Linear Dynamic Model

Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonhnear processes. If dcjdi on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets ... 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

The most direct way of obtaining an empirical linear dynamic model of a process is to find the parameters (deadtime, time constant, and damping coefficient) that fit the experimentally obtained step response data. The process being identified is usually openloop, but experimental testing of closedloop systems is also possible. 

Other approaches to genetic networks include study of small circuits with either differential equations or stochastic differential equations. The use of stochastic equations emphasizes the point that noise is a central factor in the dynamics. This is of conceptual importance as well as practical importance. In all the families of models studied, the non-linear dynamical systems typically exhibit a number of dynamical attractors. These are subregions of the system s state space to which the system flows and in which it thereafter remains. A plausible interpretation is that these attractors correspond to the cell types of the organism. However, in the presence of noise, attractors can be destabilized. 

The microkinetic models provide quite detailed description of the transients in catalyst operation. However, the number of balanced species and reaction steps is quite high for a realistic exhaust gas composition, due to the explicit consideration of all surface-deposited reaction intermediates. The models using microkinetic reaction schemes may also exhibit quite complex non-linear dynamic behavior (cf., e.g., Kubicek and Marek, 1983 Marek and Schreiber,... 

We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. 

For the low-temperature steady state y in Figure 4 (A-2) a similar analysis shows that this steady state is stable as well. However, for the intermediate steady-state temperature yi and Sy > 0 the heat generation is larger than the heat removal and therefore the system will heat up and move away from y2. On the other hand, if 5y < 0 then the heat removal exceeds the heat generation and thus the system will cool down away from 2/2 -We conclude that yi is an unstable steady state. For 2/2, computing the eigenvalues of the linearized dynamic model is not necessary since any violation of a necessary condition for stability is sufficient for instability. 

In Example 2.26, we have obtained the linear regression model for dynamic viscosity y, P, as a function of mixing speed X3, min"1 and mixing time X2, min of composite rocket propellant. To determine the conditions of minimal viscosity, a method of steepest ascent has been applied. This method has defined the local optimum region that has to be described by a second-order model. Conditions of the factor variations are shown in Table 2.146. 

One can see that the approximation of the theory, based on the linear dynamics of a macromolecule, is not adequate for strongly entangled systems. One has to introduce local anisotropy in the model of the modified Cerf-Rouse modes or use the model of reptating macromolecule (Doi and Edwards 1986) to get the necessary corrections (as we do in Chapters 4 and 5, considering relaxation and diffusion of macromolecules in entangled systems). The more consequent theory can be formulated on the base of non-linear dynamic equations (3.31), (3.34) and (3.35). 

McAvoy (1999) advanced the use of optimization calculations at the controller design stage, proposing the synthesis of plant-wide control structures that ensure minimal actuator movements. The initial work relying on steady-state models (McAvoy 1999) was recast into a controller synthesis procedure based on linear dynamic plant models (Chen and McAvoy 2003, Chen et al. 2004), whereby the performance of the generated plant-wide control structures was evaluated through dynamic simulations. 

In a direct-response model, the output of a linear dynamic model (the link model) with input c (t) drives a nonlinear static model (usually the Emax model) to produce the observed response. 

The temperature dependence of the Payne effect has been studied by Payne and other authors [28, 32, 47]. With increasing temperature an Arrhe-nius-like drop of the moduli is found if the deformation amplitude is kept constant. Beside this effect, the impact of filler surface characteristics in the non-linear dynamic properties of filler reinforced rubbers has been discussed in a review of Wang [47], where basic theoretical interpretations and modeling is presented. The Payne effect has also been investigated in composites containing polymeric model fillers, like microgels of different particle size and surface chemistry, which could provide some more insight into the fundamental mechanisms of rubber reinforcement by colloidal fillers [48, 49]. 

The usefulness of linear dynamic models is familiar to all control engineers. Linear models can be written compactly in matrix notation. 

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