16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde o far we have looked only at the R_ and S of processes. We shall find that if we wish to determine only the Rw or S(thus the y )of outputs of linear systems, all we need to know about the inputs are their R r or Sxr But what if we wanted to know the probability that the error in a dynamic system would exceed some bound? For this we need the first probability density function of the system error-an output very difficult in general n(t) y(t) System The pdf of the output y(t) satisfies the Fokker-Planck partial differential equation-also called the Kolmagorov forward equation. Applies to a continuous dynamic system driven by a white noise process Gaussian process Linear process System One case is easy: Gaussian process into a linear system, output is Gaussian Gaussian processes are defined by the property that probability density functions of all order are normal functions n(xr 41, x2,42,;.x,, t,=n-dimensional normal f(x)= LMI M is the covariance matrix for x(2) x() Thus, fn (x) for all n is determined by M-the covariance matrix for x16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 8 So far we have looked only at the Rxx and xx S of processes. We shall find that if we wish to determine only the Ryy or yy S (thus the 2 y ) of outputs of linear systems, all we need to know about the inputs are their Rxx or xx S . But what if we wanted to know the probability that the error in a dynamic system would exceed some bound? For this we need the first probability density function of the system error – an output. Very difficult in general. The pdf of the output y t( ) satisfies the Fokker-Planck partial differential equation – also called the Kolmagorov forward equation. Applies to a continuous dynamic system driven by a white noise process. One case is easy: Gaussian process into a linear system, output is Gaussian. Gaussian processes are defined by the property that probability density functions of all order are normal functions. 11 2 2 ( , ; , ;... , ) -dimensional normal n nn f xtxt xt n = 1 1 2 2 1 ( ) (2 ) TxM x n n fx e π M − − = M is the covariance matrix for 1 2 ( ) ( ) ( ) n x t x t x x t ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ M Thus, ( ) nf x for all n is determined by M - the covariance matrix for x