Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 UDK - UDC 681.511:517.9 Izvirni znanstveni članek - Original scientific paper (1.01) Programljivi logični krmilniki na temelju rešitve algebraične Riccatijeve enačbe Programmable Logic Controllers Based on the Algebraic Riccati Equation Solution Primož Podržaj - Zoran Kariž (Fakulteta za strojništvo, Ljubljana) V prispevku je predstavljena sinteza podoptimalnega krmilnika, ki sloni na rešitvi algebraične Riccatijeve enačbe (ARE). Predstavljen je numerični postopek, s katerim lahko pridemo do te rešitve. V praksi se parametri krmiljenega sistema mnogokrat precej razlikujejo od tistih v ARE. V tem primeru sta vprašljivi optimalnost in celo stabilnost krmilnega sistema. Zelo uporabno bi torej bilo, če bi lahko zasnovali prilagodljivi linearni podoptimalni krmilnik. Tak krmilnik bi bil zmožen odkriti spremembe v parametrih sistema in prirediti parametre. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: sistemi prilagodljivi, enačbe Riccatijeve, krmilniki logični, krmilniki programljivi, algoritmi numerični) This paper deals with the synthesis of a suboptimal controller, based on the solution of the algebraic Riccati equation (ARE). The numerical procedure for obtaining the solution is presented. In applications the controlled system parameters often differ from the ones used in the ARE. In this case the optimality of the control system and even its stability are questionable. Therefore, it would be very useful to design an adaptive linear suboptimal controller. Such a controller should be able to detect changes in the system parameters and adjust its parameters. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: adaptive systems, Riccati equations, programmable logic controllers, numerical algorithms) 0 UVOD Linearni časovno neodvisni sistem (LČN) z diferencialno enačbo stanj: 0 INTRODUCTION A linear time-invariant (LTI) system with the state differential equation: x&(t) = Ax(t) + Bu(t) (1), prikazan na sliki 1 je optimalen, če mu je dodana povratna zveza: shown in Figure 1 is said to be optimal if the feedback: u(t) = -F(t)x(t) u B —H ¦) x J x '+ A Sl. 1. LČN sistem Fig. 1. LTI system 852 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 tako, da je integralski kriterij: is added in such a way that the following cost functional: t1 J [xT (t)R1 (t)x(t) + uT (t)R2 (t)u(t)] -dt + xT (t1 )P1x(t1 ) najmanjši. Pri tem so: • R1(t) in R2 (t) pozitivno definitni simetrični matriki za t = t = t1, . P pozitivno poldefinitna simetrična matrika. Pokazati se da [1], da je treba za minimizacijo funkcionala rešiti naslednjo matrično Riccatijevo enačbo (MRE): is minimized. This means that . R1(t) and R2(t) are positive definite symmetrical matrices for t0 = t = t1 . P is a positive semi-definite symmetrical matrix It can be shown [1], that in order to minimize the cost functional the following matrix Riccati equation (MRE) has to be solved: & P(t) = R1 - P(t)BR2- 1BT P(t) + P(t)A+ AT (t)P Če rešitev MRE konvergira, potem je limita: If the solution of the MRE converges, then the limiting solution: P = limP(t) rešitev naslednje algebraične Riccatijeve enačbe (ARE): is the solution of the following algebraic Riccati equation (ARE): R - PBR-1BT P + PA+ AT P = 0 Krmilnik, ki je zasnovan na temelju te rešitve, se imenuje podoptimalni krmilnik. 1 OPIS PROBLEMA Podoptimalni krmilniki se veliko uporabljajo. Dejstvo pa je, da se lastnosti krmiljenega objekta, dane z matriko A, spreminjajo s časom zaradi staranja in spremenljivih delovnih razmer, čeprav se predpostavlja, da so časovno neodvisne. Matrika A je lahko časovno neodvisna, če opazujemo tipični čas prehodne funkcije krmilnega sistema, toda je časovno odvisna med dobo trajanja sistema. Če se dejanska matrika stanja AR razlikuje od tiste, ki smo jo uporabili v ARE (matrika A) za D, se lahko pojavijo težave. AR = A+D Zaradi tega se izgubi podoptimalnost sistema. V mnogih primerih je vprašljiva celo njegova stabilnost. Zaželeno bi torej bilo, da se zgradi krmilnik, ki bi lahko: . določil dejansko matriko stanja AR, . rešil ARE na podlagi dejanske matrike stanja AR. The controller based on this solution is called a suboptimal controller. 1 PROBLEM FORMULATION Suboptimal controllers are frequently used. However, it is a fact that the properties of the controlled system given in matrix A change over time, due to ageing and different working conditions etc., even though they are assumed to be time-invariant. The matrix A may be time-invariant if we consider a typical transient response time of a system, but it is time variant during the lifetime of the system. Many problems can be encountered if the real system matrix, AR, which differs from the one used in the ARE (matrix A) by the amount D, is introduced. d ¦¦¦ d Consequently, the system’s suboptimal performance is lost. Even its stability may be questionable in many cases. Therefore, it is desirable to design a controller capable of: . determining the real system matrix, A, . solving the ARE considering the real system matrix, AR, Programljivi logični krmilniki - Programmable Logic Controllers 853 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 2 DOLOČEVANJE DEJANSKE MATRIKE STANJA AR Celoten postopek določevanja dejanske matrike stanja A R je razdeljen na dva koraka, in sicer v odvisnosti od tega ali matrika stanja AR ni poznana (prvi korak), ali pa je vsaj približno poznana (drugi korak). 2.1 Prvi korak Če v enačbi (1) uporabimo dejansko matriko stanja v prvem koraku, dobimo: 2 DETERMINING THE REAL SYSTEM MATRIX, AR The whole procedure for determining the real system matrix, AR, is divided into two steps, depending on whether the system matrix, AR, is not known at all (first step) or is approximately known (second step). 2.1 First step When the real system matrix, AR1, is used in the first step, Equation (1) implies the following equality: x&(t) - Bu(t) = AR1x(t) (2). Matrika stanja AR1 pomeni linearno preoblikovanje vektorskega prostora 5R n v vektorski prostor M n in je enolično določena z baznimi vektorji prostora W. Torej je za določitev elementov sistemske matrike treba najti n linearno neodvisnih vektorjev stanj in njihove slike. Predpostavimo, da lahko merimo vse komponente vektorja stanj x in da je med fazo branja vhodni signal signal u enak 0 (sl. 2). Uporabljena je bila tudi predpostavka, da je mogoče izračunati odvode vseh komponent vektorja stanja. Postopek, ki ga lahko uporabimo v ta namen, je opisan na primer v [8] in [9]. Levo stran enačbe (2) lahko torej določimo za vsak čas t. Recimo, da krmilnik med fazo branja (u=0) za nek čas vzorčenja t prebere x(0+), x(T), x(2T) in tako naprej, dokler ne najde n linearno neodvisnih vektorjev stanj x1 , x2 , ... , x . Potem lahko oblikujemo naslednji matriki: The system matrix, AR1, represents a linear trans-formation of the space 9T into the space 9T and is uniquely determined by the images of the vectors on the basis of 9T. Therefore, in order to determine the elements of the system matrix we only need to find n linearly independent state vectors x and their images. Let us suppose that all the components of the state vector x can be measured, and that during the acquisition phase the control signal u equals 0 (Fig. 2). Another supposition, that the derivatives of all the components of the state vector x can be calculated, was also made. The procedure, which might be used in this case, is given for example in [8] and [9]. Therefore the left-hand side of Equation (2) can be determined at any time t. Now, let us say that the controller reads x(0+), x(T ), x(2T) and so on during the acquisition phase (u=0) for a specified sampling time TS, until n linearly independent state vectors x1 , x2 , ... , x are found. Then, the following matrices can be formed: X& =[x&1 x&2 L x&n] ; X=[x1 x2 L xn] Dejansko matriko stanja AR1 lahko torej določimo iz naslednje enačbe: x f x • J '------------- A R1«—' Sl. 2. Prvi korak, faza branja Fig. 2. First-step acquisition phase Therefore, the real system matrix, AR1, can be determined by the following equation: B ~+ "^ » r x "V A R1 -F1 Sl. 3. Prvi korak, delovna faza Fig. 3. First-step working phase 854 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 B +^ x —+—* J x AB2 -F, Sl. 4. Drugi korak, faza branja Fig. 4. Second-step acquisition phase B +^> x J x \ i '+ A R2 -F2 Sl. 5. Drugi korak, delovna faza Fig. 5. Second-step working phase X-X- Z določeno matriko stanja A 1 lahko pridemo do rešitve Riccatijeve enačbe. Njena rešitev P1 se nato uporabi za določitev ustrezne povratne matrike F1. S tem se faza branja konča. Sistem brez povratne zveze je spremenjen v sistem s povratno zvezo, prikazan na sliki 3. Ta postopek je treba izvesti le prvič, ko uporabimo tak tip krmilnika na sistemu, o katerem ne vemo ničesar, razen razsežnosti vektorja stanj. Če je matrika stanja AR (A R1) vsaj približno poznana, potem je pametneje uporabiti povratno matriko F1, ki ustreza tej matriki stanja in nadaljevati z drugim korakom. 2.2 Drugi korak Ko je matrika stanja določena (ali pa je uporabljena približna), je treba analizirati sistem s povratno zvezo, prikazan na sliki 4. Med fazo branja je uporabljena povratna matrika F iz prvega koraka. Dejanska matrika stanja je tekoča in ima zato indeks 2. Sistem opišemo z diferencialno enačbo: After the system matrix, AR1, is determined, the Riccatti equation can be solved. Its solution P1 is then used to find the appropriate feedback matrix, F1. At this moment the acquisition phase is ended and the working phase starts. The open-loop system is changed to the closed-loop system shown in Figure 3. But this procedure only needs to be implemented the first time we use this type of controller on a system we know nothing about, except the number of states. If the system matrix, AR ( A R1), is known approximately, then it is better to use the feedback matrix F1 associated with this system matrix and proceed immediately to the second step. 2.2 Second step Once the system matrix has been determined (or an approximate matrix has been used), the closed-loop system shown in Figure 4 should be analyzed. During the acquisition phase the feedback matrix F1 from the first step is used. The real system matrix, however, is current and therefore has index 2. The system is described by the differential equation: x&(t) = ( AR2 + BF1 )x(t) Krmilnik mora tako kot v prvem koraku prebrati x(0+), x(T), x(2T) in tako naprej. Nato sestavimo matriki X in X. Realno sistemsko matriko v drugem koraku AR2 lahko po končani fazi branja v drugem koraku določimo na podlagi naslednje enačbe: The controller has to read x(0+), x(TS), x(2TS) and so on, as in the first iteration. Then the matrices X and X& can be formed. The real system matrix in the second step, AR2, can then be determined after the acquisition phase of the second step is ended, according to the following equation: & XX-1 - BF Ko je nova povratna matrika F2 določena, se lahko začne delovna faza v drugem koraku. Končni blokovni diagram je prikazan na sliki 5. Drugi korak se nato iterativno ponavlja. After the new feedback matrix F2 is determined the working phase of the second step can begin. The associated block diagram is shown in Figure 5. The second step is then iteratively continued. Programljivi logični krmilniki - Programmable Logic Controllers 855 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 3 REŠEVANJE ARE Uporabljeni sta bili dve metodi reševanja ARE, in sicer: . Metoda z uporabo enačb Ljapunova, .Potterjeva metoda. 3.1 Metoda z uporabo enačb Ljapunova Ta metoda se je izkazala za dokaj počasno in zato ne bo opisana. Podroben opis je podan v [6] in [7]. 3.2 Potterjeva metoda Ta metoda je bila uporabljena za reševanje ARE. Sestavimo matriko M reda 2x2 [1]: M A -R Matriko R1 lahko zapišemo v obliki: 3 SOLUTION OF THE ARE Two methods for solving the ARE were considered: . a method using Lyapunov equations . Potter’s method 3.1 Method using Lyapunov equations This method turned out to be rather slow and will therefore not be described. A detailed description can be found in [6] and [7]. 3.2 Potter’s method This method was used in order to solve the ARE. Let us consider the following 2x2 matrix M [1]: -B-R-1-BT The matrix R1 can be expressed as: Če je matrična dvojica: . (A,B) ustaljiva in . (A,H) pregledljiva, R=H-HT If the matrix pair: .(A,B) is stabilizable . (A,H) is observable Im Im Im Im Re Re Re Re Sl. 6. Mogoče razporeditve lastnih vrednosti matrike reda 6x6 Fig. 6. Possible eigenvalue locations of a 6x6 matrix 856 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 potem se vse lastne vrednosti matrike M pojavljajo v dvojicah (Ai,-A). Nekaj mogočih primerov je prikazanih na sliki 6. Naj bo lastni vektor ali korenski vektor yi matrike M, ki ustreza lastni vrednosti Xi z negativno realno komponento, oblike: Potem se rešitev ARE dobi z [5]: yi P =[v ,L,v then all the eigenvalues of matrix Mr appear in pairs (li,-li). Some possible examples are shown in Figure 6. Let the eigenvector or generalized eigenvector yi of the matrix Mr corresponding to li, which has a negative real part, be: Then, the solution of the ARE is given by [5]: [] u ,L,u -1 3.2.1 Izračun lastnih vrednosti in lastnih vektorjev Da pridemo do rešitve ARE, je najprej treba dobiti lastne vektorje matrike M. Ta postopek se največkrat izvede v štirih korakih, in sicer: .z balansiranjem matrike [4], . s preoblikovanjem matrike v Hessenbergovo [3], . s QR algoritmom [3], . z izračunom lastnih vektorjev [3]. 4 ZGRADBA SISTEMA 4.1 Zgradba krmilnika Uporabljen je bil programljivi logični krmilnik SIEMENS S7-300 s funkcijskim modulom FM 356-4 in dodatnimi vhodno izhodnimi enotami (slika 7). 1. napajalnik / power supply 2. CPU 314 IFM 3. FM 356-4 funkcijski modul FM 356-4 function module 4. vhodno-izhodne enote interface modules 5. osebni računalnik PC 6. vmesnik za osebni računalnik PC adapter 7. spominska kartica memory card 8. programirna naprava programming device (Js 3.2.1 Calculation of the eigenvalues and eigenvectors In order to solve the ARE we must find the eigenvectors of the M. This procedure is usually done in four steps, by: r • matrix balancing [4], . reducing a matrix to the Hessenberg form [3], . performing the QR algorithm [3], . computing the eigenvectors [3]. 4 SYSTEM DESIGN 4.1 Controller design As an application of the introduced method the controller was built using the SIEMENS S7-300 programmable logic controller along with the FM 356-4 function module and several interface (input and output) modules (Figure 7). 3) (3) 1 7^ CD @ • j © S f (D / Sl. 7. Programljivi logični krmilnik z osebnim računalnikom Fig. 7. PLC with supporting PC Programljivi logični krmilniki - Programmable Logic Controllers 857 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 Spominska kartica EPROM se uporablja za vgradnjo in zagon operacijskega sistema ter za nalaganje uporabniškega programa. Za vgradnjo opravilnega sistema na spominsko kartico je potrebna posebna programirna naprava. Uporabniški program pa se lahko spreminja in shranjuje z uporabo osebnega računalnika in vmesnika zanj, ki je priključen na osrednjo procesno enoto (CPU). Program se naloži v CPU, ki je prek vodila (BUS) povezan s funkcijskim modulom (FM). 4.2 Modeliranje krmiljenega sistema A flash EPROM memory card is used to install and boot the operating system and to load the user software. A special programming device is needed to install the operating system on the memory card. The user software, on the other hand, can be modified and saved with the PC along with the appropriate PC adapter, which is connected to the CPU. The software is downloaded to the CPU, which is connected to the function module (FM) via a backplane BUS. 4.2 Controlled system modelling Za simulacijo krmiljenega objekta je bil A Meda 41TC iterative analogue computer uporabljen iterativni analogni računalnik Meda 41TC. was used to simulate the controlled system. 5 REZULTATI Kot primer je prikazan sistem z matrikama: 5 RESULTS As an example, a system with the following matrices: in začetnim pogojem: Matriki R1 in R2 v integralskem kriteriju sta: 114 231 425 x(0) 312 1 21 212 (3) and the initial condition: Odgovor prilagodljivega podoptimalnega krmilnika (tri komponente vektorja stanj) v prvem koraku za Ts=0,1 je prikazan na sliki 8. Dejanska matrika stanja A je bila na analognem računalniku namerno modelirana tako, da je sistem brez povratne zveze nestabilen. Zaradi tega je na sliki 8 jasno vidna faza branja. Rezultate lahko primerjamo z odgovorom linearnega podoptimalnega krmilnika, katerega povratna matrika F ustreza matriki AR (sl. 9). Očitno je, da je v primeru, ko je povratna matrika F v neprilagodljivem krmilniku točna, le-ta boljši od podoptimalnega. Do sprememb pa pride, če se matrika stanja A, za katero je izračunana povratna matrika F, le malo razlikuje od dejanske sistemske matrike AR. Recimo, da je dejanska matrika stanja AR enaka kakor v enačbi (3), povratna matrika neprilagodljivega krmilnika pa je izračunana za matriko A, definirano v enačbi (4). was used. The matrices R1 and R2 in the cost functional were: ; R2=[1] The response of an adaptive suboptimal controller (three components of the state vector) during the first step for Ts=0.1 is shown in Fig. 8. The real system matrix, AR, was deliberately modelled on an analogue computer in such a way that the open-loop system is unstable. Therefore, the acquisition phase can clearly be seen in Fig. 8. These results can be compared to the response of a linear suboptimal controller whose feedback matrix F is associated with the matrix AR (Fig. 9). It is obvious that an adaptive controller does not perform as well as a non-adaptive controller if the feedback matrix in the non-adaptive controller is accurate. The situation changes, however, if the system matrix A with which the feedback matrix of an non-adaptive controller is associated differs from the real system matrix, AR, only slightly. Let us say that the real system matrix, AR, is the same as in Eq. (3), but the non-adaptive controller ’s feedback matrix is associated with the matrix A defined in Eq. (4). 858 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 xi (V) \ 1 iti ~hh" 1\ ^ !"Txi f I \ | j~x^| f vvJ I / j ] j '"¦ čas / time (s) Sl. 8. Odgovor prilagodljivega linearnega podoptimalnega krmilnika Fig. 8. The response of an adaptive linear suboptimal controller xi (V) ________ l~x~l ¦VI i \ < \ \._____. 1 \ Ixl 'ft V .T-------- /VA4- - -\ * xTI _______X^_J^r: ¦^-—1------------------- čas / time (s) Sl. 9. Idealni odgovor neprilagodljivega linearnega podoptimalnega krmilnika Fig. 9. Ideal non-adaptive linear suboptimal controller response Odgovor neprilagodljivega krmilnika za ta primer je prikazan na sliki 10. Sklepamo lahko, da se obnašanje neprilagodljivega linearnega podoptimalnega krmilnika že pri majhnih spremembah parametrov sistema hitro poslabšuje. Kriterijski integral za vse tri primere (prilagodljivi podoptimalni (AS), idealni neprilagodljivi (IN), realni neprilagodljivi (RN)) je prikazan na sliki 11. 1,1 4,1 3,1 1,1 2 4,8 (4). The response of the non-adaptive linear controller for such a case is shown in Figure 10. We can conclude that with only a slight change in the system parameters the performance of the non-adaptive linear suboptimal controller deteriorates rapidly. The cost functionals for all three cases (adaptive suboptimal (AS), ideal non-adaptive (IN), real non-adaptive (RN)) are shown in Figure 11. 60 50 40 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 9 10 60 50 40 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 9 10 Programljivi logični krmilniki - Programmable Logic Controllers 859 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 xi (V) L ------- J ± _L i 1 ""-^ ---]---f--- ^ r r r \~T- ~™ 1 - — -1 1 —. 1 ----------—_ 1 ------- ------- ------- 01 23456789 10 čas / time (s) Sl. 10. Realni odgovor neprilagodljivega linearnega podoptimalnega krmilnika Fig. 10. Real non-adaptive linear suboptimal controller response 1 Kriterijski integral 1 l S | jf-~ i RN ------ )' ¦\- - r/ X s|aS1 Tf / -\- -----1------ —[- ------- ------ \ —r~A— ' ''' fINl / / čas / time (s) Sl. 11. Primerjava kriterijskih integralov Fig. 11. Comparison of the cost functionals Jasno je, da se obnašanje prilagodljivega linearnega podoptimalnega krmilnika spreminja v povezavi s časom vzorčenja T, še posebej, če je sistem brez povratne zveze nestabilen, kajti med fazo branja v prvem koraku le-te ni. Slika 12 prikazuje kriterijske funkcionale za različne čase vzorčenja (od T=0,025s do T=0,125s s korakom 0,025s). Jasno je, da je treba za izboljšano obnašanje zmanjšati čas vzorčenja TS. To je mogoče v teoriji in do neke mere tudi na analognem računalniku. V praksi pa lahko pride do problemov, ker se prebrani vektorji stanj zaradi It is to be expected that the performance of an adaptive linear suboptimal controller varies with sampling time, TS, especially if the open-loop system is unstable, because during the acquisition phase of the first step, there is no feedback. Figure 12 shows cost functionals depending on different sampling times (from TS=0.025s to TS=0.125s with a 0.025s step). It is obvious that in order to improve the performance of the controller we must decrease the sampling time, TS. This can be easily done in theory and to some extent on the analogue computer. In practice, on the other hand, problems may occur because the acquired 60 50 40 30 20 0 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 860 Podržaj P. - Kariž Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 Kriterijski integral i V i TS = 0,125 s — s*- i i i i i i i V^ i \ TS = 0,025 s čas / time (s) Sl. 12. Primerjava časov vzorčenja Fig. 12. Cost functional versus sampling time šumov ne izražajo v pravi dejanski sistemski matriki AR. Ti učinki se povečujejo, ko gremo s časom vzorčenja proti 0. 6 SKLEP Jasno so bile prikazane prednosti prilagodljivega linearnega podoptimalnega krmilnika pred neprilagodljivim. Te prednosti se povečujejo s povečanimi odstopanji v parametrih sistema. V tem primeru prilagoldjivi krmilnik ohrani svoje zmožnosti. Obnašanje neprilagodljivega krmilnika pa se hitro poslabša. Če matrike stanja ne poznamo, imamo med fazo branja v prvem koraku sistem brez povratne zveze. Obnašanje sistema je torej odvisno od trajanja te faze, le-to pa od časa vzorčenja. Nadaljnje raziskave je torej treba osredotočiti na določevanje optimalnega časa vzorčenja za različne jakosti šuma. state vectors during the acquisition phase do not result in the correct real system matrix, AR, due to the noise and other disturbances. These effects are magnified as the sampling time approaches 0. 6 CONCLUSION The advantages of an adaptive linear suboptimal controller over a non-adaptive controller have been clearly demonstrated. These advantages increase when deviations in the system parameters become larger. In this situation, the adaptive linear suboptimal controller maintains its capability. The performance of a non-adaptive controller, on the other hand, deteriorates rapidly. However, if the approximate system matrix is not known in advance, we are basically operating the open-loop system during the first-step acquisition phase. The overall performance is therefore dependent on the duration of this phase, which further depends on the sampling time. Further research should, therefore, focus on a determination of the optimal sampling time with regard to the presence of different levels of noise. 7LITERATURA 7REFERENCES [1] H. Kwaakernak, R. Sivan (1972) Linear optimal control systems, John Wiley & Sons Inc. [2] F. L. Lewis (1988) Optimal control, John Wiley & Sons Inc. [3] David M. Young, Robert Todd Gregory (1973) A survey of numerical mathematics, Addison—Wesley Publishing Company. [4] J. H. Wilkinson, C. Reinsch (1973) Die Grundlehren der mathematischen Wissenschaften: Linear Algebra, Springer Verlag. x 10 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Programljivi logični krmilniki - Programmable Logic Controllers 861 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)12, 852-862 [5] Katsuhisa Furuta, Akira Sano, Derek Atherton (1988) State variable methods in automatic control, John Wiley & Sons Inc. [6] E. J. Davison, F. T. Man (1968) The numerical solution of AQ+QA-C, IEEE Transactions on Automatic Control, August, str.: 448-449. [7] David L. Kleinman (1968) On an iterative technique for Riccati equation computations, IEEE Transactions on Automatic Control, February, str.: 114-115. [8] B.A.BeceKepcKHH, B.B.HapauueB (1987) CncreMH aBTOMaranecKoro ynpaBJiefflia c MHKpo3BM, «HayKa» [9] H.H.HBameHKO (1978) ABTOMaranecKoe perynnpoBafflie, «MauiuHocmpoeHue» Naslov avtorjev: dr. Primož Podržaj Authors’ Address: Dr. PrimožPodržaj doc. dr. Zoran Kariž Doc. Dr. Zoran Kariž Univerza v Ljubljani University of Ljubljana Fakulteta za strojništvo Faculty of Mechanical Eng. Aškerčeva 6 Aškerčeva 6 1000 Ljubljana 1000 Ljubljana, Slovenia primoz.podrzaj@fs.uni-lj.si primoz.podrzaj@fs.uni-lj.si zoran.kariz@fs.uni-lj.si zoran.kariz@fs.uni-lj.si Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 12.4.2005 25.10.2006 Received: Accepted: Open for discussion: 1 year 862 Podržaj P. - Kariž Z.