State Estimation Techniques

In the design of the composition and viscosity feedback controllers it is very important to establish whether the polymer reactor dynamics need to be taken explicitly into account. The choice of sampling frequency balances the requirements for good quality control versus the need to minimize analytical costs. Usually, when the reactor residence time is much shorter than the sampling frequency, integral control is appropriate, because the time between measurements is usually sufficient for the effect of an adjustment to a process variable set point to be com- 

Reaction calorimetry aims to measure heat released from a polymerization in order to infer monomer conversion and polymerization rate (as reviewed, for example, in Refs. 8, 38, and 39). Careful measurement and balancing of mass and energy flows are necessary for success of this technique. For example, the commercial Metder-Toledo RCl jacketed reactor acts as a calorimeter supplying mass balance, polymerization heat generation, and transport data. 

On-line estimation may also be accomplished using first-principles polymerization kinetic models implemented on-line in the form of an extended Kalman filter (EKF) (as illustrated for example in Refs. 8 and 40-42). It should be pointed out that the choice of techniques for on-line estimation of polymer properties is still an active area of research and is very much dependent on the specifics of the polymer chemistry and available on-line instrumentation. 

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In the classical concept of predictive control, the trajectory (or set-point) of the process is assumed to be known. Control is implemented in a discrete-time fashion with a fixed sampling rate, i.e. measurements are assumed to be available at a certain frequency and the control inputs are changed accordingly. The inputs are piecewise constant over the sampling intervals. The prediction horizon Hp represents the number of time intervals over which the future process behavior will be predicted using the model and the assumed future inputs, and over which the performance of the process is optimized (Fig. 9.1). Only those inputs located in the control horizon H, are considered as optimization variables, whereas the remaining variables between Hr+1 and Hp are set equal to the input variables in the time interval Hr. The result of the optimization step is a sequence of input vectors. The first input vector is applied immediately to the plant. The control and the prediction horizon are then shifted one interval forward in time and the optimization run is repeated, taking into account new data on the process state and, eventually, newly estimated process parameters. The full process state is usually not measurable, so state estimation techniques must be used. Most model-predictive controllers employed in industry use input-output models of the process rather than a state-based approach. 

Polymerization process control can benefit significantly from using online state estimation techniques. In general, online control of polymer properties such as molecular weight, MWD, copolymer composition, MI, density, etc. is difficult, mainly because of the lack of adequate online or in-process sensors. Therefore, many of these polymer property parameters are controlled indirectly by controlling first-level process variables such as temperature, pressure, and the flow rates of various reactants, solvents, and catalysts. When some deviations in polymer properties are detected through laboratory sample analysis, certain reactor variables need to be adjusted. Extensive plant experience might be required to make such process adjustments, or model-based online state estimator can be used. 

One alternative to the direct online measurement of polymer properties is to use a process model in conjunction with optimal state estimation techniques to predict the polymer properties. Indeed, several online state estimation techniques such as Kalman filters, nonlinear extended Kalman filters (EKF), and observers have been developed and applied to polymerization process systems. ° In implementing the online state estimator, several issues arise. For example, the standard filtering algorithm needs to be modified to accommodate time-delayed offline measurements (e.g., MWD, composition, conversion). The estimation update frequency needs to be optimally selected to compensate for the model inaccuracy. Table 5 shows the extended Kalman filter algorithm with delayed offline measurements. Fig. 2 illustrates the use of online state estimator... 

The state estimation technique can also be incorporated into the design of optimal batch polymerization control system. For example, a batch reaction time is divided into several control intervals, and the optimal control trajectory is updated online using the molecular weight estimates generated by a model/state state estimator. Of course, if batch reaction time is short, such feedback control of polymer properties would be practically difficult to implement. Nevertheless, the online stochastic estimation techniques and the model predictive control techniques offer promising new directions for the improved control of batch polymerization reactors. 

State estimation techniques provide estimates of the states of a dynamic system, which are obtained by balancing the contribution made by a deterministic dynamic process model with that given by the measurement model and the actual measurements. In formal... 

Extensions of Kalman filters and Luenberger observers [131 Solution polymerizations (conversion and molecular weight estimation) with and without on-line measurements for A4w [102, 113, 133, 134] Emulsion polymerization (monomer concentration in the particles with parameter estimation or not (n)) [45, 139[ Heat of reaction and heat transfer coefficient in polymerization reactors [135, 141, 142] Computationally fast, reiterative and constrained algorithms are more robust, multi-rate (having fast/ frequent and slow measurements can be handled)/Trial and error required for tuning the process and observation model covariance errors, model linearization required The number of industrial applications is scarce A critical article by Wilson eta/. [143] reviews the industrial implementation and shows their experiences at Ciba. Their main conclusion is that the superior performance of state estimation techniques over open-loop observers cannot be guaranteed. 

Furthermore, state estimation techniques such as the boot-strapping method developed by BenAmor et al. [14] allow one to account for fouling in real time. The method assumes that the overall heat transfer coefficient (multiplied by the surface) changes very little in the course of the time required to measure the reactor temperature. Thus, the reactor temperature is used to calculate first a heat generation rate over the course of 1-2 s, assuming constant heat transfer conditions, then a heat transfer coefficient assuming constant rate for a very short time. If fouling occurs over a reasonable timescale, it can be accounted for by the software sensor in-line. 

Data acquisition. To be able to perform the identification of the various model parameters, sufficient process data needs to be available. This data can be gathered by doing experiments specifically designed for obtaining process behavior information if this information is not readily available. In two of the three examples below, this data can be obtained by performing several simulation experiments under different operating conditions. The measurements of the state profiles can then be used as identification data. Parameter or state estimation techniques can also be used to obtain the correct information. 

The Kalman filter-based dynamic state estimation tools in combination with Monte Carlo simulation methods can be employed to estimate probability of failure in instrumented structures with performance functions encompassing unmeasured system states (Ching and Beck 2007). The variance reduction strategies developed in the context of reliability analysis when applied in conjunction with the dynamic state estimation techniques could be used to determine the updated probability of failure of the structural system. For example, the data-based extreme value analysis and the Girsanov transformation-based method can be used to determine the reliability of existing structures (Radhika and Manohar 2010 Sundar and Manohar 2013). 

Numerous other methods have been used to predict properties of gases and Hquids. These include group contribution, reference substance, approaches, and many others. However, corresponding states theory has been one of the most thoroughly investigated methods and has become an important basis for the development of correlation and property estimation techniques. The methods derived from the corresponding states theory for Hquid and gas property estimation have proved invaluable for work such as process and equipment design. 

The solid-state NMR technique may also be used in cellulose derivatives to follow the degree of substitution and degradation of the chain e.g. as found for cellulose nitrate 16). Investigations on the composition of copolymers may also been done as examplared by celluloseacetate-butyrate given in Fig. 6, 20). Here, owing to relaxation differences the spectra cannot be used for elementary analyses, but for estimating the relative number of the components. 

The reaction rate constant for each elementary reaction in the mechanism must be specified, usually in Arrhenius form. Experimental rate constants are available for many of the elementary reactions, and clearly these are the most desirable. However, often such experimental rate constants will be lacking for the majority of the reactions. Standard techniques have been developed for estimating these rate constants.A fundamental input for these estimation techniques is information on the thermochemistry and geometry of reactant, product, and transition-state species. Such thermochemical information is often obtainable from electronic structure calculations, such as those discussed above. 

Jang, S. S., Josepth, B and Mukai, H. (1986). Comparison of two approaches to on-line parameter and state estimation problem of non-linear systems. Ind. Eng. Chem. Process Des. Dev. 25, 809-814. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York. Liebman, M. J., Edgar, T. F., and Lasdon, L. S. (1992). Efficient data reconciliation and estimation for dynamic process using non-linear programming techniques. Comput. Chem. Eng. 16, 963-986. McBrayer, K. F., and Edgar, T. F. (1995). Bias detection and estimation on dynamic data reconciliation. J Proc. Control 15, 285-289. 

In this chapter, the general problem of joint parameter estimation and data reconciliation was discussed. First, the typical parameter estimation problem was analyzed, in which the independent variables are error-free, and aspects related to the sequential processing of the information were considered. Later, 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, was stated. Alternative solution techniques were briefly discussed. Finally, joint parameter-state estimation in dynamic processes was considered and two different approaches, based on filtering techniques and nonlinear programming techniques, were discussed. 

One problem encountered in solving Eq. (11.12) is the modeling of the prior distribution P x. It is assumed that this distribution is not known in advance and must be calculated from historical data. Several methods for estimating the density function of a set of variables are presented in the literature. Among these methods are histograms, orthogonal estimators, kernel estimators, and elliptical basis function (EBF) estimators (see Silverman, 1986 Scott, 1992 Johnston and Kramer, 1994 Chen et al., 1996). A wavelet-based density estimation technique has been developed by Safavi et al. (1997) as an alternative and superior method to other common density estimation techniques. Johnston and Kramer (1998) have proposed the recursive state... 

Liquid-Phase Models. Theoretical models of the liquid state are not as well established as those for gases consequently, the development of general equations for the description of liquid-phase equilibrium behavior is not far advanced. Cubic equations of state give a qualitative description of liquid-phase equilibrium behavior, but do not generally yield good quantitative results (3). For engineering calculations, equations and estimation techniques developed specifically for liquids must normally be used. 

Estimation Techniques for Phase Equilibria state fugacity is used for both activities) to obtain... 

The equilibrium constant can be calculated from thermodynamic data. Robie et al.9 give the standard state thermodynamic data at T = 298.15 K, which we summarize in Table 15.1. The value of C° m for COj" (aq) has not been determined experimentally. Nordstrom and Munoz10 used estimation techniques to obtain the value of —403.3 J K 1 mol-1 given in the table. 

Generally, sorption estimates are based on equilibrium conditions only however, incorporation of kinetic considerations into sorption estimation techniques is likely to be an important area of future work. For example, the assumption of equilibrium sorption in dynamic field systems may result in calculating too much pesticide in the sorbed state. 

In order to perform the on-line optimization strategy, the knowledge of current state variables and/or parameters in the process models is required. Due to the fact that some of these variables cannot be known exactly or sometime can be measured with time delay, it is essential to include an on-line estimator to estimate these process variables using available process measurements as well. The sequence of an estimation and optimization procedure is known as an estimation-optimization task [6], As in several estimation techniques, an Extended Kalman Filter (EKF) has become increasingly popular because it is relatively easy to implement. It has been found that the EKF can be applied to a number of chemical process applications with great success. Once the estimate of unknown process variables is deter-... 

The estimation was made by putting different data together, which included thermal transition data obtained by dynamic mechanical analysis and spectroscopic data quantified mainly by solid-state NMR techniques [43-48], The binary compositions situated in an enclosed area, denoted by connection of dotted lines in the scale list, can be assumed to be a highly compatible state of mixing. a Obtained by hydrolysis treatment of MC/P4VPy... 

The success of MPC is based on a number of factors. First, the technique requires neither state space models (and Riccati equations) nor transfer matrix models (and spectral factorization techniques) but utilizes the step or impulse response as a simple and intuitive process description. This nonpara-metric process description allows time delays and complex dynamics to be represented with equal ease. No advanced knowledge of modeling and identification techniques is necessary. Instead of the observer or state estimator of classic optimal control theory, a model of the process is employed directly in the algorithm to predict the future process outputs. 

An abundance of CJ-state data is available in the cited literature (Refs. 1-6) but seldom at the densities of interest to us. Also, much of the data are actually calculated or estimated, and we do not know how accurate their methods are. Therefore, we should arm ourselves with a tool kit of estimating techniques for which we understand and know the accuracy and limitations. 

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