The parameters of a model describing the dynamic behavior of a process change when the operation of the process varies significantly. If a process model is developed for normal process operation, the model parameters can be re-estimated when new data are collected and compared with nominal values of model parameters. Significant deviations in the values of model parameters indicate the presence of faults or disturbances or modification of the operating point of the process. If the last two possibilities are eliminated, then changes in parameter values indicate faults.

The model of the process may be constructed from first principles. Then the parameters that depend on process operation should be determined and those parameters should be estimated when new data are collected [251], Another alternative is to develop a time series type of model (Section 4.3.1). Then, changes in model parameters over time are monitored for FDD.

A procedure suggested for implementing this approach in a deterministic framework is outlined [251]. Consider a process model described by linear input/output differential equations with constant coefficients any[n) (i) + • ■ ■ + Oitf(t) + y(t) = 6mu(m) («) + ■•• + bQu(t) (8.98)

where y(") indicates the nth derivative of y. The model parameters are collected in a vector 6

Determine relationships between model parameters Oi and physical parameters tpj e = f(<t>). (8.100)

Identify model parameters 0 from process data (u, y). Then, determine the physical parameter values using the inverse relationship 4> — f1 (0) and compute changes in cf>, A<j). Use threshold logic or other tools to determine the magnitude of changes in A(f> and presence of faults.

A more general framework can be established for modeling the changes in the eigenstructure of a data-based model in state space form (F matrix of discrete-time equation such as Eq. (8.49)) or time series form (AR or ARMA model). The version discussed below will provide detection of change in univariate systems. Extension to multivariable processes has been developed [45]. Additional steps are necessary for diagnosis if multiple faults are possible. For the case of additive changes, the cumulative sum to be computed becomes

where p$1 reflects the change of magnitude <5 at time r and the stacked output values are yfc-1 = [Vk-1 Vk-2 ■■■ yi]T ■ (8.102)

The GLR is

1 <r<k 0i and the GLR test becomes

Significant savings in computation time can be generated by using a two-model approach [50]. For illustration, consider a two-model approach for on-line detection of change in scalar AR models

¿=i where e/> is Gaussian white noise with variance af^ and for i = 1, • • • ,p

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