MODELLING OF DYNAMICAL SYSTEMS

Transcription

1 IV. Éflyam 4. zám december Szablc Róbert [email protected] MODEING OF DYNAMICA SYSTEMS Abztrakt/Abtract A zerző célja, hgy kmplex módzereket mtan be a dnamka rendzerek mdellezéére. Jelen írá az elektrm, mechanka é elektr-mechank rendzerekre, agy a mechatrnka rendzerekre fókzál. A matematka mdellezé a legfntabb fáz az atmatk rendzerek elemzéében é azk előzete terezéében. Szerző a zámítógépe elemzé é terezé területere kncentrál, amely rán zám példát mtat be a mdellezé é rányítá prblémák köréből. Prpe f the athr t ge a cmplex et f methd appled fr mdelng f the dynamcal ytem. The mre attentn pad fr electrcal, mechancal, and electr-mechancal ytem,.e. fr mechatrncal ytem. Mathematcal mdelng the mt mprtant phae n atmatc ytem analy, and prelmnary degn. Athr f the paper dete attentn t cmpter aded analy and degn. Example fr th are taken frm the de branch f mdelng and cntrl prblem. Klczaak/Keyrd: dnamk rendzerek, rányítá, mdellezé ~ dynamcal ytem, cntrl, mdelng I. INTRODCTION Mdelng f the dynamcal ytem n the fc f attentn f centt nce many decade. Tday there a man mtatn t hae neceary nfrmatn fr atmated cntrl f the dynamcal ytem. Dynamcal ytem can be techncal e.g. electrcal ytem, mechancal ytem etc., blgcal e.g. bld prere cntrl, nln cntrl etc. ne, and al they can repreent many ther branche f ecnmc, cety, cence etc. Dynamcal mdelng neceary fr cmpter aded prelmnary degn, t. There are many perfl tl t degn a cntrl ytem fr the frt pble cheme. The degned ytem mt be teted fr degn crtera, and n cae f necety mt be re-degned. Prpe f the athr t preent a typcal et f pble dynamcal ytem appled n Mechatrnc, and, n Rbtc, t. There are many fam clacal example f 34

2 Mechanc, thery f electrcty, Electrtechnc beng nled n th paper t h h t get a cmplex et f dynamcal charactertc f them. Sltn f the analy tak pprted by MATAB, by SIMINK ftare, and by ther tlbxe. II. BRIEF HISTORY & ITERATRE OVERVIEW Mathematcal backgrnd fr mdelng f determntc dynamcal ytem are gen detaled n [, 3], and tchatc ytem are dced n [1]. The thery f the cntrl ytem bth fr SISO and MIMO applcatn are tlned n [4, 5, 7, 8, 9]. In general, ytem and gnal are netgated by Pkrád,. [1]. Mathematcal mdel f the trblent ar are dced n [6, 10, 11, 14]. Cmpter aded mlatn pprted by MATAB [13]. III. MODEING DYNAMICA SYSTEMS 3.1. MODEING OF THE MASS-SPRING-DAMPER MECHANICA SYSTEM. The dynamcal mdel f the ma-prng-damper mechancal ytem can be een n Fgre 3.1. [, 5, 7]. Fgre 3.1. The ketch f the ma-prng-damper mechancal ytem. Eqatn f mtn f the free bratn ytem can be dered a [, 5]: d m dt F. 3.1 Frm Fgre 3.1. t ealy can be een that the reltng dampng frce a fll [, 3, 5]: dx F kx. 3. dt In Eq 3.: kx the prng frce ppng t tranlatnal mtn f the cntant ma, m; dx the c damper all frctn frce. Reltng eqatn f mtn f the dt mechancal ytem a gen bel: d x dx m kx dt dt 343

3 The ltn f the ecnd rder dfferental eqatn 3.3 th cntant parameter gen n [, 3] t be a fll: 1t t x t Ae Be, 3.4 here A and B are cntant dered by the ntal cndtn,, and are ltn f the 1 charactertc plynmal f Slng eq 3.5 yeld t the fllng rt []: m k k k 1 ;. 3.6 m m m m m m et ppe that 0. Fr frther dcn e ppe that A >4 k m. In th partclar cae ltn f the eq 3.5 are real and negate ne,.e. et ppe ytem parameter t be a fll: 1 <0, <0, > Cae a A ; B ; 1;. 1 1 Cae b A ; B ; 1;. 1 1 Cae c A ; B ; 1; Fr ytem defned by eqatn cmpter aded mlatn a dne, and the relt can be een n Fgre 3., Fgre 3.3, and fnally, Fgre

4 Fgre 3.. Cae a A ; B ; 1; 1 1 Fgre 3.3. Cae b A ; B ; 1;

5 Fgre 3.4. Cae c A ; B ; 1; 1 1 Frm Fgre t ealy can be dered that the mechancal ytem behae aperdcally hang expnental repne. B <4 k m. In th partclar cae ltn f the charactertc plynmal 3.5 are cmplex cnjgate ne hang negate real part. It ell-knn that ltn f eq 3.4 can be dered a [, 5]: x t C e c k m m t D e n t t m m k m m t In eq 3.11 C and D are nty cntant parameter dependng n ntal cndtn. et the mechancal ytem ha parameter a gen bel: k 1;, m 1kg 3.1 m m Sbtttng eqatn 3.1 nt eqatn 3.11 yeld t the next frmla: t t e n 1,75 t 0,5 t 0,5 x t e c 1, Relt f the cmpter mlatn f eqatn 3.13 can be een n Fgre

6 Fgre 3.5. Dynamcal Sytem Repne C=1, D=1 3.. MODEING OF THE MASS-SPRING MECHANICA SYSTEM CONSTRAINED TO SINSOIDA INPT SIGNA The ma-prng mechancal ytem can be een n Fgre 3.6, here l the prng length n the teady-tate ptn; x the ncreae f the prng length, m the ma f bdy. Fr ndamped mechancal ytem eqatn f mtn can be dered a fll [, 5]: m d x kx a n t 3.14 dt Frm Fgre 3.6. crdnate f the ma meared frm t bac leel ealy can be determned a: n t l x 3.15 a ng Netn Secnd a dynamc eqatn 3.14 can be rertten a: r n ther manner: d m a n t l x kx 0, 3.16 dt d x m kx ma n t F n t 3.17 dt 347

7 Fgre 3.6. The ma-prng ytem. Sltn f the dynamcal eqatn 3.14 can be dered a m f the ltn f the hmgene eqatn f the lnear ytem 3.14, and a partclar ltn f the nhmgene lnear eqatn et fnd partclar ltn n the frm f the next frmla: Sbtttng eqatn 3.18 nt eqatn 3.17 relt n x C n t mc kc Frm eqatn 3.19 e hae F C. 3.0 k m k et dente the renance peak freqency f the mechancal ytem beng m netgated. Th, partclar ltn 3.18 can be rertten a: F n t F n t x 3.1 m k 1 Sltn f the dynamcal eqatn can be fnd a F F n t x Ac t Bn t, 3. m here cntant A, and B can be fnd ng ntal cndtn. et fr t 0 ntal cndtn are a fll: dx x 0 0; dt t0 Fnally, reltng eqatn f 3. can be dered a [, 5]: n t F n t x n t n t 3.4 m Dynamcal ytem defned by eqatn 3.4 a cntraned t cmpter mlatn. Relt can be een n Fgre

8 k= k=10 k=0 k=30 Fgre 3.7. Ma-prng ytem dplacement Fr partclar cae f partclar ltn f 3.1 de nt ext, and ntead f e try t fnd th ltn n the next frm: x C t c t 3.5 Sbtttng eq 3.5 nt eqatn 3.14 cntant C can be dered a Sppe that Cm n t Cmt c t kctc t F n t 3.6 m k, eqatn 3.6 can be mplfed t that f F C 3.7 m Th, fnal frm f the ltn f eqatn 3.14 can be dered a: F x Ac t Bn t t c t 3.8 m The prng-ma ytem ften pplemented th c damper,.e. eqatn f mtn 3.14 can be re-rtten a fll: m d x dx kx F n t dt dt 349

9 et fnd partclar ltn f the nhmgene eqatn f 3.19 n the fllng manner: x a n t b c t. 3.0 Cntant a and b can be fnd th btttn f eq 3.0 nt 3.19, and t yeld: k m a b F, k m b a Slng ytem f eqatn 3.1, and 3. fr ceffcent a and b, and btttng them nt eqatn 3.0 ge the fllng frmla: k m x F n t F0 c k m k m 0 t. 3.3 It ell-knn that fnal ltn f eqatn 3.1 can be dered a m f eqatn 3.3, and 3.4. Frm 3.4 t edent that 3.4 part ge t zer hle tme ge t nfnty,.e. behar f the dynamcal ytem determned by eq 3.3. IV. MATHEMATICA MODES OF THE STOCHASTIC CONTINOS ATMOSPHERIC DISTRBANCES There are t perfl mathematcal mdel f the cntn gt repreentatn. The frt, the -called n Kármán pectrm, hch better ft regtratn f the trblent ar recrd. The n Kármán per pectral denty PSD fnctn gen bel a fll [1, 6, 10, 11]: 8 1 1,339 3 Kármán, / 6 1 1,339 here [m] the gt aelength, 1 0 [rad/m] patal freqency, [rad/] the bered anglar freqency, and fnally, [m/] r.m.. gt elcty. The ecnd ne, the mre fared PSD fnctn the Dryden PSD fnctn, hch can be prgrammed mre ealy then the n Kármán-mdel. If there n trctral analy perfrmed the e f Dryden PSD fnctn permble. The Dryden PSD fnctn can be defned a gen bel [1, 6, 10, 11, 14]: 1 3 Dryden Hang gal t analyze hypthetcal arcraft mathematcal mdel th n nteret n netgatn f the trctral behar and ppng arcraft t be rgd ne, the mplet mathematcal frm f the PSD fnctn defned by eqatn f 4. e ll e n th artcle. Regardng bac reference f [6, 10, 11, 14] ne can defne PSD fnctn f the cmpnent peed f the trblent ar alng bdy ax ytem f the arcraft,.e.: g g

10 g 4.5 here r d,, 0. Snce frmla f may be rertten a fll: / 1 1 g, 4.6 / 1 / 3 1 g, 4.7 / 1 / 3 1 g. 4.8 Fr generatng randm gnal th the reqred ntenty, cale length, and PSD fnctn fr gen peed and heght f the flght, a hypthetcal de-band ne generatr th PSD fnctn f N mt be ed t prde gnal th the lnear flter, chen ch that t ha an apprprate freqency repne that the tpt gnal frm the lnear flter ll hae a PSD fnctn f ee Fgre 4.1. [6]: N j N j G G G. 4.9 If the hte ne rce chen that t per pectrm mlar t that f called hte ne ne can rte that 1 N Fgre 4.1. Blck Dagram fr Generatng Stchatc Sgnal. Sbtttng eqatn 4.10 nt eqatn f 4.9 relt the fllng frmla j N j G G G The lnear flter tranfer fnctn f G are gen n [6] t be: g K G, g K G, g K G, 4.1 here: K, 3 K, 3 K, , 3, 4.14,, It ealy can be dered that btttn eqatn nt eqatn 4.9 relt n the PSD fnctn f the Dryden-mdel PSD-fnctn f If the ar

11 trblence mdel ed fr analy f t effect n flght f the mall AV arcraft let the ntal parameter be a they are gen bel: H 100 m 38,084 feet; 0 5 m / 90 km / h Frm eqatn t edent that fr deratn f tranfer fnctn f the lnear flter defned by eqatn 4.1 t neceary t kn trblence cale f, and trblence ntenty f, meared alng apprprate ax f the gen crdnate ytem. et cnder NASA-parameter taken frm [6, 10] t be a fll: alng lngtdnal OX ax: 3,4 m / 0,85 m / 4.17 alng lateral OY ax:,8 m / 0,7 m /, 4.18 alng ertcal OZ ax: 1,8 m / 0,45 m / Fr extreme eather cndtn thndertrm MCEAN [6] gget trblence ntente a they gen bel: 7 m/. 4.0 Trblence ntegral cale length f the l alttde trblence mdel fr 10 feet h 1000 feet can be dered ng fllng frmla [6, 10]: h, 1, 0,5 h ,177 0,00083 h Regardng MCEAN, fr extreme eather cndtn thndertrm ne can apply fllng ntegral cale length gen n [6]: 580 m. 4. Cntant peed cmpnent f the trblent ar are gen n mltary tandard f [4, 7] a fnctn f ther exceedance. Fr the l alttde randm trblence mdel ntenty f the trblence, can be meared a [6, 11, 14]: 0,1, here 0 cntant lngtdnal cmpnent peed f the trblent ar meared at the alttde f h 0 feet. ng eqatn f ntegral cale length f the ar trblence ere fnd and they are mmarzed n Table 1. Table 1. Integral cale length at alttde f H 100 m 38,084 feet. Scale length, [m] Nmnal Nm Extreme Thndertrm 86, feet 6, m 580 0,5 431, feet 131, m ng eqatn f trblence ntente ere fnd and they are mmarzed n Table. 1 ft 0,3048 m 1 m 3,8084 feet 35

12 Table. Trblence ntente. Trblence ntente NASA-Mn Mn NASA-Max Max Extreme Thndertrm, [m/] 0,85 3,4 7, [m/] 0,7,7 7, [m/] 0,45 1,8 7 Cntant lngtdnal cmpnent peed f the trblent ar, called 0, ere fnd ng mltary tandard f [10], and ng eqatn f Cntant peed f 0 are mmarzed n Table 3. Table 3. Cntant peed f 0. Trblent Ar Charactertc NASA-Mn Mn NASA-Max Max Extreme Thndertrm 0,1, [m/] 0,45 1, , [m/] [km/h] 4,5 16, 18 64, near tranfer fnctn defned by eqatn 4.1 hang parameter gen by eqatn f , and atfyng cndtn dered by eqatn , and cnderng eather cndtn gen by Table 1., and Table, can be determned, and they can be fnd n the fllng table gen bel [11, 14]: Table 3. Parameter f the lnear flter prdng lngtdnal peed cmpnent f the ar trblence, g t. Flter Parameter Weather Cndtn K 1 m 3 NASA-Mn 0, , NASA-Max 0, , Extreme Thndertrm 1, ,

13 Weather Cndtn Table 4. Parameter f the lnear flter prdng lateral peed cmpnent f the ar trblence, g t. K Flter Parameter m 3 3 NASA-Mn 0, , , NASA-Max 1, , , Extreme Thndertrm 8, , , Weather Cndtn Table 5. Parameter f the lnear flter prdng ertcal peed cmpnent f the ar trblence, g t. K Flter Parameter m 3 3 NASA-Mn 0, , ,5 NASA-Max 1, , ,5 Extreme Thndertrm, , , ng parameter f Table 3, Table 4, Table 5, tranfer fnctn f the lnear flter defned by eqatn 4.1 can be dered a fll [11, 14]: 0,0918 G Mn g, 0, ,8367 G Max g, 0, ,15956 G Extr g, 0, ,10984 G Mn g 0,9837 0,3805 0, ,10984 G Max g 1,15087, 0,3805 0, ,0488 G Extr g, ,0860 0, ,8867 G Mn g 0,31094, 0, ,8867 G Max g 1,4377 0,

14 0,0488 G Extr g 1, ,0860 0,00185 ng lnear tranfer fnctn mdel f eqatn t eay t generate randm tme ere th gen tattcal parameter, hch can be appled bth fr dentfcatn, mdelng, analy and degn prpe [6, 10, 11, 14] RESTS OF THE COMPTER SIMATION ng prncple dered by Fgre 1., and ng tranfer fnctn f the lnear flter defned fr eeral eather cndtn ne can generate cmpter cde fr ltn f th prblem. In r prelmnary tdy e hae ed MATAB R009 [13]. Regardng mathematcal mdel f the randm ar tlned n Chapter 4 all cmpnent f the peed f the trblent ar meared alng axe f the arcraft bdy-ax ytem, and they ll be preented n the next ectn RANDOM ONGITDINA SPEED COMPONENT OF THE TRBENT AIR The lngtdnal peed cmpnent ery mprtant frm the pnt f e f the bac flght cndtn,.e. arcraft flght lmted th t mnmm lngtdnal peed f, ay, mn. Frm Chapter 3 t knn that eqlbrm peed f the hypthetcal AV arcraft 5 m/. Relt f the cmpter mlatn can be een n Fgre 4.. Frm Fgre 4., t ealy can be determned that n tme dman f ecnd, n ther rd, n the rt f the trblent zne, the mean ale f the lngtdnal peed apprxmately, mean 4, m/, hch 16,8 % f that f the eqlbrm ne. There a qetn arng frm analy f the charactertc f the lngtdnal peed cmpnent drectn,.e. t can be cncdng ne t that f the mean drectn f the flght, r t can ppe arcraft flght In ther rd, lngtdnal peed cmpnent f the trblent ar can be called fr headnd, r, tal nd. Gng that ay, lngtdnal peed f the arcraft flyng thrgh atmpherc trblence can be dered a fll: fr head-nd : fr tal nd : head tal 5 4, 0,8 m/, 4.7 mean 5 4, 9, m/. 4.8 mean 355

15 Fgre 4.. ngtdnal Speed Cmpnent f the Stchatc Ar RANDOM ATERA SPEED COMPONENT OF THE TRBENT AIR. ng the ame manner a t a hn n pre ectn, cmpter cde fr randm lateral peed cmpnent f the trblent ar a generated, and relt f the cmpter mlatn can be een n Fgre 4.3. Frm Fgre 4.3. t ealy can be een that n the tme dman f abt ecnd, the mean ale f the lateral peed are: max 1,7 m /, 0,5 m /. 4.3 mn If t ppe eather cndtn hang tattcal parameter beteen eather cndtn f NASA-Mn, and NASA-Max, t can be pped that mean ale f the lateral peed, apprxmately, f 1 m/. Fgre 4.3. ateral Speed Cmpnent f the Stchatc Ar. 356

16 It mean that drng flght arcraft change t lateral crdnate fr abt 4 m n ne ecnd. If t take nt cnderatn the free-flght f the arcraft, r een f n nrmal flght arcraft plt de nt crrect the lateral crdnate, n 50 ecnd tme perd, beng netgated abe, arcraft mantan dtance f 150 m, changng t lateral crdnate fr 00 m. It b, that there a trng need t cmpenate lateral deatn meared frm the flght drectn RANDOM VERTICA SPEED COMPONENT OF THE TRBENT AIR. Randm ertcal peed f the trblent ar ery mprtant frm many apect f the alttde cntrl f the arcraft, frm the pnt f e f the mdelng f the aerelatc trctral mtn f the felage, and ng. There are many ther rean hghlghtng mprtance f the knledge f the tchatc ertcal peed f the atmpherc trblence. Relt f the cmpter mlatn ncldng NASA-Mn, and NASA-Max eather cndtn can be een n Fgre 4.4. Frm Fgre 4.4. t ealy can be een that n the tme dman f abt ecnd, the mean ale f the ertcal peed are a fll: max 0,7 m /, 0, m /. 4.4 mn It t take mean ale f the ertcal randm peed f the nd t be f 0,5 m/, drng flght arcraft change t alttde fr 1,8 m per ecnd. Fr the free-flght f the arcraft, r een f n nrmal flght arcraft plt de nt crrect the heght f the flght, n 50 ecnd tme perd, beng netgated abe, arcraft mantan dtance f 150 m, changng t heght f the flght fr 90 m, t that f the ntal f H 100 m. It mean that hang n cntrl n arcraft alttde, n trblent ar arcraft nearly dplcate t heght f the flght. It b, that heght f the flght mt be cntrlled, and alttde mt be kept at t cntant ale. Fgre 4.4. Vertcal Speed Cmpnent f the Stchatc Ar. 357

17 RESTS OF THE COMPTER SIMATION OF THE ATMOSPHERIC TRBENCES FOR THE NASA-MIN WEATHER CONDITIONS ng relt f the pre cmpter mlatn, fr NASA-Mn eather cndtn all apprprate tme ere f the lngtdnal, lateral, and ertcal cmpnent f the randm ar ere plt n ne, cmmn crdnate ytem, and they can be een n Fgre 4.5. Fgre 4.5. Relt f the Cmpter Smlatn fr NASA-Mn Weather Cndtn RESTS OF THE COMPTER SIMATION OF THE ATMOSPHERIC TRBENCES FOR THE NASA-MAX WEATHER CONDITIONS ng relt f the cmpter mlatn made befre, fr NASA-Max eather cndtn all apprprate tme ere f the lngtdnal, lateral, and ertcal cmpnent f the randm ar ere plt n ne, cmmn crdnate ytem, and they can be een n Fgre

18 Fgre 4.6. Relt f the Cmpter Smlatn fr NASA-Max Weather Cndtn. Frm Fgre 4.6. t ealy can be dered that lngtdnal peed cmpnent, g t, f the atmpherc trblence ha larget mean ale. It edent that fr head-nd eather cndtn, there ext a maxmm ale f the lngtdnal randm peed, t, hch alled t ad tallng f the arcraft. g max RESTS OF THE COMPTER SIMATION OF THE ATMOSPHERIC TRBENCES FOR THE EXTREME THNDERSTORM WEATHER CONDITIONS Relt f thee cmpter mlatn are manly hypthetcal, heer, t neceary t kn h extreme ar mae are mng. Thee relt are ery mprtant althgh frm the pnt f e f the flght acheed beynd al range fr large dtance, hen there are bg dfference beteen eather cndtn at arral and departre arfeld. Relt f the cmpter mlatn can be een n Fgre 4.6. The mt mprtant relt that atmpherc trblence ha larget ale n the mean f lateral cmpnent f the trblent ar. The ther mprtant tatement cmng frm th analy, that f t cnder maxmm ale f the lngtdnal head-nd t be f g _ head t 5 m /, th maxmm ale reached at abt 5 ecnd f the cmpter-aded mlatn. It mean that t ad tallng f the arcraft t neceary t cmpenate decreae f the lngtdnal peed f the arcraft ncreang thrttle, r t neceary t mantan maneer t keep gen flght parameter n the defned flght enelpe f the gen type f the arcraft. 359

19 Fgre 4.6. Relt f the Cmpter Smlatn fr Extreme Weather Cndtn. The mt mprtant relt that atmpherc trblence ha larget ale n the mean f lateral cmpnent f the trblent ar. The ther mprtant tatement cmng frm th analy, that f t cnder maxmm ale f the lngtdnal head-nd t be f g _ head t 5 m /, th maxmm ale reached at abt 5 ecnd f the cmpter-aded mlatn. It mean that t ad tallng f the arcraft t neceary t cmpenate decreae f the lngtdnal peed f the arcraft ncreang thrttle, r t neceary t mantan maneer t keep gen flght parameter n the defned flght enelpe f the gen type f the arcraft. V. SMMARY Mathematcal mdel are dely ed drng prelmnary analy, dentfcatn, and degn f the atmatc cntrl ytem. They can be bth determntc and randm ne, regardng ther tme dman behar. They pprt ytem decrptn hether t cntn, r dcrete. Mdelng f dynamcal ytem pprted by many cmpter package e.g. MATAB, SIMINK, and ther tlbxe. OPS CITATM [1] Krn, G. A. Randm-Prce Smlatn and Mearement, McGra-Hll Bk Cmpany, Ne Yrk Trnt ndn Sydney, [] Kármán, T. Bt, A. M. Matematka módzerek műzak feladatk megldáára, Műzak Könykadó, Bdapet, [3] Krn, G. A. Krn, T. M. Matematka módzerek műzakaknak, Műzak Könykadó, Bdapet,

20 [4] K, B. C. Atmatc Cntrl Sytem, Prentce-Hall, Engled Clff, Ne Jerey, 198. [5] Ogata, K. Mdern Cntrl Engneerng, Prentce-Hall Internatnal Inc., Engled Clff, Ne Jerey, [6] Mcean, D. Atmatc Flght Cntrl Sytem, Prentce-Hall, Int., Ne Yrk ndn Trnt Sydney Tky Sngapre, [7] Drf, R. C. Bhp, R. H. Mdern Cntrl Sytem, Prentce Hall Internatnal, pper Saddle Rer, Ne Jerey, 001. [8] Stefan, R. T. Shahan, B. Saant Jr., C. J. Htetter, G. H. Degn f Feedback Cntrl Sytem, Oxfrd nerty Pre, Ne Yrk-Oxfrd, 00. [9] Ne, N. S. Cntrl Sytem Engneerng, Jhn Wley & Sn, Inc., 004. [10] MI STD 1797A, Ntce 3, Flyng Qalte f Plted Arcraft, Department f Defene, Interface Standard, 004. [11] Szablc, R. Mathematcal Mdel fr Gt Mdelng Appled n Atmatc Flght Cntrl Sytem Degn, CD-ROM Prceedng f the 5 th Internatnal Cnference n the Feld f Mltary Scence 007, Nember 007, Bdapet, Hngary. [1] Pkrád,. Jelek é rendzerek mdellezée, Camp Kadó, Debrecen, 008. [13] MATAB 7 Gettng Started Gde, The MathWrk, Inc., 009. [14] Szablc, R. Stchatc Ne Affectng Dynamc Perfrmance f the Atmatc Flght Cntrl Sytem, Ree f the Ar Frce Academy, N1/009, pp3-30, ISSN , Bra, Rmana. 361