Modelling uncertainties

The MPC control problem illustrated in Eqs. (8-66) to (8-71) contains a variety of design parameters model horizon N, prediction horizon p, control horizon m, weighting factors Wj, move suppression factor 6, the constraint limits Bj, Q, and Dj, and the sampling period At. Some of these parameters can be used to tune the MPC strategy, notably the move suppression faclor 6, but details remain largely proprietary. One commercial controller, Honeywell s RMPCT (Robust Multivariable Predictive Control Technology), provides default tuning parameters based on the dynamic process model and the model uncertainty. 

Preliminary Analysis The purpose of the preliminary analyses is to develop estimates for the model parameter values and to estabhsh the model sensitivity to the underlying database and plant and model uncertainties. This will estabhsh whether the unit test will actually achieve the desired results. 

Unstructured model uncertainty relates to unmodelled effects such as plant disturbances and are related to the nominal plant CmCv) as either additive uncertainty (s)... 

Consider a Nyquist contour for the nominal open-loop system Gm(iLu)C(iuj) with the model uncertainty given by equation (9.119). Let fa( ) be the bound of additive uncertainty and therefore be the radius of a disk superimposed upon the nominal Nyquist contour. This means that G(iuj) lies within a family of plants 7r(C(ja ) e tt) described by the disk, defined mathematically as... 

The disk contains over 120 models in files that may contain source and executable code, sample input lilcs, other data files, sample output files, and in many cases, model documentation in WordPerfect, ASCII text or other formats. The disk contains IMES with information on >clecting tin appropriate model, literature citations on validation of models in actual applications, and a demonstration of a model uncertainty protocol. 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

Uncertainty on tlie other hand, represents lack of knowledge about factors such as adverse effects or contaminant levels which may be reduced with additional study. Generally, risk assessments carry several categories of uncertainly, and each merits consideration. Measurement micertainty refers to tlie usual eiTor tliat accompanies scientific measurements—standard statistical teclmiques can often be used to express measurement micertainty. A substantial aniomit of uncertainty is often inlierent in enviromiiental sampling, and assessments should address tliese micertainties. There are likewise uncertainties associated with tlie use of scientific models, e.g., dose-response models, and models of environmental fate and transport. Evaluation of model uncertainty would consider tlie scientific basis for the model and available empirical validation. 

Source terms for dispersion and other models Uncertainties in effects modeling -Animal data inappropriate for humans (especially for toxicity) Mitigating effects may be omitted... 

Model Uncertainty Estimates. In spite of the complexity of the 3 models compared by Dodge, (CB4, CAL and RADM employ 81, 132, and 154 chemical steps and 34, 51 and 57 chemical species, respectively), it is possible to gain useful information about radical interrelationships by considering approximate... 

Sinnar, R., Impact of model uncertainties and nonlinearities on modem controller design In Chemical Process Control, CPC-III. (Morari, M. and McAvoy, T. J., eds.), p. 53 CACHE-Elsevier, 1986. 

Because the rules or procedures in expert systems are heuristic they are often not well-defined in a logical sense. Nevertheless, they are used to draw conclusions. A conclusion can be uncertain because the truth of the rules deriving it cannot be established with 100% certainty or because the facts or evidence on which the rule is based are uncertain. Some measure of reliability of the obtained conclusions is therefore useful. There are different approaches used in expert systems to model uncertainty. They can be divided into methods that are based on... 

All these methods pretend to represent the intuitive way an expert deals with uncertainty. Whether this is true remains an open question. No method has yet been evaluated thoroughly. Modelling uncertainty to obtain a reasonable reliability measure for the conclusions remains one of the major unsolved issues in expert system technology. Therefore, it is important that in the expert system a mechanism is provided to define its boundaries, within which it is reasonably safe to accept the conclusions of the expert system. 

Obtaining a good quality QSAR model depends on many factors, such as the quality of biological data and the choice of descriptors and statistical methods. As a consequence, the uncertainty of the QSAR predictions is a combination of experimental uncertainties and model uncertainties. QSAR methods have to be applied to individual chemicals, not on mixtures. If the QSAR demands it, the components of the mixture have to be addressed separately and individually - in case of unknown compounds, QSAR cannot identify the toxicity risk and is therefore not useful. 

As seen previously for some specific applications such as wastewater treatment plants, software sensors can be envisaged to provide on-line estimation of non-measurable variables, model parameters or to overcome measurement delays [81-83]. Software sensors have been developed mainly for monitoring bioprocesses because the control system design of bioreactors is not straightforward due to [84] significant model uncertainty, lack of reliable on-line sensors, the non-linear and time-varying nature of the system or slow response of the process. 

Finally, a Monte Carlo method coupled with the Latin Hypercube Sampling (LHS) was used to assess the overall model uncertainty. The 2a standard deviation of the model was estimated to be 30-40% for OH and 25-30% for HO2, which is comparable to the instrumental uncertainty. 

American Petroleum Institute (API), Publication 4546. Hazard Response Modeling Uncertainty (A Quantitative Method) Evaluation of Commonly Used Hazardous Dispersion Models, Volume 2, API, Washington, D.C., 1992. 

The A function is a measure of the model uncertainty. If we plot A as a function of fiequency (see Fig. 16.11) we can see that its magnitude is quite small at low fiequencies, hut increases as frequency is increased. The larger the uncertainty (the bigger 5), the lower the frequency at which the magnitude of the imcer-tainty becomes large. 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

J.L. Gouze, O. Bernard, and Z. Hadj-Zadok. Observers with modelling uncertainties for the wastewater treatment process. In Journees thematiques Au-tomatique et Environnement , Nancy, France, March 2000. 

ROBUST FEEDBACK CONTROL OF COMBUSTION INSTABILITIES WITH MODEL UNCERTAINTY... 

The use of feedback-control techniques to modulate combustion processes in propulsion systems has recently received extensive attention [1-3]. Most of the previous studies involved direct implementation of existing control methods designed for mechanical devices, with very limited effort devoted to the treatment of model and parametric uncertainties commonly associated with practical combustion problems. It is well established that the intrinsic coupling between flow oscillations and transient combustion responses prohibits detailed and precise modeling of the various phenomena in a combustion chamber, and, as such, the model may not accommodate all the essential processes involved due to the physical assumptions and mathematical approximations employed. The present effort attempts to develop a robust feedback controller for suppressing combustion instabilities in propulsion systems. Special attention is given to the treatment of model uncertainties. Various issues related to plant... 

Hoo and /u control Generic combustion instability [20] 1. Observer-based controller with robust property valid for intensity-unknown disturbance 2. Can regulate frequency domain property 3. Accommodate model uncertainty... 

In view of this, a robust scheme based on the Hoo control theory [24] is developed in the present work. The algorithm guarantees both stability and performance for a family of perturbed plants with model uncertainties and exogenous inputs (i.e., chamber disturbances and sensor noises) over a wide range of operating conditions, an advantage especially desired for combustion dynamics problems. 

The system dynamics uncertainty A(s) contains parametric and model uncertainties, and its L2 gain bounded as A(s) oo < 1/7- Based on the L2"gain control theory, the first task of a robust controller for stabilizing perturbed plants is to endow the closed-loop system with the following property ... 

There is a trade-off between the affordable model uncertainty bound 1/7 and the maximum time delay 6t Fig. 22.5 shows this relationship. 

Figures 22.6 and 22.7 present the results. The control scheme developed in the present work indeed guarantees robust performance for a wide variety of perturbed systems with significant model uncertainties.

A comprehensive framework of robust feedback control of combustion instabilities in propulsion systems has been established. The model appears to be the most complete of its kind to date, and accommodates various unique phenomena commonly observed in practical combustion devices. Several important aspects of distributed control process (including time delay, plant disturbance, sensor noise, model uncertainty, and performance specification) are treated systematically, with emphasis placed on the optimization of control robustness and system performance. In addition, a robust observer is established to estimate the instantaneous plant dynamics and consequently to determine control gains. Implementation of the controller in a generic dump combustor has been successfully demonstrated. 

Figure 22.7 Time response of perturbed system with 50% model uncertainty... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

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