The aim of parameter estimation is to find the values of the unknown model parameters that minimize the distance between model simulations and experimental data. To that purpose, the following elements are required:

Let us denote by

$\theta $ the set of unknown model parameters and by

y the observable. In this work, the cost function is defined as the Root Mean Square Error (RMSE) value. The parameter estimation problem is formulated as, find:

subject to the model dynamics presented in previous section. In Equation (

20),

${n}_{e}$ is the number of experiments for estimation purposes,

${n}_{s,k}$ is the number of sampling times in experiment

k and

${\overline{y}}_{mod}\left({t}_{i,k}\right)$ represents the model prediction of the observable at time

${t}_{i}$ in experiment

k.

The control variables, represented in

Figure 2 as arrows entering the subsystems, correspond to those process variables that can be directly manipulated to actuate into the process. Three controls are available in this pilot plant: condenser pressure (

${P}_{c}$), chamber pressure (

${P}_{ch}$) and shelf temperature (

${T}_{sh}$). These variables are also measured so the real value can be used for parameter estimation tasks. Arrows leaving the subsystems in

Figure 2 represent the state variables. The chamber vapor pressure (

${P}_{ch}^{v}$) is the only state variable computed in the condenser model. State variables in the product subsystem are the front position (

x) and velocity (

w) as well as product temperature (

${T}_{p}$), the latter distributed along the product height. Both subsystems, condenser and product, are coupled through two state variables,

${P}_{ch}^{v}$ and

w.

In this system, Pirani pressure (

${P}^{p}$) and total chamber pressure (

${P}_{ch}$) are also measured. Since chamber vapor pressure is related to

${P}^{p}$ and

${P}_{ch}$ through the formula [

36]:

we can consider that

${P}_{ch}^{v}$ is indirectly measured. These indirect measurements can be used, in the product model, so that the condenser model can be neglected in the estimation of the product model parameters. In this regard, the estimation task will proceed in two steps:

The following freeze-drying experimental protocol was applied is the different tests:

#### 3.1. Parameter Estimation in the Product/Chamber Subsystem

The subset of model parameters to be estimated is

$\theta =[{\kappa}_{dr},{k}_{2},{h}_{L,2}]$, i.e., the heat conductivity of the dried region; product/chamber mass resistance coefficient—see Equation (

11); and heat transfer coefficient—see Equation (

6).

The observable in this subsystem is the temperature at the bottom of the product. Because of limitations on the equipment, the precise location of such thermocouple is not known with precision. Therefore, a mean value of the temperatures in the bottom region of the product is used as the observable $y={T}_{p,meas}$. Such region is 1 $\mathrm{m}\mathrm{m}$ long.

In this subsystem, Equation (

20) may be rewritten as:

where

${\overline{T}}_{p,mod}\left({t}_{i,k}\right)$ represents the mean value of the temperatures in the bottom region of the product.

The experimental data employed in the parameter estimation were obtained from five experiments. Each experiment differs from the others in the control variables, i.e., shelf temperature (${T}_{sh}$) and chamber pressure (${P}_{ch}$). Besides, although the set point for the freezing temperature was set to −50 ${}^{\xb0}\mathrm{C}$, the refrigerant group was not able to reach such temperatures. Experimental measurements showed that the actual values turn out to range between −37 and −43 ${}^{\xb0}\mathrm{C}$, depending on the experiment. Therefore, initial conditions for primary drying also differ from one experiment to another.

A cross validation procedure was applied [

47] to robustly estimate the parameters. In this procedure, part of the experimental data is saved for validation purposes, i.e., such data is not used in the estimation task. In this case, we use four of the five experiments for the parameter estimation task, leaving the remaining one for validation. To increase robustness of the results, we have performed five estimations, each one excluding a different experiment.

The product model, together with the measured values of the temperature at the bottom of the product (${T}_{p,meas}$) in the experiments considered for estimation, were implemented in the AMIGO2 toolbox. RMSE values were obtained by comparing experimentally measured values of the product temperature with the simulation results.

The estimation results, which include optimal parameter values as well as RMSE values for both estimation and validation tests, are summarized in

Table 2.

The values found for the parameters are in accordance with the expected physical range. Besides, variability in the parameter estimates of the cross validation procedure for ${h}_{L,2}$, ${k}_{2}$ and ${\kappa}_{dr}$ is 4.4%, 14.7% and 23.7%, respectively. While the variability of ${\kappa}_{dr}$ is much larger than the variability of the other two parameters, the results are considered to be robust.

Figure 3 illustrates the predictive capabilities of the model.

Four subplots are shown, each one corresponding to the validation experiment in the cross validation procedure. Blue line represents the shelf temperature (control variable) evolution in each experiment whereas dashed lines correspond to model predictions of the product temperature evolution at different spatial points. The figure shows that model predictions at the bottom of the product (continuous black line) reproduce the experimental behavior (black dots). In particular, the end of the primary drying stage as well as the product temperature during such stage are predicted by the model with a reasonable degree of accuracy. Temperature during secondary drying is also accurately described by the model. However, some mismatch between predicted and measured temperatures can be observed in the transition from primary to secondary drying (highlighted in

Figure 3 by two vertical black lines). In fact, the model seems to predict a faster temperature increase than the one measured by the thermocouple.

So far, no clear explanation has been found to such disagreement although it most probably can be attributed to an interphase structure which is more complex than the sharp transition between a frozen and a dried region as it is considered by the model. Nevertheless, as mentioned on

Section 2.2.3, this does not have a significant impact on the product quality. Besides, product temperature predicted by the model in the transition is usually larger than the experimental measurements. Therefore, as it will be described in

Section 4.1, operation policies that satisfy safety constraints according to model predictions will also satisfy such constraints in the experiments.

#### 3.2. Parameter Estimation for the Condenser Model

The unknown model parameter in the condenser model is coefficient

$\theta =\beta $, see Equation (

14). In this plant, it is possible to experimentally measure the total chamber and Pirani pressures so that the chamber vapor pressure (

${P}_{ch}^{v}$) can be obtained, see Equation (

21).

$y={P}_{ch}^{v}$ is used as the observable in the estimation task.

In this case, the optimization problem defined by Equation (

20) reads as:

The same experiments used in the previous section are employed here to estimate the condenser parameter $\beta $. In this case we use both the product model, with the parameter values computed in previous section, and the condenser model.

As in previous section, cross validation is used to evaluate the performance of the model. The estimation results, including optimal value of

$\beta $ and the RMSE values corresponding to both the estimation and validation experiments, are presented in

Table 3.

The estimated values of

$\beta $ are within the expected physical range and close to the values presented in the literature [

36]. Besides, the variability in the parameter estimates for the different cases in the cross validation is lower than

$4\%$ which evidences the estimation robustness.

Model predictive capabilities, in terms of chamber pressure prediction, are illustrated in

Figure 4. As in the previous case, the four subplots correspond to the validation experiment in the cross validation tests.

Only primary drying is represented in the figures since during secondary drying vapor pressure is almost zero. As mentioned in

Section 2.2.4, the transition between primary and secondary drying is assumed to be instantaneous. Simulation results for the chamber vapor pressure are represented by continuous black lines whereas black circles correspond to experimental measurements. As shown in the figure, the model is able to reproduce the experimental behavior with a high degree of accuracy.