Design of the Controller-Observer Scheme

The model-based controller-observer scheme requires to solve online the system of differential equations of the observer. The phenol-formaldehyde reaction model is characterized by 15 differential equations, and it is, thus, unsuitable for online computations. To overcome this problem, one of the reduced models developed in Sect. 3.8.1 can be adopted. In order to be consistent with the general form of nonchain reactions (2.27) adopted to develop the controller-observer scheme, the reduced model (3.57) with first-order kinetics has been used to design the observer. The mass balances of the reduced model are given by 

Gains Model-free RB FI-based Model-based Adaptive model-based 

The model-based observer requires tuning of 6 parameters, i.e., the nonzero values in matrix L and y . As for the model-free observer defined by (5.29), (5.30), and (5.34), the dynamics of the reaction is not required, and only two gains (the main diagonal of matrix Le) and two update gains (y0 and yq) are needed. Finally, the observer (5.36) requires tuning of the two gains k and lq. All the above gains have been tuned via a trial-and-error procedure and are summarized in Table 5.2. 

As stated in Remark 5.3, the parameters of the two-loop controller (gpir, gpj) and yc have been chosen independently from the adopted observer via a trial-and-error procedure. Therefore, they are identical for all the control schemes and are reported in Table 5.3. In the controller-observer scheme without adaption (i.e., using the nominal estimate of 0), both y0 and yc have been set to zero. 

It can be argued that the differences between the compared schemes are mainly due to the different estimation accuracy of the quantity aq (Fig. 5.6). It can be seen that, after the initial transient phase in which the model-free observers present an inverse response, both the adaptive (model-based and model-free) approaches achieve very good estimates. As for the parameter estimate, since both the adaptive observers (0O) and the controller (0C) estimates converge to the true value of 0 (see Fig. 5.7), it is possible to argue that the persistency of excitation condition is fulfilled. 

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