Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints

This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gauß-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems

Integration of design and NMPC-based control of processes under uncertainty

The implementation of a Nonlinear Model Predictive Control (NMPC) scheme for the integration of design and control demands the solution of a complex optimization formulation, in which the solution of the design problem depends on the decisions from a lower tier problem for the NMPC. This formulation with two decision levels is known as a bilevel optimization problem. The solution of a bilevel problem using traditional Linear Problem (LP), Nonlinear Problem (NLP) or Mixed-Integer Nonlinear Problem (MINLP) solvers is very difficult. Moreover, the bilevel problem becomes particularly complex if uncertainties or discrete decisions are considered. Therefore, the implementation of alternative methodologies is necessary for the solution of the bilevel problem for the integration of design and NMPC-based control. The lack of studies and practical methodologies regarding the integration of design and NMPC-based control motivates the development of novel methodologies to address the solution of the complex formulation. A systematic methodology has been proposed in this research to address the integration of design and control involving NMPC. This method is based on the determination of the amount of back-off necessary to move the design and control variables from an optimal steady-state design to a new dynamically feasible and economic operating point. This method features the reduction of complexity of the bilevel formulation by approximating the problem in terms of power series expansion (PSE) functions, which leads to a single-level problem formulation. These functions are obtained around the point that shows the worst-case variability in the process dynamics. This approximated PSE-based optimization model is easily solved with traditional NLP solvers. The method moves the decision variables for design and control in a systematic fashion that allows to accommodate the worst-case scenario in a dynamically feasible operating point. Since approximation techniques are implemented in this methodology, the feasible solutions potentially may have deviations from a local optimum solution. A transformation methodology has been implemented to restate the bilevel problem in terms of a single-level mathematical program with complementarity constraints (MPCC). This single-level MPCC is obtained by restating the optimization problem for the NMPC in terms of its conditions for optimality. The single-level problem is still difficult to solve; however, the use of conventional NLP or MINLP solvers for the search of a solution to the MPCC problem is possible. Hence, the implementation of conventional solvers provides guarantees for optimality for the MPCC’s solution. Nevertheless, an optimal solution for the MPCC-based problem may not be an optimal solution for the original bilevel problem. The introduction of structural decisions such as the arrangement of equipment or the selection of the number of process units requires the solution of formulations involving discrete decisions. This PhD thesis proposes the implementation of a discrete-steepest descent algorithm for the integration of design and NMPC-based control under uncertainty and structural decisions following a naturally ordered sequence, i.e., structural decisions that follow the order of the natural numbers. In this approach, the corresponding mixed-integer bilevel problem (MIBLP) is transformed first into a single-level mixed-integer nonlinear program (MINLP). Then, the MINLP is decomposed into an integer master problem and a set of continuous sub-problems. The set of problems is solved systematically, enabling exploration of the neighborhoods defined by subsets of integer variables. The search direction is determined by the neighbor that produces the largest improvement in the objective function. As this method does not require the relaxation of integer variables, it can determine local solutions that may not be efficiently identified using conventional MINLP solvers. To compare the performance of the proposed discrete-steepest descent approach, an alternative methodology based on the distributed stream-tray optimization (DSTO) method is presented. In that methodology, the integer variables are allowed to be continuous variables in a differentiable distribution function (DDF). The DDFs are derived from the discretization of Gaussian distributions. This allows the solution of a continuous formulation (i.e., a NLP) for the integration of design and NMPC-based control under uncertainty and structural decisions naturally ordered set. Most of the applications for the integration of design and control implement direct transcription approaches for the solution of the optimization formulation, i.e., the full discretization of the optimization problem is implemented. In chemical engineering, the most widely used discretization strategy is orthogonal collocation on finite elements (OCFE). OCFE offers adequate accuracy and numerical stability if the number of collocation points and the number of finite elements are properly selected. For the discretization of integrated design and control formulations, the selection of the number of finite elements is commonly decided based on a priori simulations or process heuristics. In this PhD study, a novel methodology for the selection and refinement of the number of finite elements in the integration of design and control framework is presented. The corresponding methodology implements two criteria for the selection of finite elements, i.e., the estimation of the collocation error and the Hamiltonian function profile. The Hamiltonian function features to be continuous and constant over time for autonomous systems; nevertheless, the Hamiltonian function shows a nonconstant profile for underestimated discretization meshes. The methodology systematically adds or removes finite elements depending on the magnitude of the estimated collocation error and the fluctuations in the profile for the Hamiltonian function. The proposed methodologies have been tested on different case studies involving different features. An existent wastewater treatment plan is considered to illustrate the implementation of back-off strategy. On the other hand, a reaction system with two continuous stirred reaction tanks (CSTRs) are considered to illustrate the implementation of the MPCC-based formulation for design and control. The D-SDA approach is tested for the integration of design, NMPC-based control, and superstructure of a binary distillation column. Lastly, a reaction system illustrates the effect of the selection and refinement of the discretization mesh in the integrated design and control framework. The results show that the implementation of NMPC controllers leads to more economically attractive process designs with improved control performance compared to applications with classical descentralized PID or Linear MPC controllers. The discrete-steepest descent approach allowed to skip sub-optimal solution regions and led to more economic designs with better control performance than the solutions obtained with the benchmark methodology using DDFs. Meanwhile, the refinement strategy for the discretization of integrated design and control formulations demonstrated that attractive solutions with improved control performance can be obtained with a reduced number of finite elements