Variable state

Since the errors are the differences between setpoints and controlled variables 

We will use this matrix of transfer function representation extensively in the rest of our work with multivariable processes. 

The time-domain differential equation description of systems can be used instead of the Laplace-domain transfer function description. Naturally the two are related, and we will derive these relationships later in this chapter. State variables are very popular in electrical and mechanical engineering control problems which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representations are more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. 

The states of a dynamic system are simply the variables that appear in the time differential. For example, if we have a chemical reactor in which the concentration of reactant Ca and the temperature T change with time, the material balance for component A and the energy balance would give two differential equations  

State variables appear very naturally in the differential equations describing chemical engineering systems because our mathematical models are based on a number of first-order differential equations component balances, energy equations, etc. If there are N such equations, they can be linearized (if necessary) and written in matrix form 

Systems at equilibrium have measurable properties. A property of a system is any quantity that has a fixed and invariable value in a system at equilibrium. If the system changes from one equilibrium state to another, the properties therefore have changes that depend only on the two states chosen, and not on the manner in which the system changed from one to the other. This dependence of properties on equilibrium states and not on processes is reflected in the alternative name for them, state variables. Recall from the discussion of Euler s theorem in Chapter 2 that extensive variables are proportional to the quantity of matter being considered—for example, volume and (total) heat capacity. Intensive variables are independent of quantity, and include con- 

Since state variables have fixed values in equilibrium states and have changes between equilibrium states that do not depend on how the change is carried out, it follows that the differentials of state variables will always be exact differentials, according to our definitions in Chapter 2. 

This analysis by Knapp is useful in defining and clarifying the local equilibrium problem in a quantitative way. Unfortunately, despite the rather drastic simplification, most of the parameters required to define the problem in real situations at the present time are poorly known. The quantitative results are then of questionable significance in any practical sense, but they are worth reflecting on. All applications of thermodynamics assume local equilibrium, but defining just what that is has proven difficult. 

Many physical properties, such as the volume and various energy terms, come in two forms - the total quantity in the system and the quantity per mole or per gram of substance considered. We use different fonts for these total and molar properties. For example, water has a volume per mole (V) of about 18.0686 cm mol , so if we have 30 moles of water in a beaker, its volume (V) is 542.06 cmT This relationship for a pure substance such as HjO is Z = Z/u , where Z is any total property, Z is the corresponding molar property, and Hi is the number of moles of the substance. In our water example, above, 542.06/30 = 18.068. In more complex systems where more than one substance is present, total and molar properties are related in the same way. A beaker containing, for example, a kilogram of water (55.51 moles H2O) and 1 mole of NaCl occupies 1019.9 cm. The molar volume of the system is then Z = Z/Ei i, or 1019.9/(1-1-55.51)= 18.05 cm moE.  

These two types of state variables have been given names  

In science, molar properties, such as molar volumes, molar energies, are most commonly used. In engineering on the other hand, specific properties are more common. Specific properties are mass-related rather than mole-related. Thus the specific volume of water at 25 °C is 1.0029 cm g. Molar and specific properties are of course related by the molar mass (or so-caUed gram formulas weight, gfw) of the substance. That for water is 18.0153, so 1.0029cm g x 18.0153gmoE = 18.068cm moE.  

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In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

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. 

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... 

Since G is a state variable and forms exact differentials, an alternative expression for dG is... 

While this is an easy calculation to make, Eq. (3.7) does little to clarify exactly what AS means. Phenomenological proofs that AS as defined by Eq. (3.7) is a state variable often leave us with little more than a lament for the inefficiency of spontaneous processes. 

Reaction (5. EE) is particularly useful for the discussion of thermodynamic considerations because of the way differences in thermodynamic state variables are independent of path. Accordingly, if we know the value of AG for reaction (5. EE), we have characterized the following ... 

In process simulation it is necessary to calculate enthalpy as a function of state variables. This is done using the following formulas, derived from the above relations by considering S and H as functions of T and p. 

Fiypothesis unable to explain the values of the state variables observed... 

Incorrect readings of appropriate state variables State variables which are considered appropriate are constant State variables which are considered appropriate may change within the batch and batch-to-batch... 

K set of referential internal state variables I velocity gradient tensor... 

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. 

Specific applications of the theory are not considered in this chapter. Only one example, that of small deformation classical plasticity, is worked out in Section 5.3. The set of internal state variables k is taken to be comprised of... 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

Motivated by the qualitative observations made above, a set of internal state variables deseribing the internal strueture of the material will be intro-dueed ab initio, denoted eolleetively by k. Their physieal meaning or preeise properties need not be established at this point, and they may inelude sealar, veetor, or tensor quantities. The following eonstitutive assumptions are now made ... 

Inelastic Loading. The strain lies on the elastic limit surface = 0, and the tangent to the strain history points in a direction outward from the elastic limit surface > 0. The material is said to be undergoing inelastic loading, and k is assumed to be a function of the strain s, the internal state variables k, and the strain rate k... 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

The remainder of this section will be concerned with a particular case in which normality conditions hold. The constitutive equation for the internal state variables (5.11) involves the constitutive function a, and the normality conditions (5.56) and (5.57) involve an unknown scalar factor y. In some circumstances, a may be eliminated and y may be evaluated by using the consistency condition. These circumstances arise if b is nonsingular so that the normality condition in strain space (5.56j) may be solved for k... 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

Prager s rule of kinematic hardening is expressed by a = ce where c is a constant. Generalizing these concepts, the evolution equations for the internal state variables will be taken in the form... 

The choice (5.77) for the evolution equation for the plastic strain sets the evolution equations for the internal state variables (5.78) into the form (5.11) required for continuity. The consistency condition in the stress space description may be obtained by differentiating (5.73), or directly by expanding (5.29)... 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

In Section 5.2 the set of internal state variables k was introduced. In the referential theory, a similar set of referential internal state variables K will be introduced in the same way without further physical identification at this stage. It will merely be assumed that each member of the set K is invariant under the coordinate transformation (A.50) representing a rigid rotation and translation of the coordinate frame. 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

P.S. Follansbee and U.F. Kocks, A Constitutive Description of the Deformation of Copper Based on the Use of the Mechanical Threshold Stress as an Internal State Variable, Acta Metall. 36, 81-93 (1988). 

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