## M

actuator noise

input . sensor noise (vm)

process faults disturbances ( "p) (d)

measured inputs

process noise (vP)

process faults disturbances ( "p) (d) output -

sensor faults

output.

sensor noise

measured outputs

Figure 8.8. Additive faults and disturbances .

Consider a multiple-input, multiple-output system described by y(t) - G(g)u(t) (8.44)

where G(q) denotes the multivariable input-output transfer function matrix (TFM) and q is the shift operator defined in Section 4.3.1. Denoting the actual inputs and outputs of the process with superscript they can be expressed as:

u °c(t) = uc(t) + <5uc(t) + vc(t) u°M{t) = UM(i) - SuM(t) - vM(i) (8.45)

The relation in Eq. (8.44) is between the nominal inputs and outputs. Expanding this relationship to show explicitly the faults, noises and process disturbances:

y(t) = G(q)u(t) + SF(q)i(t) + SD{q)d(t) + S N(q)u(t) (8.46) where SF(q) is the combined fault FTM, Sat (q) is the combined noise TFM,

Sd(ç) is the process fault TFM, and f(t) = [Su&t) -6uTM(t) SuTp(t) Sy(t)T }

T iT

with G c{q) denoting the actuator (controlled input) fault TFM, G m (q) the input sensor fault TFM, SPF(q) the plant fault TFM and SPN(q) the plant noise TFM.

This framework with additive faults has to be augmented to consider multiplicative faults. Multiplicative faults may reflect a parametric fault resulting from the change in process operation (hence the model is not accurate anymore) or a modeling error such as inaccuracy in model structure or parameters resulting from approximating a nonlinear process with a linear model or a high ordered process with a low order model. The input-output model representation Eq. (8.46) is further expanded to incorporate multiplicative faults where Np(t) denotes the matrix of time-varying coefficients of multiplicative parametric faults, Njvi(t) the coefficient matrix of multiplicative modeling faults, (j)P the parametric faults and <j)M the modeling errors. An important difference between additive and multiplicative faults is that the TFMs of additive faults are constant while the TFMs of multiplicative faults are time dependent. Whereas additive fault vectors are time dependent, multiplicative fault vectors are constant . The remainder of this section will focus mainly on additive faults. Multiplicative faults are equally important, but the model-based techniques for addressing them are active research issues that can not be adequately treated in the framework of this text.

State-space relations for systems subject to faults. The relationship between y and u can also be written in state-space form. The state-space form equivalent to input-output relations in Eq. (8.46) is given in Eq. (8.85). Most early methods based on observers and Kalman filters use a simplified state-space representation that ignores noise as a separate factor, resulting in y (t) = G(q)u(t) + SF(q)f(t) + S D(q)d(t) + S^M*)

xfc+1 = Fxfc + Gufc + Epffc + Eßdfe yfc = Cxfe + FFffc + FDdfc where component and actuator faults are modeled by Ejrffe and sensor faults are modeled by F^fThe unknown inputs affecting the actuators and process dynamics are introduced by E^dfe and unknown inputs to sensors are introduced by F^d^. This modified representation is used in illustrating the use of Kalman filters and observers in subsequent sections.