Abstract
This work presents a methodology to represent logical decisions in differential algebraic equation simulation and constrained optimization problems using a set of continuous algebraic equations. The formulations may be used when state variables trigger a change in process dynamics, and introduces a pseudo-binary decision variable, which is continuous, but should only have valid solutions at values of either zero or one within a finite time horizon. This formulation enables dynamic optimization problems with logical disjunctions to be solved by simultaneous solution methods without using methods such as mixed integer programming. Several case studies are given to illustrate the value of this methodology including nonlinear model predictive control of a chemical reactor using a surge tank with overflow to buffer disturbances in feed flow rate. Although this work contains novel methodologies for solving dynamic algebraic equation (DAE) constrained problems where the system may experience an abrupt change in dynamics that may otherwise require a conditional statement, there remain substantial limitations to this methodology, including a limited domain where problems may converge and the possibility for ill-conditioning. Although the problems presented use only continuous algebraic equations, the formulation has inherent non-smoothness. Hence, these problems must be solved with care and only in select circumstances, such as in simulation or situations when the solution is expected to be near the solver’s initial point.
Highlights
In dynamic optimization, models are ideally formulated as a set of continuous equations with continuous derivatives, so that solutions can be efficiently obtained using gradient-based solution algorithms, such as Newton’s method
Using the collocation scheme combined with the logical MPCC framework developed in the previous section, dynamic systems of equations with logical conditions can be simulated using only a set of continuous algebraic equations
As a demonstration of the value of integrating logic into a simultaneous solution method, a nonlinear model predictive control (NMPC) problem is solved for a continuous stirred tank reactor (CSTR), which carries out the reaction: A + B −→ 2C
Summary
Models are ideally formulated as a set of continuous equations with continuous derivatives, so that solutions can be efficiently obtained using gradient-based solution algorithms, such as Newton’s method. In many systems, the need frequently arises to include operators that may be discontinuous (such as the signum operator) or have discontinuous first derivatives (such as the absolute value operator) The introduction of such discontinuities into a model can have adverse impacts on the solver’s ability to efficiently obtain an accurate solution due to the introduction of non-smooth gradients. The equations are formulated in such a way, that only binary (0 or 1) solutions are obtained for certain variables at the solution These pseudo-binary variables are used to represent logical conditions within the model. This work does not present a detailed explanation of the convergence properties of problems with complementarity constraints, but rather puts forward a novel formulation that can be used by practitioners to represent logical statements within a continuous process model. In certain circumstances, natural discontinuities in the process require specialized techniques for representing these conditions in the model
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