Water Recycling System Prognostics

# Faulty System Modeling

Faults are modeled as unexpected changes in the system parameters. We assume the faults are persistent. Typical degradation modes of the WRS include clogged membranes, clogged filters, and sensor faults. In particular, Filter 1, Filter 2, the FO membrane, and the RO membrane all get clogged over time due to buildup of solids. These clogging faults, denoted by R^{–}_{Filt1}, R^{–}_{Filt2}, A^{–}_{FO}, and A^{–}_{RO}, respectively, are represented as gradual decrease in the coefficient of flow through the filters, R_{Filt1} and R_{Filt2}, and the membrane permeabilities A_{FO} and A_{RO}, respectively. Sensors faults can include abrupt bias and gradual drift fault.

## Modeling Filter Clogging Fault

For Filter i, the gradual decrease in R_{Filti} is represented as dR_{Filti}/dt = 0 when t\f, and dR_{Filti}/dt = ΔR_{Filti} otherwise, where t_{f} is the time of fault occurrence, and Δ_{RFilt1} is the fault parameter.

## Modeling Membrane Clogging Fault

For membrane j, the gradual decrease in A_{i} is represented as dA_{i}/dt = 0 when t\f, and dA_{i}/dt = ΔA_{i} otherwise,
where ΔA_{i} is the fault parameter.

## Modeling Sensor Bias Fault

A bias fault in sensor S is indicated as S^{(b;ΔS)}, and is modeled as an abrupt addition of a constant bias b added to the sensor value from the point of fault injection t_{f} , i.e., S = S when t\f, and S = S + ΔS otherwise.

## Modeling Sensor Drift Fault

A drift fault in sensor S is indicated as S^{(d;ΔS)}, and is modeled as a gradual addition of a constant drift d to the sensor value at each time step from the point of fault injection t_{f} , i.e., dS/dt = 0 when t\f, and dS/dt = ΔS otherwise.