nteracton matrce, then the correpondng SISO controller K; hould be et equal to zero, nce that peudo-ngular value and the correpondng column of U and V are uncontrollable [4,50,73].

9 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49± Symmetrc nomnal model Somewhat broader uncertanty type than thoe condered n Theorem 3 and 4 are applcable to heet and lm procee wth ymmetrc nomnal model. More pec cally, n th cae the reult hold for dagonal uncertante of any of the form lted n Table. Corollary (Robut optmalty wth LTI for ymmetrc nomnal model). Aume the condton of Theorem 3 wth the addtonal condton that U ˆ V. Then the reult of Theorem 3 hold for any combnaton of uncertante of the followng form: () one full block uncertanty of any type, () any number of repeated dagonal uncertante of the form lted n Table, () an ndependent dagonal addtve uncertanty of the form (7). Corollary 3 (Robut optmalty wth SLTV or LTI for ymmetrc nomnal model). Aume the condton of Theorem 4 wth the addtonal condton that U ˆ V. Then the reult of Theorem 4 hold for any combnaton of uncertante of the followng form: () multple full block uncertante and repeated dagonal uncertante of the form lted n Table, () an ndependent dagonal addtve uncertanty of the form (7) Remark All of the reult n th ecton yeld controller that are uperoptmal [74±76], that, the H norm mnmzed n n drecton. Th n contrat to the H controller computed by commercal oftware package, whch only mnmze the H norm n the wort-cae drecton [5]. From a practcal pont of vew, th mean that the uperoptmal H wll gve much better cloed loop repone to mot dturbance, although t wll have the ame overall H -norm a a non-uperoptmal controller. The controller degn theorem n Secton 3 and 4 yeld controller of the form K ˆV K U T. The robutne for the overall ytem by mnmzng the robutne margn for the SISO control problem. The peudo-ngular value of P CD that are nearly zero cannot be relably controlled [4,50]. Separate relatonhp for the SISO controller K; are gven dependng on the magntude of P; 0. For P; 0 cloe to zero the correpondng SISO controller K; et equal to zero. Otherwe, K; computed accordng to the approprate theorem from Secton 3 or 4. The parameter de ne the boundary between controllable and e ectvely uncontrollable peudo-ngular value, and can be computed from expermental data ung a Monte Carlo algorthm [50]. The SISO robut control problem aocated wth the uncontrollable peudo-ngular value hould not be ncluded n the robutne margn calculaton, that, the multvarable performance pec caton only appled to the controllable plant drecton. It wa hown n prevou work that explct contrant handlng not alway needed when robut control degn method are ued [4,50,77]. Th becaue drecton correpondng to low gan are not manpulated by the SVD controller. Alo, degnng of the SVD controller to be robut prevent overly large dynamc excuron n the manpulated varable. A recent paper provde explct crtera for determnng when contrant-handlng neceary [78]. In cae where contrant-handlng needed, any of the well-etablhed multvarable ant-wndup procedure can be appled [79±83]. Th reult n a mple controller mplementaton (ee the end of the next ecton for more detal). When ued for controller degn va DK-teraton, the theorem n Secton 3 may yeld controller of unacceptably hgh order. In practce, low order controller are often derable. Low order controller can be acheved by ung model reducton technque to reduce the controller order or by xng the controller order n the ynthe tep. The theorem provded above are utable for the former approach, whle the theorem n the next ecton are utable for the latter. Fxng the controller order n the ynthe tep lead to further mpl caton n robut controller degn. A wll be een n the example ecton, th mpl caton can be wth a mall lo n cloed loop performance. 4. Algorthm for low order robut controller degn The reult of the prevou ecton can be ued to compute robut uboptmal controller ung the DKteraton method. It unlkely, however, that any controller degn method, rrepectve of complexty, wll produce a controller that gve precely the dered tablty and performance for all dturbance and all operatng condton (for example, durng tartup or grade change). Th motvate the development of controller whch have parameter that can be tuned (or detuned) on-lne when neceary. Secondly, controller produced by DK-teraton tend to have very hgh order, whle low order controller are eaer to mplement. That an SVD controller optmze robut performance for a varety of uncertanty type ugget that uch low order tunable controller hould be elected to have the SVD tructure. In th way, the low order tunable controller wll have the optmal drectonalty. The algorthm for low order robut controller degn for the LTI uncertanty type condered n Theorem 3 requre le computaton and are preented rt, followed by the algorthm for the uncertanty type condered n Theorem,, and 4±6.

10 58 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49± Low order robut controller degn for LTI uncertanty For the LTI uncertanty type covered by Theorem 3, the followng reult how that any SVD controller (4) decouple the multvarable robut control ynthe nto ndependent SISO control problem. Corollary 4 (Robutne analy wth LTI ). Conder the condton and notaton n Theorem 3. Then up ˆj! D F l G ; K n ˆ up max ˆj! D^ F l G~ ; K; ( ) ˆ max up ˆj! o D^ F l G~ ; K; hold for any controller of the form K ˆV K U T. 7 The robutne for the overall ytem optmzed by mnmzng the robutne margn for the SISO control problem. A low order multvarable controller can be degned by degnng low order SISO controller K;. The controller K; for each SISO problem can be degned by any robut controller degn method; here we decrbe the ue of nternal model control (IMC) tunng [46] for calar dynamc decrbed by rt order plu tme delay (th by far the mot commonly ued model for decrbng heet and lm proce dynamc [0,84], for more complex model ee [46]: The SISO controller K; are tacked up a the dagonal element of a matrx K, wth the overall SVD controller computed from (4). The number of tate n K contructed ung the IMC-PID form (9) le than or equal to n, wherea ung the IMC-PI form (3) reult n K havng not greater than n tate. The IMC tunng parameter l can be elected ether a fat a poble whle mantanng robut tablty [43], or to maxmze robut performance. If the l are ued to optmze robut performance, then care mut be taken to enure that the combned uncertanty-performance decrpton not too conervatve. The IMC tunng rule ued n (9) and (3) are known to provde poor load dturbance uppreon for procee whch have the open loop tme contant larger than the dered cloed loop tme contant l [85]. For mot heet and lm procee, the tme delay domnate the open loop dynamc and relatvely mall, o that l wll be greater than for a robut control ytem [0]. For thoe rare heet and lm procee where robut performance allow l <, the IMC-tunng rule ued n (9) hould be replaced by the mod ed IMC-PID rule [86]. 4.. Low order robut controller degn for the SLTV, NLTV, NLTI, LTV uncertante Here we conder low order controller degn for the uncertanty type condered by Theorem,, and 4. p ˆ e : 8 Corollary 5 (Robutne analy wth SLTV ). Conder the condton and notaton n Theorem. Then Wthout lo of generalty, the teady-tate gan of p ha been caled o that p 0 ˆ. The nternal model control-proportonal ntegral dervatve (IMC-PID) control form K; ˆ 8 D >< P; 0 F; >: I l f P; 0 > 0 f P; I ˆ ; D ˆ ; F; ˆ l l ; 30 where the tolerance a decrbed n Secton 3.3. If a lower order controller dered, the IMC-PI form 8 < f K; ˆ P; 0 > l P; 0 3 : 0 f P; 0 4 D D nu ˆ D F l G ; K D d k kˆ;...;u d max D^ F l G~ ; K; D^ hold for any controller of the form K ˆV K U T ; 3 Corollary 6 (Robutne analy wth NLTV, NLTI, and LTV ). Conder the condton and notaton n Theorem. Then DF l G ; K D ˆ max D^ F l G~ ; K; D^ d k d kˆ;...;u DD nu ; 33

11 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 59 hold for any controller of the form K ˆV K U T, where D nu the et of contant matrce wth the ame tructure a D nu. Corollary 7 (Robutne analy wth SLTV or LTI ). Conder the condton and notaton n Theorem 4. Then D D nu ˆ D F l G ; K D 8 >< >: D^ max D^ f D^ f 6 4 D^ d D^ d 6 4 D^ d d 3 7 5F l G~ ; K; D^ f D^ f >= hold for any controller of the form K ˆV K U T. >; 34 If de ne a robut performance objectve (wth IO repreentng the performance pec caton), then low order tunable controller can be degned by olvng the approprate optmzaton problem (3), (33), or (34), wth the K; retrcted to be a low order controller, uch a (9) or (3). A procedure mlar to DK-teraton can be ued to compute a hgh qualty uboptmal oluton to the nonconvex optmzaton problem. In the K tep, the H optmzaton over the controller replaced by an optmzaton over the l. The optmzaton over the l are ndependent, and can be ealy automated. Moreover, nce the SISO control problem are nearly decoupled, each l behave mlarly a n tunng a SISO IMC controller. In partcular, for reaonable uncertanty and performance weght, the SISO robut performance objectve wll be large when l ether mall (poor tablty robutne) or large (poor performance). Extenve experence wth IMC tunng of tme delay procee ndcate that the optmzaton of the upper bound over l wll uually have a unque mnmum. Alo, gven that the P; have the ame dynamc wth a nearly contnuou range of gan from low to hgh ngular value of P CD, the mnmzng l for one optmzaton can be ued a an ntal condton for the adjacent optmzaton (l ). In the D tep, ttng the D-cale at each frequency to a tranfer functon unneceary, nce the IMC-PI/PID K; ' are not computed from the tranfer functon d, but only from ther value at each frequency. Thu the mod ed DK-teraton procedure avod both the D- ttng and the H -ynthe procedure, whch are the tep n tandard DK-teraton that can caue numercal naccurace [3]. An alternatve to the mod ed DK-teraton procedure wll be to drectly optmze the overall upper bound over the l ung a generc optmzaton procedure. Th would requre re-computng the D-cale every tme the l are updated. The mod ed DK-teraton procedure, on the other hand, requre a lmted number of D-cale computaton f properly ntalzed. The ndependent degn procedure n Secton 4. can be ued to ntalze the algorthm. If the robutne meaure de ne a robut tablty objectve (wthout nvere multplcatve nput or output uncertante), then t dered to elect the IMC tunng parameter l a fat a poble whle mantanng robut tablty. Th optmzaton problem can be poed a: ( ) max D^ D^ K; K D^ F l G~ ; K; D^ : 35 A mod ed DK-teraton procedure mlar to that decrbed n the prevou ecton can be ued to olve th optmzaton problem. The robut tablty objectve achevable f and only f the optmal value of the objectve functon n (35) zero (n practce, ome tolerance cloe to zero ued). If the optmal value of the objectve functon n (35) greater than zero, then the uncertanty et mut be reduced (for example, through ncreaed data collecton [77]) Low order robut controller degn for multplcatve or addtve LTI uncertante Here we conder low order controller degn for the uncertanty type condered by Theorem 5 and 6. Corollary 8 (Robutne analy wth multplcatve LTI uncertante). Conder the condton and notaton n Theorem 5. Then up ˆj! D F l G ; K ˆ max ( ) D~ F l G~ ; K; ; 36 up ˆj! hold for any controller of the form K ˆV K U T. Furthermore, all SISO controller K; can be contructed from a ngle SISO controller degn problem. Let the low order controller degned be denoted. The other controller are gven by k;;opt ˆK;;opt P; P; 37

12 60 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 Corollary 9 (Robutne analy wth addtve LTI uncertante). Conder the condton and notaton n Theorem 6. Then up ˆj! D F l G ; K ˆ max ( ) D~ F l G~ ; K; ; 38 up ˆj! hold for any controller of the form K ˆV K U T. Furthermore, all SISO controller K; can be contructed from a ngle SISO controller degn problem. Let the low order controller degned be denoted. The other controller are gven by K;;opt ˆK;;opt P; P; 39 For the uncertanty decrpton treated by Theorem 5 and 6, a controller of the form K ˆV K U T decouple the proce nto n ndependent SISO problem. If low order controller are dered, K; may be elected to have the form of (9) or (3) and only one K; need to be ynthezed. The other controller are contructed a multple of that one controller Implementaton SVD controller (4) can be mplemented n the form of a tatc decoupler U T n ere wth a dagonal dynamc matrx K n ere wth another tatc decoupler V. The mplementaton for the PI and PID SVD controller partcularly mple Ð the technology for mplementng tatc decoupler and nonnteractng PI/PID controller ha been avalable for over two decade. Sheet and lm procee uually have mn-max and econd-order patal contrant on ther manpulated varable to prevent exceve tree (uch a n a de or lce lp) or ow ntablte [0]. Thee contrant can be addreed by applyng any of the well-etablhed multvarable ant-wndup procedure [79±83] to the SVD controller. The SVD controller wth ant-wndup are mplementable n real tme on large cale heet and lm procee ung extng hardware [0]. 5. Applcaton Here the robut controller degn theorem developed n the prevou ecton are appled to a model developed from ndutral data that capture many of the realte of an ndutral paper machne. 5.. Paper machne model Many of the feature of th model are common to other heet and lm procee (e.g. contant nteracton matrx, calar dynamc, edge e ect). The model wa developed from ndutral dent caton data reported by Heaven et al. [84] who tuded the lce lp to weght pro le tranfer functon of a ne paper machne (ee [3] for detal): e y ˆ 0:533 Pu : 40 where y the vector of meaurement of ba weght (n lb), u the vector of actuator poton (n ml), and P CD the nteracton matrx (wth unt of lb/ml). The actuator are motor whch change the lce lp openng and the weght pro le meaured by a cannng enor at the reel of the machne. The nteracton matrx P CD of the form P CD ˆ C G where the matrx 4 3 c c 7 c c c c 6 c c c 0 c 5 c 0 c c c 4 c 9 c c c 3 c 8 c 33 c c 3 c c 7 c 3 c c 4 c c 6 c 3 c c 5 c 0 c 5 c 30 c c 6 c c 6 c 9 c c 7 c c 7 c 8 c 33 c c 35 c 30 c 5 c 0 c 5 c 0.. c 36 c 3 c 6 c c 4 c C ˆ c 37 c 3 c 7 c c 3 c 8... c 38 c 33 c 8 c 3 c c c 34 c 9 c 4 c c c 35 c 30 c 5 c 0 c c c c 3 c c 4 c c 5 c c 6 c c 7 c 4

13 8 0:3 ˆ 0; ;...; 0 9 : >< : c ˆ : : >= 9: >: >; 0:9736 4:308 ˆ ; ;...; repreent the nteracton between 30 actuator and 650 downtream meaurement locaton, and the dagonal matrx G capture the varaton of the actuator gan acro the machne: 8 7: >= : ˆ ; ;...; 0 >; 0:305 : : >< 4: L G; ˆ 5: >= >: : :75 :05 >; ˆ ; ;...; 43 0:469 ˆ 44; 45;...; 87 L G;30 ;30 ˆ 88; 89;...; We conder the cae where there uncertanty n both the nput and the output of the proce (ee Fg. 4). Th uncertanty nclude naccurace n the actuator and enor, a well a uncertanty aocated wth the actual proce. The operator I and O are unty norm bounded and aumed to be lnear tme nvarant (LTI). The magntude of the uncertanty et by the weght W I and W O. Each uncertanty weght (W I, W O ) wa choen to repreent up to 0% teady tate error and up to 00% dynamc error. The uncertanty weght alo J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 6 cover model error due to replacng the tme delay wth a 3rd order Pade approxmaton. The performance weght elected to enure le than 0.4% teady-tate error and a cloed loop tme contant of p ˆ 5 mn. Eq. (5) ndcate that the maxmum dturbance ampl caton wll be le than at all frequence, and that the bandwdth of the cloed loop ytem wll be at leat 0.. Rearrangng the block dagram n Fg. 4 and ncludng a performance block reult n the generalzed plant matrx W I W G ˆ O P 0 0 W O W P P W P W P W P P 5 45 P I I P where 0: 0 W I ˆ W O ˆ I W P ˆ 0:5 p p 0:00 I p ˆ Controller degned to be robut to the uncertanty decrpton wll alo be nentve to meaurement noe, a the uncertanty pec caton requre a rollo of the complementary entvty functon The nadequacy of commercal oftware The commercal oftware package for degnng robut controller are the Matlab -toolbox [5] and the Robut Control Toolbox [53]. It mpoble to even form the G matrx (45) for the large cale paper machne n Matlab on a Sparc Ultra 00 computer wth 64 MB of RAM and 40 MB of wap pace Ð the computer run out of memory. It ntructve however to etmate the tme requred to degn a robut controller ung the tandard DKteraton procedure [43,46,5,53,87] f t were poble to perform thee calculaton. For only 0 actuator, one DK-teraton tep took 77 mn. One H ynthe tep took 0 mn, analy took 57 mn for 50 frequency Fg. 4. Block dagram wth both nput and output uncertanty.

14 6 J.G. VanAntwerp et al. / Journal of Proce Control (00) 49±77 pont, and the D- ttng tep took. Aumng that calng up to 30 actuator follow an On 3 ncreae n computaton tme, and that x DK-teraton tep are neceary, then DK-teraton for 30 actuator would requre more than 000 h of computaton. Note that aumng an On 3 ncreae n computaton tme a lower bound Ð t lkely that a hgher order would occur n practce. For example, for 40 actuator the analy tep took more than 30 mn per frequency pont. The conervatve tmng etmate above are for the cae where the uncertante are all full block. DK teraton for repeated calar uncertante not mplemented n the commercal oftware package. If t were mplemented, the D- ttng tep for repeated calar uncertante would take much longer, a n th cae the number of degree of freedom to be computed grow very rapdly (quadratcally) a a functon of plant nputoutput dmenon. Th hgh computatonal expene lkely why the D- ttng tep for repeated calar uncertante not mplemented n commercal package. Even f a upercomputer wth GB of RAM and/or wap pace were avalable, and f tme to compute the robut optmal controller wa not a concern, the paper machne control problem ha a large enough dmenonalty that the DK teraton algorthm would lkely produce hghly uboptmal reult (the algorthm would have d culty convergng). Th behavor ha been demontrated on much maller problem n pat work [3]. Alo, the reultng controller would be of very hgh order and would be expenve to mplement. Th motvate the robut controller degn procedure preented n th manucrpt. The dmenonalty reducton theorem gven here allow robut controller to degned for ytem n whch no other degn technque are utable. The total computaton tme of the followng algorthm on the order of mnute on a Sun Worktaton or Pentum II. The multvarable robut control problem decouple nto ndependent SISO robut control problem a de ned n (9), where G~ ; K; ˆ F l w I K; P; K; P; w O P; K; P; 6 4 w P P; K; P; w I K; P; K; w O P; K; K; P; w P K; P; 3 w I K; P; K; w O P; K; K; P; 7 w P 5 K; P; 50 Snce the number of uncertante n the SISO problem (49) le than four, equal to t upper bound. DKteraton can be ued to compute a -uboptmal oluton for the SISO controller degn problem. Ung the -toolbox [5], DK-teraton wa appled to one of the SISO robut control problem de ned n (49). The frequency-dependent D cale, D, were allowed to be up to thrd order. The DK-teraton procedure wa topped after x tep, at whch pont the maxmum value of wa DK-teraton for th SISO ytem requred about 0 of computaton per teraton on Sparc Ultra 00. The tate pace matrce for the SISO controller are gven n the Appendx. The other robut SISO controller K;jj were contructed a hown n (5), and the robut multvarable controller contructed a hown n (4). The value of for the SISO problem equal to for the multvarable ytem (ee Fg. 5). 5.. Full order controller degn for repeated calar nput and output uncertante If D I and D O are treated a beng repeated calar, then the nput±output uncertanty decrpton at e the condton of Theorem 5, and the robut controller degn problem reduce to the degn of a ngle K;. The n lower dmenonal tranfer functon G~ are contructed a hown n (3): w I w G~ ˆ O P; 0 0 w O P; w P P; w P w P w P P; 5 ; P; P; I 0 0 ^ ˆ 4 0 O P 3 5: 49 Fg. 5. a a functon of frequency for the full order controller wth repeated calar uncertante (dahed), the low order controller wth repeated calar uncertante (old), and the full order controller wth full block uncertante (dotted).