Dynamic parameters estimation algorithm

The dynamic parameters estimation (DPE) algorithm proposed by Velardi et al. (2008) solves the energy balance for the frozen layer to get the temperature profile in the product taking into account the different dynamics of the temperature at the interface and at the vial bottom. The energy balance in the frozen layer during the PRT can be described by the following equations ... 

Other recent developments in the field of adaptive control of interest to the processing industries include the use of pattern recognition in lieu of explicit models (Bristol (66)), parameter estimation with closed-loop operating data (67), model algorithmic control (68), and dynamic matrix control (69). It is clear that discrete-time adaptive control (vs. continuous time systems) offers many exciting possibilities for new theoretical and practical contributions to system identification and control. 

An adaptive controller continually and automatically readjusts itself for proper operation in the presence of changing system dynamics or noise characteristics. It combines a parameter estimator and a control scheme that changes the control algorithm as needed. A block diagram of an adaptive system can be seen in Figure 4.4.6. Adaptive control is based on a linear differential equation with nonconstant coefficients, and is often used in drug-delivery systems (Woodruff, 1995) or where patient-to-patient variation is particularly wide. 

If the sublimation step is not yet finished, it could be interesting to estimate the time left to the endpoint. This can be easily done using a mathematical model that describes the dynamics of the process (e.g., the same as in the DPE algorithm) and uses the process parameters estimated by DPE. 

Another important role of the noise model in the iterative algorithm is to ensure whiteness of the residuals. This allows us to estimate the covariance of the FSF model parameter estimates and then to develop statistical confidence bounds for the corresponding step response estimates. In order to apply these results, it is important that the bias error in the model arising due to unmodelled dynamics be small relative to the variance error caused... 

As mentioned in Chapter 4, although this is a dynamic experiment where data are collected over time, we consider it as a simple algebraic equation model with two unknown parameters. The data were given for two different conditions (i) with 0.75 g and (ii) with 1.30 g of methanol as solvent. An initial guess of k =1.0 and k2=0.01 was used. The method converged in six and seven iterations respectively without the need for Marquardt s modification. Actually, if Mar-quardt s modification is used, the algorithm slows down somewhat. The estimated parameters are given in Table 16.1 In addition, the model-calculated values are... 

In this section we deal with estimating the parameters p in the dynamical model of the form (5.37). As we noticed, methods of Chapter 3 directly apply to this problem only if the solution of the differential equation is available in analytical form. Otherwise one can follow the same algorithms, but solving differential equations numerically whenever the computed responses are needed. The partial derivations required by the Gauss - Newton type algorithms can be obtained by solving the sensitivity equations. While this indirect method is... 

Dynamics of the dissipative two-state system. Reviews of Modern Physics 59 (l) l-85. Louisell, W. H. 1973. Quantum Statistical Properties of Radiation. New York John WUey. Marquardt, D. W. 1963. An algorithm for least-squares estimation of nonlinear parameters. 

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