Accurate solution of differential-algebraic optimization problems

Abstract

Differential algebraic optimization problems arise often in chemical engineerting processes. Current numerical methods for differential algebraic optimization problems rely on some form of approximation in order to pose die problem as a nonlinear program. Here we explore an appropriate discretization and formulation of this optimization problem by considering stability and error properties of implicit Runge * Kutta (IRK) methods for differential • algebraic equation (DAE) systems. From these properties we are able to enforce appropriate error constraints and method orders in a collocation based nonlinear programming (NLP) formulation. After demonstrating the IRK properties on a small DAE system, we show from variational conditions that optimal control problems can have the same difficulties as higher index DAE systems. This is illustrated for a number of small chemical engineering optimization examples that exhibit higher index characteristics. For these cases the NLP formulation in this paper yields efficient and accurate solutions. L Introduction The determination of optimal control profiles is of major importance for process applications. Examples within chemical engineering include problems in reactor design, process startup, batch process operation, etc. However, solution of optimization problems with differential and algebraic equation modehstiD remains a diflBcniltproble optimization problems with T n ^n algebraic equations can be solved in a straightforward way as nonlinear programs. On die other hand, unconstrained problems with differential equation models can be handled through die calculus of variations. However, models that combine both of these features are currently optimized by imposing some level of approximation to the problem. The purpose of this paper is to develop and discuss a nonlinear programming formulation that leads to the accurate solution ( within an e tolerance) of the general differential-algebraic optimal control problem. Current methods for handling these problems either apply an approximation to the_ control variable profile or to both the state and control profiles. A straightforward approach adopted by Sargent and Sullivan (1977) is to parameterize the control profile (e.g. piecewise constant) over variable-length finite elements and to solve the differential equations with this parameterization. A nonlinear programming algorithm is then applied to the control parameters in an outer calculation loop. Similar strategies have been proposed by Ray (1981) and Morshedi (1986). This approach requires the repeated and expensive solution of the differential-algebraic eqautions. Also, state variable inequality constraints cannot be handled in a straightforward way. Finally, the quality of the solution is strongly dependent on die parameterization of the control profile. Early studies with the second approach, parameterization of both the state and control profiles, were reported by Neuman and Sen (1972), Tsang et al (1974) and Lynn et al (1971). Here state and control profiles and die differential equations were parameterized using some method of weighted residuals (eg. orthogonal collocation). This leads to a large nonlinear program (NLP) with algebraic equality constraints. However, since NLP algorithms were less developed at that time, this approach was either inefficient when compared to feasible path methods, or was restricted to specialized (e.g. linear) problems. With advances in NLP methods through the development of Successive Quadratic Programming (SQP) and MINOS, these NLP's could be solved more efficiently and could handle nonlinear state and control profile constraints in a straightforward manner. Biegler (1984) demonstrated this approach on a small batch reactor problem. Renfro et al (1987) solved much larger problems with orthogonal collocation on finite elements and piecewise constant approximations to the control profile. In order to obtain accurate finite element solutions, however, Cuthrell and Biegler (1986,1989) imposed additional constraints in the NLP formulation in Older to enforce accurate state profiles. They classified the role of finite elements in terms of knot locations (over which die error was equidistributed, hence minimiTfri) and breakpoints that allowed for control profile discontinuities. This led to a formulation that enforced the accurate solution of the differential equations and allowed for a general description of the control profile. In this paper we explore the theoretical development of these finite element constraints and present a formulation that leads to arbitrarily accurate state variable and control variable profiles. Here finite elements serve as decision variables in the optimization problem and are simultaneously required to satisfy approximation error constraints and to locate control profile discontinuities. This formulation will be considered from the perspective of a discretized Differential-Algebraic Equation (DAE) system. Recent approximation error and stability results by Petzold and coworkers will be tailored to optimal control problems and incorporated into the NLP. The next section will review the equivalence between the variatibnal conditions for general optimal control problems and die Kuhn-Tucker conditions for the corresponding NLP formulation. Section 3 then discusses recent stability and approximation error results for Runge-Kutta methods (including collocation methods) ipplied to DAE systems. In particular we will discuss the appropriate selection of collocation methods for higher index (i.e., more difficult) DAE systems. The following section then discusses how these higher index DAE systems arise in optimal control problems with path constraints and singular arcs. Section 5 presents the solution of a number of higher index optimal control examples with our approach. Here it is shown that arbitrarily accurate solutions can be found with our NLP formulation. Finally, section 6 summarizes die paper and discusses approaches to dealing with large-scale optimal control problems. 2. Analysis of the Opdmalhy CotKlitwns for Optimal Control ProW In this section we briefly review the equivalence between the calculus of variations and the math programming approach. Special cases for optimal control problems such as singular arcs and path constraints will be discussed after this section. Consider the following general problem: Min ¥ ( z ( b ) ) + J *( z ( t ) , i i ( t ) ) dt u(t).x(t) S.L 2(0 g(u(t),: « F(z(t), e(t)) ^ 0 gf(z(b)) £ 0 z(a) •= z(t) u(t) z0 £ z(t) ^ ^ u(t) £ u(t)) z(t) u(t) where: Y( z(b)) * component of objective function due to final conditions I ( z ( t ) f u ( t ) ) d t = component of objective function due to integral of state and control vectors g « inequality design constraint vectors z(t) « state profile vector u(t) « control profiles gj « final conditions inequality constraints ZA K frwtifti condition for ttfttf vector z(t)f z(t) U « sate profile bounds u(t) , u(t) * control profile bounds The variational conditions for Ais problem are: du du du (b) (c) (d) (c) d<D dz g ( M(t) z(t) dF dz u(i). z( g(2(t = F A + t ) ) )) = ( 2(0 , & M + dz £ 0 • o , u(0) , A(t) = 0 M(t) £ 0 z(a) = z0 \J) A.(v) = -[ + Mr) t m h dz dz ' where M(t) and A(t) are adjoint functions for the constraint g (u(t), z(t)) < 0 ,and the ODE model respectively. Note that these conditions form a DAE system. Here the algebraic relation (a) is used to determine the optimal control profile. Also, when constraints (d) are active, these additional algebraic conditions can cause an additional degree of difficulty in the solution of the DAE system. This difficulty is classified by the index of the system and is considered later. Finally, if (a) is not explicitly a function of u, then singular arcs can be encountered for the DAE system. Kreindler (1982) showed that the above equations are stronger necessary conditions than those presented in Bryson and Ho (1975). Cuthrell and Biegler (1987) showed the similarity between the solution solved with a nonlinear programming formulation and the corresponding variational conditions of the optimal control problem. The Kuhn Tucker conditions for the DAE's discretized with finite element collocation are considered next. Here we include the integration lengths, Actj f as decision variables in order to find the breakpoints for control profile discontinuities. Later, in section 4 we also impose constraints for the approximation error. The nonlinear program to be solved by applying collocation on finite elements now has the following form: