Yilda boshqaruv nazariyasi , holatga o'tish matritsasi koeffitsienti vektorga ega bo'lgan matritsa x { displaystyle x} t 0 { displaystyle t_ {0}} x { displaystyle x} t { displaystyle t}
Lineer tizim echimlari
Umumiy holatga o'tish matritsasi umumiy echimni topish uchun ishlatiladi davlat-kosmik vakolatxonasi a chiziqli tizim quyidagi shaklda
x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) siz ( t ) , x ( t 0 ) = x 0 { displaystyle { dot { mathbf {x}}} (t) = mathbf {A} (t) mathbf {x} (t) + mathbf {B} (t) mathbf {u} (t) ), ; mathbf {x} (t_ {0}) = mathbf {x} _ {0}} qayerda x ( t ) { displaystyle mathbf {x} (t)} siz ( t ) { displaystyle mathbf {u} (t)} A ( t ) { displaystyle mathbf {A} (t)} B ( t ) { displaystyle mathbf {B} (t)} matritsa funktsiyalari va x 0 { displaystyle mathbf {x} _ {0}} t 0 { displaystyle t_ {0}} Φ ( t , τ ) { displaystyle mathbf { Phi} (t, tau)} [1] [2]
x ( t ) = Φ ( t , t 0 ) x ( t 0 ) + ∫ t 0 t Φ ( t , τ ) B ( τ ) siz ( τ ) d τ { displaystyle mathbf {x} (t) = mathbf { Phi} (t, t_ {0}) mathbf {x} (t_ {0}) + int _ {t_ {0}} ^ {t } mathbf { Phi} (t, tau) mathbf {B} ( tau) mathbf {u} ( tau) d tau} Birinchi atama nolinchi javob va ikkinchi atama sifatida tanilgan nol holatidagi javob .
Peano-Beyker seriyasi
Eng umumiy o'tish matritsasi Peano-Beyker seriyasida berilgan
Φ ( t , τ ) = Men + ∫ τ t A ( σ 1 ) d σ 1 + ∫ τ t A ( σ 1 ) ∫ τ σ 1 A ( σ 2 ) d σ 2 d σ 1 + ∫ τ t A ( σ 1 ) ∫ τ σ 1 A ( σ 2 ) ∫ τ σ 2 A ( σ 3 ) d σ 3 d σ 2 d σ 1 + . . . { displaystyle mathbf { Phi} (t, tau) = mathbf {I} + int _ { tau} ^ {t} mathbf {A} ( sigma _ {1}) , d sigma _ {1} + int _ { tau} ^ {t} mathbf {A} ( sigma _ {1}) int _ { tau} ^ { sigma _ {1}} mathbf {A } ( sigma _ {2}) , d sigma _ {2} , d sigma _ {1} + int _ { tau} ^ {t} mathbf {A} ( sigma _ {1 }) int _ { tau} ^ { sigma _ {1}} mathbf {A} ( sigma _ {2}) int _ { tau} ^ { sigma _ {2}} mathbf { A} ( sigma _ {3}) , d sigma _ {3} , d sigma _ {2} , d sigma _ {1} + ...} qayerda Men { displaystyle mathbf {I}} identifikatsiya matritsasi . Ushbu matritsa bir xil va mutlaqo mavjud va noyob echimga yaqinlashadi.[2]
Boshqa xususiyatlar
Davlat o'tish matritsasi Φ { displaystyle mathbf { Phi}}
1. U uzluksiz va uzluksiz hosilalariga ega.
2, bu hech qachon birlik emas; Aslini olib qaraganda Φ − 1 ( t , τ ) = Φ ( τ , t ) { displaystyle mathbf { Phi} ^ {- 1} (t, tau) = mathbf { Phi} ( tau, t)} Φ − 1 ( t , τ ) Φ ( t , τ ) = Men { displaystyle mathbf { Phi} ^ {- 1} (t, tau) mathbf { Phi} (t, tau) = I} Men { displaystyle I}
3. Φ ( t , t ) = Men { displaystyle mathbf { Phi} (t, t) = I} t { displaystyle t} [3]
4. Φ ( t 2 , t 1 ) Φ ( t 1 , t 0 ) = Φ ( t 2 , t 0 ) { displaystyle mathbf { Phi} (t_ {2}, t_ {1}) mathbf { Phi} (t_ {1}, t_ {0}) = mathbf { Phi} (t_ {2}, t_ {0})} t 0 ≤ t 1 ≤ t 2 { displaystyle t_ {0} leq t_ {1} leq t_ {2}}
5. Differentsial tenglamani qondiradi ∂ Φ ( t , t 0 ) ∂ t = A ( t ) Φ ( t , t 0 ) { displaystyle { frac { kısalt mathbf { Phi} (t, t_ {0})} { qismli t}} = mathbf {A} (t) mathbf { Phi} (t, t_ { 0})} Φ ( t 0 , t 0 ) = Men { displaystyle mathbf { Phi} (t_ {0}, t_ {0}) = I}
6. Holat-o'tish matritsasi Φ ( t , τ ) { displaystyle mathbf { Phi} (t, tau)}
Φ ( t , τ ) ≡ U ( t ) U − 1 ( τ ) { displaystyle mathbf { Phi} (t, tau) equiv mathbf {U} (t) mathbf {U} ^ {- 1} ( tau)} qaerda n × n { displaystyle n times n} U ( t ) { displaystyle mathbf {U} (t)} asosiy echim matritsasi bu qondiradi
U ˙ ( t ) = A ( t ) U ( t ) { displaystyle { dot { mathbf {U}}} (t) = mathbf {A} (t) mathbf {U} (t)} U ( t 0 ) = Men { displaystyle mathbf {U} (t_ {0}) = I} 7. Davlatga berilgan x ( τ ) { displaystyle mathbf {x} ( tau)} τ { displaystyle tau} t { displaystyle t}
x ( t ) = Φ ( t , τ ) x ( τ ) { displaystyle mathbf {x} (t) = mathbf { Phi} (t, tau) mathbf {x} ( tau)} Davlat-o'tish matritsasini baholash
In vaqt o'zgarmas holda, biz aniqlay olamiz Φ { displaystyle mathbf { Phi}} matritsali eksponent , kabi Φ ( t , t 0 ) = e A ( t − t 0 ) { displaystyle mathbf { Phi} (t, t_ {0}) = e ^ { mathbf {A} (t-t_ {0})}}
In vaqt varianti holat, holatga o'tish matritsasi Φ ( t , t 0 ) { displaystyle mathbf { Phi} (t, t_ {0})} siz ˙ ( t ) = A ( t ) siz ( t ) { displaystyle { dot { mathbf {u}}} (t) = mathbf {A} (t) mathbf {u} (t)} siz ( t 0 ) { displaystyle mathbf {u} (t_ {0})} [ 1 , 0 , … , 0 ] T { displaystyle [1, 0, ldots, 0] ^ {T}} [ 0 , 1 , … , 0 ] T { displaystyle [0, 1, ldots, 0] ^ {T}} [ 0 , 0 , … , 1 ] T { displaystyle [0, 0, ldots, 1] ^ {T}} n { displaystyle n} Φ ( t , t 0 ) { displaystyle mathbf { Phi} (t, t_ {0})} Φ ( t , τ ) = Φ ( t , t 0 ) Φ ( τ , t 0 ) − 1 { displaystyle mathbf { Phi} (t, tau) = mathbf { Phi} (t, t_ {0}) mathbf { Phi} ( tau, t_ {0}) ^ {- 1} } t 0 ≤ τ ≤ t { displaystyle t_ {0} leq tau leq t}
Shuningdek qarang
Adabiyotlar
^ Baake, Maykl; Schlaegel, Ulrike (2011). "Peano Beyker seriyasi". Steklov nomidagi Matematika instituti materiallari . 275 : 155–159. ^ a b Rugh, Wilson (1996). Lineer tizim nazariyasi . Yuqori Saddle River, NJ: Prentice Hall. ISBN 0-13-441205-2 ^ Brokett, Rojer V. (1970). Sonlu o'lchovli chiziqli tizimlar . John Wiley & Sons. ISBN 978-0-471-10585-5 Qo'shimcha o'qish
![{ displaystyle [1, 0, ldots, 0] ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3580cffe2bd91c872d764504aed41994da942d50)
![{ displaystyle [0, 1, ldots, 0] ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/501377442a7e92022254de32e0cd25192ac734ac)
![{ displaystyle [0, 0, ldots, 1] ^ {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ffe0074f9f159b8c6daa47484f6d55d1a4923e)
