Algebraic constraints

This section deals with optimal control problems constrained by algebraic equalities and inequalities. 

Since this scenario is particularly important for process control, in this chapter we will introduce this approach applied to algebraic constraints only. 

Conversely, relevant cases that involve also differential or differential-algebraic constraints are discussed in-depth in Vol. 4 - Buzzi-Ferraris and Manenti (in press). 

It is useful to adopt parametric optimization if the following conditions are verified  

1) The constraints play a predominant role in the solution of the problem. 

2) The algebraic equations contain simultaneously constants, c, parameters, z, and variables, x. 

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Differential-Algebraic Systems Sometimes models involve ordinary differentia equations subject to some algebraic constraints. For example, the equations governing one equihbrium stage (as in a distillation column) are... 

FIG. 8-46 Diagram for selection of optimization techniques with algebraic constraints and objective function. 

Eq. (122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161],... 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

Several researchers [e.g., Tjoa and Biegler (1992) and Robertson et al. (1996)] have demonstrated advantages of using nonlinear programming (NLP) techniques over such traditional data reconciliation methods as successive linearization for steady-state or dynamic processes. Through the inclusion of variable bounds and a more robust treatment of the nonlinear algebraic constraints, improved reconciliation performance can be realized. 

This problem can be solved using a combined optimization and constraint model solution strategy (Muske and Edgar, 1998) by converting the differential equations to algebraic constraints using orthogonal collocation or some other model discretization approach. 

By discretizing the differential algebraic equations model using some standard discretization techniques (Liebman et al., 1992 Alburquerque and Biegler, 1996) to convert the differential constraints to algebraic constraints, the NDDR problem can be solved as the following NLP problem ... 

For these time periods, the ODEs and active algebraic constraints influence the state and control variables. For these active sets, we therefore need to be able to analyze and implicitly solve the DAE system. To represent the control profiles at the same level of approximation as for the state profiles, approximation and stability properties for DAE (rather than ODE) solvers must be considered. Moreover, the variational conditions for problem (16), with different active constraint sets over time, lead to a multizone set of DAE systems. Consequently, the analogous Kuhn-Tucker conditions from (27) must have stability and approximation properties capable of handling ail of these DAE systems. 

The system of equations in the Von Mises form leads to a coupled system of nonlinear differential-algebraic equations. The transport equations (Eqs. 7.59 and 7.62) have parabolic characteristics, with the axial coordinate z being the timelike direction. The other three equations (Eqs. 7.60, 7.61, and 7.63) are viewed as algebraic constraints—in the sense that they have no timelike derivatives. 

An alternative to the standard-form representation is the differential-algebraic equation (DAE) representation, which is stated in a general form as g(r, y, y). The lower portion of Fig. 7.3 illustrates how the heat equation is cast into the DAE form. The boundary conditions can now appear as algebraic constraints (i.e., they have no time derivatives). For a problems as simple as the heat equations, this residual representation of the boundary conditions is not necessary. However, recall that implicit boundary-condition specification is an important aspect of solving boundary-layer equations. 

A extremely simple example can serve to illustrate why consistent initial conditions are essential when using the method of lines to solve DAEs. Consider the problem illustrated in Fig. 7.5, which has one ordinary differential equation and one algebraic constraint. Obviously the solution for y is a straight line beginning at the origin with a slope of one. The solution for y2 is simply that > 2 = 1 always. The difficulty occurs if the initial condition is for y2 f 1, which is an inconsistent initial condition. 

As a bit of an aside, one can think of the algebraic constraint as an infinitely stiff problem. Referring to the stiff model problem (Section 15.2), stiff problems are characterized by a fast transient and a slowly varying solution. Regardless of the initial condition, a stiff problem will always decay to the slowly varying solution, and the stiffer the problem, the faster will be the decay (e.g., Fig. 15.1). The situation in a problem like that in Fig. 7.5 is that there is no transient in the y2 component because it is a constraint, and not a differential equation. If, however, the y2 equation is modeled as y 2 = — X(y2 — 1), then y2 = (y2(0) — l)e Xl. As A. becomes larger, the differential equation becomes stiffer, and as X —> oo, the differential equation becomes an algebraic constraint. 

The vector function g can be, and generally is, nonlinear. It may also have components that do not involve y (i.e., algebraic constraints). 

The site-fraction constraint (Eq. 16.64) means that all the s in Eq. 16.63 are not independent. Therefore only Ks — 1 of Eq. 16.63 are solved. Solving the plug-flow problem requires satisfying the algebraic constraints represented by Eqs. 16.63 and 16.64 at every point along the channel surface. The coupled problem is posed naturally as a system of differential-algebraic equations. 

The continuity equation at the inlet boundary can be viewed as a constraint equation. Referring to the difference stencil (Fig. 17.14), it is seen that this first-order equation itself is evaluated at the boundary and no explicit boundary condition is needed. Moreover, since the inlet temperature, pressure, and composition are specified, the density is fixed and thus dp/dt = 0. Therefore, at the boundary, the continuity equation (Eq. 17.15) has no time derivative it is an algebraic constraint. There is no explicit boundary condition for A. At the inlet boundary, the value of A must be determined in such a way that all the other boundary conditions are satisfied. Being an eigenvalue, A s effect is felt through its influence on the V velocity in the radial momentum equation, and subsequently by V s influence on u through the continuity equation. 

It is possible and beneficial to reduce the system to index-one by replacing A with a new dependent variable , where A = 3/31 [13], The initial condition for is arbitrary, since itself never appears in the equations—a suitable choice is 4> = 0. Anywhere A appears, it is simply replaced with 3/31, which is conveniently done in the DAE software interface. The index reduction can be seen from the following procedure The continuity at the inlet boundary (an algebraic constraint) can be differentiated once with respect to t to yield an equation for dV/dt. Then dV/dt is replaced by substitution of the radial-momentum equation. This substitution introduces A = 34>/31, which makes the continuity equation (at the inlet boundary) an independent differential equation for 4>. Thus the modified system is index-one. This set of substitutions is not actually done in practice—it simply must be possible to do them to achieve the index reduction. 

The user specified subroutines allow for connections to various other programs such as process simulators and ordinary differential equation solvers. Currently, MINOPT is connected to the DASOLV (Jarvis and Pantelides, 1992) integrator, and can solve MINLP models with differential and algebraic constraints. 

The representation of different types of reactor units in the approach proposed by Kokossis and Floudas (1990) is based on the ideal CSTR model, which is an algebraic model, and on the approximation of plug flow reactor, PFR units by a series of equal volume CSTRs. The main advantage of such a representation is that the resulting mathematical model consists of only algebraic constraints. At the same time, however, we need to introduce binary variables to denote the existence or not of the CSTR units either as single units or as a cascade approximating PFR units. As a result, the mathematical model will consist of both continuous and binary variables. 

Constraints in optimization problems often exist in such a fashion that they cannot be eliminated explicitly—for example, nonlinear algebraic constraints involving transcendental functions such as exp(x). The Lagrange multiplier method can be used to eliminate constraints explicitly in multivariable optimization problems. Lagrange multipliers are also useful for studying the parametric sensitivity of the solution subject to the constraints. 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

Note that the system (2.45) is a DAE system of nontrivial index, since z cannot be evaluated directly from the algebraic equations. A solution for the variables z must be obtained by differentiating the constraints k(x) = 0. For most chemical processes, such as reaction networks (Gerdtzen et al. 2004), reactive distillation processes (Vora 2000), and complex chemical plants (Kumar and Daoutidis 1999a), the z variables can be obtained after just one differentiation of the algebraic constraints ... 

The algebraic variable 2 can be computed after differentiating the algebraic constraints (2.69) ... 

Remark 3.1. In contrast to the theory presented thus far (Section 2.3), the algebraic constraints of (3.12) incorporate a set of (unknown) manipulated inputs, u1. The equilibrium manifold described by (3.12) is thus referred to as control-dependent. 

In this case, however, the algebraic constraints in the DAE system describing the slow dynamics (3.22) will explicitly involve manipulated input variables (i.e., y p), and the direct application of the methods in Section 2.3 for the derivation of a state-space realization of the slow dynamics is not possible.2... 

Note that, owing to the underlying algebraic constraints in the DAE system that describes the slow dynamics, the holdups Mb, Me, and Mr are not independent (there are only two linearly independent constraints among the three holdups, i.e., 0 = wr — and 0 = k2w2 — r, where u, U2, and kr are determined by the proportional control laws in Equation (3.35)). Thus, controlling one of the holdups (e.g., Mb) amounts to regulating the total material holdup in the process. 

The model of the slow dynamics of the system consists therefore of a set of coupled DAEs of nontrivial index, since the variables z (that physically correspond to the net material flows of the system in the slow time scale) are implicitly fixed by the quasi-steady-state constraints, rather than explicitly specified in the dynamic model. Also, note that the DAE model (4.27) has a well-defined index only if the flow rates u1 which appear in the algebraic constraints are specified as functions of the state variables x. This is typically accomplished via a control law u (x). 

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... 

The expressions of the algebraic constraints in Equations (6.57) have been obtained by substituting the definitions in Equations (6.49)-(6.52) into Equations (6.56). In order to determine the algebraic variables 2j, we must differentiate these algebraic constraints and, to this end, we must provide an expression for Hvec, the duty of the FEHE. 

We turn to the slow time scale t to obtain a description of the slow dynamics. In particular, on multiplying Equations (7.18) by e and considering the limit i —> 0, we obtain a set of algebraic constraints that need to be satisfied in the slow time scale ... 

Finally, let us consider the limit gq —> 0 in the slow time scale t, where the ratios lim l o C// i become indeterminate. By denoting these unknown, yet finite terms by Zi, i = 0,..., N + 1, we obtain a description of the slow dynamics of the column that captures its slow input-output behavior. This description is in the form of a DAE system of nontrivial index, since the algebraic constraints Ci = 0 are singular with respect to the algebraic variables zq a state-space... 

It is straightforward to verify that the algebraic constraints in Equations (7.34) are generically linearly independent and hence they can be solved for the quasi-steady-state values Q (M,C) = [T, TR,T ] of the variables [T,TR,TC], Substituting the value for T, we then obtain... 

The common underlying principle in the approaches for characterizing the solvability of a DAE system is to obtain, either explicitly, or implicitly, a local representation of an equivalent ODE system, for which available results on existence and uniqueness of solutions are applicable. The derivation of the underlying ODE system involves the repeated differentiation of the algebraic constraints of the DAE, and it is this differentiation process that leads to the concept of a DAE index that is widely used in the literature. For the semi-explicit DAE systems (A. 10) that are of interest to us here, the index has the following definition. 

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