Parameter Estimation Methods for Fault Monitoring

Abstract

Fault detection via parameter estimation relies on the principle that possible faults in the monitored process can be associated with specific parameters and states of a mathematical model of a process given in general by an input-output relation,

$$y(t) = f(u,e,\theta ,x)$$

(3.1)

where y(t) represents the vector output of the process, u(t) the vector input, x(t) the partially measurable state variables, θ the nonmeasurable process parameters likely to change and e(t) unmodeled or noise terms affecting the process. It is obvious therefore, that it is necessary to have an accurate theoretical dynamic model of the process in order to apply parameter estimation methods. This is usually derived from the basic balance equations for mass, energy, and momentum, the physico—chemical state equations and the phenomenological laws for any irreversible phenomena. The models will then appear in the continuous or discrete time domain, in the form of ordinary or partial differential or difference equations. Their parameters θ _i are expressed in dependence on process coefficients p _j, like storage or resistance quantities, whose changes indicate a process fault. Hence, the parameters θ _i of continuous time models have to be estimated. In this case there is a minimum number of independently measurable quantities which permit the estimation of various states and parameters. As an example consider a simple dynamic process model with lumped parameters, linearized about an operating point, which may be described by the differential equation

$$y(t) + ... + {a_n}{y^{(n)}}(t) = {b_0}u(t) + {b_1}{u^{(1)}}(t) + ... + {b_m}{u^{(m)}}(t)$$

(3.2)