Simple model state variables

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

A full-order state observer estimates all of the system state variables. If, however, some of the state variables are measured, it may only be neeessary to estimate a few of them. This is referred to as a redueed-order state observer. All observers use some form of mathematieal model to produee an estimate x of the aetual state veetor x. Figure 8.8 shows a simple arrangement of a full-order state observer. 

The internal structure of an observer is based on the model of the considered system. Of course, the model can be extremely simple or reduced to a simple algebraic relationship binding available measurements. However, when the model is of the dynamical type, the value of a variable is no longer influenced uniquely by the inputs at the considered moment but also by the former values of the inputs as well as by other system variables. These phenomena are then described by differential equations. Since these models carry information on the interactions between the inputs and the state variables, they are used to estimate unmeasured variables from the readily available measurements. 

In reply to these criticisms, Hanna addressed the objections point by point. He stated that the use of regionally averaged variables is a necessary first step and has no special limitations. He asserted that the simple model formulation does not assume steady-state conditions, but... 

A suitable response variable is selected. This variable should be chosen such that it has a homoscedastical error and results in simple models. For reasons stated below is chosen (see Section 6.2.10). 

Unfortunately, many transformations purported to linearize the model also interchange the role of dependent and independent variables. Important examplE are the various linearization transformations of the simple steady-state Michaelis-Menten model... 

Assuming that all the events are homogeneous in all vesicles, and using the proper dimensionless state variables and parameters, we consider the behavior for a single synaptic vesicle as described by this simple two-compartment model, where (I) and (II) denote the two compartments. 

The relatively simple two-enzymes/two-compartments model is thus represented in (4.101) via the above set of eight coupled ordinary nonlinear differential equations (4.103) to (4.106). This system of IVPs has the eight state variables hj(t), sy(t), S2j(t), ssj(t) for j = 1,2 that depend on the time t. The normalized reaction rates rj t) are given in equations (4.107) and (4.108). The system has 26 parameters that describe the dynamics for all compounds considered in the two compartments. A specific list of validated experimental parameter values follows in Section 4.4.5. 

The simple fermentor model has three state variables that change with time during the batch. The first is the concentration of cells X (grams of cells per liter of reactor liquid) that grow during the batch from some initial small value. This is provided by a seed fermentor that is itself a small batch fermentor in which a small number of cells are grown. 

M.F. Horstemeyer et al A multiscale analysis of fixed-end simple shear using molecular dynamics, crystal plasticity, and a macroscopic internal state variable theory. Modell. Sim. Mat. Sci. Eng. 11, 265-286 (2003)... 

The interaction of these two processes can be described by a simple isothermal model, which is based on balances of mass and charge. The model describes the extent of the reforming and oxidation reactions along the anode channel. The essential simulation results can easily be displayed in a conversion diagram which is a phase diagram of the two dynamic state variables, namely the extents of two reactions. 

Biomass concentration is of paramount importance to scientists as well as engineers. It is a simple measure of the available quantity of a biocatalyst and is definitely an important key variable because it determines - simplifying - the rates of growth and/or product formation. Almost all mathematical models used to describe growth or product formation contain biomass as a most important state variable. Many control strategies involve the objective of maximizing biomass concentration it remains to be decided whether this is always wise. 

The belief is that the statistical method used (such as PLS, PCR, MLR, PCA, ANNs) will extract from the data those variables which are most important, and discard irrelevant information. Statistical theory shows that this is incorrect. In particular, the principle of parsimony states that a simple model (one with fewer variables or parameters), if it is just as good at predicting a particular set of data as a more complex model, will tend to be better at predicting a new, previously unseen data set [153-155]. Our work has shown that this principle holds. 

A major limitation of the present work is that it deals only with well-defined (and mostly unidirectional) flow fields and simple homogeneous and catalytic reactor models. In addition, it ignores the coupling between the flow field and the species and energy balances which may be due to physical property variations or dependence of transport coefficients on state variables. Thus, a major and useful extension of the present work is to consider two- or three-dimensional flow fields (through simplified Navier-Stokes or Reynolds averaged equations), include physical property variations and derive lowdimensional models for various types of multi-phase reactors such as gas-liquid, fluid-solid (with diffusion and reaction in the solid phase) and gas-liquid-solid reactors. 

Figure 3.2 Radial runs of various disk variables according to the simple steady-state toy disk model described in the main text for three different global accretion rates. The central star is Sun-like. We have assumed a gray opacity of k = 1 in regions where Tm < 1500K. In regions where Tm > 1500 K we switched to k = 0.01 to mimic the effect of dust evaporation. Since dust evaporation reduces the mid-plane temperature there will be a region where dust is only partly evaporated to keep Tm at 1500K. Dust evaporation acts as a thermostat here.

State feedback control is commonly used in control systems, due to its simple structure and powerful functions. Data-driven methods such as neural networks are useful only for situations with fully measured state variables. For this system in which state variables are not measurable and measurement function is nonlinear, we are dependant on system model for state estimation. On the other hand, as shown in figure 2, in open-loop situations, system has limit cycle behavior and measurements do not give any information of system dynamics. Therefore, we use model-based approach. 

Complex Model. A complex model is a set of statistically derived equations that relate fuel properties to vehicle emissions. This model became mandatory in 1998. The simple model calculates emission based on a fuel s RVP, oxygen, aromatic, and benzene content whereas the complex model adds four more variables (sulfur, olefin, and the 200 F and 300°F distillation volume fractions) to the equation. This model is based on the data collected from programs conducted around the United States. The database was made up of over 200 test fuels, 500 automobiles, and 5000 emission test-ings. The complex model can be divided into two portions exhaust and nonexhaust. The nonexhaust VOC was derived directly from the simple model approach where the nonexhaust benzene was modeled as a weight fraction of nonexhaust VOC from the headspace model of General Motors. The exhaust model was based on 19 different test programs. ... 

Figures 2-7 show the phase behavior for the van der Waals like EoS where hard sphere diameter depends on the state variables. It was detected an appearance of third critical point with repulsive term (2) that surprisingly broadens the possibilities of very simple EoS model. It allows considering the liquid state as a mixture of the two corresponding fluid phases, LDL and HDL.

The time-domain method presented here tackles head-on the problem of finding the parameter variations necessary to cause model matching for all k recorded variables. In some cases, the derivative of the measured state will itself be a state in the model, in which case the behaviour of two state variables may be found from a single record using the simple procedure of differentiation, which may be carried out to very good accuracy off-line. For example, suppose both position, X, and velocity, v = dxjdt, are state variables... 

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