Adaptive Optimal Control of Ship Steering Autopilots

Copy igh @IFAC 12 h T iennial Wo ld Cong ess, Sydney, Aus alia, 1993 ADAPTIVE OPTIMAL CONTROL OF SHIP STEERING AUTOPILOTS F.R. Rublo· and M..J. L6pez·· ·Escuela Supe io h /"ge1lie os h Se illa A da, Reina Me cedes s/". 41012 Se illa. Spai" ··Dplo . /"ge"ie la h Sis emas y Au omOlica. clDwque h Ndje a. 10. 11003 Cddiz. Spai" Abs ac : This pape desc ibes he applica ion o an adap i e LQG/LTR con olle o au oma ic s ee ing o ships. The con olle is based on Nomo o's model and on he inno a ion model o iden i y he disc e e ime sys em. The bene i s o his con olle a e demons a ed by simula ion. Keywo ds: Ship Con ol, Adap i e Con ol, Loop T ans e Reco e y, Robus Con ol. 1. INTRODUCTION A con en ional au opilo o ship s ee ing is based on he PID algo i hm. In se e al cases manual ad- jus men s o he egula o a e necessa y because he dynamics o a ship a y wi h speed, im and loading. Also dis u bances in e m o wind, wa es, cu en s, e c., mus be ake in o accoun . Fo all his, i is o in e es o ha e adap i e au opilo s. One o he possible solu ions o con olling a plan wi h pa ame e s which a y wi h ime is by means o a sel - uning egula o . The basic sel - uning con olle consis s o a sui able combina ion o a ecu si e pa ame e es ima ion algo i hm com- bined wi h a linea con olle whose pa ame e s a e compu ed om he p ocess pa ame e es i- ma es . Fo his i is possible o implemen a wide a ie y o sel - uning algo i hms (As om 1983). This a icle p esen s a sel - uning egula o in which he con olle is an LQG/LTR. Tha is a LQG con olle wi h a mechanism o Loop T ans e Reco e y (LTR), o leads a mo e obus con olle (Doyle 1979). The con ol s uc u e ha we used is dec ibed in sec ion 2. In sec ion 3 we ilus a ed he p oposed me hod wi h an example by simula ion and inally he majo conclusions o be d awn a e gi en. 2. CONTROL STRUCTURE The e a e wo di e en basic ope a ions o con- olling a ship: cou se changing and cou se keeping and in gene al wo di e en con olle s uc u es will be necessa y. Fo he second s uc u e (cou se keeping) wi h small a ia ion abou he ope a ing poin , i is possible o ob ain a 3 d o de model o he sys em and in mos cases his model can be simpli ied o a second o de sys em (Nomo o's 1017 model). I make he ela ion be ween he heading ('I/;), and he udde (6). I is con enien o add a pu e ime delay o model he s ee ing engine dynamics and unmodelled dy- namics. Wi h his, he ans e unc ion o he sys em is , G(s) _ 'I/;(s) _ J( T6 -6(s) -s(s + l/T) e- (1) The sampled e sion o he sys em wi h he model o he dis u bances due o wind, wa es and mea- su emen e o s, desc ibed as andom p ocess, co esponds o he ollowing equa ions (As om 1983). wi h: A(z-l) B(z-l) C(z-l) 1 + alz- 1 + a2z-2 + a3z-3 b1z-1 + b2 z- 2 + b3 z- 3 (2) 1 + C1Z- 1 + C2Z-2 + C3Z-3 This model o he sys em is used o con ol p o- pose and i is iden i ie by ELS in each sampling pe iod. I has been e i ied by ex ensi e expe i- men s on many ships, ha he model can indeed cap u e he essencial cha ac e is ics o ship s ee - ing dynamics (As om 1983). The adap i e con ol s uc u e used, as we can be seen in igu e 1, co esponds o ha o a sel - uning egula o (STR) which, in b ie , consis s in calcula ing he pa ame e s o he egula o sup- posing ha he plan pa ame e s a e hose gi en by means o an iden i ica ion algo i hm. Recu si e ex ended leas squa e as iden i ica ion algo i hm has been used. In each sampling pe iod he sel - uning egula o consis s o he ollowing s eps: a) PIOCiSS MmIlS Figu e 1: Diag am o he sel - uning con olle An es ima ion o he pa ame e s o a linea model by measu ing he inle and ou le alues o he p ocess. b) The adjus men o he pa ame e s o he egula o . c) The calcula ion o he con ol signal. In his case, he egula o co espond o a LQG/LTR con olle . The con olle pa ame e s a e calcula ed on he basis ha he es ima ed pa- ame e s a e he ue pa ame e s. This implies ha closed loop p ocess beha io depends, o a g ea ex en , on he accu acy o he pa ame e es- ima e and on he obu ness o he con olle o changes in he pa ame e s o he sys em. Linea Quad a ic Gaussian Con olle Conside ing he p ocess model, I can be w i en (As om 1989), in s a e space o m as, A= x(k + 1) y(k) ( -a1 -an 1 0 Ax(k) + Bu(k) + Kope(k) ---- ",(k) Cx(k) + e(k) '- -" ",(k) 0 C = ( 1 0 o ) K~ = ( Cl - a1 Cn - an ) (4) (5) Kop is he op imal s eady-s a e gain in he Kalman il e . This model is called he inno a ion model. This s uc u e o he model has he ad an age o ob aining Kop, di ec ly, wi hou ha ing o sol e he Ricca i equa ion. In oducing he loss unc- ion, 00 :J = L (y2(k) + pu 2(k)) (6) k=O he op imal egula o is gi en by, u(k) = -Kcx(k) 1018 whe e Kc = (Rc + BT P B)-l BT PA and P is ob- ained om he well known Ricca i equa ion: AT PA-P-A T P B(Rc+BT P B)-l BT PA+Qc = 0 wi h Qc = cTc and Rc = P Conside ing plan model equa ion 4, we ha e, Mo = a [ 1(k), 2(k)] Qo = a [ 1(k), 1(k)] Ro = a [ 2(k), 2(k)] Kopu; Kopu;K~ (7) u2 e wi h e(k) = y(k) - j(k) and u; = a [e(k), e(k)] The p oblem o inding op imal Kop b ings abou he so called disc e e- ime op imal obse e p ob- lem o he KBF p oblem p edic i e e sion (Kwak- e naak 1972.) This p oblem is sol ed by a ecu - ancy. The il e e sion can be used wi h ma ix Ko! being ob ained om he ela ionship: Ko! = A -1 K op , so ha he comple e solu ion o he LQG p oblem, is gi en by, x(k + 1) y(k) {(k) u(k) Ax(k) + Bu(k) + Kop[y(k) - j(k)] Cx(k) x(k) + Ko! [y(k) - j(k)] (8) -Kc{(k) I has been obse ed ha he Linea Quad a ic Gaussian Con olle (LQG) me hod wo ked well when e y p ecise ma hema ical models we e used, bu he me hod was ex emely sensi i e o imp e- cisions in he pa ame e s and o s uc u al modi- ica ions. The idea is o y o ecupe a e he open loop ans e unc ion (LTR) which is p o ided by he applica ion o he con ol law alone, because in his way s abili y is assu ed, he e is li le sen- si i i y, and he empo a y speci ica ions a e ul- illed. This can be achie ed, in heo y, ac ing on he pa ame e s o he Kalman il e so ha he open loop ans e unc ion, when he Kalman il- e has been in oduced, app oxima es he o iginal open loop ans e unc ion. Fo con inuous ime sys ems, his can be achie ed by making he K alman il e gains depend on a de e mined pa ame e q and he open loop LQG asymp o ically app oxima es he open loop LQR (Doyle 1979). I is also possible o make he ecu- pe a ion, in a dual o m, by ac ing on he pa am- e e s o he loss unc ion in he LQR p oblem in acco dance wi h a sensi i i y eco e y p ocedu e due o Kwake naak (1972). In he case o disc e e sys ems (Maciejowski 1985), i can be shown ha o minimum phase sys ems, de (CB) ::j:. 0 and using he il e ed e sion o he Kalman il e as obse e , pe ec ecupe a ion is ob ained ac ing on he pa ame e s o he LQR. In con inuous sys ems he e is a comple e dual- i y be ween he s a e ec o eedback LQR and he s a e es ima ion KBF, which allows o he ecu- pe a ion o he wo op ions, gi en he duali y o bo h p oblems. Howe e in he case o disc e e sys ems his duali y is no comple e. The s a e ec o eedback scheme is he dual one o he p e- dic i e e sion o he obse e . The e o e, i is no possible o achie e an exac ecupe a ion in he case o he Kalman il e modi ica ions. Howe e he use o he Kalman il e ecupe a ion op ion equen ly yields use ul esul s. Then he LQG/LTR me hod wo ks well o min- imum phase sys ems bu canno be elied on o non-minimum phase ones. In se e al cases i wo ks well also o non-minimun phase sys ems, ha has been p o ed by simula ion s udies (Maciejowski 1985, Lopez 1992). Based in his idea, we p opose o modi y he Kalman il e co a iance ma ices ( ha is, o change J( op and J( 0/ ), in o de o ob ain he ecu- pe a ion . To do his he K alman il e is designed wi h some ic i ious co a iance ma ices. The ol- lowing a e used: R= Ro whe e Qo and Ro (eq. 7), a e nominal co a iance ma ices and q a pa ame e . The Kalman il e is calcula ed om he modi ied co a iance ma ix . In his way he g ea e alue o he pa ame e q he nea e i is o he open loop ans e unc ion o he LQR. By doing his, p ecision in he s a e es ima ion is los because he K alman il e is calcula ed by ic i ious co a i- ances. Howe e obus ness is gained. Sel - uning Con olle An explici sel - uning con ol algo i hm was de- eloped inco po a ing he LGQ/LTR design me hod o he p e ious sec ion, he pa ame e s o he sys- em model being de e mined on-line ia ecu si e ex ended leas squa es es ima ion. Fo his, conside he p ocess model o he equa- ion 3, ha o es ima ion pu poses, i can be ex- p essed as: [-y(k -1), -y(k - 2),·· ., -y(k -n), u(k -1), u(k - 2) , ... , u(k -n), e(k -1),e(k -2), ·· ·,e(k - n)] The pa ame e s iden i ie is a e y impo an pa o he sel - uning con olle s. The e a e a ious ypes o iden i ie s which a e deal wi h in he el- a i e li e a u e. Gene ally, he mos used me hods is he ecu si e ex ended leas squa es a e, because o i s simplici y and i s good con e gence cha ac- e is ics. The algo i hm is pe o med by he ol- lowing s eps: 1. Selec he ini ial alues o P(k) and O(k). 1019 2. Read he new alues o y(k + 1) and u(k + 1). 3. Calcula e he a p io i e o : e(k + 1) = y(k + 1) -I{)T(k + I)O(k) 4. Calcula e L(k + 1) gi en by he exp ession: P(k)l{)(k + 1) L(k + 1) = c(k) + I{)T(k + I)P(k)l{)(k + 1) 5. Calcula e he new pa ame e es ima ed gi en by: O(k + 1) = O(k) + L(k + l)e(k + 1) 6. Ac ualize he co a iance ma ix. P(k + 1) = (I -L(k + 1)I{)T (k + 1)) ~g; 7. Calcula e he new o ge ing ac o c(k + 1) . e(k+l)2 c(k+ 1) = 1- (1_I{)T (k+ I)L(k+ 1)) So I c(k + 1) < Cmin Then c(k + 1) = Cmin 8. Ac ualize he measu emen s ec o I{)(k + 2) . 9. Make k = k + 1 and e u n o s ep 2. In o de o educe he memo y o he iden i ie we use a a iable o ge ing ac o (c( k)) (Fo escou 1981), as has been desc ibed p e iously. I is known ha he adap i e con ol is nonlin- ea and ime a ying. Bu i is desi able o make he sys em as linea ime in a ian as possible (Lamai e 1991), because in his way, he con- ol sys em is mo e obus o dis u bances han a highly nonlinea adap i e con olle . I he con olle is calcula ed in equen ly, he adap i e con olle educes o a obus con ol law , ha is, he adap i e con olle becomes simply he bes obus linea ime in a ian con ol law ha one could design based only on a p io i in o ma- ion. Fo all his, he implemen a ion o he me hod de- sc ibed in he p e ious sec ion, has been make in wo ime scales. E e y sampling ime, we calcu- la e he con ol law and he iden i ie is unnig, bu he con olle a e edesigned only e e y N c sampling ime. 3. SIMULATION STUDIES In o de o s udy he p oposed me hod he heo- e ic model o a la ge ship, o wo di e en con- di ion o speed, has been chosen, whe e o small de ia ions o he angle o he udde i can be ap- p oxima ed by Nomo o's second o de model so ha : 0.001880 -5. G1(s) = s(s + 0. 006450) e (9) 0.011085 C2(s) = s(s + 0.01567) (10) These models co esponds o a ca go o 161 m wi h wo di e en speeds and i is ob ained om (As- om 1989). They ha e been iden i ied by ELS us- ing a sampling ime o 10 seconds. In he s a ing phase he sys em is iden i ied by ELS algo i hm, based on a non- ecu si e me hod ( obus mul is ep algo i hm), and wi h his, he con olle is calcula ed. A e hen, he plan is iden i ied by ecu si e ELS me hod and is changed om Cl o C2. In igu e 2, whe e he ixed con- 40~----~----~----~--~ 20 Ou pu o -20 -40~----~----~----~----~ o 100 200 300 400 k (cides) 40~----~----~----~----~ 20 Con ol -20 -40~----~----~----~----~ o 100 200 300 400 Figu e 2: Sys em esponse wi h ixed egula o olle calcula ed in he s a ing phase is used, we can see ha he con ol e o is high and he ou - pu o he sys em is nea ly uns able. Figu e 3 shows he esul s o applying he adap i e con- olle explained in he p e ious sec ion and, as can be seen, i is qui e sa is ac o y. 4. CONCLUSIONS A sel - uning egula o wi h a LQG/LTR con olle o non-minimum phase sys ems wi h no p io knowlege o exis ing noise has been de eloped. The ELS iden i ica ion me hod has been used and he inno a ions model has been conside ed o be he sys em model. The p oposed me hod has been applied o he model o a ship and he ad an ages o said me hods ha e been shown by he simula- ions ca ied ou . Acknowledgemen : The au ho s would like o hank o CICYT o suppo ing his wo k unde g and ROB89-0614-C03-01. 1020 10~----~----~----~----~ Ou pu 5 o -5 -10L-----~----~----~----~ o 100 200 300 400 k (cides) 4~----~------~----~------. 2 Con ol o -2 _4L-----~----~------~--~ o 100 200 300 400 Figu e 3: Sys em esponse wi h adap i e egula o 5. REFERENCES As om, K.J., Theo y and Applica ions o Adap- i e Con ol: A Su ey, Au oma ica, Vol 19-5, 1983, pp 471-486. As om, K.J. and B. Wi enma k, Adap i e Con- ol, A.W. 1989. Doyle, J .C . and G. S ein (1979), Robus ness wi h Obse e s, IEEE T ans. on Au oma ic Con- ol, Vol AC-24, num . 4, pp 607-611. Fo escue, T .R., L. S. Ke shenbaum, B. E. Yds ie, Implemen a ion o sel - uning egula o s wi h a iable o ge ing ac o s, Au oma ica Vol. 17, 1981, pp. 831-835. Kwake naak, H. and R. Si an, Linea Op i- mal Con ol Sys ems, Wiley-In e cience, New Yo k, 1972. Lamai e, R.O., L. Vala ani, M. A hans and G. S ein, A F equency-domain Es ima o o Use in Adap i e Con ol Sys em, Au oma ica Vol 27-1, pp 23-38, 1991. Lopez, M.J. and F.R. Rubio (1992), LQG/LTR Con ol o Ship S ee ing A u opilo s, IEEE In e na ional Symposium on In elligen Con- ol, (ISIC-92), Glasgow, Sco land, U.K. Maciejowski, J .M. (1985), Asymp o ic Reco e y o Disc e e- Time Sys ems, IEEE T ans. on Au oma ic Con ol, Vol. AC-30-6, pp. 602- 605 .