The fault detection and diagnosis approach follows the scheme presented in Fig。 1。 Measurements of the setpoint u, the output y, as well as intermediate process states x are taken and are supplied to both the control level as well as the supervisory level。 The protective level encompasses simple fault detection methods and shall not be in the focus of  the  paper。 They typically encompass limit   and

978-3-902661-46-3/09/$20。00 © 2009 IFAC 1097 10。3182/20090630-4-ES-2003。0247

Fig。 1。 Scheme of an Integrated Fault Management System, see Isermann [2005b]

threshold checking and are solely used to detect potentially harmful states。

The supervision with fault diagnosis employs model-based fault detection  and diagnosis methods  allows  to   detect

tiny,  incipient  faults,  which  do  not  yet  cause  harm  to i i

health and wealth。 Also, the methods allow a much more detailed diagnosis。 Neural net models are used for the feature generation by means of parity equations as will be explained later。 The features are then supplied to a  fault

detection stage, where the deviation of certain features from their nominal range leads to symptoms that can then be subject to classification or inference methods to map the symptoms to faults。 The exact diagnosis could then be used to initiate counter actions, which brings up the topic of fault tolerance, which is treated in detail in the papers Muenchhof et al。 [2009a,b]。

2。NEURAL NET LOLIMOT

The LOLIMOT net, see Nelles [2000] and Nelles and Fink [2002] is a local model approach, which can be classified according to three typical features of local models:

① Partitioning Principle / Structure Identification: The pision of the input space can be carried out by dif- ferent methods。 The most prominent approaches are a grid structure or a recursive partitioning。 LOLIMOT employs an axis-orthogonal, recursive partitioning al- gorithm, where always the submodel with the worst performance is split into two submodels。 This selec- tion principle is assumed to increase the model fidelity as much as possible。 See Fig。 2a。 for a possible parti- tioning result and the resulting membership functions。 An axis oblique partitioning algorithm for LOLIMOT has been suggested by Töpfer [2002]。

② Transition between Submodels : The transition be- tween the submodels can be hard or soft。 LOLIMOT uses a soft transition, since a hard transition is rather artificial and will seldom be encountered in nature。 Hard transitions would only make sense in the presence of hard (e。g。 switching) nonlinearities。 LOLIMOT uses a soft transition, but allows to make

Fig。 2。 Scheme of the LOLITMOT  Net

this transition more abrupt by changing an internal parameter and thus is also capable of modeling harder nonlinearities。 The inpidual membership functions are furthermore normalized to sum up to one at every point of the input space, see Fig。 2b。

③ Local Model Structure: The local models typically have polynomial structure。 Due to the computational expense, constant or linear models are typically pre- ferred, sometimes however quadratic models are cho- sen。 LOLIMOT uses a linear submodel。 The inpid- ual linear submodels are shown in Fig 2c。

The output of the LOLIMOT net can be calculated   by

M

yˆ(u)= 。 Φm(u)gm(u)。 (1)

m=1

Here, the local submodels are given by gm and are weighted by the corresponding membership function Φm。 The out- put of the sample neural net is graphed in Fig。 2d。 Instead of supplying the entire set of input variables u to both the partitioning algorithm and the local linear models, the regressors will now be pided into the regressors x for the