16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde 0 Hx+y The filter is 文=A+Gm+K(x-压) P= AP+ pa+ BNB'-PhR-HP where K= Ph R Steady state properties If the system is time invariant and the random processes are stationary (A, B, H, N, R all constants), then the differential equation for P may approach a constant steady state value as time increases Some characteristics of the steady state solution: D)If(A, H) is detectable and(A, B) is stabilizable, the solution of the ricatti differential equation approaches a steady state value Po which independent of the initial condition Po 2) Under the conditions of (1), this P is the unique positive semidefinite solution of the algebraic Ricatti equation AP+PA+BNB'-PHR-HP=o 3)The steady state filter i=(A-KH)i+ Gu+K2 is asymptotically stable if and only if(A, H)is detectable at least and(A, B)is stabilizable at least with K=ph r- Page 8 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 9 z xv [1 0] Hx v = + = + The filter is: ( ) 1 ˆˆ ˆ T TT x Ax Gu K z Hx P AP PA BNB PH R HP − =++ − =+ + −   where T 1 K PH R− = . Steady state properties If the system is time invariant and the random processes are stationary ( ABH N R ,, , , all constants), then the differential equation for P may approach a constant steady state value as time increases. Some characteristics of the steady state solution: 1) If (, ) A H is detectable and (, ) A B is stabilizable, the solution of the Ricatti differential equation approaches a steady state value P∞ which is independent of the initial condition P0 . 2) Under the conditions of (1), this P∞ is the unique positive semidefinite solution of the algebraic Ricatti equation 1 0 T TT AP PA BNB PH R HP − ++ − = 3) The steady state filter xˆ ˆ =− + + ( ) A KH x Gu Kz  is asymptotically stable if and only if ( A H, ) is detectable at least and ( ) A B, is stabilizable at least with T 1 K PH R− = ∞