Estimation of noise covariances and identification of disturbance structure
Systematic methods and tools for managing the complexity
Process Control (T4-8)
Keywords: Kalman filter, covariance estimation, state estimation, disturbance model
In state estimation the state of a system is reconstructed from process measurements. State estimation has important applications in control, monitoring and fault detection of chemical processes. The Kalman filter and its counterpart for nonlinear systems, the extended Kalman filter, are well-established techniques for state estimation. However, a well-known drawback of Kalman filters is that knowledge about process and measurement noise statistics is required from the user. In practical applications the noise covariances are generally not known. Tuning the filter, i.e. choosing the values of the process and measurement noise covariances such that the estimation error is minimized, is a challenging task. If performed manually in an ad hoc fashion it represents a considerable burden for the user. Since better estimates result in better control there is a motivation for developing methods for Kalman filter tuning.
The filter tuning problem is essentially a covariance estimation problem where the Kalman filter gain is computed based on the estimated covariances. A promising technique for covariance estimation is the autocovariance least-squares method proposed recently by Odelson and co-workers for linear time-invariant systems (Odelson et al. 2006). This method is based on the estimated autocovariance of the output innovations, which is used to compute a least-squares estimate of the noise covariance matrices. The approach has the advantage that routine operating data from the process can be used. The estimation problem can be stated in the form of a convex semidefinite program, which can be solved by interior-point methods which guarantee positive semidefiniteness of the covariance matrices. The estimation method has been generalized to systems where the process noise and the measurement noise are mutually correlated.
The structure of the noise that enters the system is likely to be unknown as well. Generally there are only a few independent disturbances affecting the states and by minimizing the rank of the process noise covariance matrix the noise structure can be identified. The trace is used as an approximation of the rank and is added as an objective to the least-squares problem. The minimum rank can be determined by plotting the trace versus the fit to data for different values of a weight parameter (Rajamani and Rawlings 2006). To extract information about the number of disturbances and the structure of the disturbances, singular value decomposition can be applied to the combined covariance matrix for both process and measurement noise. The number of disturbances is the number of nonzero singular values and the noise shaping matrices for process and measurement noise are obtained from the orthogonal matrices containing the singular vectors.
The performance of the noise covariance estimation method and the disturbance identification technique is demonstrated on numerical examples from the chemical process industry.
References:
1. Odelson, B. J., M. R. Rajamani and J. B. Rawlings (2006). A new autocovariance
least-squares method for estimating noise covariances, Automatica 42(2): 303-308.
2. Rajamani, M. R. and J. B. Rawlings (2006). Estimation of noise covariances and disturbance structure from data using least squares with optimal weighting, AIChE Annual Meeting, San Francisco, CA, USA.
Presented Tuesday 18, 16:20 to 16:40, in session Process Control (T4-8).
