Matrix equations in multivariable control

Ma ix Equa ions in Mul i a iable Con ol ROMAN PROKOP, JIŘÍ KORBEL Tomas Ba a Uni e si y in Zlín Nám. T.G.M. 5555, 760 01 Zlín CZECH REPUBLIC [email p o ec ed] h p://www.u b.cz/ ai Abs ac : - The con ibu ion is ocused on a con ol design and simula ion o mul i inpu ou pu (MIMO) linea con inuous- ime sys ems. Sui able and e icien ools o desc ip ion and con olle de i a ion a e algeb aic no ions as ings, polynomial ma ices, and Diophan ine equa ions. The gene alized MIMO PI con olle design is s udied o s able and uns able sys ems. A uni ied app oach h ough ma ix Diophan ine equa ion can be applied in bo h cases. All s abilizing eedback con olle s a e ob ained ia solu ions o a ma ix Diophan ine equa ion. The me hodology allows de ining scala pa ame e s (one o mo e) o uning and in luencing o con olle pa ame e s. A Ma lab-Simulink p og am implemen a ion was de eloped o simula ion and e i ica ion o he s udied app oach. Illus a i e examples show he e ec i eness and lexibili y o he p oposed me hod o some simple MIMO sys ems. Key-Wo ds: - Polynomial ma ices, Diophan ine equa ions, Mul i a iable sys ems, S abiliza ion. 1 In oduc ion The s udy o mul i inpu –mul i ou pu (MIMO) sys ems has a ac ed scien i ic a en ion o decades. Analysis and con ol me hods and ools ha e been de eloped in many monog aphs (e.g. [1], [2], [4], [8], [12], [14]) as well as in jou nal and con e ence con ibu ions (e.g. [3], [9], [10], [16], [18]) o in p og am oolboxes, e.g. [17]. Mul i a iable sys ems ep esen an in e es ing esea ch ield also om ma hema ical poin o iew. Many no ions, me hods and ools o single inpu – ou pu (SISO) sys ems canno be simply and i ially gene alized in o mul i a iable cases. The main p oblem ela es o he ma ix non-commu a i e mul iplica ion. Howe e , many algeb aic no ions and ools can be success ully u ilized also in he non-commu a i e case. The main ool o con inuous- ime sys ems is he Laplace ans o m and b ie ly speaking, mul i a iable linea con inuous- ime sys ems a e desc ibed and exp essed by a se o linea di e en ial equa ions. So, scala polynomials desc ibing single inpu – ou pu linea sys ems a e eplaced by polynomial ma ices. Algeb aic no ions and modules emain a sui able and e ec i e ool o analysis and con ol design o MIMO sys ems. T ans e unc ions as a a io o wo polynomials a e in MIMO cases conside ed as ma ix ac ions and due o non- commu a i e ma ix mul iplica ion he denomina o can be in he le o igh side o he ma ix ac ion ([2], [8], [12]) in disc e e and con inuous- ime case. Also, a scala linea Diophan ine equa ion is gene alized in o a ma ix one, see e.g. [3], [14], [15]. The con ibu ion is scheduled as ollows. The basic no ions a e men ioned in sec ion II, he sys em desc ip ion o MIMO sys ems is in oduced in sec ion III. Sec ion IV deals wi h ma ix Diophan ine equa ions and he nex sec ion ou lines and summa izes a con ol design. Some i s o de examples and de i a ions a e p esen ed in sec ion VI. The p oposed me hodology b ings one o se e al scala which can une and in luence he con ol beha io in an easy way. Simula ions a e p esen ed in Sec ion VII, he las sec ion concludes he con en s o he con ibu ion. 2 Polynomial Ma ices Polynomial ma ices a e called l x m ma ices whe e all elemen s o ma ices a e polynomials in an inde e mina e s. This inde e mina e can be conside ed in linea sys ems as he Laplace ope a o and he se o polynomial ma ices is Rlm(s). I l = m, hen he se o polynomial ma ices cons i u es a non-commu a i e ing. A uni in his ing (an in e se elemen exis s in he ing) is a ma ix wi h eal nonze o de e minan and all uni s a e called unimodula . Gene ally, l  m se Rlm(s) is no mo e a ing. I A = BC hen B is a le di iso o A and A is a igh mul iple o B, while C is a igh di iso o A and A is a le mul iple o C. Simila ly, g ea es common le and igh di iso s a e in oduced. Two ma ices A, B a e le ( igh equi alen , i A = U1 B (A = B U2) wi h unimodula U1, U2. When A = U1 B WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 320 Volume 10, 2015 U2 hen A, B a e simply called equi alen . Ma ices wi h he same numbe o columns a e le cop ime i hei all le di iso s a e unimodula ma ices and ma ices wi h he same numbe o aw a e igh cop ime i hei all igh di iso s a e unimodula ones. The known ex ended (scala ) Euclidean algo i hm o can be gene alized in mul i a iable cases. A g ea es le common di iso G1(s) can be calcula ed o A, B wi h he same numbe o aw by 1 1 1 11 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) 0, A s P s B s Q s G s A s R s B s S s   (1) Mo eo e , L = AR1 = - BS1 is he leas common igh mul iple o A, B. A g ea es igh common di iso G2(s) can be calcula ed o A, B wi h he same numbe o columns by 2 2 2 22 ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) 0, P s A s Q s B s G s R s A s S s B s   (2) Also, L = R2A = - S2B is he le common mul iple o A, B. Rela ions (1), (2) a e he basic algeb aic no ions o Diophan ine equa ions, see e.g. [1], [3], [10]. 3 Sys em Desc ip ion A linea con inuous- ime mul i a iable (MIMO) sys em is desc ibed by a se o linea di e en ial equa ions and hen i can be easily exp essed by he Laplace ans o m echnique in he o m ( ) ( ) ( ) ( ),A s Y s B s U s (3) whe e ( ), ( )A s B s a e polynomial ma ices in he Laplace ans o m a iable s. Fo con ol design, i is use ul o cha ac e ize MIMO linea ime-in a ian sys ems in e ms o hei ans e unc ion ma ices. The gene aliza ion o single inpu -ou pu linea sys em o MIMO ones is e y simple in he s a e space desc ip ion ( ) ( ) ( ), ( ) ( ) ( ) x Fx u y Hx Lu     (4) The sys em wi h l inpu s ad m ou pu s in (4) has he s a e ec o x( ) wi h alues in Rn and eal ma ices F,  , H, L ha e dimensions (nxn), (nxl), (mxn), (mxm), espec i ely. Any decomposi ion 1 ( ) ( )G s H sI F L     (5) de ines a a ional sys em´s ans e unc ion ma ix. A ealiza ion (5) is minimal when he s a e ec o dimension n is as small as i can be and his alue is called he MacMillan deg ee and i ep esen s he o de o he sys em. So, G(s) in (5) is a a ional ma ix unc ion, i means ha all en ies a e a ional unc ions o s. This ma ix unc ion can be hen exp essed by he le o igh ma ix ac ion 11 ( ) ( ) ( ) ( ) ( ) RR G s A s B s B s A s   (6) whe e A, B, AR, BR a e polynomial ma ices, mo e de ails can be ound i.e. in [8]. No e, ha bo h ma ices A(s), AR(s) a e squa ed bu no necessa ily o he same dimension. In he case o sys ems wi h l inpu s and m ou pu s, he le denomina o A(s) has dimension lxl, while he igh denomina o AR(s) has he dimension mxm. Howe e , bo h ma ices a e associa es and he cha ac e is ic polynomial ollowing om he s a e-space desc ip ion (4) is also associa es. I means ha all oo s o he men ioned polynomials a e same. I means de ( ) de ( ) de ( ) R A s A s sI F (7) whe e F is he squa ed sys em ma ix in (4). Wi h ela ion (3) he no ion o s abili y is closely connec ed. A linea sys em is asymp o ic (in e nal s able), i all de e minan s in (3) a e s able, o con inuous- ime sys ems i means ha all oo s lie in he open le hal o he complex plane. 4 Ma ix Diophan ine Equa ions Diophan ine equa ions de ined in commu a i e ings a e linea equa ions o he o m ,ax by c (8) whe e a, b, c a e known gi en en ies and x, y a e unknown ones in he ing. I is well known (see e.g. [1], [2], [8]) ha equa ion (8) has a solu ion i and only i he g ea es common di iso o a, b di ides c, b ie ly gcd(a,b) / c. Mo eo e , i x0, y0 is a pai o pa icula solu ions o (8) , hen all x, y gi en by 00 00 , , x x b y y a   (9) WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 321 Volume 10, 2015 whe e is an a bi a y elemen o he ing and a0=a/gcd(a,b), b0=b/gcd(a,b).Then, wi hou loss o gene ali y, equa ion (8) can be supposed wi h cop ime a, b and he solu ion o (8) exis s o any c. The si ua ion is mo e complex in non- commu a i e ings, such is a se o polynomial ma ices. Due o he non-commu a i i y o ma ix mul iplica ion, equa ion (8) is spli in o h ee kinds o linea ma ix equa ions o e he ing. A na u al gene aliza ion o his equa ion is ei he he equa ion 1 1 1,A X BY C (10) o he equa ion 2 2 2,XA YB C (11) Bo h equa ions a e called unila e al ones. The las equa ion is called a bila e al one and i has he o m 3 3 3,A X YB C (12) In he case o equa ion (10), ma ices (A1, B1, C1) ha e he same numbe o ows, while in equa ion (11) ma ices in he iple (A2, B2, C2) ha e he same numbe o columns. The sol abili y o equa ions (10) – (12) is s udied e.g. in [1], [2], [8]. The esul s can be b ie ly o mula ed in he enginee ing pa lance as ollows: a) Equa ion (10) has a solu ion i and only i he g ea es le common di iso o ma ices A, B is a le di iso o C. b) Equa ion (11) has a solu ion i and only i he g ea es common igh di iso o ma ices A, B is a igh di iso o C. c) Equa ion (12) has a solu ion i and only i he ma ices 0, 00 A A C BB             (13) a e equi alen . This case is ou o he in e es o his con ibu ion and some de ails can be ound in [2]. I a pa icula solu ion o a gi en linea Diophan ine equa ion exis s, he e exis a se o all solu ions. In he case o (10), (11) he se s o solu ions a e gi en 0 1 0 1 ,,X X BT Y Y AT    (14) whe e X0, Y0 a e pa icula solu ions o (10) and T is an a bi a y polynomial ma ix o he app op ia e dimension and 1 1 1 1.AB B A (15) Solu ions o (11) a e 0 2 0 2 ,,X X TB Y Y TA    (16) and again X0, Y0 a e pa icula solu ions o (11), T is an a bi a y polynomial ma ix o he app op ia e dimension and 2 2 2 2.B A A B (17) Rela ions (15), (17) a e no hing else han he opposi e ma ix ac ion. Sui able and con enien ools o he solu ion o linea ma ix equa ions a e o e ed by a Polynomial oolbox [17] which con ains a se o use iendly Ma lab unc ions o a ious con ol sys em pu poses. As a simple example sol e equa ion (10) o ma ices 1 1 1 1 2 1.5 1 0 ,, 3 2 2 0 1 s A B C ss                        (18) Ma lab unc ion AXBYC in Polynomial oolbox gi es he pa icula solu ion   0 0 0 0.33 , 2 0.67 0.67 2 0.67 0.67 Xs Ys        (19) All solu ion a e gi en in he o m (14) wi h 11 2 1 2 0.94 0.47 ,. 3 2 2.4 2.8 0.94 ss AB s s s                  (20) The ee polynomial ma ix T has he o m   12 ( ) ( )T s s (21) wi h 1(s), 2(s) a bi a y polynomials. Really, he p oduc 1 1 1 1 AB B A gi es he same esul in he o m WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 322 Volume 10, 2015 2 23 5.7 4.2 1.4 7.5 1.9 4.7 0.94 ss s s s        (22) which con i ms equa ion (15). 5 Con ol Design The mos equen scheme o a basic eedback con ol sys em is depic ed in Fig. 1. All signals in he MIMO case a e ec o ones. Inpu signals o he eedback sys em in Fig. 1 is a e e ence (se poin ) signal w = Fw-1(s) Gw(s) and a load dis u bance signal d = Fd-1(s) Gd(s) de ined by hei ma ix le ma ix ac ions. Fig. 1 one deg ee o eedom (1DOF) con ol sys em All s abilizing con olle s o he 1DOF eedback sys em in Fig. 1 a e gi en by any solu ion o ma ix Diophan ine equa ion ( ) ( ) ( ) ( ) ( ), RR A s P s B s Q s M s (23) whe e P-1(s)Q(s) = QR(s) PR-1(s) is a le and igh ma ix ac ion o he con olle C and A-1(s)B(s) = BR(s)AR-1(s) is a le and igh ma ix ac ion o he con olled plan G. Mo e de ails ca be ound e.g. in [1], [5], [12], [15], [16]. Howe e , o asymp o ic acking and dis u bance ejec ion mus be ul illed u he condi ions. B ie ly speaking, denomina o o he con olle mus be di isible by he denomina o s o inpu signals. I is a eason o a p e-compensa o F in Fig. 2 which ep esen s he condi ions o di isibili y. In he case o asymp o ic acking only, i is F=Fw. In he case o simul aneous asymp o ic acking and dis u bance ejec ion F=FwFd. The basic s abili y and asymp o ic acking in he sense o Fig. 2 is hen he con olle QR(s)PR-1(s) gi en by he solu ion o ma ix Diophan ine equa ion ( ) ( ) ( ) ( ) ( ) ( ), RR A s F s P s B s Q s M s (24) whe e M(s) is a s able polynomial ma ix wi h p esc ibed poles o i s de e minan . Resul ing ma ices PR, QR ep esen he igh ma ix ac ion 11 ( ) ( ) ( ) ( ) RR P s Q s Q s P s   (25) Fig. 2 eedback 1DOFcon ol sys em wi h p e- compensa o The con ol law is hen go e ned by he equa ion 1 ( ) ( ) ( ) ( )( ( ) ( )),P s F s U s Q s W s Y s  (26) which can be easily ew i en in o di e en ial equa ions. Now, i is necessa y o p opose he me hod o solu ion o ma ix equa ion (2). Fo simple cases, he solu ion can be ound by means o elemen a y column ope a ion, acco ding o he scheme elemen a y column ope a ions  (27) Elemen a y column ope a ions (27) may always be lead in he way ha he polynomial ma ix PR(s) emains as uni ma ix and he con e sion (25) is i ial and also a uni one. Then no in e sion in (26) is necessa y and he ealiza ion o he con ol law is e y simple. In mo e complex cases, he s anda d echniques based on Euclidean algo i hms can be used, see [2], [8], [15]. Mo e complex ma ix polynomial equa ions can be con enien ly sol ed by Polynomial oolbox [17] as i is shown in Sec ion IV. 6 Illus a i e Examples Illus a i e examples 1 - 3 in his con ibu ion a e i s o de s able, uns able and in eg a ing ones wo inpu – wo ou pu (TITO) sys ems a e ep esen ed by he ma ix equa ion 1 2 1 2 11 3 4 3 4 22 ( ) ( ) ( ) ( ) s a a b b Y s U s a s a b b Y s U s                           (28) The s abiliza ion ma ix Diophan ine equa ion (24) akes he o m 1 2 0 R R M PZ QZ      0 0 AF B I I      WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 323 Volume 10, 2015 1 2 1 2 1 2 3 4 3 4 3 4 2 1 0 5 4 0 2 3 2 7 6 0 0 0 ( ) 0 0 ( ) s a a p p b b s a s a p p b b s q s q q s q sm q s q q s q sm                                  (29) Example 1: Le a TITO linea con inuous- ime sys em be exp essed by he Laplace ans o m echnique in he o m 1 1 2 1 2 2 2 1 1 2 '( ) 2 ( ) 0.8 ( ) 5 ( ) 6 ( ) '( ) 1.5 ( ) 0.6 ( ) 2 ( ) 3 ( ) y y y u u y y y u u         (30) The Laplace ans o m o equa ions (6) gi es ma ices A, B 2 0.8 5 6 ( ) , ( ) 0.6 0.6 2 3 s A s B s s                (31) The sys em desc ibed in (31) is e iden ly s able because de A = s2 + 2.6s + 0.72 is a s able polynomial. Then he scheme (27) can be applied and he esul is in he o m o gene alized PI con olle : 1 1 1 0 1 5 2 4 2 2 3 1 2 1 7 2 6 2 ( ) ( ) ( ) ( ) u q e q e d q e q e d u q e q e d q e q e d                   (32) whe e con olle pa ame e s we e ob ained by elemen a y column ope a ions acco ding scheme (5) in he o m: 10 2 00 30 2 20 2 0.8 41 33 2 3 qm qm qm qm       50 2 40 70 2 60 4 2.2 2 10 5.9 33 5 3 qm qm qm qm       (33) In (9) ei a e na u ally acking e o s and m0>0 is a uning pa ame e in luencing con ol beha iou . Example 2: Le an uns able TITO linea con inuous- ime sys em can be exp essed by di e en ial equa ions 1 1 2 1 2 2 2 1 1 2 '( ) ( ) ( ) ( ) 0.5 ( ) '( ) 0.5 ( ) 2 ( ) 0.8 ( ) 2 ( ) y y y u u y y y u u         (34) and he ma ix exp ession has he o m 11 22 ( ) ( ) 1 1 1 0.5 ( ) ( ) 2 0.5 0.8 2 Y s U s s Y s U s s                           (35) Ma ix equa ion (27) gi es he con olle ma ices PR, QR 1 0 5 4 3 2 7 6 10 01 RR q s q q s q P and Q q s q q s q          (36) whe e pa ame e s a e 10 2 00 30 2 20 2.5 0.65 1.25 0.75 0.5 qm qm qm qm       50 2 40 70 2 60 0.6 1.1 0.3 1.25 0.19 0.625 qm qm qm qm       (37) The o m o he con ol law (32) is again a gene alized PI con olle . Example 3: Le an in eg a ing (also uns able) TITO linea con inuous- ime sys em can be exp essed by di e en ial equa ions 1 2 1 2 2 1 1 2 '( ) ( ) ( ) 0.5 ( ) '( ) 0.5 ( ) 0.6 ( ) 1.5 ( ) y y u u y y u u       (38) De e minan o A(s) = s2 – 0.5 is e iden ly an uns able one. The con olle is de i ed in a simila way bu a he igh hand side o (29) is he s able ma ix M(s) in he o m 2 112 2 2 ( ) 0 ( ) , , 0 0 ( ) sm M s m m sm       (39) The choice o di e en mi > 0 gi es he possibili y o di e en dynamics in bo h con olled ou pu s. The con ol law is again in he o m o (32) wi h he ollowing se o pa ame e s qi: WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 324 Volume 10, 2015 11 2 01 31 2 21 2.5 0.21 1.25 0.42 0.5 qm qm qm qm       52 2 42 72 2 22 0.83 1.25 0.42 0.5 1.67 0.83 qm qm qm qm       (40) The las example ep esen s a simple asymme ical case, a wo inpu – one ou pu sys em o he i s o de gi es also a kind o a gene alized PI con olle . Example 4: A con olled sys em is a wo inpu – single ou pu (TISO) sys em desc ibed by he di e en ial equa ion 1 1 1 2 '( ) 0.5 ( ) 0.5 ( ) 1.2 ( )y y u u    (41) Sys em (41) is e iden ly an uns able one. The ini ial and inal s a e o he scheme o s abilizing equa ion (24) is and he con ol law akes he o m o wo equa ions which can be also conside ed as a gene alized PI con olle 1 0 1 2 2 0 1 ( ) (4 1) ( ) ( ) 0.833 ( ) u m e u m e d    (42) An impo an ema k is ha he e exis an in ini e numbe o easible s abilizing con olle s. I depends how o choose elemen a y column ope a ions in educ ion (27). Con ol law (42) ep esen s p opo ional con olle in u1( ) and an in eg a ing one in he u2( ) con ol loop. Polynomial oolbox [17] gi es a simila solu ion bu no necessa ily he same one. 7 Simula ion Resul s Ma lab and Simulink o e a sui able en i onmen o modelling and simula ion o dynamic sys ems. The Simulink scheme o wo inpu – wo ou pu uns able sys em (7) wi h con olle (10) is depic ed in Fig. 3. Con ol esponses o s able TITO sys em (Example 1) o uning pa ame e m0=1.5 and m0=3 a e shown in Fig. 4. The con ol esponses o he uns able TITO sys em (example 2) o uning pa ame e m0=1.5 and m0=3 a e shown in Fig. 5. Fig. 3 simulink scheme o eedback uns able sys em Examples 1 and 2 illus a e ha uning pa ame e m0 > 0 in luences he dynamical beha io o he con olled a iable in he s able as well as in he uns able case. The pa ame e m0 > 0 ep esen s a mul iple pole o he eedback cha ac e is ic polynomial. Fig. 4 con ol esponses o m0=1.5 and m0=3 (Example 1) 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 Time (s) w, y w1 y1 (m=1.5) y2 (m=1.5) w2 y1 (m=3) y2 (m=3) 22 200 0 2 0 2 0.5 0.5 1.2 0.5 1.2 1 1 0 0 0 0 (4 1) 0 1 0 1 0 0 0 1 0 1 0.833 s m m ss ms m                 WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 325 Volume 10, 2015 Fig. 5 con ol esponses o m0=1.5 and m0=3 (Example 2) In some cases, i can be use ul e e y con olled a iable in luence in di e en dynamics. I is easily ob ained by a di e en choice o poles in eedback loops. The si ua ion is shown in Fig. 6 and Fig. 7 o TITO in eg a ing sys em. While he esponse in Fig. 6 is o m1 = m2 = 1, he esponses in Fig. 7 a e o he choice m1 = 1.5, m2 = 2. Fig. 8 and Fig. 9 illus a e con ol esponses o he wo inpu – single ou pu sys em sol ed in Example 4. Tuning pa ame e m0 > 0 again in luences he con ol beha iou and dynamics. Fig. 6 con ol esponses (Example 3) o in eg a ing sys em o uning pa ame e s m1 = 1, m2 = 1. Fig. 7 con ol esponses (Example 3) o in eg a ing sys em o uning pa ame e s m1 = 1.5, m2 = 2. Fig. 8 con ol esponses (Example 4) o TISO sys em o uning pa ame e s m0=0.5. Fig. 9 con ol esponses (Example 4) o TISO sys em o uning pa ame e s m0=1. 0 5 10 15 20 25 30 35 40 -0.5 0 0.5 1 1.5 2 2.5 Time (s) w, y w1 y1 (m=1.5) y2 (m=1.5) w2 y1 (m=3) y2 (m=3) 0 5 10 15 20 25 30 35 40 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Time (s) w, y, u w1 y1 y2 w2 u1 u2 0 5 10 15 20 25 30 35 40 -2 -1 0 1 2 3 4 5 Time (s) w, y, u w1 y1 y2 w2 u1 u2 0 5 10 15 20 25 30 35 40 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Time (s) w, y, u w y u1 u2 0 5 10 15 20 25 30 35 40 -2 -1 0 1 2 3 4 5 Time (s) w, y, u w y u1 u2 WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 326 Volume 10, 2015 8 Conclusion The pape deals wi h mul i a iable con ol o simple con inuous- ime linea sys ems. The con olle design is pe o med h ough a solu ion o a ma ix Diophan ine equa ion. This app oach enables o de ine one o a couple scala uning pa ame e s o in luencing o con ol beha iou . The uning pa ame e s ep esen poles o he cha ac e is ic eedback equa ion. In he i s o de cases, he solu ion and a inal con olle can be ob ained in simple and explici o m pe o ming by elemen a y column ope a ion o he gi en ma ices. Resul ing con olle s hen a e o gene alized PI con olle s. All simula ions and esul s a e clea ly demons a ed in he Ma lab-Simulink en i onmen . Acknowledgemen This wo k was suppo ed by he Eu opean Regional De elopmen Fund unde he p ojec CEBIA-Tech No. CZ.11.05./2.1.00/03.0089. Re e ences: [1] Kuče a, V. Disc e e Linea Con ol. The Polynomial Equa ion App oach. P ague: Academia, 1979. [2] Kuče a, V. Analysis and Design o Disc e e Linea Con ol sys ems. P ague: Academia, 1991. [3] Kuče a, V. "Diophan ine equa ions in con ol - A su ey," Au oma ica, Vol. 29, (1993), pp. 1361-75. [4] Kaczo ek, T. (1985). Two-dimensional Linea Sys ems. Sp inge -Ve lag. Be lin. [5] M.J.G imble and V. Kuče a, Polynomial Me hods o Con ol Sys ems Design. London:Sp inge , 1996. [6] A. O´Dwye , Handbook o PI and PID con olle uning ules. London: Impe ial College P ess, 2003. [7] R. Ma ušů and R. P okop, Va ious App oaches o Sol ing an Indus ially Mo i a ed Con ol P oblem: So wa e Implemen a ion and Simula ion. In e na ional Jou nal o Ma hema ics and Compu e s in Simula ions, 2012, oč. 6, č. 1, s. 161-168. ISSN 1998-0159. [8] T. Kaila h, Linea sys ems, P en ice Hall 1980. [9] R. P okop, N. Volko a, Z. P okopo á, Uns able sys ems- acking and dis u bance a enua ion. WSEAS T ansac ions on Sys ems and Con ol, 2011, oč. 6, č. 3, s. 69-78. ISSN 1991-8763. [10] R. P okop, N. Volko a, Z. P okopo á, T acking and dis u bance a enua ion o uns able sys ems: An Algeb aic App oach, In: Recen Resea ch in Social Science, Digi al Con e gence, Manu ac u ing &Tou ism. WSEAS P ess, 2011, p. 161-166. [11] R. P okop and J. Ko bel, Mul i a iable Con ol o Uns able sys ems - A Ma ix Diophan ine Equa ions, in P oc. 18 h In . Con . on Sys ems (CSCC´14), San o ini, G eece, p. 438-442, 2014. [12] Rosenwasse , E.N. and B.P. Lampe, Mul i a iable Compu e -con olled Sys ems. Sp inge - e lag, Be lin, 2006. [13] K.J. ?s öm and R.M. Mu ay, Feedback Sys ems. Resea ch T iangle Pa k, NC: Ins umen al Socie y o Ame ica, 1995. [14] S. Skoges ad and I. Pos le hwai e: Mul i a iable Feedback Con ol. Analysis and Design. John Wiley and Sons, 1997. [15] Vidyasaga , M. Con ol sys em syn hesis: a ac o iza ion app oach. MIT P ess, Camb idge, M.A., 1987. [16] Volko a, N. and R. P okop, R. "Ma ix equa ion app oach o MIMO con ol design," in P oc. DAAAM Con ., p.225-226, 2011. [17] PolyX, L d. Polynomial Toolbox. 1998. [18] M. Kubalčík and V. Bobál, Compu a ion o P edic ions in Mul i a iable P edic i e Con ol, in P oc. 13 h In . Con . on Au oma ic Con ol, Modelling and Simula ion(ACMOS), pp.15-20, 2011. WSEAS TRANSACTIONS on SYSTEMS and CONTROL Roman P okop, Jiří Ko bel E-ISSN: 2224-2856 327 Volume 10, 2015