State-variable formulation

The following state-variable formulation of equation (7.4.2) is used Defining... 

TECHNIQUES FOR PROBLEMS WITH BOTH CONVECTIVE AND DIFFUSION EFFECTS THE STATE-VARIABLE FORMULATION... 

For the pure convective case U2 = 0), the Crank-Nicolson formulation suffered rather severe oscillations, while the centered-difference (which is equivalent to the state-variable formulation for this case) performed well. 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Modularity. Since we would like to use the sufficient theory in a variety of contexts and problems, we need a theory that was easy to extend and modify depending on the context. In our state-space formulation the sufficient theory is couched in terms of constraints on variables. This theory gives us the opportunity to modularize its representation, partitioning the information necessary to prove the looseness of one type of constraint from that required to prove the looseness of a different constraint type. The ability to achieve modularity is a function not only of the theory but also of the representation, which should have sufficient granularity to support the natural partitioning of the components of the theory. 

Over the years two ML estimation approaches have evolved (a) parameter estimation based an implicit formulation of the objective function and (b) parameter and state estimation or "error in variables" method based on an explicit formulation of the objective function. In the first approach only the parameters are estimated whereas in the second the true values of the state variables as well as the values of the parameters are estimated. In this section, we are concerned with the latter approach. 

Implicit estimation offers the opportunity to avoid the computationally demanding state estimation by formulating a suitable optimality criterion. The penalty one pays is that additional distributional assumptions must be made. Implicit formulation is based on residuals that are implicit functions of the state variables as opposed to the explicit estimation where the residuals are the errors in the state variables. The assumptions that are made are the following ... 

At each point on the critical locus Equations 40a and b are satisfied when the true values of the binary interaction parameters and the state variables, Tc, Pc and xc are used. As a result, following an implicit formulation, one may attempt to minimize the following residuals. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Following the steps for formulation of a CFD model introduced earlier, we begin by determining the set of state variables needed to describe the flow. Because the density is constant and we are only interested in the mixing properties of the flow, we can replace the chemical species and temperature by a single inert scalar field (x, t), known as the mixture fraction (Fox, 2003). If we take = 0 everywhere in the reactor at time t — 0 and set / = 1 in the first inlet stream, then the value of (x, t) tells us what fraction of the fluid located at point x at time t originated at the first inlet stream. If we denote the inlet volumetric flow rates by qi and q2, respectively, for the two inlets, at steady state the volume-average mixture fraction in the reactor will be... 

Let us formulate the dynamic mass balance equation of the chemical in the SMSL. Fig.23.5 summarizes all processes. At this point we have to select the variable which shall characterize the SMSL. (Remember for the open water box we have chosen the total concentration Ctop.) Due to the large solid-to-water ratio of sediments rss, chemicals with moderate to large distribution coefficients (Kd > 0.1 m3kg ) are predominantly sorbed to the solid phase. Therefore, C offers itself as the natural choice for the second state variable. 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

As this book is mainly intended for undergraduates, we only treat those chemical/biological processes and problems that can be modeled with ODEs. PDEs are significantly more complicated to understand and solve. However, we will often have to solve systems of ODEs, rather than one single ODE. Such ODE systems contain several DEs in one and the same independent variable, but they generally involve several functions or state variables in their formulation. In fact, systems of ODEs (and matrix DEs) occur quite naturally in chemical/biological engineering problems. 

One advantage in the sequential approach is that only the parameters that are used to discretize the control variable profile are considered as the decision variables. The optimization formulated by this approach is a small scale NLP that makes it attractive to apply for solving the optimal control with large dimensional systems that are modeled by a large number of differential equations. In addition, this approach can take the advantage of available IVP solvers. However, the limitation of the sequential method is a difficulty to handle a constraint on state variables (path constraint). This is because the state variables are not directly included in NLP. 

The inelastic strain as an internal state variable is obtained from a linear evolution equation formulated with respect to the intermediate configuration ... 

Next to metals, probably the synthetic polymer-based composites have been modeled most by hierarchical multiscale methods. Different multiscale formulations have been approached top-down internal state variable approaches, self-consistent (or homogenization) theories, and nanoscale quantum-molecular scale methods. 

In Equations 74 and 78 data fit to a model that considers the concept of excess substrate in culture, as set previously in Equation 36 (Zeng, 1995). However, this formulation needs modification in order to include high cell density conditions. In Equations 75 and 79, as seen before, the mass of nutrient available per cell (S/Xv) is the state variable to take this into account. 

Unstructured models, as detailed in Sections 8.3.1 and 8.3.2, are formulated by a series of kinetic and differential non-linear equations that represent the dynamics of all the state variables during the process. Thus, to simulate a model that consists of parameters and state variables, it is necessary to attribute values to the parameters. 

Theoretical description has to be performed within the framework of nonlinear dynamics (16). In brief, knowledge about the microscopic elementary steps for a particular system permits formulation of a set of rate equations for the variation of the surface concentrations of the species involved in the overall reaction, xh (the state variables) with time, in terms of the external parameters pk (temperature, partial pressures) and also of all state variables (17) ... 

In this section we limit ourselves to equilibrium. The time evolution that is absent in this section will be taken into consideration in the next two sections. We begin the equilibrium analysis with classical equilibrium thermodynamics of a one-component system. The classical Gibbs formulation is then put into the setting of contact geometry. In Section 2.2 we extend the set of state variables used in the classical theory and introduce a mesoscopic equilibrium thermodynamics. 

Let the state variables (1) be replaced by a state variable xeM corresponding to a more detailed (more microscopic) view than the one taken in classical equilibrium thermodynamics. Several examples of x are discussed below in the examples accompanying this section. Following closely Section 2.1, we shall now formulate equilibrium thermodynamics that we shall call a mesoscopic equilibrium thermodynamics. 

The relationship between the different state variables of a system subjected to no external forces other than a constant hydrostatic pressure can generally be described by an equation of state (EOS). In physical chemistry, several semiempirical equations (gas laws) have been formulated that describe how the density of a gas changes with pressure and temperature. Such equations contain experimentally derived constants characteristic of the particular gas. In a similar manner, the density of a sohd also changes with temperature or pressure, although to a considerably lesser extent than a gas does. Equations of state describing the pressure, volume, and temperature behavior of a homogeneous solid utilize thermophysical parameters analogous to the constants used in the various gas laws, such as the bulk modulus, B (the inverse of compressibUity), and the volume coefficient of thermal expansion, /3. 

The question of constrained optimisation is answered in a standard manner by Euler-Lagrange optimisations. By formulating the problem with optimal control theory, Johannessen and Kjelstrup explained that the Hamiltonian of the problem was constant in a study of chemical reactors. The total entropy production for a plug flow reactor was written as a function of a position-dependent state variable vector x(z) and the control variable u(z) ... 

The major challenge in the model formulation is the representation strategy adopted for the tank cycle. Normal operation considers that each tank is filled up completely before settling. After the settling period, the tank is released for clients satisfaction, until it is totally empty. These procedures are usually related to the product quality, where it isn t desired to mix products from several different batches. This implies that they are formulated four states for each tank i) full, ii) delivering product to clients, iii) empty and iv) being filled up with product from the pipeline. Each one of the states has a corresponding state variable, related to tank inventory ID), and has to be activated or deactivated whenever a boundary situation occurs (Eq. 1) the maximum UB) and minimum LB) capacities of the tank are met. For this purpose, the state variable y, binary) will have to be activated whenever both inequalities ( < and > ) hold (Eq. 2) ... 

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