16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 3 Lecture 23 Last time: (, ) () (, ) (,) d tAtt dt I τ τ ττ Φ=Φ Φ= So the covariance matrix for the state at time t is () () () () () () () () 0 0 0 00 1 111 0 2 2 2 2 00 002 2 2 2 () () () () () () () , () , ()() () , () () , ,()() , ,()()() , T T tt TT TTT T T t T TT t Xt xt xt xt xt xtxt Ettxt tBndxt t n B t d tt xtxt tt tt xt n B t d t τ τττ τ τ τ τ ττ ττ ???? =? ? ???? = ? ? =Φ +Φ Φ + Φ ??? ? ??? ? =Φ Φ +Φ Φ +Φ ∫∫ ∫ %% %%%% %% %% () () () 0 00 1110 0 1 12 1 11 2 2 2 , ( )( )() , ,()()()()(,) t T T t tt TT T tt Bnxt ttd dd t Bnn B t τττ τ ττ τ τττ τ τ Φ +Φ Φ ∫ ∫∫ %% %% The two middle terms are zero: - For 0 tτ > , ()n τ% and 0 ()x t% are uncorrelated because ()n τ% is white (impulse correlation function) - For 0 tτ = , ()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. () () () ( ) () () () 00 0 00 0 12 111212 2 00 0 () , ( ) , , ( ) ( ) ( ) (, ) ,(), ,()()()(,) tt T TT tt t T TT t Xt tt Xt tt d d t B N B t tt Xt tt t B N B t d τ ττττδτττ τ ττττ 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 Page 2 of 3 () () () () () () () () 0 0 00 0 00 0 () () , ( ) , ,(), () () , ( ) ( ) ( ) , ,()()() , () () () () () () () () () () () () T T T t T T t t T TT t T TT d Xt At tt Xt tt dt tt Xt tt At At t B N B t d tBNB t Atd BtNtBt d Xt AtXt XtAt BtNtBt dt τ τττ ττ τ τττ 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 0t = . 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 (0,0)X which are variances or covariances involving the states of the system are zero. - All elements of (0,0)X 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 Page 3 of 3 , System states only System and filter states System and filter states Filter states only 00 (0,0) ()0 T T ss s T T s vv X xx xv vx vv X X ?? = ?? ?? ?? =?? ?? ?? = ?? ∞ ?? where System states Shaping filter states s x x v ?? == ?? ???? With this initialization, , () vv X t will remain constant – which it should do if we think of ()x t as a member of a stationary process.