When no instrument is available to measure the desired variable on-line at a reasonable cost and within a reasonable time, we have to devise an alternative to keep track of relevant unmeasured variables. For example, it is not possible to measure product concentrations on-line fast enough to be included in a control loop. This is so since these components are typically absorbed in the solid matrix and they must be extracted before analytical determination. Hence the whole measuring procedure can last several hours. Soft-sensors or "state observers" are a useful alternative in these cases (Montague 1997). Here, a process model and measurements that are somehow related with the unmeasured variable are used to provide an online estimation. These are then processed by a Kalman Filter.
The Kalman Filter is widely used in submerged bioreactors for indirect measurements of specific consumption or production rates, yields, and heat loads. In SSF it has been mainly applied in lab scale bioreactors for biomass, bed water content and heat load estimations. Although this filter was originally developed for linear models, it has been extended to deal with nonlinear systems also.
The Kalman Filter is an on-line data processing algorithm that provides optimal estimation of output process variables, as shown in Fig. 26.3. The main reasons why we need a data processing algorithm are:
• measurements are contaminated by errors and they do not describe completely the state of the process;
• process models provide only an approximation to the real process behavior;
• the process is subjected to disturbances, which cannot be modeled or controlled.
Therefore, given (1) an approximate dynamic linear process model, (2) incomplete and noisy measurements, and (3) statistical information regarding measurement and model errors, the Kalman Filter provides the best linear estimation of the true values of the process outputs, even if some outputs are not measured or the process is subjected to disturbances. The obtained estimation is a compromise between the estimation provided by the model and the measured value. If we have confidence in the model, the optimal estimation will be obtained by putting more weight on the model outputs; on the contrary, if we trust the measurements more, the optimal estimation will be obtained by putting more weight on the measured values. The derivation of the algorithm is rather involved (Welch and Bishop 2003), therefore we will describe only its most basic principles.
Let us consider the following linear dynamic model relating several inputs with several outputs:
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