16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 14 Last time: w(t, r)=w(t-T) ()=」v(t-r)x( Let y(r=w(r)x(t-t)dr For the differential system characterized by its equations of state, specialization to invariance means that the system matrices A, B, C are constants. =Cx For AB. constant. y(1)=Cx(D) x()=Φ(t-1)x()+」c(t-)B(ldr The transition matrix can be expressed analytically in this ca ap(t, r)= A(t, r), where (r,r) This is a matrix form of first order constant coefficient differential equation. The solution is the matrix exponential +A(t-)+A(t-r)2+…+4(1-r) Useful for computing a(t) for small enough t-T Page 1 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 6 Lecture 14 Last time: wt wt (, ) ( ) τ ⇒ −τ 0 () ( ) ( ) ' Let: () ( ) ( ) t yt wt x d t d d yt w xt 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 ABC , , are constants. x Ax Bu y Cx = + =  For ABC , , constant: 0 0 0 () () () ( ) ( ) ( ) ( ) t t y t Cx t x t t t x t t Bu d τ τ τ = =Φ − + Φ − ∫ The transition matrix can be expressed analytically in this case. ( , ) ( , ), where ( , ) d t At I dt Φ =Φ Φ = τ τ ττ 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 At k k t e e I At A t A t k τ τ τ ττ τ − − Φ = =+ − + − + + − + Useful for computing Φ( )t for small enough t −τ