lecture14

# lecture14 - 16.322 Stochastic Estimation and Control Fall...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

lecture14 - 16.322 Stochastic Estimation and Control Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online