Simulations nonlinear models

In addition to the mechanistic simulation of absorptive and secretive saturable carrier-mediated transport, we have developed a model of saturable metabolism for the gut and liver that simulates nonlinear responses in drug bioavailability and pharmacokinetics [19]. Hepatic extraction is modeled using a modified venous equilibrium model that is applicable under transient and nonlinear conditions. For drugs undergoing gut metabolism by the same enzymes responsible for liver metabolism (e.g., CYPs 3A4 and 2D6), gut metabolism kinetic parameters are scaled from liver metabolism parameters by scaling Vmax by the ratios of the amounts of metabolizing enzymes in each of the intestinal enterocyte compart-... 

Off-line analysis, controller design, and optimization are now performed in the area of dynamics. The largest dynamic simulation has been about 100,000 differential algebraic equations (DAEs) for analysis of control systems. Simulations formulated with process models having over 10,000 DAEs are considered frequently. Also, detailed training simulators have models with over 10,000 DAEs. On-line model predictive control (MPC) and nonlinear MPC using first-principle models are seeing a number of industrial applications, particularly in polymeric reactions and processes. At this point, systems with over 100 DAEs have been implemented for on-line dynamic optimization and control. 

Transient simulations using the full, nonlinear model show that under the conditions studied concentration profiles reach a quasi steady state quite rapidly (often within 3 to 5 sec), whereas the thermal response of the reactor bed is much slower22 due to the large heat capacity of the reactor bed and thermal well. An example of this phenomenon is shown in Fig. 18, where the transient responses of the solid temperatures, thermal well temperatures, and concentrations are shown for a major step change in the inlet gas temperature and inlet CO concentration. In this example, the effect of the step change is nearly immediate on the concentration profiles, with the major effect being within the first 10 sec. However, Fig. 18a shows that the thermal well temperatures and the catalyst temperatures take up to 10 times as long as the... 

Simulations show negligible differences in the transient temperature and concentration profiles as a result of this quasi-steady-state approximation. The major advantage of this assumption should be apparent in control system design, where a reduction in the size of the state vector is computationally beneficial or in the time-consuming simulations of the full nonlinear model. 

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

Figure 21 shows the simulated dynamic behavior of the gas temperatures at various axial locations in the bed using both the linear and nonlinear models for a step change in the inlet CO concentration from a mole fraction of 0.06 to 0.07 and in the inlet gas temperature from 573 to 593 K. Figure 22 shows the corresponding dynamic behavior of the CO and C02 concentrations at the reactor exit and at a point early in the reactor bed. The axial concentration profiles at the initial conditions and at the final steady state using both the linear and nonlinear simulations are shown in Fig. 23. The temporal behavior of the profiles shows that the discrepancies between the linear and nonlinear results increase as the final steady state is approached. Even so, there are only slight differences (less than 2% in concentrations and less than 0.5% in temperatures) in the profiles throughout the dynamic responses and at the final steady state even for this relatively major step-input change.

Other important characteristics of the converter are the reflected ripple attenuation and the turn-on characteristics. It is expected that the turn-on characteristics will be difficult to simulate because of the nonlinear characteristics of a saturating core. A nonsaturating core is simply described by Faraday s law, and it can be easily modeled by any of the SPICE simulators. The model used for the EMI filter is shown in Fig. 3.66, and the results of each of the simulators output and the measured impedance plots are shown in Figs. 3.67 to 3.70. 

Figure 3.72 IsSpice results of nonlinear model for inrush current simulation.

The two-compartment model and the model of the enzymatic reaction (cf. Sections 9.1.2 and 8.5.1, respectively) will be presented as typical cases for linear and nonlinear models, respectively. For these simulations, the model parameters were set as follows ... 

While the nonlinearities are eliminated, it is clear the number of discrete and continuous variables is increased as well as the number of constraints. Also, in the general case the definition of the matrix of coefficients and the right-hand sides of problem (MAPP) requires an a priori evaluation or simulation of nonlinear models. 

The model predictive control used includes all features of Quadratic Dynamic Matrix Control [19], furthermore it is able to take into account soft output constraints as a non linear optimization. The programs are written in C++ with Fortran libraries. The manipulated inputs (shown in cm Vs) calculated by predictive control are imposed to the full nonlinear model of the SMB. The control simulations were made to study the tracking of both purities and the influence of disturbances of feed flow rate or feed composition. Only partial results are shown. 

This approach is called the first-order (FO) method in NONMEM. This is the most widely used approach in PPK and PD data analysis and has been evaluated by simulation. The use of the first-order Taylor series expansion to approximate the nonlinear model in p, and possibly e by a linear model in these parameters is the greatest limitation of the FO approach. 

Donaldson and Schnabel (1987) used Monte Carlo simulation to determine which of the variance estimators was best in constructing approximate confidence intervals. They conclude that Eq. (3.47) is best because it is easy to compute, and it gives results that are never worse and sometimes better than the other two, and is more stable numerically than the other methods. However, their simulations also show that confidence intervals obtained using even the best methods have poor coverage probabilities, as low as 75% for a 95% confidence interval. They go so far as to state confidence intervals constructed using the linearization method can be essentially meaningless (Donaldson and Schnabel, 1987). Based on their results, it is wise not to put much emphasis on confidence intervals constructed from nonlinear models. 

The same caveats that apply to linear models when the predictor variables are measured with error apply to nonlinear models. When the predictor variables are measured with error, the parameter estimates may become biased, depending on the nonlinear model. Simulation may be used as a quick test to examine the dependency of parameter estimates within a particular model on measurement error (Fig. 3.14). The SIMEX algorithm, as introduced in the chapter on Linear Models and Regression, can easily be extended to nonlinear models, although the computation time will increase by orders of magnitude. 

Alternative 2 can be used in very few cases, whereas alternatives 1 and 3 are, in principle, always feasible. In this chapter we discuss the computer simulation of nonlinear processes very briefly because it is a subject to be covered primarily in a course on numerical analysis. More emphasis will be given on the approximation of nonlinear models by linear ones. It should be noted that all the theory for the design of control systems, available from past work, is based on linear systems,... 

The FSSW process simulation involved modeling the coupled thermoelastoplastic response of the tool-workpiece system, in which the constitutive model of the material and the nonlinear temperature-dependent transient heat-transfer response produce both plastic deformations and a temperature distrihution as the material flows and stirs, forming the weld. 

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