Error estimate dynamic models

The semi-empirical bond polarization model is a powerful tool for the calculation of, 3C chemical shift tensors. For most molecules the errors of this model are in the same order of magnitude as the errors of ab initio methods, under the condition that the surrounding of the carbon is not too much deformed by small bond angles. A great advantage of the model is that bond polarization calculations are very fast. The chemical shift tensors of small molecules can be estimated in fractions of a second. There is also virtually no limit for the size of the molecule. Systems with a few thousand atoms can be calculated with a standard PC within a few minutes. Possible applications are repetitive calculations during molecular dynamics simulations for the interpretation of dynamic effects on 13C chemical shift distribution. 

Implementation of dynamic simulators has led to interesting research issues. For example, many have been implemented in a sequential modular format. To carry out the integration correctly from the point of view of correctly assessing integration errors, each unit model can receive as input a current estimate for the state variables (variables x), the unit input stream variables, and any independent input variables specified versus time... 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

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... 

Any measurement transducer output contains some dynamic error an estimate of the error can be calculated if transducer time constant t and the maximum expected rate of change of the measured variable are known. For a ramp input, x(t) = at, and a first-order dynamic model (see Eq. 9-15), the transducer output y is related to x by ... 

The next task is to seek a model for the observer. We stay with a single-input single-output system, but the concept can be extended to multiple outputs. The estimate should embody the dynamics of the plant (process). Thus one probable model, as shown in Fig. 9.4, is to assume that the state estimator has the same structure as the plant model, as in Eqs. (9-13) and (9-14), or Fig. 9.1. The estimator also has the identical plant matrices A and B. However, one major difference is the addition of the estimation error, y - y, in the computation of the estimated state x. 

On the other hand, the main characteristic of interval observers is the use of the aforementioned cooperativity property, which must hold on the estimation error dynamics. Now, let hypotheses Hlf-g be verified. That means that, on the one hand, some bounds are now available on the initial conditions and, on the other hand, b t) is considered in the following as unmeasured, but some lower and upper bounds — possibly time varying — are known. In such a situation, notice that model (16) may not be observable. Consequently, it may not possible to design an asymptotic observer such as (19). Nevertheless, its basic exponentially stable structure and its property of being independent of the nonlinearities may be used. The main idea developed in the following... 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

The second stage realizes a two-step procedure that re-calculates the ozone concentration over the whole space S = (tp, A, z) (, A)e l 0atmospheric boundary layer (zH 70 km), whose consideration is important in estimating the state of the regional ozonosphere. These two steps correspond to the vertical and horizontal constituents of atmospheric motion. This division is made for convenience, so that the user of the expert system can choose a synoptic scenario. According to the available estimates (Karol, 2000 Kraabol et al., 2000 Meijer and Velthoven, 1997), the processes involved in vertical mixing prevail in the dynamics of ozone concentration. It is here that, due to uncertain estimates of Dz, there are serious errors in model calculations. Therefore the units CCAB, MFDO, and MPTO (see Table 4.9) provide the user with the principal possibility to choose various approximations of the vertical profile of the eddy diffusion coefficient (Dz). 

Let us now look at several results of this investigation. In the nuclear war scenario, for example, the SSMAE shows that Arctic environmental stability would be disturbed 3 months after the impact. From other scenarios, it follows that variations in the velocity of vertical advection from 0.004cm/s to 0.05 cm/s does not affect the Arctic environmental state. An error of 32% in ice area estimate leads to a variation in simulation results of 36%. When this error is more than 32%, simulation results become less stable and can vary by several times. The problem exists of finding the proper criterion to estimate SSMAE sensitivity to variations in model parameters. As Krapivin (1996) showed, a survivability function J(t) reflecting the dynamics of the total biomass of living elements would enable this sensitivity to be estimated. In this instance... 

In the following, the model-based controller-observer adaptive scheme in [15] is presented. Namely, an observer is designed to estimate the effect of the heat released by the reaction on the reactor temperature dynamics then, this estimate is used by a cascade temperature control scheme, based on the closure of two temperature feedback loops, where the output of the reactor temperature controller becomes the setpoint of the cooling jacket temperature controller. Model-free variants of this control scheme are developed as well. The convergence of the overall controller-observer scheme, in terms of observer estimation errors and controller tracking errors, is proven via a Lyapunov-like argument. Noticeably, the scheme is developed for the general class of irreversible nonchain reactions presented in Sect. 2.5. 

Remarks 5.1 and 5.2 on the exponential stability of the estimation error dynamics can be extended to the overall controller-observer scheme as well. Hence, robustness with respect to effects due to modeling uncertainties and/or disturbances is guaranteed. Moreover, the following remarks can be stated. 

Very often a mixture of these two approaches is used to determine the values of the parameters. Good examples are Dano et al. [78] and Chassagnole et al. [79]. In these studies many parameters were taken from the literature and, in a parameter estimation approach, were allowed to vary within experimental error to fit the unknown parameters. When considering dynamics, the boundary conditions of the network have to be supplied as explicit functions of time, and therefore they have to be measured in order to give good values for parameter with parameter estimation [74, 79]. Many detailed and core models can be interrogated online at JWS online (www.jjj.bio.vu.nl) [80]. 

Then, given a model for data from a specific drug in a sample from a population, mixed-effect modeling produces estimates for the complete statistical distribution of the pharmacokinetic-dynamic parameters in the population. Especially, the variance in the pharmacokinetic-dynamic parameter distributions is a measure of the extent of inherent interindividual variability for the particular drug in that population (adults, neonates, etc.). The distribution of residual errors in the observations, with respect to the mean pharmacokinetic or pharmacodynamic model, reflects measurement or assay error, model misspecification, and, more rarely, temporal dependence of the parameters. 

An estimator (or more specifically an optimal state estimator ) in this usage is an algorithm for obtaining approximate values of process variables which cannot be directly measured. It does this by using knowledge of the system and measurement dynamics, assumed statistics of measurement noise, and initial condition information to deduce a minimum error state estimate. The basic algorithm is usually some version of the Kalman filter.14 In extremely simple terms, a stochastic process model is compared to known process measurements, the difference is minimized in a least-squares sense, and then the model values are used for unmeasurable quantities. Estimators have been tested on a variety of processes, including mycelial fermentation and fed-batch penicillin production,13 and baker s yeast fermentation.15 The... 

---