Syudi perenapravlyayetsya zapit Prostir staniv Na cyu temu potribna okrema stattya Prostir staniv u teoriyi keruvannya odin z osnovnih metodiv opisu povedinki dinamichnoyi sistemi Ruh sistemi v prostori staniv vidbivaye zminu yiyi staniv Zmist 1 Viznachennya 1 1 Linijni neperervni sistemi 1 2 Diskretni sistemi 1 3 Nelinijni sistemi 1 3 1 Linearizaciya 2 Prikladi 2 1 Model u prostori staniv dlya mayatnika 2 2 Linearizaciya modeli mayatnika 3 Div takozh 4 Literatura 5 PosilannyaViznachennya RedaguvatiProstir staniv zazvichaj nazivayut fazovim prostorom dinamichnoyi sistemi a trayektoriyu ruhu sho zobrazhaye tochki v comu prostori fazovoyu trayektoriyeyu B 1 B 2 A 1 U prostori staniv stvoryuyetsya model dinamichnoyi sistemi sho vklyuchaye nabir zminnih vhodu vihodu i stanu pov yazanih mizh soboyu diferencialnimi rivnyannyami pershogo poryadku yaki zapisuyutsya v matrichnij formi Na vidminu vid opisu u viglyadi peredavalnoyi funkciyi ta inshih metodiv chastotnoyi oblasti prostir staniv dozvolyaye pracyuvati ne tilki z linijnimi sistemami i nulovimi pochatkovimi umovami Krim togo v prostori staniv vidnosno prosto pracyuvati z MIMO sistemami Linijni neperervni sistemi Redaguvati nbsp Strukturna shema neperervnoyi linijnoyi sistemi opisanoyi u viglyadi zminnih stanuDlya vipadku linijnoyi sistemi z p displaystyle p nbsp vhodami q displaystyle q nbsp vihodami i n displaystyle n nbsp zminnimi stanu opis maye viglyad x t A t x t B t u t displaystyle dot mathbf x t A t mathbf x t B t mathbf u t nbsp y t C t x t D t u t displaystyle mathbf y t C t mathbf x t D t mathbf u t nbsp de x t R n displaystyle x t in mathbb R n nbsp y t R q displaystyle y t in mathbb R q nbsp u t R p displaystyle u t in mathbb R p nbsp dim A n n displaystyle operatorname dim A cdot n times n nbsp dim B n p displaystyle operatorname dim B cdot n times p nbsp dim C q n displaystyle operatorname dim C cdot q times n nbsp dim D q p displaystyle operatorname dim D cdot q times p nbsp x t d x t d t displaystyle dot mathbf x t d mathbf x t over dt nbsp x displaystyle x cdot nbsp vektor stanu elementi yakogo nazivayut stanami sistemi y displaystyle y cdot nbsp vektor vihodu u displaystyle u cdot nbsp vektor keruvannya A displaystyle A cdot nbsp matricya sistemi B displaystyle B cdot nbsp matricya keruvannya C displaystyle C cdot nbsp matricya vihodu D displaystyle D cdot nbsp matricya pryamogo zv yazku Chasto matricya D displaystyle D cdot nbsp ye nulovoyu ce oznachaye sho v sistemi nemaye yavnogo pryamogo zv yazku Diskretni sistemi Redaguvati Dlya diskretnih sistem zapis rivnyan u prostori gruntuyetsya ne na diferencialnih a na riznicevih rivnyannyah x n T T A n T x n T B n T u n T displaystyle mathbf x nT T A nT mathbf x nT B nT mathbf u nT nbsp y n T C n T x n T D n T u n T displaystyle mathbf y nT C nT mathbf x nT D nT mathbf u nT nbsp Nelinijni sistemi Redaguvati Nelinijnu dinamichnu sistemu n go poryadku mozhna opisati u viglyadi sistemi z n rivnyan 1 go poryadku x 1 f 1 x 1 t x n t u 1 t u m t displaystyle dot x 1 f 1 x 1 t ldots x n t u 1 t ldots u m t nbsp displaystyle vdots nbsp x n f n x 1 t x n t u 1 t u m t displaystyle dot x n f n x 1 t ldots x n t u 1 t ldots u m t nbsp abo v kompaktnishij formi x t f t x t u t displaystyle mathbf dot x t mathbf f t mathbf x t mathbf u t nbsp y t h t x t u t displaystyle mathbf y t mathbf h t mathbf x t mathbf u t nbsp Pershe rivnyannya ce rivnyannya stanu druge rivnyannya vihodu Linearizaciya Redaguvati U deyakih vipadkah mozhliva linearizaciya opisu dinamichnoyi sistemi dlya okolu robochoyi tochki x u displaystyle mathbf tilde x mathbf tilde u nbsp U stalomu rezhimi u c o n s t displaystyle mathbf tilde u const nbsp dlya robochoyi tochki x c o n s t displaystyle mathbf tilde x const nbsp spravedlivij takij viraz x f x u 0 displaystyle mathbf dot x mathbf f mathbf tilde x mathbf tilde u mathbf 0 nbsp Vvodyachi poznachennya d u u u displaystyle delta mathbf u mathbf u mathbf tilde u nbsp d x x x displaystyle delta mathbf x mathbf x mathbf tilde x nbsp Rozklad rivnyannya stanu f x t u t displaystyle mathbf f mathbf x t mathbf u t nbsp v ryad Tejlora obmezhenij pershimi dvoma chlenami daye takij viraz f x t u t f x t u t d f d x d x d f d u d u displaystyle mathbf f mathbf x t mathbf u t approx mathbf f mathbf tilde x t mathbf tilde u t frac delta mathbf f delta mathbf x delta mathbf x frac delta mathbf f delta mathbf u delta mathbf u nbsp Pri vzyatti chastkovih pohidnih vektor funkciyi f displaystyle mathbf f nbsp za vektorom zminnih staniv x displaystyle mathbf x nbsp i vektorom vhidnih vpliviv u displaystyle mathbf u nbsp vihodyat matrici Yakobi vidpovidnih sistem funkcij d f d x d f 1 d x 1 d f 1 d x n d f n d x 1 d f n d x n d f d u d f 1 d u 1 d f 1 d u p d f n d u 1 d f n d u p displaystyle frac delta mathbf f delta mathbf x begin bmatrix frac delta mathbf f 1 delta mathbf x 1 amp cdots amp frac delta mathbf f 1 delta mathbf x n vdots amp ddots amp vdots frac delta mathbf f n delta mathbf x 1 amp cdots amp frac delta mathbf f n delta mathbf x n end bmatrix quad frac delta mathbf f delta mathbf u begin bmatrix frac delta mathbf f 1 delta mathbf u 1 amp cdots amp frac delta mathbf f 1 delta mathbf u p vdots amp ddots amp vdots frac delta mathbf f n delta mathbf u 1 amp cdots amp frac delta mathbf f n delta mathbf u p end bmatrix nbsp Analogichno dlya funkciyi vihodu d h d x d h 1 d x 1 d h 1 d x n d h q d x 1 d h q d x n d h d u d h 1 d u 1 d h 1 d u p d h q d u 1 d h q d u p displaystyle frac delta mathbf h delta mathbf x begin bmatrix frac delta mathbf h 1 delta mathbf x 1 amp cdots amp frac delta mathbf h 1 delta mathbf x n vdots amp ddots amp vdots frac delta mathbf h q delta mathbf x 1 amp cdots amp frac delta mathbf h q delta mathbf x n end bmatrix quad frac delta mathbf h delta mathbf u begin bmatrix frac delta mathbf h 1 delta mathbf u 1 amp cdots amp frac delta mathbf h 1 delta mathbf u p vdots amp ddots amp vdots frac delta mathbf h q delta mathbf u 1 amp cdots amp frac delta mathbf h q delta mathbf u p end bmatrix nbsp Z oglyadu na d x x x x displaystyle delta mathbf dot x mathbf dot x mathbf dot tilde x mathbf dot x nbsp linearizovanij opis dinamichnoyi sistemi v okoli robochoyi tochki nabude viglyadu de A d f d x B d f d u C d h d x D d h d u displaystyle mathbf A frac delta mathbf f delta mathbf x quad mathbf B frac delta mathbf f delta mathbf u quad mathbf C frac delta mathbf h delta mathbf x quad mathbf D frac delta mathbf h delta mathbf u nbsp Prikladi RedaguvatiModel u prostori staniv dlya mayatnika Redaguvati Mayatnik ye klasichnoyu vilnoyu nelinijnoyu sistemoyu Matematichno ruh mayatnika opisuye take spivvidnoshennya m l 8 t m g sin 8 t k l 8 t displaystyle ml ddot theta t mg sin theta t kl dot theta t nbsp de 8 t displaystyle theta t nbsp kut vidhilennya mayatnika m displaystyle m nbsp zvedena masa mayatnika g displaystyle g nbsp priskorennya vilnogo padinnya k displaystyle k nbsp koeficiyent tertya v pidshipniku pidvisu l displaystyle l nbsp dovzhina pidvisu mayatnikaU takomu vipadku rivnyannya v prostori staniv matimut viglyad x 1 t x 2 t displaystyle dot x 1 t x 2 t nbsp x 2 t g l sin x 1 t k m x 2 t displaystyle dot x 2 t frac g l sin x 1 t frac k m x 2 t nbsp de x 1 t 8 t displaystyle x 1 t theta t nbsp kut vidhilennya mayatnika x 2 t x 1 t displaystyle x 2 t dot x 1 t nbsp kutova shvidkist mayatnika x 2 t x 1 t displaystyle dot x 2 t ddot x 1 t nbsp kutove priskorennya mayatnikaZapis rivnyan stanu v zagalnomu viglyadi x t x 1 t x 2 t f t x t x 2 t g l sin x 1 t k m x 2 t displaystyle dot mathbf x t left begin matrix dot x 1 t dot x 2 t end matrix right mathbf f t x t left begin matrix x 2 t frac g l sin x 1 t frac k m x 2 t end matrix right nbsp Linearizaciya modeli mayatnika Redaguvati Linearizovana matricya sistemi dlya modeli mayatnika v okoli tochki rivnovagi x 1 0 displaystyle left tilde x 1 0 right nbsp maye viglyad d f d x 0 1 g l cos x 1 k m 0 1 g l k m displaystyle frac delta mathbf f delta mathbf x left begin matrix 0 amp 1 frac g l cos tilde x 1 amp frac k m end matrix right left begin matrix 0 amp 1 frac g l amp frac k m end matrix right nbsp Za vidsutnosti tertya v pidvisi k 0 otrimayemo rivnyannya ruhu matematichnogo mayatnika x g l x displaystyle ddot x frac g l x nbsp Div takozh RedaguvatiTeoriya keruvannya Fazovij prostir Kriterij stijkosti v prostori staniv Prostir ponyat Sistema vidliku Modalne keruvannyaLiteratura Redaguvatiknigi Andronov A A Leontovich E A Gordon I M Majer A G Teoriya bifurkacij dinamicheskih sistem na ploskosti M Nauka 1967 Andronov A A Vitt A A Hajkin S E Teoriya kolebanij 2 e izd pererab i ispr M Nauka 1981 918 s statti Fejgin M I Proyavlenie effektov bifurkacionnoj pamyati v povedenii dinamicheskoj sistemy arh 30 listopada 2007 ros Sorosovskij obrazovatelnyj zhurnal zhurnal 2001 T 7 3 S 121 127 Posilannya RedaguvatiVihidni diferencialni rivnyannya SAR ros Otrimano z https uk wikipedia org w index php title Prostir staniv teoriya keruvannya amp oldid 37832134
