Adaptive parameter estimation

The adaptation of the original LJ optimization procedure to parameter estimation problems for algebraic equation models is given next. 

Based on the above, we can develop an "adaptive" Gauss-Newton method for parameter estimation with equality constraints whereby the set of active constraints (which are all equalities) is updated at each iteration. An example is provided in Chapter 14 where we examine the estimation of binary interactions parameters in cubic equations of state subject to predicting the correct phase behavior (i.e., avoiding erroneous two-phase split predictions under certain conditions). 

The adaptive parameters in the model were estimated by nonlinear and multiresponse regression, performed using the Fortran subroutine BURENL23 based... 

Laboratory data collected over honeycomb catalyst samples of various lengths and under a variety of experimental conditions were described satisfactorily by the model on a purely predictive basis. Indeed, the effective diffusivities of NO and NH3 were estimated from the pore size distribution measurements and the intrinsic rate parameters were obtained from independent kinetic data collected over the same catalyst ground to very fine particles [27], so that the model did not include any adaptive parameters. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

An adaptive control of the batch racton-HI Simplified parameter estimation (with W.H. Ray). Automatica 3, 53-71 (1965). 

The architecture of the self-tuning regulator is shown in Fig. 7.99. It is similar to that of the Model Reference Adaptive Controller in that it also consists basically of two loops. The inner loop contains the process and a normal linear feedback controller. The outer loop is used to adjust the parameters of the feedback controller and comprises a recursive parameter estimator and an adjustment mechanism. 

Remark 5.1 As usual, in direct adaptive estimation and/or control schemes, the convergence to zero of the parameter estimation error 0O is not guaranteed, unless the persistency of excitation condition is fulfilled [5, 35], In detail, if there exist three scalars A.i > 0, A.2 > 0, and r > 0 such that... 

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. 

The above result is usual in direct adaptive estimation and/or control schemes. The exponential convergence to zero of both the state estimation error and the parameter estimation error is guaranteed only in the presence of the persistency of excitation condition [1,3]. This, in turn, implies that they keep bounded in the presence of bounded uncertainties. However, since persistency of excitation may be difficult to guarantee in practice, a modified parameters update law can be adopted, according to the concept of projection operator [2], In detail, adoption of the following update law instead of (6.28),... 

The aim of parameter estimation is an adaptation of the model function to the observations made to gain model parameters which describe the observed data best. In NONMEM this is done by the minimization of the extended least square objective Oels function, which provides maximum likelihood estimates under Gaussian conditions [13]. The equation calculating the Oels function is given in the following ... 

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. 

Radenkovic, M. S., and Ydstie, B. E., Using persistent excitation with fixed energy to stabilize adaptive controllers and obtain hard bounds for the parameter estimation error, SIAM J. Contr. and Optimization 33(4), 1224—1246 (1995). 

A key, yet difficult, decision in the model-update step is to select the parameters to be updated. These parameters should be identifiable, represent actual changes in the process, and contribute to approach the process optimum also, model adequacy proves to be a useful criterion to select candidate parameters for adaptation [8]. Clearly, the smaller the subset of parameters, the better the confidence in the parameter estimates, and the lower the required excitation. But too low a number of adjustable parameters can lead to completely erroneous models, and thereby to a false optimum. 

Parameter identification is complicated by several factors (i) the complexity of the models and the nonconvexity of the parameter estimation problems, and (ii) the need for the model parameters to be identifiable from the available measurements. Moreover, in the presence of structural plant-model mismatch, parameter identification does not necessarily lead to model improvement. In order to avoid the task of identifying a model on-line, fixed-model methods have been proposed. The idea therein is to utilize both the available measurements and a (possibly inaccurate) steady-state model to drive the process towards a desirable operating point. In constraint-adaptation schemes (Forbes and Marlin, 1994 Chachuat et al., 2007), for instance, the measurements are used to correct the constraint functions in the RTO problem, whereas a process model is used to... 

Image analysis is well adapted to estimate the porosity of a supported membrane, this parameter is often difficult to obtain by other techniques. Up to now this method has been applied to a large extent to study the porous texture of organic membranes. 

The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

Finally, control equations of the form proposed by Stolwijk have two characteristics the use of a set point temperature for each layer tends to hold temperatures within a rather narrow range of values, and the model contains a very large number of adjustable parameters. Thus, one should be able to fit a limited amount of data rather well even if the model is invalid. Devising definitive experiments and adapting modem techniques for parameter estimation to this problem present a real challenge for engineers and scientists. 

Adaptive control systems have been applied in chemical processes. The range of their applicability has expanded with the introduction of digital computers for process control. Several theoretical and experimental studies have appeared in the chemical engineering literature, while the number of industrial adaptive control mechanisms increases continuously. Most of the adaptive control systems require extensive computations for parameter estimation and optimal adjustment of controller parameters, which can be performed on-line only by digital computers. Therefore, we will delay any discussion on the quantitative design of such systems until Chapter 31, after we have studied the use of digital computers for control. 

Show qualitatively that the structure of a self-tuning regulator can be derived from that of a model-reference adaptive control if the parameter estimation is done by updating the reference model. 

In this chapter we have presented a rather simplistic view of the on-line adaptive control systems. There are a number of very important questions which have not been addressed, such as whether the parameter estimates are biased, the interplay between estimation and control, and the stability characteristics of the adaptive controller. A thorough examination of these questions is beyond the scope of this text. The interested reader can consult the relevant references at the end of Part VII. 

The parameter estimation methodology was demonstrated on an alternative dynamic stmctured model [8], adapted from Rotboll and Jorgensen [9] to simulate a tower bioreactor for ethanol production by immobilized Saccharomyces cerevisiae. The model contains 34... 

DeNOx - Scope of the model analysis was to evaluate on a quantitative basis the effective dependence of the intrinsic activity of the monoliths on the thermal sintering, and separate it from the contributions of inter-phase mass transfer and the effect of morphological modifications on intra-porous diffusion. When excess ammonia is present, as in the case of the experiments herein analyzed, then the Ealy-Rideal kinetic expression which is contained in the model of the SCR reactor reduces to a first order dependence on NO concentration under such conditions, an unique adaptive parameter, kc, accounts for the DeNOx intrinsic activity. Estimation of kc for the three calcined catalysts was obtained by fitting the model to each set of experiments. Input data included the operating conditions, the geometrical... 

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. 

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