New State Variable

yo t ) is equal to /, as can be verified by integrating the above differential equation. Next, we define the Hamiltonian, 

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The next step in the construction of the particular realization of (55) is the specification of the kinematics of (85) (i.e., specification of the operator L). The necessity to satisfy the degeneracy requirement (52) suggests to begin the search for the operator L by passing to new state variables... 

An open system can exchange mass with its surroundings, and mole numbers of the chemical substances are the new state variables required for the specification of the system. Consider the Gibbs energy of an open system. In addition to its natural state variables T and p, there are the state variables, ni,n2... . i.A> where stands for mole number of species i, and n is the number of chemical species in the system. 

This is sufficient to suggest the existence of a state variable equal to ( )rev since this function is conserved in heat engine cycles carried out reversibly. This new state variable is of course the entropy, S, where... 

Applying the First and Second Laws (of Thermodynamics) The equation dU = 8Qe + generally serves as a first step toward the calculus of thermodynamics to be created. It is considered an application of the First Law where the new state variable U can be constructed with the help of the two measurable process quantities 2e and Wg. At first, we will limit ourselves to simple, closed systems, meaning systems without any exchange of substance with the surroundings and in which temperature and pressure are the same everywhere. Except as heat, energy can only be transferred in or out, without friction, by changes to the volume 8We = pdV and therefore... 

Any non-autonomous differential equation can be transformed into a set of autonomous differential equations by introducing additional state variables. For example, with the introduction of the new state variable... 

The problem at the moment is that these new state variables S and G will have no feeling of reality for a reader new to the subject. That is, what is entropy or Gibbs energy, and how does one measure these things Only by actually using these concepts will one become familiar with them. The next chapter is a first attempt at describing these variables in more familiar terms. 

De Donder defined a new state variable called afiOnity [2, 3] ... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The last step in the preceding argument, the use of our knowledge about flowshop scheduling, turns what had been a mainly syntactic criterion over the tree structure of the example, into a criterion based on state variables of (jc, y). The state variable values, the completion times of the various flowshop machines, are accessible before the subtrees beneath jc and y have been generated. Indeed, they determine the relationships between the respective elements of the subtrees (jcm, yu). If we can formalize the process of showing that the pair (jc, y) identified with our syntactic criterion, satisfies the eonditions for equivalence or dominance, wc will in the process have generated a new equivalence rule. 

Notice that the left-hand side of this rule contains two types of clauses. The first type is the variable values of the current state and those necessary to compute the new state, while the second, represented by = computes the value of the variable in the new state. This last clause enables the procedural information about how to compute the state variables to be attached to the reasoning. We must, however, be careful about how much of the computation we hide procedurally, and how much we make explicit in the rules. The level to which computation can be hidden will be a function of the theories we employ to try to obtain new dominance and equivalence conditions. If we do not hide the computation, we will be able to explicitly reason about it, and thus may find simplifications or redundancies in the computation that will lead to more computationally efficient procedures. 

The focus of the representation so far, has been on giving the form of rules, which enable us to reason about the values of state variables. This, however, is only one part of the overall reasoning task. We must also represent the theoreies we are going to use to derive the new control knowledge. 

You may notice that nothing that we have covered so far does integral control as in a PID controller. To implement integral action, we need to add one state variable as in Fig. 9.2. Here, we integrate the error [r(t) -, (t) to generate the new variable xn+1. This quantity is multiplied by the additional feedback gain Kn+1 before being added to the rest of the feedback data. 

The mole fraction X in the previous equation is replaced with a new unitless variable at, the species activity. The standard potentials pt° are defined at a new standard state a hypothetical one-molal solution of the species in which activity and molality are equal, and in which the species properties have been extrapolated to infinite dilution. 

System stability can also be analysed in terms of the linearised differential model equations. In this, new perturbation variables for concentration C and temperature T are defined. These are defined in terms of small deviations in the actual reactor conditions away from the steady-state concentration and temperature Css and Tss respectively. Thus... 

Both quantities are usually written as m x r elasticity matrices e and n, respectively. In contrast to the local elasticities, the control coefficients describe the global or systemic properties of the system, that is, the response to the perturbation after all variable shave relaxed to the new state. 

The decoupling structure is depicted in Figure 11. The global transfer matrix is diagonal. It is clear that the decoupling, as stated, is not always possible, as the transfer function of the different blocks should be stable and physically feasible. If it is possible, the new input variable u will control the concentration and U2 will control the temperature, and both control loops could be tuned independently. Sometimes, a static decoupling is more than enough. 

The purpose of this example is to show how to design the control system by using the pole-placement technique and the use of integrators. The integrators can be represented by introducing a new set of state variables u(t), so the equations of the global system are the following ... 

The independent variable used here for the absorption function is Ei. This is the energy the photon has in excess of that is required to promote to the Fermi level a X-electron of the element in its metallic state. This excess energy, Ex, of the photon, is transferred to the photoelectron, predominantly as kinetic energy. To this approximation, and usingJEquation (2), Ex is proportional to 1/X . A new independent variable, - /Ex or 1/X, is thus analogous to the (sin 6)/X in the diffraction function. 

Equation (6.35b) shows that the new intensive variable, chemical potential pi, as introduced in this chapter, is actually superfluous for the case c = 1, because its variations can always be expressed in terms of the old variations dT dP. Thus, as stated in Inductive Law 1 (Table 2.1), only two degrees of freedom (independently variable intensive properties) suffice to describe the thermodynamic variability of a simple c = 1 system. This confirms (as expected) that chemical potential pu only becomes an informative thermodynamic variable when chemical change is possible, that is, for c > 2 chemical components. 

Let us refer to Figure 5-7 and start with a homogeneous sample of a transition-metal oxide, the state of which is defined by T,P, and the oxygen partial pressure p0. At time t = 0, one (or more) of these intensive state variables is changed instantaneously. We assume that the subsequent equilibration process is controlled by the transport of point defects (cation vacancies and compensating electron holes) and not by chemical reactions at the surface. Thus, the new equilibrium state corresponding to the changed variables is immediately established at the surface, where it remains constant in time. We therefore have to solve a fixed boundary diffusion problem. 

In the DAE system (5.25), the variables z IRC+TO 1 are implicitly fixed by the algebraic constraints, rather than specified explicitly, and thus the index of the system is again nontrivial (i.e., higher than 1). Also, note that, as in the previous reduction step, the index of (5.25) is well-defined only if the flow rates us are specified as a function of the state variables (in this case, expressed in the new coordinates ), i.e., us = us( ). Specifying these flow rates via feedback control laws allows z to be determined through differentiation of the algebraic constraints in Equation (5.25). Differentiating these constraints once yields... 

In this example we keep the same state variables (7) as in the previous example but change the projection (8). The new relation y = y(x) is... 

The quantity q has the physical interpretation of the free volume. It is the state variable used in the Simha-Somcynski equilibrium theory of polymeric fluids (Simha and Somcynski, 1969). The new variable p that we adopt has the meaning of the velocity (or momentum) associated with q. 

Now we proceed to specify the kinematics of (58). In order that the eta-function h (that remain unspecified at this point) and the number of moles n be preserved in the nondissipative time evolution (i.e., the time evolution governed by the first term on the right-hand side of (55)), the matrix L has to be such that Lhx = 0 and Lnx = 0. This degeneracy can be discussed more easily if we pass from the state variables x to a new set of state... 

The physical systems considered in this example are the same as in Example 2.2.4. We also choose the same state variables (41). What is new now is that we shall follow explicitly the time evolution initiated by... 

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