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lecture14 - 16.322 Stochastic Estimation and Control Fall...

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Unformatted text preview: 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 6 Lecture 14 Last time: ( , ) ( ) w t w t τ τ ⇒ − ( ) ( ) ( ) ' Let: ( ) ( ) ( ) t y t w t x d t d d y t w x t d τ τ τ τ τ τ τ τ τ τ −∞ ∞ = − = − ⎧ ⎨ ′ − = ⎩ ′ ′ ′ = − ∫ ∫ For the differential system characterized by its equations of state, specialization to invariance means that the system matrices , , A B C are constants. x Ax Bu y Cx = + = ¡ For , , A B C constant: ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t y t Cx t x t t t x t t B u d τ τ τ = = Φ − + Φ − ∫ The transition matrix can be expressed analytically in this case. ( , ) ( , ), where ( , ) d t A t I dt τ τ τ τ Φ = Φ Φ = This is a matrix form of first order, constant coefficient differential equation. The solution is the matrix exponential. ( ) ( ) 2 2 ( , ) 1 1 ( ) ( ) ... ( ) ... 2 ! A t A t k k t e e I A t A t A t k τ τ τ τ τ τ − − Φ = = + − + − + + − + Useful for computing ( ) t Φ for small enough t τ − . 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 6 The solution is ( ) ( ) ( ) ( ) ( ) ( ) ( ) t A t t A t t y t Cx t x t e x t e B u t d τ τ − − = = + ∫ For t → ∞ : ( ) ( ) ( ) ( ) t A t A x t e B u d e Bu t d τ τ τ τ τ τ − −∞ ∞ ′ = ′ ′ = − ∫ ∫ and for a single input, single output (SISO) system, ( ) T At w t c e b = If ( ) j t x t e ω = for all past time ( ) ( ) ( ) ( ) ( ) ( ) j t j j t y t w e d w e d e F x t ω τ ω τ ω τ τ...
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lecture14 - 16.322 Stochastic Estimation and Control Fall...

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