Dynamic parameters estimation

The dynamic parameter estimation problem is fully discretized by Radau collocation over finite elements, and it is formulated in GAMS [4]. We have solved the problem with 10 and 30 finite elements, respectively. State variable profiles are shown in Figures 14.12 and 14.13. It can be seen that the finer discretization gives better profiles for state variables with 574 algebraic variables in the NLP (211 variables for 10 FE) and the same in CPU time (0.032 s). The use of 10 finite elements is... 

Estrada, V., Parodi, E., and Diaz, M.S. (2009) Determination of biogeochemical parameters in eutrophication models as large scale dynamic parameter estimation problems. Computers Chemical Engineering, 33, 1760-1769. 

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 ... 

Velardi, S. A., Rasetto, V., Barresi, A. A., 2008. Dynamic Parameters Estimation Method advanced Manometric Temperature Measurement approach for freeze-drying... 

After having proved the principles a dynamic test facility has been constructed. In this facility it is possible to inject 3 tracers in a flownng liquid consisting of air, oil and water. By changing the relative amounts of the different components it is possible to explore the phase diagram and asses the limits for the measurement principle. Experiments have confirmed the accuracy in parameter estimation to be below 10%, which is considered quite satisfactorily for practical applications. The method will be tested on site at an offshore installation this summer. 

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

Let us first concentrate on dynamic systems described by a set of ordinary differential equations (ODEs). In certain occasions the governing ordinary differential equations can be solved analytically and as far as parameter estimation is concerned, the problem is described by a set of algebraic equations. If however, the ODEs cannot be solved analytically, the mathematical model is more complex. In general, the model equations can be written in the form... 

The steps that should be followed to determine the best grid point in the operability region for precise parameter estimation of a dynamic system are given below ... 

If instead of precise parameter estimation, we are designing experiments for model discrimination, the best grid point of the operability region is chosen by maximizing the overall divergence, defined for dynamic systems as... 

As an example for precise parameter estimation of dynamic systems we consider the simple consecutive chemical reactions in a batch reactor used by Hosten and Emig (1975) and Kalogerakis and Luus (1984) for the evaluation of sequential experimental design procedures of dynamic systems. The reactions are... 

The design procedures depend heavily on the dynamic model of the process to be controlled. In more advanced model-based control systems, the action taken by the controller actually depends on the model. Under circumstances where we do not have a precise model, we perform our analysis with approximate models. This is the basis of a field called "system identification and parameter estimation." Physical insight that we may acquire in the act of model building is invaluable in problem solving. 

Dynamic Joint State-Parameter Estimation A Filtering Approach 173... 

All of the previous ideas are developed further in Chapter 8, where the analysis of dynamic and quasi-steady-state processes is considered. Chapter 9 is devoted to the general problem of joint parameter estimation-data reconciliation, an important issue in assessing plant performance. In addition, some techniques for estimating the covariance matrix from the measurements are discussed in Chapter 10. New trends in this field are summarized in Chapter 11, and the last chapter is devoted to illustrations of the application of the previously presented techniques to various practical cases. 

In this chapter, the general problem of joint parameter estimation and data reconciliation will be discussed. The more general formulation, in terms of the error-in-variable method (EVM), where measurement errors in all variables are considered in the parameter estimation problem, will be stated. Finally, joint parameter and state estimation in dynamic processes will be considered. 

DYNAMIC JOINT STATE-PARAMETER ESTIMATION A FILTERING APPROACH... 

The problem of state-parameter estimation in dynamic systems is considered in terms of decoupling the estimation procedure. By using the extended Kalman filter (EKF) approach, the state-parameter estimation problem is defined and a decoupling procedure developed that has several advantages over the classical approach. 

The experimental setup, described in Example 8.1, for calculating the bias in a dynamic environment will be used here to discuss the parameter estimation methodology. In this case both the surface heat transfer coefficient (h) and the thermal conductivity (A) of the body in the condition of natural convection in air are considered (Bortolotto et al., 1985). 

The multichannel procedure introduces an alternative approach to the problem of dynamic state-parameter estimation. The decoupling of the state estimator from... 

In this section the extension of the use of nonlinear programming techniques to solve the dynamic joint data reconciliation and parameter estimation problem is briefly discussed. As shown in Chapter 8, the general nonlinear dynamic data reconciliation (NDDR) formulation can be written as ... 

Allows no description of dynamic properties Estimation of kinetic parameters not feasible... 

From the resulting reactions a set of coupled differential equations can be derived describing the deactivation of P, L and PI and the reaction rate constants can be derived from storage stability data by the use of parameter estimation methods. The storage stability data give the concentration of P+PI (it is assumed that the inhibitor fully releases the protease during analysis due to fast dynamics and the extensive dilution in the assay) and L as a function of time. 

The observed transients of the crystal size distribution (CSD) of industrial crystallizers are either caused by process disturbances or by instabilities in the crystallization process itself (1 ). Due to the introduction of an on-line CSD measurement technique (2), the control of CSD s in crystallization processes comes into sight. Another requirement to reach this goal is a dynamic model for the CSD in Industrial crystallizers. The dynamic model for a continuous crystallization process consists of a nonlinear partial difference equation coupled to one or two ordinary differential equations (2..iU and is completed by a set of algebraic relations for the growth and nucleatlon kinetics. The kinetic relations are empirical and contain a number of parameters which have to be estimated from the experimental data. Simulation of the experimental data in combination with a nonlinear parameter estimation is a powerful 1 technique to determine the kinetic parameters from the experimental... 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

The experimental setup and the testing procedures specifically needed for the dynamic kinetic investigation will be discussed in the following sections, as well as the relevant methods for data analysis and for parameter estimation. 

Step 4 Model Development The dynamic model is developed from the plant test data by selecting a model form (e.g., a step response model) and then estimating the model parameters. However, first it is important to eliminate periods of test data where plant upsets or other abnormal situations have occurred. Decisions to omit portions of the test data are based on visual inspection of the data, knowledge of the process, and experience. Parameter estimation is usually based on least squares estimation. 

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