Fault diagnosis (FD) for industrial processes requires reliable models for effective malfunctioning detection. As linear models are seldom effective in describing complex industrial processes, more complicated non linear models should be used for this purpose. The construction of structured non-linear model from input-output data is currently under investigation in several institutes and laboratories all over the world but is far to be fully established, especially when we consider that in industrial environment the acquired data are always affected by noise. A further problem is to consider that the identification procedure should be coupled to a fault diagnosis algorithm.
In order to establish a methodology applicable in a wide class of industrial plants, we can observe that in many cases the processes can be described using simple model having a local validity around an operating condition. Therefore, instead of exploiting complicated non-linear models obtained by modelling techniques, it is also possible to describe the plant by a collection of affine models. Each submodel approximates the system locally around a working condition and a selection procedure determines which particular submodel has to be used. Such a multimodel structure will be called multiple model approach. At each operating point, the behaviour of the multiple model is described by a local affine dynamic model.
This approach is currently explored by several researchers, among them we can cite the works by Billings and Leonaritis , Takagi and Sugeno , Branicky and Benveniste [3,4]. In particular this technique has been studied in coupling with FD algorithm by Patton et al. .
However a few attention has been paid to the problem of noise affecting plant measurements, which can significantly decrease the suitability of the model in the context of FD, in particular producing false alarms. As in practical condition the measured data are always affected by noise, this problem is really important to be carefully considered.
Some preliminary study have been already carried on this stream,
particularly in the context of linear models, by Kalman and Beghelli
[6,7,8], and recently for non linear models, by Beghelli, Fantuzzi,
Rovatti and Simani [9-13].
Multiple model approach
The construction of the multiple model from only one set of global input-output noisy measurements is a non-trivial problem since the model structure, a switching function and the local model parameters have to be identified.
The technique we aim to develop concerns the estimate of the operating point regions, the identification of the structure and parameters of the piecewise affine system based on local linear models from input-output data affected by noise. A non-linear dynamic process is, in fact, described as a composition of several local submodels selected according to the process operating conditions. This project addresses a method for the identification and the optimal selection of the local submodels from a sequence of noisy measurements acquired from the process.
In particular, in this project, a novel non-linear identification technique [9-12] is combined with the model-based method  to formulate a fault detection and isolation (FDI) tool exploiting the multimodel approach for residual generation. The model for non-linear dynamic systems is described by a number of local linear models. Each submodel approximates the system locally around an operating point and a selection procedure determines which particular submodel has to be used.
Under such a new identification method, a number of local linear models are designed and the estimate of outputs is given by a combination of local outputs. The diagnostic signal (residual) is the difference between the estimated and actual system output.
The key idea of model-based approaches for FD is, in fact, the generation of signals, termed residuals, obtained by using observers, parameter estimation or parity equations  designed on the basis of mathematical models of the monitored system.
Residual are signals representing inconsistencies between the model and the actual system being monitored. Any inconsistency will indicate a fault in the system. Residual must, therefore, be different from zero when a fault occurs and zero otherwise. However, the deviation between the model and the plant is influenced not only by the presence of the fault but also the modelling error.
The main disadvantage of model-based methods is that, being based on the mathematical model, it can be very sensitive to modelling errors, parameter variations, noise and disturbances, etc. The success of the model based method is heavily dependent on the quality of models.
Instead of exploiting complicated non-linear models obtained by modelling techniques, the problem is overcome describing the plant by a collection of local linear models obtained by the non-linear identification method presented above.
The contribution of this research is two fold. First, it is shown how to integrate the well-established Frisch Scheme  method for the identification of affine algebraic systems within a general procedure for non-linear dynamic system. Second, some interesting properties of such a Scheme can enhance the solution of the optimisation problem as well as of the continuity constraint fulfilment [8-11].
In a first stage of this project, the non-linear dynamic system will be assumed piecewise affine on the same region of the model so to explore the problem of noise rejection under the same assumption as in the affine theory [5,6]. Such a multiple model is piecewise affine with non-smooth boundary transition. In order to ensure a smooth transition between models, continuity constraints among local affine models have to be forced. In the project, this problem will be solved by using an optimisation technique . This observation allows to overcome the problems due to Takagi-Sugeno stuctures, in particular, the well-known model mismatch in transition regions studied by Babuska and Verbruggen [15,16].
The methodology we are keen to develop will be used in the fault
detection of a high power induction motor located in a power plant in
Sardinia (Italy) owned by Enichem S.p.A.
 Leonaritis, I. J. and Billings, S. A., "Input-output parametric models for non-linear systems", International Journal of Control. 41, pp 303-344. 1985.
 Takagi, T. and Sugeno, M., "Fuzzy identification of systems and its application to modelling and control", IEEE Trans. Sys. Man. & Cyber., 15(1), pp.116-132, 1985.
 Branicky, M. S., "Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems", IEEE Trans. AC., 43(4), pp. 475-482. 1998.
 Benveniste A., "Compositional and Uniform Modelling of Hybrid Systems", IEEE Trans. AC., 43(4), pp. 479-583. 1998.
 Chen, J. and Patton, R. J., "Robust Mode-Based Fault Diagnosis for Dynamic Systems", Kluver Academic Publishers, 1999.
 Kalman, R. E., "System Identification from Noisy Data", Dynamical System II, Academic Press, New York. Pp. 135-164. 1982.
 Kalman, R. E., "Nine Lectures on Identification", Lecture Notes on Economics and Mathematical System, Springer-Verlag, Berlin. 1990.
 Beghelli, S., Guidorzi, R. P. and Soverini, U., "The Frisch scheme in dynamic system identification, Automatica, 26(1), pp171-176. 1990
 Simani, S., Fantuzzi, C., Rovatti, R. and Beghelli, S., "Noise rejection in parameter identification for piecewise linear fuzzy models", FUZZ-IEEE Int. Conf., 1998.
 Rovatti, R. and Fantuzzi, C. and Simani, S. and Beghelli, S. "Parameters identification for piecewise linear models with weakly varying noise", CDC'98, Tampa (FL). U.S.A. Pp. 4488-4489. 1998
 Simani, S. and Fantuzzi, C. and Rovatti, R and Beghelli, S., "Noise rejection in parameters identification for piecewise linear fuzzy models", in WCCI '98, FUZZ-IEEE '98. 1998 IEEE International Conference on Fuzzy Systems. Ancorage, Alaska. 1998. May, 5-9.
 S. Simani, C. Fantuzzi, R. Rovatti and S. Beghelli, "Parameter Identification for Piecewise Linear Fuzzy Models in Noisy Environment", in International Journal of Approximate Reasoning, 1999. (Accepted)
 Simani, S.,"Fuzzy multiple inference identification and its application to fault diagnosis of industrial processes", in ISAS'99/SCI'99. The Fifth Conference of the ISAS (Information Systems Analysis and Synthesis)/The Third Conference of the SCI (Systemics, Cybernetics and Informatics). Orlando, Florida, USA. July 31, August 4, 1999.
 Frisch, R., Statistical Confluence Analysis by Means of Complete Regression Systems, University of Oslo, Economic Institute. Publication n. 5. 1934.
 Babuska, R. and Verbruggen, H. B., "Identification of composite linear models via fuzzy clustering", Proc. 3rd ECC'95, ,1207-1212. 1995.
 Babuska, R. and Keizer, J. and Verhaegen, M., "Identification of nonlinear dynamic systems as a composition of local linear parametric or state space models". Proc. of SYSID'97, Fukuoka, Japan. 1997.
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