4.1. Estimation Structure
The choice of the estimation structure has been carried out considering: (i) condition number and the minimum singular value of the Jacobian matrix
(or
) for a different choice of measurements and innovated states and (ii) evaluating the responses of the reconstructed states for a given trajectory. Temperature and dissolved oxygen measurements have always been considered available, according to the laboratory and industrial practice. On the other hand, sensors suited for ethanol measurements as well as substrate and biomass are not always available for large scale real-time applications [
4]. According to the analysis reported in [
3], two possible scenarios have been considered: (i) biomass concentration in the reactor is measured online or (ii) substrate concentration in the reactor is measured online. Using the representation in (9), the considered cases are reported in Equation (19):
where
represents the measured output vector.
According to Equation (13), it is easy to demonstrate that no combination of indexes
satisfies the observability property for the output vector in Equation (19a). This implies that a full order observer is possible if the substrate concentration is measured online and therefore when using the output configuration reported in Equation (19b). In this case, the nonlinear estimation maps satisfying Equation (13a) are reported in Equation (20)
A first comparison of the two structures can be carried out by considering the values of condition number and minimum singular value for the Jacobian of the maps (20), calculated by averaging along the trajectories obtained with input step changes T1 and T2 (
Table 4) and reported in
Table 5. The structure
seems to be more robust (lower condition number), but it shows a lower minimum singular value, indicating that changes in the states should affect the outputs to a lesser extent.
The reconstruction capabilities of the two structures using the geometric observer are therefore calculated, using the input variations T1 and T2, reported in
Table 4. Results are shown in
Figure 1,
Figure 2,
Figure 3 and
Figure 4, only for the unmeasured variables, which are ethanol and biomass concentration. It is worth noticing that also the state values calculated only with the model used in the estimation algorithm (open-loop model), but without innovation are reported in order to better highlight the correction provided by the estimation algorithm.
It is possible to observe that using the map
, allows a good reconstruction of the biomass behavior (
Figure 1), while there is a large mismatch between the ethanol concentration obtained with the virtual plant and the reconstructed one (
Figure 2).
When using the second configuration, results worsen, both for biomass (
Figure 3) and for ethanol (
Figure 4) concentration. It is worth noticing that the state’s values estimated with map
are more corrupted by the measurement noise because in this case a greater observer gain has been used to decrease the offset.
The two full order structures are not able to adequately estimate the product of the reactor, therefore a different solution is required to improve ethanol concentration. Using the same measured outputs, it is possible to improve estimation performance by reducing the order of the observer using only one Lie’s derivative [
22]. The maps reported in Equation (21) lead to five observable states and only one detectable.
The rank of the Jacobian of the maps
(
i = 3, 4) depends on the choice of the non innovated state (
) between the two that are not measured, which are ethanol and biomass concentration. It can be verified that the map
can be inverted only if
is innovated and
is not. On the other hand, the Jacobian of map
always has a rank equal to five, regardless of the choice of the innovated states. Recalling Equation (15), the following partitions are considered:
The map
can be used with the partition in Equation (22a), while the map
can be used with both partitions in Equation (22a,b). Therefore, two different solutions are identified:
for partition (22b) and
for partition (22a). A first analysis of the possible configurations can be obtained by considering the minimum singular value and condition number reported in
Table 6. The indexes’ values are comparable; therefore, the evaluation of the best structure has been performed analyzing the reconstruction performance.
Figure 5 and
Figure 6 represent the estimation of the unmeasured states (ethanol and biomass concentration) for the input step change T1 and T2 described in
Table 4. The best reconstruction capabilities are shown by configuration
for both the states. This result may suggest that conditions calculated with Equation (14) are informative when the magnitude between the different configurations is significantly different, otherwise, it is necessary to evaluate the estimation capabilities by evaluating the estimator response for given input changes.
4.2. Validation
The analysis carried out in the previous section indicates the best estimation structure with four measured outputs. In order to validate the obtained results, a new test was carried out considering as reference trajectory the variation of the input temperature (
Tin) as shown in
Table 4 (Case T3).
Figure 7 shows the dynamic behavior of biomass and product concentration and confirms that the proposed structure can effectively reconstruct the unmeasured states also with different process conditions. It is worth noticing that the ethanol concentration is not innovated, and the correction of the other states also has a positive impact on its estimation.
Using the same number and choice of measured outputs Equation (19b) and partition between innovated and not innovated states Equation (22a), the estimation task has been addressed using the extended Kalman filter (
Figure 8). The main reason for using another algorithm as a measurement processor is to demonstrate that the estimator performance depends on the structure selection rather than estimation algorithm. EKF has been preferred for this validation because it is usually preferred in the industrial practice as it is easy to implement and robust if adequately calibrated [
25,
26].
Results show that EKF can effectively reconstruct the unmeasured states, revealing that estimator structure design is the key step for a successful achievement of the estimation goals. The only difference between the two approaches is that the biomass calculated with the geometric observer is more affected by noise. This behavior can be explained by the presence of the Lie derivative in GO, which implies a higher sensitivity to measurement noise with respect to the EKF.