Multivariable Robust LTR-i Controller for a Ship

256 yay and oll changes. Ca go, passenge and na al essels usually employ s ee ing and s abilisa ion sys ems in o de o p o ide imp o ed manoeu e - ing cha ac e is ics and mo ion con ol. Roll is ce - ainly he mos se e e angula mo ion expe ienced by a ship. La ge oll angles can make wo king on he ship di icul and can lead o mo ion sickness. The easons o in oducing ac i e oll s abilisa- ion sys ems in ships a e basically: 1) secu i y con- di ions, 2) anspo cos s educ ion, 3) passenge com o , 4) pe sonnel e iciency; and addi ionally in na al essels: 5) s able weapon pla o m main- enance and 6) s able pla o m o helicop e land- ing on he ship. I only he mo ions o oll. sway, yaw and su ge a e conside ed, he sys em is educed o a p oblem o ou deg ees o eedom. The ship model desc ibed by Kalls om and O osson (1982) has been used in he simula ions ca ied ou in his wo k. This model has demons a ed o be o g ea u ili y o e alua ing he con ol algo i hms by simula ion, as a p e ious phase o sea ials (Kalls om and O osson, 1982; Messe and G imble, 1992). The ship model is a non-linea mul i a iable model, and he mo ion equa ions a e (Kalls om and O - osson, 1982): [ all 0 0 0 0 a22 a23 a2-1 0 an a33 a3'! 0 a-u a'!3 aH 'Whe e " o " indica es he o al o ces and o ques ac ing on he ship, due o he ollowing e ec s: hy- d odynamics , wind, wa es and cu en . The s a e a iables a e espec i ely: .El = ~~ ( ans e sal speed), .E2 =~, .E3 = -0, .E4 = </J and .E;, = 1l :. Ac ua o s dynamic a e modelled as: b = (be -b)/TH , e = (Qc -Q)/Tl- , I e I~ e ma% , I Q I~ Qmu The con ol magni udes a e aC ) and be ), he angles o he ins and udde espec i ely, and he magni udes o be con olled a e </J( ) and 1l !( ), he angles o oll and heading. To de- sign he con olle , a linea ized model has been chosen o nominal condi ions o c uising speed F = lO . 8m/ s. Fo ces and o ques exp essions, hy- d odynamic de i a i es and coe icien s a e aken om Kalls om and O osson (1982). Fig , L Feedback con ol con igu a ion 3 CONTROL ALGORITHM Conside he con ol sys em o ig. 1. I consis s o he plan (C), con olle (K), p e-compensa o P, e e ence signal (T), measu emen noise (I ), and dis u bancies (d;, do). All signals a e mul- i a iable, and nominal ma hema ical models o C, 1(, P a e LT!. The con ol obje i es can be exp essed a di e en le els o demanding: 1) Nominal s abili y (NS): bounded ou pu s o all bounded dis u bancies, and bounded e e ence in- pu s. 2) Nominal pe o mance (NP): small e o s in he p esence o dis u bancies d;, do and e e - ence inpu s T. 3) Robus s abili y (RS): conside he eedback sys em in ig. 1. Suppose ha he plan is no p ecisely known, and is modelled as belonging o a class o possible ans e ma ices 9. A con olle / ,- sa is ies he obus s abili y condi ion i K s abilizes all C' E 9. 4) Robus pe o mance (RP): his equi emen is said o be me i he pe o mance speci ica ions a e sa is ied o all possible plan s C' E 9. The LTR (Loop T ans e Reco e y) design me hodology seeks o de ine he Mn IO compen- sa o K( s) so ha he s abili y obus ness and pe o mance speci ica ions a e me o he possible g ea es ex en . This in ol es wo basic s eps: 1) Ve gene a e a MIMO a ge loop ans e unc- ion (TLTF) . 2) A special compensa o K( s) is used, so ha pe o mance o he eedback sys- em in ig. 1 app oxima es he pe o mance o he TLTF es ablished in s ep one. The deg ee o app oxima ion (o eco e y) depends on cha - ac e is ics o he plan . I he plan is minimum phase, hen he deg ee o eco e y o he TLTF can be a bi a ily good (S ein and A hans, 1987). I he plan is nonminimum phase and he equen- cies o he uns able ze os a e beyond he band- wid h o he TLTF, he eco e y will ake place in low equencies, and o all p ac ical pu poses he p esence o a -away non minimum phase ze os does no deg ade he low equency cha ac e is ics o he design. Di e en app oaches ha e been sugges ed in he con ol li e a u e. ob ain he TLTF . One o hese is based on K alman il e echniques (which gen- e a es he LTR-o p ocedu e (A hans , 1986). An- Fig. 2. TLTF syn hesis o he one is based on h e l·inea qu( d a -I.c egula o (LQa, o also known as LQSF: linea quad a ic s a e eedback) heo y, and i gene a es he LTR- i p ocedu e (Zhang and F eudenbe g, 1990; Ma- ciejowski, 1989). In his wo k we ha e employed he la e one: LTR-i. Ta ge Loop T ans e Func ion Syn hesis Conside he plan model (which includes he scal- ing o he a iables and augmen a ion dynamics ha he designe has appended o mee speci ica- ions): .i;( ) Ax( ) + Bu( ) y = Cx( ) The ans e unc ion ma ix o he plan is: G(s) = Ci >(s)B, whe e i >(s) = (sI -A)-I, and we assume ha [A, B] is s abilizable and ha [A, C] is de ec able. The s uc u e o he TLTF is shown in ig. 2. I is simply de ined by he pa ame e s Band i > (s) o he plan model , and by a cons an ma ix Kc (op imal s a e eedback ma ix). I we b eak he loop a he inpu o he plan we ob ain he TLTF: Fo s abili y obus ness o hold, in he ace o mul- iplica i e unce ain ies a he inpu o he plan (G ' = (I + E)G, 0'( E) < e( u:) , he in e connec ion sys em (Mo a i and Za i iou, 1989) is in his case M (s) = Tc (s)), he ollowing inequali y mus be ue o all J.-. (small gain heo em): o /1 . [T c (j:..:)] < 1j e (J.-·) in he case o s uc u ed unce ain ies (diagonal s uc u e); whe e 0' is he maximum singula alue and /1 . ep esen s he s uc u ed singula alue [MoZa89]. Con ol demand , command- ollowing and dis u bance- ejec ion can be e alua ed om ig. 2 o he ma ix Kc ob ained . F equency-domain analysis is made and he empo al esponses o he sys em a e ob ained by simula ing he TLTF in ig. 2, in o de o p o e i design speci ica ions a e sa is ied. To ob ain ma ix Kc we sol e he LQR p oblem, which consis s o mee ing he con ol signal 'which will minimize he cos : wi h: Q = QT ~ O,R c = R~ > O, Qc = MTQM . The solu ion is u = -Kcx, and Kc is gi en by: whe e Pc = PI' ~ 0 sa is ies he algeb aic Ricca i e qua ion: Some ema kable cha ac e is ics o he TLTF ob- ained in hi s way a e: 1) op imal con ol law, 2) O'(T c) :S 2, 3) O'(Sc) :S 1, 4) a leas 60° o phas e ma gin in each inpu channel, and in ini e gain ma gin ; i he loop is condi ionally s able i has a ma gin o a leas 6dB agains gain educ ions (S ein and A hans, 1987; Maciejowski, 1989). LTR p ocedu es Once he TLTF has been ob ained, we can ask ou sel es i ""ould be possible o cons uc a com- pensa o K(s) in ig. 1 wi h he p ope y ha he eedback sys em o ig. 1 app oxima es he beha iou o he TLTF in ig. 2. This would happen i he ollowing equali y we e ue (whe e K(s)G(s) is he loop ans e unc ion LTF): K(s)G(s) = Hc{s). Howe e , o he pu poses o design i is no necessa y o us o ha e exac equali y. Indeed, i we a e in e es ed in inding K(s) so ha he app oxima e ela ion o e he band o in e es equencies is sa is ied. This is he poin o iew o he LTR-i me hod p esen ed in his wo k. 'Ve now examine wo p ocedu es o ob ain he LTR con olle K(s) , one obse e based , and he o he non obse e based. The espec i e s uc- u es a e shown in ig. 3 and ig. 4. As we c an see ig . 3 shows he con en ional LQG obse e based con olle s s uc u e (OBC), and ig. 4 il- lus a es he compensa o s uc u e de eloped by 257 258 Fig . .3 . LTR-i (OBe) s uc u e Fig. 4. LTR-i (NOBC) s uc u e Chen e al. (1991) (NOBC). The di e ence be- ween hem is ha he NOBC emo es he link om he con ol signal 11 o he obse e ia he con ol dis ibu ion ma ix B, which is ou side he ealm o obse e heo y and hence he sepa a ion p inciple is no longe alid. In his case o gua - an ee he closed-loop s abili y J(o mus be such ha A - J(0C' has all i s eigen alues in he le complex hal plan. The espec i e con olle s a e: The p ocedu e o ob ain he ma ix I{o is he same in bo h cases. One way is ha p oposed by Doyle and S ein (1981), and is based on he Kalman il- e p oblem (KBF). Fo his he ollowing alge- b aic Ricca i equa ion is sol ed: whe e: and he Kalman il e gain ma ix is ob ained om: I we ob ain J(o( q) by choosing he co a iance ma- ix Qo as: i can be p o ed [DoS 81] o he minimum phase plan ha lim J((s)G(s) = Hc(s) q- oc · The e o e : LTF q~ , TLTF The NO BC cha ac e is ics o q ~ qo ( he alue o qo mus be calcula ed in each case) a e ha (Chen e aI., 1991, 1992): 1) The compensa o is open- loop s able , 2) closed-loop s abili y is gua an eed and abo e all c) much smalle alues o gain e- co e y gain q a e equi ed han he con en ional OBC o he same deg ee o eco e y. This ac implies ha he compensa o band-wid h is much smalle han ha o he con en ional con olle and hus we ha e he ad an age o a oiding, in some ci cums ancies, he sa u a ion o he ac ua- o s as well as an imp o emen in he insensi i i y o noise o o he high- equency dis u bances. The app oach ollowed in his wo k is based on he ollowing poin s: 1) 'We a e only in e es ed in a pa ial eco e y in he in e es equency ange (low and medium equencies). 2) A high e- quencies he singula alues o Hc(j:..:) oll-o a - 20 dB/dec , , hile hose o J((j:..:)G(j:.,;) oll-o a - 40 dB/dec. Thus, LTR loops o e some addi ional obus ness o high equency unmodelled dynam- ics as compa ed o he TLTF. 3) The command- ollowing and dis u bance ejec ion pe o mance in he low equency egion be ween he TLTF and he LTF wi h LTR will be essen ially he same. 4 Sn"IULATION STUDIES Fi s we design a LTR-i con olle o achie e ad- equa e esponses o changes in he e e ence sig- nal. Fo his we use he linea ized nominal model o he ship o Y = 7. 72 m/sand we employ he ollowing design pa ame e s: signi ican wa e heigh o 4m wi h 40° ela i e o ship e e ence cou se is chosen in he simula ions. -Ve can see ha he e is a ema kable imp o e- men in oll damping wi h he LTR-i MIMO con- olle . Figu e 11 shows heading and oll o non- nominal speed condi ions (9. 0m/s and 8m/s) ; we can see ha he beha iou is adequa e, which is ano he p oo o he con olle obus ness. In o - de o imp o e pe o mance cha a c e is ics a gain scheduling con olle can be used , i h he speed o he ship as he auxilia y a iable. Due o plan and egula o s dynamics, we can implemen he con olle di ec ly in a digi al compu e wi h a sample ime o 0.1 seconds, wi hou explici ly ak- ing in o accoun he sample - da a cha ac e o he sys em. All he algo i hm implemen a ions used in he simula ions wi h he non-linea model o he ship a e ealized in his way. ,) CONCLUDING REl lARKS Mul i a iable con olle s based on LTR-i (Loop T ans e Reco e y a he inpu o he plan ) ha e been de eloped: a) o cou se changing, wi h con- side able dec ease in he coupling oll angle and b) o ship s ee ing and oll egula ion , wi h a con- side able dec ease in oll angle due o wa es. 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