Constrained Kalman filtering via density function truncation for turbofan engine health estimation

Abstract

Kahnan filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically_ For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This article develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter truncates the probability density function (PDF) of the Kalman filter t'Stimate at the known constraints and then computes the constrained filter estimate as the mean of the truncated PDF. The incorporation of state variable constraints increases the computational effort of the filter but also improves its estimation accuracy. The improvement is demonstrated via simulation results obtained from a turbofan engine modeL It is also shown that the truncated Kalman filter may provide a more accurate way of incorporating inequality constraints than other constrained filters (e.g. the projection approach to constrained filtering).

Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

“…After all the constraints are applied, the normalization process is reversed to obtain the constrained state estimate. Details of the algorithm are given in [2,20]…”

“…The constrained state estimate is equal to the mean of the truncated PDF [2,19,20] This method is complicated when the state dimension is more than one. In that case the state estimate is normalized so that its components are statistically independent of each other.…”

“…Section 2 considers linear systems and linear constraints. The various ways of enforcing linear constraints in the linear Kalman filter include model reduction [15], perfect measurements [8 -10], estimate projection [13,16], gain projection [17, 1S], probability density function (PDF) truncation [2,19,20], and system projection [21]. Under certain conditions, all these approaches result in the same state estimate.…”

Gas-Path Health Estimation for an Aircraft Engine Based on a Sliding Mode Observer

“…server designing process. The coordinate (4) transformation, the measurement matrix (5) ordinates, Equation (3) can be described ) ) t (6) in the form of where ∈ ) is added with modeling uncertainties as ( , , ) t ξ x u (7) ution matrix.…”

“…Luppold et al [4] proposed an algorithm based on a KF concept to estimate in-flight engine performance variations. Simon et al [5] applied the constrained KF approach, along with constraint tuning on the basis of measurement residuals, to estimate engine In order to estimate health parameters via observer techniques, health parameters is required to be considered as state variables, thus the following augmented system is created: …”