Validation of fuzzy Kalman Filters using the local statistical approach to fault diagnosis
Gerasimos G. Rigatos, Pierluigi Siano · 2012
To obtain accurate estimates with the linear Kalman Filter, previously a tuning process is required. A question that arises is about which state estimates can be considered as reliable. There is need for systematic methods showing when the Kalman Filter is not performing optimally and when its retuning, either in terms of the used dynamic or kinematic model or in terms of the covariance matrices, should be performed. Several tests can be applied to test the consistency of the linear Kalman Filter, according to the characteristics of the measurement residuals. These include the normalized error square test, the autocorrelation test and the normalized mean error test [1-2]. The above mentioned tests can be applied to the residuals of the Kalman Filter for checking the consistency of the obtained estimation and the filter’s optimality. If the tests are not satisfied then this means that the Kalman Filter is not running optimally, and the filter has to be retuned, or that the filter’s design has to be reconsidered. The paper aims at developing a systematic method for the validation of nonlinear estimators, such as fuzzy Kalman Filters. The fuzzy Kalman Filter is designed based on the coverage of the state space by several local linear estimators according to the concept of fuzzy local linearization. Actually, the fuzzy Kalman Filter consists of local linear Kalman estimators which are weighted by fuzzy membership functions. It can be shown that the fuzzy Kalman Filter can be written in the form of a Takagi-Sugeno fuzzy model, which stands for local ARMA models weighted by fuzzy membership functions [3-5]. It is necessary to validate periodically the accuracy of such estimators, i.e. to evaluate if there are discrepancies between the model of the system used by the state estimator and the real system kinematics or dynamics. Validation of nonlinear state estimators is important for several applications (e.g. target tracking and motion estimation through the processing of measurements provided by one single or multiple sensors). Application domains for which this validation procedure would be particularly useful are autonomous navigation systems and defense systems. The paper proposes the local statistical approach to fault diagnosis for validating the fuzzy Kalman Filter [6-8]. Validation of the fuzzy Kalman filter with the Local Statistical Approach has two significant advantages: i) it provides a credible criterion (2 test) to detect if the fuzzy Kalman Filter is acceptable or not, no matter what the distribution of the training data is. This criterion is more efficient than the aforementioned normalized square error and mean error tests since it employs the modeling error derivative and records the tendency for change. Thus early change detection for the filter’s parameters becomes possible ii) it recognizes the parameters of the fuzzy Kalman Filter that are responsible for the deviation of the filter’s estimates from the real output of the monitored dynamical system. Thus the retuning or redesign of the fuzzy Kalman Filter can focus on a small number of its parameters. References [1] M. Basseville and I. Nikiforov, Detection of Abrupt changes, Prentice Hall, 1993. [2] J.L. Crassidis and J.L. Junkins, Optimal estimation of dynamic systems (2nd Edition), CRC Press, 2012. [3] C.J. Harris and Q. Gan, State estimation and multi-sensor data fusion using data-based neurofuzzy local linearization process models, Information Fusion, Elsevier, vol. 2, pp. 17-29, 2001. [4] S. McGinnity and G. Irwin, Nonlinear Kalman Filtering Using Fuzzy Local Linear Models, Proc. of the American Control Conference Albuquerque, New Mexico June 1997. [5] G.G. Rigatos, Modelling and control for intelligent industrial systems: adaptive algorithms in robotics and industrial engineering, Springer, 2011. [6] Q. Zhang, M. Basseville, A. Benveniste, Fault Detection and Isolation in Nonlinear Dynamic Systems: A Combined Input-Output and Local Approach, Automatica, Elsevier, vol.34, no. 11, pp. 1359-1373, 1998. [7] G. Rigatos and Q. Zhang, Fuzzy Model Validation using the Local Statistical Approach, Publication Interne IRISA No 1417, Rennes, France, 2001. [8] G. Rigatos and Q. Zhang, Fuzzy model validation using the local statistical approach, Fuzzy Sets and Systems, Elsevier, vol. 60, no.7, pp. 882-904, 2009.