Estimation Formulation

Three different factors affect the estimation formulation objective function: the error between the measurement residuals and the model, the change in parameter estimates across a given window in time, and the change in parameter values from one time step to another. The current formulation solves the following problem:

$$\begin{array}{c} \min \\ i=k-h...k \end{array}\vert\vert m_{... ...\vert _{2}+\vert\vert m_{3}\Delta _{T}\Theta (i)\vert\vert _{2}$$

where k is the current time, y_i is the process measurement vector at time i, is the vector of process model estimates at time i, m₁ is a scaling vector for weighting and normalizing the measurements, m₂ is a scaling vector for weighting and normalizing the change in the parameter estimates in the given horizon window, m₃ is a scaling value for normalizing the change in parameter estimates across time, h is the length of the moving horizon, and is the vector of parameter estimate at time i. The value for is given by:

$$\Delta \Theta (i)=\Theta (i)-\Theta (i-1)$$

is the value of the parameter estimate from the previous horizon window. An impulse response formulation is used to calculate the response of the process model estimate, to changes in the system parameters, . An individual system parameter (or fault) may be described at time i as . The value for is shown as:

$$\Delta \Theta _{T}(i)=\Theta (i)-\Theta _{k-1}(i+1)$$

Here, parameter estimates from the previous horizon's optimization result are used to minimize the change in a parameter from one time step to another. The value is shifted in time so that the current estimate corresponds to the value in the previous estimation window.

The impulse response formulation for is:

$$\widehat{y}(i)=\sum ^{F}_{j=1}M_{j}\theta _{j}(i)+...+M_{j}\theta _{j}(i)$$

where M_j are the impulse response coefficients for model j. The value F is the total number of disturbances modeled in the formulation. The index j represents the number of possible faults (or disturbances).

The formulation up to this point only includes continuous variables and few constraints. Solving this formulation without additional constraints typically yields an under specified problem that can match the measurement values with the estimated measurement values exactly. One may make the assumption that only a limited number of disturbances can affect the system during a single horizon. This leads one to use binary decision variables to represent whether or not a fault has occurred in the current horizon window. The following constraints are added:

$$\vert\theta _{j}(i)\vert\, \leq \, Mf_{j}\, \, \, \, \, \forall i,\, \forall j$$

$$\sum ^{F}_{j=1}f_{j}\leq S$$

for all fault parameters j and all horizon indices (times) i. The value M is a large number that ensures whenever is nonzero, f_j switches from 0 to 1. S is the total number of faults that can occur in a horizon window. In the presented results, response to positive and negative changes in a parameter are treated as separate fault events.