Aircraft Turbofan Engine Health Estimation Using Constrained Kalman Filtering

Kalman 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 paper develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. The incorporation of state variable constraints increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation results obtained from application to a tiu-bofan engine model. This model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new algorithms provide improved performance in this example over unconstrained Kalman filtering. INTRODUCTION For linear dynamic systems with white process and measurement noise, the Kalman filter is known to be an optimal estimator. However, in the application of Kalman filters there is often known model or signal information that is either ignored or dealt with heuristically [1]. This paper presents a way to generalize the Kalman filter in such a way that known inequaHty constraints among the state variables are satisfied by the state variable estimates. The method presented here for enforcing inequality constraints on the state variable estimates uses hard constraints. It is based on a generalization of the approach presented in [2], which dealt with the incorporation of state variable equality constraints in the Kalman filter. Inequality constraints are inherently more compHcated than equality constraints, but standard quadratic programming results can be used to solve the Kalman filter problem with inequality constraints. At each time step of the constrained Kalman filter, we solve a quadratic programming problem to obtain the constrained state estimate. A family of constrained state estimates is obtained, where the weighting matrix of the quadratic programming problem determines which family member forms the desired solution. It is stated in this paper, on the basis of [2], that the constrained estimate has several important properties. The constrained state estimate is unbiased (Theorem 1 below) and has a smaller error covariance than the unconstrained estimate (Theorem 2 below). We show which member of all possible constrained solutions has the smallest error covariance (Theorem 3 below). We also show the one particular member that is always (i.e., at each time step) closer to the true state than the unconstrained estimate (Theorem 4 below). Finally, we show that the variation of the constrained estimate is smaller than the variation of the unconstrained estimate (Theorem 5 below). The application considered in this paper is turbofan engine health parameter estimation [3]. The performance of gas turbine engines deteriorates over time. This deterioration can affect the fuel economy, and impact emissions, component Ufe consumption, and thrust response of the engine. Airlines periodically collect engine data in order to evaluate the health of the engine and its components. The