State-variable form

To solve these equations in MATLAB we put them into state variable form (this subject is discussed more fully in Chapter 12). 

Using the system shown in Fig. 11.29, one recognizes that it is a fourth-order system. Considering it has two degrees of freedom, two second-order differential equations can be obtained by simply applying Newton s law. Once the system is represented in state variable form (first-order form), then four first-order differential equations, one for each state variable, would be generated. 

Often, an nth-order differential equation is placed in state-variable form. This is a set of n first-order equations. When deriving equations via material and energy balances, this state-variable form arises naturally. By developing the equations in state-variable form, we must solve the simultaneous sets of linear first-order differential equations. The state variable form is... 

We now place equation (7.6.7) in state-variable form by introducing a new variable... 

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

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... 

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. 

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)... 

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... 

In equation (8.93), r(t) is a vector of desired state variables and K is referred to as the state feedback gain matrix. Equations (8.92) and (8.93) are represented in state variable block diagram form in Figure 8.7. 

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. 

As we have seen earlier, the thermodynamic variables p, V, T, U, S, H, A, and G (that we will represent in the following discussion as W, X, T, and Z) are state functions. If one holds the number of moles and hence composition constant, the thermodynamic variables are related through two-dimensional Pfaffian equations. The differential for these functions in the Pfaff expression is an exact differential, since state functions form exact differentials. Thus, the relationships that we now give (and derive where necessary) apply to our thermodynamic variables. 

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. 

When the output vector (measured variables) are related to the state variables (and possibly to the parameters) through a nonlinear relationship of the form y(t) = h(x(t),k), we need to make some additional minor modifications. The sensitivity of the output vector to the parameters can be obtained by performing the implicit differentiation to yield ... 

Let us consider the special class of problems where all state variables are measured and the parameters enter in a linear fashion into the governing differential equations. As usual, we assume that x is the n-dimemional vector of state variables and k is the p-dimemional vector of unknown parameters. The structure of the ODE mode is of the form... 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

The discretized reservoir model can be written in the general form presented in Section 10.3. The state variables are the pressure and the oil, water and gas satu-... 

The fifteen layers of constant permeability and porosity were taken as the reservoir zones for which these parameters would be estimated. The reservoir pressure is a state variable and hence in this case the relationship between the output vector (observed variables) and the state variables is of the form y(t,)=Cx(t,). 

The water-oil ratio is a complex time-dependent function of the state variables since a well can produce oil from several grid cells at the same time. In this case the relationship of the output vector and the state variables is nonlinear of the form y(t,)=h(x(t,)). 

Since enthalpy is a state variable, the integral on the right side of equation 10.2.6 is independent of the path of integration, and it is possible to rewrite this equation in a variety of forms that are more convenient for use in reactor design analyses. One may evaluate this integral by allowing the reaction to proceed isothermally at the initial temperature from extent 0 to extent and then heating the final product mixture at constant pressure and composition from the initial temperature to the final temperature. 

Constraint Programming (CP) is a further optimization approach where relations between variables are stated in form of constraints in order to better solve specifically hard bounded integer optimization problems such as production scheduling... 

Parameter estimation problems result when we attempt to match a model of known form to experimental data by an optimal determination of unknown model parameters. The exact nature of the parameter estimation problem will depend on the mathematical model. An important distinction has to be made at this point. A model will contain both state variables (concentrations, temperatures, pressures, etc.) and parameters (rate constants, dispersion coefficients, activation energies, etc.). 

The simplest case of this parameter estimation problem results if all state variables jfj(t) and their derivatives xs(t) are measured directly. Then the estimation problem involves only r algebraic equations. On the other hand, if the derivatives are not available by direct measurement, we need to use the integrated forms, which again yield a system of algebraic equations. In a study of a chemical reaction, for example, y might be the conversion and the independent variables might be the time of reaction, temperature, and pressure. In addition to quantitative variables we could also include qualitative variables as the type of catalyst. 

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... 

Now denoting the state variables as the dependent variables v, we form Z by... 

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

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