Modeliranje i simulacija sistema
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Sadržaj Modelovanje i simulacija sistema ............................................................................................................. 1 Osnovni pojmovi modeliranja i simulacije. Model i teorija ................................................................... 1 Detaljnost modela. Neformalan i formalan opis modela ........................................................................ 1 Klasifikacije Klasifikacije modela modela ............................... ................................................ .................................. .................................. .................................. ................................. .......................... .......... 2 .......................................... ......... 3 Primer procesa dobijanja matematiĉkog modela. Pojednostavljenje modela ................................. Verifikacija i valjanost modela. Stepeni valjanosti ................................................................................ 4 .................................................. ............................... ...............4 Analitiĉko i simulaciono rešenje. Simulacija u širem smislu .................................. Modeli sistema ................................. ................................................. ................................. ................................. .................................. .................................. ................................. ..................... .... 5 promenljive, elementi i zakonitosti ................................................... 5 Translatorni mehaniĉki mehaniĉki sistemi – promenljive, dobijanje modela sistema .................................................................. 6 Translatorni mehaniĉki sistemi – dobijanje promenljive, elementi i zakonitosti ....................................................... 6 Rotacioni mehaniĉki sistemi – promenljive, dobijanje modela sistema ...................................................................... 9 Rotacioni mehaniĉki sistemi – dobijanje promenljive, elementi i zakonitosti ......................................................................... 9 Termiĉki sistemi – promenljive, dobijanje modela sistema ...................................................................................... 11 Termiĉki sistemi – dobijanje Sistemi sa fluidima – promenljive, promenljive, elementi i zakonitosti ................................................................... 11 Sistemi sa fluidima – dobijanje dobijanje modela sistema .................................................................................. 13 promenljive, elementi i zakonitosti .......................................................... 13 Elektromehaniĉki Elektromehaniĉki sistemi – promenljive, .................................................. ................................. ................................. ................................. ................... .. 15 Matematiĉki model u prostoru stanja .................................. ......................................... ....... 15 Matematiĉki model u prostoru stanja. Koncept i izbor promenlji vih stanja .................................. ............................ 16 Numeriĉko rešavanje rešavanje odziva modela modela (rešavanje (rešavanje obiĉnih diferencijalnih diferencijalnih jednaĉina) . ............................ .................................................. ................................. ................18 Linearan matematiĉki model (u prostoru stanja). Osobine ................................. Linearizacija nelinearnih modela i formiranje linearnog modela modela u prostoru stanja ................... .......... ................... ............ 19 Digitalan model. Kvantovanje. Teorema o odabiranju ........................................................................ 20 Vremenski diskretan odziv linearnog modela ..................................................................................... 21 Analogije Analogije ................................ ................................................. .................................. ................................. .................................. .................................. ................................. ........................... .......... 23 Analogije elemenata i parametara ....................................................................................................... 23 Matlab .................................. .................................................. ................................. .................................. .................................. ................................. ................................. .............................. ............. 24 Matlab-a ................................. ................................................. ..................... ..... 24 Rešavanje obiĉnih diferencijalnih jednaĉina upotrebom Matlab-a Skript datoteka i funkcija Matlab funkcija Matlab-a....... -a........................ .................................. ................................. ................................. ................................. ........................... ........... 24 Kontrola toka
Matlab
25
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Matlab-u ................................. ................................................. ................................. ........................... .......... 31 Grafiĉka predstava rezultata simulacije u Matlab-u Control System Toolbox ........................................................................................................................ 33 Namena Control System Toolbox-a Toolbox -a (CST (CST ) ................................ ................................................. .................................. ................................. ........................ ........ 33 -u .............................. ................................................ .................................. ................................. ................... .. 33 Naĉini predstavljanja predstavljanja modela sistema sistema u CST -u Analiza simulacionog modela upotrebom CST .............. CST .............................. .................................. .................................. ................................. ................... .. 34 -u ................................ ................................................ ................35 Postupno formiranje složenih linearnih simulacionih modela u CST -u -u............ ............ ... 36 Formiranje složenog linearnog simulacionog modela uptrebom matrice veza u CST -u..................... Simulink Simulink ................................. .................................................. .................................. ................................. .................................. .................................. ................................. ........................... .......... 37 Namena Simulink -a i naĉini upotrebe simulacionog modela................................................................ 37 Formiranje simulacionog modela u Simulink -u. -u. S-funkcija ................................................................. 37 Analiza modela upotrebom upotr ebom Simulink -a, zadavanje poĉetnog stanja i ulaza ............................... .......................................... ........... 39 Pokretanje simulacije u Simulink -u. -u. Parametri simulacije ................................................................... 40 Linearizacija modela upotrebom upotr ebom Simulink -a -a ................................ ................................................ .................................. .................................. ..................... ..... 41 .................................................. .................................. ...................... .... 42 OdreĊivanje stacionarnih stanja upotrebom Simulinka .................................. Identifikaci Identifikacija ja ................................. .................................................. ................................. ................................. .................................. ................................. .................................. ...................... .... 43 Zadeh-ov Zadeh-ov opis problema identifikacije ................................................................................................ 43 ................................................. ................... .. 43 Primena identifikacije i naĉini sprovoĊenja. Postupak identifikacije ................................ Parametarska identifikacija i metoda najmanjih kvadrata ( LS LS algoritam) algoritam) ............................................ 44 Osobine procene parametara. Osobina identifiabilnosti....................................................................... 45 Identifikacija parametara «jednostavnog» vremenski diskretnog modela (C(z)=1) ................... .......... ................... ............ 46 Identifikacija parametara vremenski diskretnog modela (C(z) 1). (TS algoritam) ................... .......... .................. ......... 47 Identifikacija promenljivih parametara. Rekurzivni metod najmanjih kvadrata .................. ......... ................... ................. ....... 48 modeliranju u ................................. ................................................. ................................. .................................. ........................... .......... 49 Upotreba neuronskih mrež a u modeliranj ................................................ .................................. ................................. ................ 49 Model veštaĉkog neurona i aktivacione funkcije ............................... ................................................ .................................. ................................. ................................. ................... .. 49 Modeli veštaĉkih nauronskih mreža ............................... ................................................ .................................. .................................. ........................... .......... 51 Obuĉavanje veštaĉkih neuronskih mreža ............................... ................................................ .............................. ............. 52 Uloga veštaĉkih neuronskih mreža u modeliranju i simulaciji ............................... ................................................. ................................. ................ 53 Veštaĉka nauronska mreža kao model dinamiĉkog sistema ................................ ekanja......................................... .................................. ................................. ................................. .................................. ................................. ........................... ........... 54 Redovi ĉekanja........................ ............................... 54 54 Komponente jednostavnog procesa sa redovima ĉekanja. Karakteristiĉni procesi ............................... ............................................... ................ 54 Ciljevi teorije redova ĉekanja. Parametri sistema. Rezultati simulacije ...............................
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Modelovanje i simulacija sistema Osnovni pojmovi modeliranja i simulacije. Model i teorija Modeliranje i
simulaciju ĉine niz aktivnosti za pravljenje modela realnog sistema i njegovu simulaciju na
raĉunaru. Osnovni elementi koji figurišu u modeliranju i simulaciji su realan sistem, model i raĉunar.
radi odreĊenog saznanja. Time se izbegava opasnost eksperimenta nad realnim sistemom. Model ne treba da reprodukuje stvarnost u potpunosti, već treba da iskaže deo strukture ili ponašanja realnog sistema.
Cilj modeliranja je upotreba modela umesto realnog sistema,
Realan sistem je ureĊen i meĊu meĊuzavisan skup kompone nti, koje formiraju celinu i deluju zajedniĉki da bi ostvarili cilj ili funkciju. On predstavlja deo r ealnog sveta koji je od interesa i predstavlja i zvor podataka o ponašanju, potrebnih za formiranje modela. Model je pogodan naĉin predstavljanja predstavljanja ukupnog ukupnog ĉovekovog iskustva iskustva i njegovog njegovog naĉina razmišljanja razmišljanja o
sistemu koji istražuje. Model predstavlja rezultat modeliranja, apstrakciju realnosti (uprošćenu, idealizovanu, ne obuhvata sve aspekte) i sadrži izabrane elemente i karakteristike sistema. Simulacioni mo del je program za generisanje ponašanja sistema. Nastaje na osnovu modela (tipiĉno
matematiĉkog modela) i omogućava simulacione simulacione eksperimen e ksperimente te (simulacije). Teorija je opšti iskaz iskaz principa, principa, izveden iz posmatranja posmatranja sistema i podataka dobijenih dobijenih posmatranjem. posmatranjem. Teorija je neophodan element koji povezuje model i sistem. Ona objašnjava ponašanje sistema i
omogućava predviĊanje zakljuĉaka koji se mogu proveriti. Model je samo predoĉena teorija, konkretan je i omogućava da se teorija proveri na delu. teorija mora biti prethodno formulisana. Model ne može post ojati ako nema teorije – teorija
Detaljnost modela. Neformalan i formalan opis modela
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Inženjerski pristup: Model treba što što vernije da preslikava preslikava stvarnost stvarnost u skladu sa traženom traženom složenosti složenosti i cenom razvoja. razvoja.
Model koji ukljuĉuje sve promenljive i veze meĊu njima je bazni model. Apsolutno je valjan i ne može se praktiĉno realizovati. realizovati. Pojednostavljen bazni model je sažeti ili grubi model i odnosi se na izabran eksperimentalni okvir. Neformalan opis daje osnovne osnovne pojmove o modelu.
model treba da obezbedi veću preciznost i potpunost opisa sistema. Opisuje sistem na jasan i nedvosmislen naĉin i koristi poznate metodologije.
Formalan
model se brzo i lako formira. Uvodi objekte, opisne promenljive i ponašanje i pravila interakcije objekata. Objekti i opisne promenljive ĉine strukturu modela. modela. Neformalan model model je najĉešće
Neformalan
nejasan, nekompletan i nekonzistentan.
Klasifikacije modela Mode li se klasifikuju po više kriterijuma: priroda pr iroda promenljivih promenljivih i opseg o pseg vrednosti, opseg vrednosti
vremena, vremenska vremenska zavisnost modela, modela, determinizam, linearnost, formalan opis modela, “opipljivost” modela, stanje ravnoteže, … Klasifikacija modela po raznim kriterijumima: o
o
o
Skup vrednosti koje prima promenljiva “vreme” može biti: - neprebrojiv – vremenski kontinualan model - prebrojiv – vremenski diskretan model Opisne promenljive uzimaju vrednosti: - opsega realnih brojeva – modeli modeli sa kontinualnim stanjima - iz skupova ĉ iji su elementi diskretne diskret ne vrednosti – modeli modeli sa diskretnim stanjima stanjima. Postoje i modeli sa mešovitim stanjima.
Da li model sadrži sluĉajne promenljive? -
o
o
ako su izlazi modela uvek isti za iste ulaze i stanje modela – deterministički model deterministički model suprotno – stohastički model (model je stohastiĉan ako ima barem jednu stohastiĉku promenljivu)
Da li se model menja sa vremenom? - ukoliko se struktura modela ili pravila interakcije menjaju sa vremenom – varijantan (vremenski promenljiv) model - suprotno – invarijantan (vremenski nepromenljiv) model Statički modeli daju izlaze modela za sistem u ravnoteži – tzv. stacionarno st acionarno stanje. Opisuje se
algebarskim jednaĉinama.
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o
modeli su opštiji, ali i složeniji. Linearni modeli menjaju stanja i daju izlaze poštujući linearne transformacije. Tipiĉno nastaju
Nelinearni
postupkom linearizacije, linearizacije, radi r adi jednostavnije upotrebe upotrebe modela. o
o
Po postojanju opisnih promenlljivih (stanja, ulaznih i izlaznih) modeli se dele na: - modeli bez memorije (trenutne funkcije) – (ne, (ne, da, da) - modeli sa memorijom (autonomni i neautonomni) – ( ( (da, ne, da/ne), (da, da, da/ne) ) Fizički modeli zakona. Apstraktni
su materijalne reprezentacije istraživanog sistema, zasnovane na analogiji fiziĉkih
modeli su simboliĉna, verbalna i matematiĉko -logiĉka reprezentacija sistema.
Primer procesa dobijanja matematiĉkog modela. Pojednostavljenje modela Primer procesa dobijanja modela:
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Verifikacija i valjanost modela. Stepeni valjanosti
Provera da li se model ponaša onako kako je t o zamislio autor. Proveravaju se program(i) i podaci. Definiše se stepen podudaranja (apsolutno podudaranje je nemoguće, a razlike su posledica aproksimacija). Stepeni valjanosti modela:
replikativna valjanost (najniži stepen) - porede se izlazi modela i sistema
prediktivna valjanost - model proizvodi dobre vrednosti na izlazima pre nego pre nego što se m ogu izmeriti u realnom sistemu - omogućava istraživanje situacija k oje nisu posmatrane u sistemu
strukturna valjanost - model u potpunosti odslikava naĉin na koji realan sistem funkcioniše - omogućava se istraživanje operacija sistema koje se ne mogu meriti
Analitiĉko i simulaciono rešenje. Simulacija u širem smislu Analitičko rešavanje koristi deduktivne postupke matematiĉke matematiĉke analize. Daje opšte rešenje u ob liku azne kombinacije ulaza i parame tara. Treba ga koristiti uvek kada je to moguće. formule. Važi za r azne
Postoje i neka ograniĉenja: - sistem i njegovi odnosi nosu dovoljno poznati da se opišu matematiĉki - složeno se sprovodi, a ĉesto je i nemoguće. Simulacija je odreĊivanje odreĊivanje ponašanja ponašanja modela na osnovu osnovu vrednosti vrednosti ulaza (ili skupa skupa opisnih promenljivih) promenljivih) .
Simulacioni model je program za generisanje ponašanje sistema i on na osnovu zadatih ulaza i parametara modela modela izraĉunava izlaze izlaze modela. modela. Raĉunarska simulacija podrazumeva eksperimente na raĉunaru. Ukljuĉuje i izgradnju apstraktnog modela, programiranjem.
Raĉunar se upotrebljava za formiranje (razvoj) modela i numeriĉke proraĉune na osnovu modela.
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Modeli sistema
Translatorni mehaniĉki sistemi – promenljive, promenljive, elementi i zakonitosti Osnovne promenljive:
Dodatne promenljive:
x ras rastojan janje [ m] v brzina [m / s ] a ubrzanje [ m / s 2 ] f sila [ N ]
v
dx
w energija [ J ]
dt
p snaga [W ]
a
dv dt
d 2x dt 2
p f v p
t
w(t ) w(t0 ) p( t )dt t 0
Elementi i njihovi zakoni: -
Masa tela M tela M [kg ]
d
( M v ) f dt
za M const M
dv dt
f
(II Njutnov zakon)
Energija može biti: - kinetiĉka
wk
1 2
M v2
- potencijalna w p Mgh -
Trenje – sila – sila trenja se javlja kada se dva tela dodiruju i kreću razliĉitim brzinama.
f f ( v)
v v2 v1
Linearizovana zavisnost:
f c v c – koeficijent koeficijent trenja (viskoznosti) [ Ns / m]
Karakteristika trenja:
Elastiĉnost
dw dt
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2. Zakon akcije i reakcije (III Njutnov zakon) zakon)
3. Zak on on pomeraja: suma pomeraja duž zatvorene putanje je 0.
( x)
i
0
i
Translatorni mehaniĉki sistemi – dobijanje dobijanje modela sistema Kombinuju se zakonitosti elemenata i zakonitosti interakcije (meĊusobnih veza) elemenata. Primer Za svako telo posmatramo sile koje na njega deluju.
Na osnovu Dalamberovog Dalamberovog zakona, zakona, pišemo pišemo jednaĉine.
Rotacioni mehaniĉki sistemi – promenljive, promenljive, elementi i zakonitosti
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Elementi i njihovi zakoni: -
Moment inercije
d
( J ) dt
za J const J
d dt
Njutnov zakon) (II Njutnov
J – moment inercije [kgm2 ] – moment moment sile koji deluje na osu rotacije – moment
Energija može biti: - kinetiĉka
wk
1 2
J 2
- potencijalna w p mgh
h – visina – visina težišta u odnosu na ref. taĉku.
w p se ne menja kada telo rotira oko težišne ose
Steiner-ova teorema:
J J o ma2 J o - moment inercije za osu koja
prolazi kroz centar centar mase (težišnu (težišnu osu) J J - moment inercije za osu paralelnu prethodnoj -
Trenje usled rotacije – (rotaciono (rotaciono trenje) je algebarska veza momenta sile i relativne ugaone
brzine izmeĊu izmeĊu dve površi. površi. Opšta, nelinearna veza: Linearna veza:
f f ( )
2 1
f c c – koeficijent koeficijent trenja [ Nms]
-
Elastiĉnost usled uvrtanja – uvrtanja – (rotaciona (rotaciona elastiĉnost) je algebarska veza momenta sile i relativnog ugaonog pomeraja.
Opšta, nelinearna veza: Linearna veza:
f ( )
2 1
k
k – koeficijent elastiĉnosti [ Nm] – koeficijent
1
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Suma momenata oko obrtne taĉke je 0. (kada je masa 0) L1 f1 L2 f 2 0 -
f1
L2 L1
f 2
Zupĉanici Idealni zupĉanici nemaju moment inercije, trenje, unutrašnju energiju i zubci im savršeno naležu. enos N – odnos broja zubaca Zupĉasti pr enos N – odnos z 1 , z 2 - broj zubaca
r11 r2 2 N
r2 r1
z 2 z 1
1 2
r 2 r 1
N
1 2
Prenošenje momenta sile: - za levi zupĉanik važi:
f c r 1 1 0
- za desni zupĉanik važi:
f c r 2 2 0 f c -
sila koja se prenosi na drugi zupĉanik
- moment sile primenjen na zupĉanik 2
r 2
N
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Rotacioni mehaniĉki sistemi – dobijanje dobijanje modela sistema Naznaĉimo Naznaĉimo (usvojimo) pozitivan smer za promenljive. promenljive. (smer porasta , , nekog tela je isti) Upotrebom zakona ugaonih pomeraja, izbegavamo višak simbola potrebnih za opis kretanja. Za svaku masu ili spojnu taĉku (ĉije je kretanje nepoznato) crtamo dijagram koji pokazuje sve momente sila, ukljuĉujući inericjalni moment sile. Sve momente sila, sem p obudnih (ulaznih) izražavamo preko , , upotrebom zakona elemenata. Primenjujemo Dalamberov zakon na svaki dijagram. Primer
Termiĉki sistemi – promenljive, promenljive, elementi i zakonitosti
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Pasivni elementi: -
Termiĉka kapacitivnost Postoji algebarska zavisnost izmeĊu temperature tela i akumulirane toplote u njemu q Ako nema promene agregatnog stanja i se men ja u uskim granicama, zavisnost zavisnost se može posmatrati kao linearna.
(t )
1
t
qin (t ) qout ( t)
C
(t ) (t0 )
1
C q
in
(t ) qout ( t) dt
t 0
C – toplotni kapacitet tela t ela [ J / K ] – toplotni
C m – specifiĉna ĉna toplota to plota tela – specifi -
Termiĉka otpornost Toplota se prenosi na 3 naĉina: provoĊenjem, strujanjem ili zraĉenjem. Ovde posmatramo samo provoĊenje toplote. ProvoĊenje toplote sa jednog tela na drugo je srazmerno razlici temeperatura dva tela . q(t )
1 R
1 (t ) 2 (t)
q(t )
1 R
R – termiĉka – termiĉka otpornost [ Ks / J ]
R
d A A
R zavisi R zavisi od karakteristika materijala (debljina (d (d ), ), površina
popreĉnog preseka preseka ( A), termiĉka provodljivost ( )). Termiĉka otpornost se može upotrebiti samo ako material (telo) (te lo) ne akumulira energiju.
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Termiĉki sistemi – dobijanje dobijanje modela sistema Kao promenljive stanja se uzimaju temperature svakog tela koje ima toplotni kapacitet. Prenošenje toplote u telo t elo sa toplotnim kapacitetom zavisi od izvora toplote i prenošenja toplote preko termiĉkih otpornosti. Primer
Ovaj model se može na kraju linearizovati, izražavanjem promenljivih temperature i provoĊenja , kao zbirove nominalnih i inkrementalnih vrednosti, i uvoĊenjem smena. Zatim se eliminišu konstantni ĉlanovi i dobijamo model izražen samo preko inkrement alnih vrednosti. Primer (nastavak (nastavak )
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Postoje dve karakteristike koje se mogu predstaviti pojednostavljenim elementima: -
Kapacitet Kapacitet posude za teĉnosti Kod teĉnosti smeštene u otvoren sud postoji algebarska ralacija izmeĊu zapremine teĉnosti i pritiska u osnovi suda. h
Zapremina:
V A(h )dh
površina na visini h A( h) - površina
o
Apsolutni pritisak:
p pgh pa
za A(h ) const
p pg
V A
pa
V c
pa
hidrauliĉk i kapacitet c – reciproĉna – reciproĉna vrednost nagiba krive za datu radnu taĉku p i v: Veze izmeĊu p i
-
Otpor pri proticanju
Kada teĉnost protiĉe kroz cev, ventil ili otvor, dolazi do pada pritiska. Pad pritiska se objašnjava gubitkom energije i obiĉno je posledica nelinearne zavisnosti q f ( p) . Npr. kod ventila: ventila:
q k p
k – karakteristika cevi, ventila ili otpora – karakteristika
opisuje turbulentno tur bulentno kretanje kroz cevovod p - dobro opisuje
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Sistemi sa fluidima – dobijanje dobijanje modela sistema Primer Napisati linearan model promene p1 i odrediti funkciju prenosa, koja povezuje inkrementalne promenljive p1 i qi (t ) . ˆ
ˆ
Nakon formiranja, formiranja, predstavimo predstavimo linearne ĉlanove ĉlanove kao sume nominalnih nominalnih i inkrementalnih inkrementalnih vrednosti. Linearizujemo nelinearne ĉlanove razvojem u Tejlorov red i skratimo konstantne ĉlanove. Na kraju opet dobijamo izraz u kom figurišu samo inkrementalni ĉlanovi.
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Moment sile usled proticanja struje: Indukovana ems: ems:
e f ea (i 2 Ns B)a
em (2 Ns ) v B (2 Ns) ( a ) B J (t ) c(t ) k (t ) (2 NBsa) i(t )
Ravnoteža momenata i elektriĉno kolo:
L
di(t ) dt
Ri(t ) (2 ( 2 NBsa) (t ) ei (t )
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Matematiĉki model u prostoru stanja Matematiĉki model u prostoru stanja. Koncept i izbor izbor promenljivih stanja Model sistema ĉine (Košijeva teorem t eorema): a): -
Sistem diferencijalnih jednaĉina prvog reda x1 (t ) f1 ( x1, x2 , ..., xn , u1, u2 , ..., ur , t), ), x1(0) x10 x2 (t ) f 2 ( x1, x2 , ..., xn , u1, u2 , ..., ur , t), ), x2 (0) x20
xn (t ) f n ( x1 , x2 , ..., xn , u1 , u2 , ..., ur , t ), xn (0) xn 0 -
Sistem algebarskih jednaĉina y1 (t ) g1 ( x1, x 2 , ..., x n , u1, u 2 , ..., u r , t ) y2 (t ) g 2 ( x1, x 2 , ..., x n , u1, u 2 , ..., u r , t )
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Numeriĉko rešavanje odziva modela (rešavanje obiĉnih diferencijalnih diferencija lnih jednaĉina) Rešavanje diferencijalne jednaĉine 1. reda Polazni problem predstavlja obiĉna diferencijalna jednacina 1. reda sa poznatom vrednosti vrednosti zavisno promenljive y promenljive y u po ĉetnom trenutku t 0.
dy dt
), f ( y, t ),
y (t 0 ) 0
Numeriĉko rešavanje polazi od poznatih ( t 0, y0) i sukcesivno raĉuna parove (t ( t i, yi) za i=1,2,… Iterativan raĉun se sprovodi “korak -po-po-korak”, gde se ( t i+1, yi+1) raĉuna na osnovu prethodno sraĉunatog (t (t i, yi). Ojlerov (Euler) postupak Vrednost y Vrednost yi+1 se može odrediti na osnovu razvoja u Tejlorov red u okolini tacke ( ti, ti, yi) yi)
yi 1 yi y yi
dyi dt
h
d 2 yi h2 dt 2 2!
...
h ti 1 t i
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Metoda Runge-Kuta
yi 1 yi ck1 ck 2
Kreće se od izraza:
k1 f ( yi , ti )h k2 f ( yi a2 k1 , ti a 2h)h
Razvojem funkcije u Tejlorov red
i njegovom aproksimacijom, aproksimacijom, dobija do bija se izraz koji se uvrštava u vrštava u poĉetni. Daljim sreĊivanjem dobijenog izraza dobijamo: yi 1 yi ( c1 c2 )
dyi dt
h c2a 2
d 2 yi 2
dt
Da bi ova jednaĉina bila jednaka jednaĉini yi 1 yi
dyi dt
h
d 2 yi h2 dt 2 2!
potrebno je da je:
c1 c2 1
c2a 2
1 2
...
h 2 O ( h3 )
O(h 3 ) - ostali sabirci razvoja u Tejlorov red
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Diferencijalna jednaĉina višeg reda
Jednaĉina y ( n ) f (t , y, y , y , y , ..., y (n 1) ) se svodi na sistem n diferencijalnih jednaĉina prvog reda. To se
naĉešće se koriste pomoćne promenljive: yi y ( i 1) , i 1,2,...
može uraditi na nekoliko naĉina, a
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Vektorski format linearnog matematiĉkog modela u prostoru stanja: x (t ) A x( t ) B u( t ), x(0) x0 y (t ) C x(t ) D u( t )
a11 A an1
ann
b11
b1r
a1n
matrica stanja
c11 C cm1
d11
c1n
cmn
d 1r
matrica izlaza
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Koraci linearizacije:
1. Odrediti radnu taĉku – pisanjem pisanjem i rešavanjem odgovarajućih algebarskih algebarskih jednaĉina 2. Prepisati sve linearne ĉlanove kao sume -
nominalne i
x(t ) x x(t )
-
inkrementalne vrednosti
x(t ) - nominalna vrednost x(t ) - inkrementalna vrednost
ˆ
ˆ
3. Zameniti sve nelinearne ĉlanove sa prva 2 sabirka razvoja u Tejlorov red 4. S kratiti konstantne
ĉlanove u di ferencijalnim jednaĉinama jednaĉinama
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Odabiraĉ
Kolo zadrške nultog nultog
reda
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Odziv u trenutku t kT je kT
x (kT ) e
AkT
x(0) e A ( kT ) B u( )d 0
Period [kT [kT , kT +T ] je dovoljano kratak da možemo smatrati da je u(kT ) nepromenljivo tokom tog perioda. se SreĊivanjem izraza za trenutak t kT T dobija
x (kT T ) e
AT
T At x(kT ) e dt B u(kT ) , 0
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