Kalman filter constraint tuning for turbofan engine health estimation

Original publisher: Cleveland, Ohio : National Aeronautics and Space Administration, Glenn Research Center, [2005] OCLC Number: (OCoLC)648242596 Excerpt: ... we can accomplish constraint tuning one state at a time. The constraint tuning algorithm can be summarized as follows. 1. We are given the following system with n states, q measurements, and s constraints. x ( k + 1 ) = Ax ( k ) + Bu ( k ) + w ( k ) ( 15 ) y ( k ) = Cx ( k ) + e ( k ) D ( k ) x ( k ) · d ( k ) We initialize the Kalman ¯ lter quantities x ^ ( 0 ), x ~ ( 0 ), and § ( 0 ). 2. At each time step k = 0; 1; ¢ ¢ ¢, perform the following. ( a ) Run the unconstrained and constrained Kalman ¯ lters as follows. T T ¡ 1 K ( k ) = A § ( k ) C ( C § ( k ) C + R ) ( 16 ) x ^ ( k + 1 ) = Ax ^ ( k ) + Bu ( k ) + K ( k ) ( y ( k ) ¡ Cx ^ ( k ) ) T § ( k + 1 ) = ( A § ( k ) ¡ K ( k ) C § ( k ) ) A + Q T min [ x ~ ( k + 1 ) ¡ x ^ ( k + 1 ) ] W ( k + 1 ) [ x ~ ( k + 1 ) ¡ x ^ ( k + 1 ) ] x ~ ( k + 1 ) such that D ( k + 1 ) x ~ ( k + 1 ) · d ( k + 1 ) where W ( k ) is our weighting matrix ( see Section 2.1 ). This gives us an unconstrained estimate x ^ ( k + 1 ) and a constrained estimate x ~ ( k + 1 ). ( b ) Compute the theoretical residual covariance S ( k + 1 ) from ( 12 ). ( c ) For i = 1; ¢ ¢ ¢; n, perform the following. i. Find the rows with the ¹ largest magnitudes in the ith column of the ¢ matrix. Label these row numbers M ( j = 1; ¢ ¢ ¢; ¹ ). ji ii. Examine the ¹ residuals that correspond to measurement numbers M ( j = 1; ¢ ¢ ¢; ¹ ). If all ¹ of these residuals have been smaller than ji ® S ( k + 1 ) ( where r = M ) for · consecutive time steps, then use rr ji x ^ ( k + 1 ) as the estimate of the ith state. Otherwise, use x ~ ( k + 1 ) i i as the estimate of the ith state NASA / TM - 2005-213962 11