16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 23 ast time d(t,r)=A(1)(t,r d(r,z)=1 So the covariance matrix for the state at time t is x(1)=x(1)-x(1)x(t)-x(r) =x(1)x(1) =Ea(4)x4)+j()B(7厘(xM1y(24)+J(yB()()dt =Φ(14)x0)x()d(t,) +∫a()x(6)(z)B(z)(z)d2 +∫a(.x)B()x)7a(4)dr +∫d可Jdr(t.x)B(x)(工(可)B(2)(x)y The two middle terms are zero For t>to, n(r) and x(to) are uncorrelated because n(r) is white(impulse correlation function For T=to, n(t) has a finite effect on x(to) because n(r) is white. But the integral of a finite quantity over one point is zero X()=(4)X()(5)+ dr dt2(x)B()N()5(z2-z)B(z2yo(x2 =(4)X((4)+」o(t:)B)N()(ry(r)yar This is an integral expression for the state covariance matrix. But we would prefer to have a differential equation. So take the derivative with respect to time Page 1 of 3
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 3 Lecture 23 Last time: (, ) () (, ) (,) d t At t dt I τ τ τ τ Φ =Φ Φ = So the covariance matrix for the state at time t is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 1 1 11 0 2 2 2 2 0 0 002 2 2 2 () () () () () () () , ( ) , ( ) ( ) () , ( ) ( ) , , () () , , ()( ) ( ) , T T t t T TT T T t t T T t T T T t Xt xt xt xt xt xtxt E tt xt t B n d xt tt n B t d tt xtxt tt tt xt n B t d t τ τ ττ τ τ τ τ τ τ ττ =− − ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ = ⎡ ⎤⎡ ⎤ = Φ +Φ Φ + Φ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ =Φ Φ +Φ Φ + Φ ∫ ∫ ∫ % % % %% % % % % % ( ) ( ) ( ) 0 0 0 1 1 10 0 1 12 1 11 2 2 2 , ( )( )() , , ( ) ( ) ( ) ( ) (, ) t T T t t t TT T t t B n xt tt d dd t Bn n B t τ ττ τ ττ τ τττ τ τ Φ +Φ Φ ∫ ∫ ∫ % % % % The two middle terms are zero: - For 0 τ > t , n%( ) τ and 0 x%( ) t are uncorrelated because n%( ) τ is white (impulse correlation function) - For 0 τ = t , n%( ) τ has a finite effect on 0 x%( ) t because n%( ) τ is white. But the integral of a finite quantity over one point is zero. () () ( ) ( ) ( ) ( ) () 0 0 0 0 0 0 1 2 1 1 1 21 2 2 00 0 () , ( ) , , ( ) ( ) ( ) (, ) , ( ) , , ( ) ( ) ( ) (, ) t t T T T t t t T T T t Xt tt Xt tt d d t B N B t tt Xt tt t B N B t d τ τ τ τ τ δτ τ τ τ τ τ ττ ττ =Φ Φ + Φ − Φ =Φ Φ + Φ Φ ∫ ∫ ∫ This is an integral expression for the state covariance matrix. But we would prefer to have a differential equation. So take the derivative with respect to time
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde x(1)=A(y(,40)X(4)Φ(t4) +(t,4)X()dp(40)4) +A((L, r)B()N(o)B(r)'(,t)dr +o(s, r)B(0)N(a)B(r)'d(i, r)"A(dr +B(1)N(t)B() A x(=A(0)X(o+X(O)A(0+B(ON(OB( This defines the first and second order statistics of the state Initial conditions Often we wish to compute the time evolution of the statistics of a system which starts from rest at time zero. If the input to this real system is being formed by a shaping filter, then not all elements of X are zero at t=0 We want to model x(t) as a stationary process This situation is equivalent to aping u(t) stem Filter where the white noise input has been applied for all past time. Thus at time zero All elements of X(o,o) which are variances or covariances involving the states of the system are zero All elements of X(0, O)which are variances or covariances involving onl states of the shaping filter are at their steady state values for the shaping filter alone driven by the white noise Page 2 of 3
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 3 () () () () () () () () 0 0 00 0 00 0 () () , ( ) , , ( ) , () () , ( ) ( ) ( ) , , ( ) ( ) ( ) , () () () () () () () () () () () () T T T t T T t t T T T t T T T d X t At tt X t tt dt tt X t tt At At t B N B t d t B N B t At d BtNtBt d X t At X t X t At Bt N t Bt dt τ τ ττ τ τ τ τ ττ τ τ =Φ Φ +Φ Φ +Φ Φ +Φ Φ + =+ + ∫ ∫ This defines the first and second order statistics of the state. Initial conditions Often we wish to compute the time evolution of the statistics of a system which starts from rest at time zero. If the input to this real system is being formed by a shaping filter, then not all elements of X are zero at t = 0. We want to model x( )t as a stationary process. This situation is equivalent to: where the white noise input has been applied for all past time. Thus at time zero: - All elements of X (0,0) which are variances or covariances involving the states of the system are zero. - All elements of X (0,0) which are variances or covariances involving only states of the shaping filter are at their steady state values for the shaping filter alone driven by the white noise
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde X System states only System and filter states System and filter states Filter states only X(0,0) 0X,,(∞) The System states v Shaping filter states With this initialization X.(t) will remain constant- which it should do if we think of x(t) as a member of a stationary process
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 3 , System states only System and filter states System and filter states Filter states only 0 0 (0,0) 0 ( ) T T s s s T T s v v X x x x v vx vv X X ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ∞ where System states Shaping filter states s x x v ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ With this initialization, , ( ) Xv v t will remain constant – which it should do if we think of x( )t as a member of a stationary process
