Multi-parametric programming : novel theory and algorithmic developments

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International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Versatile Multi-Contact Planning and Control for Legged Loco-Manipulation

Loco-manipulation planning skills are pivotal for expanding the utility of robots in everyday environments. These skills can be assessed based on a system's ability to coordinate complex holistic movements and multiple contact interactions when solving different tasks. However, existing approaches have been merely able to shape such behaviors with hand-crafted state machines, densely engineered rewards, or pre-recorded expert demonstrations. Here, we propose a minimally-guided framework that automatically discovers whole-body trajectories jointly with contact schedules for solving general loco-manipulation tasks in pre-modeled environments. The key insight is that multi-modal problems of this nature can be formulated and treated within the context of integrated Task and Motion Planning (TAMP). An effective bilevel search strategy is achieved by incorporating domain-specific rules and adequately combining the strengths of different planning techniques: trajectory optimization and informed graph search coupled with sampling-based planning. We showcase emergent behaviors for a quadrupedal mobile manipulator exploiting both prehensile and non-prehensile interactions to perform real-world tasks such as opening/closing heavy dishwashers and traversing spring-loaded doors. These behaviors are also deployed on the real system using a two-layer whole-body tracking controller

On the controllability of fermentation systems

This thesis concerns the controllability of fermentation processes. Fermentation processes are often described by unstructured process models. A control system can be used to reduce the effect of the uncertainties and disturbances. A process is called controllable if a control system satisfying suitably defined control objectives can be found. Controllability measures based on linear process models are identified. The idealised control objective for perfect control allows fast evaluation of the controllability measures. These measures are applied to compare different designs of a continuous fermentation process by identifying the controllability properties of the process design. The operational mode of fed batch fermentations is inherently dynamic. General control system design methods are not readily applicable to such systems. This work presents an approach for the design of robust controllers suitable for these processes. The control objective is to satisfy a set of robustness constraints for a given set of model uncertainties and disturbances. The optimal operation and design problems are combined into a single optimal control problem. The controller design is integrated into the process design problem formulation. In this way the control system and the process are designed simultaneously. Different problem formulations are investigated. The proposed approach is demonstrated on complex fermentation models. The resulting operating strategies are controllable with respect to the aims of control

Special Bilevel Quadratic Problems for Construction of Worst-Case Feedback Control in Linear-Quadratic Optimal Control Problems under Uncertainties

Almost all mathematical models that describe processes, for instance in industry, engineering or natural sciences, contain uncertainties which arise from different sources. We have to take these uncertainties into account when solving optimal control problems for such processes. There are two popular approaches : On the one hand the so-called closed-loop feedback controls, where the nominal optimal control is updated as soon as the actual state and parameter estimates of the process are available and on the other hand robust optimization, for example worst-case optimization, where it is searched for an optimal solution that is good for all possible realizations of uncertain parameters. For the optimal control problems of dynamic processes with unknown but bounded uncertainties we are interested in a combination of feedback controls and robust optimization. The computation of such a closed-loop worst-case feedback optimal control is rather difficult because of high dimensional complexity and it is often too expensive or too slow for complex optimal control problems, especially for solving problems in real-time. Another difficulty is that the process trajectory corresponding to the worst-case optimal control might be infeasible. That is why we suggest to solve the problems successively by dividing the time interval and determining intermediate time points, computing the feedback controls of the smaller intervals and allowing to correct controls at these fixed intermediate time points. With this approach we can guarantee that for all admissible uncertainties the terminal state lies in a given prescribed neighborhood of a given state at a given final moment. We can also guarantee that the value of the cost function does not exceed a given estimate. In this thesis we introduce special bilevel programming problems with solutions of which we may construct the feedback controls. These bilevel problems can be solved explicitly. We present, based on these bilevel problems, efficient methods and approximations for different control policies for the combination of feedback control and robust optimization methods which can be implemented online, compare these approaches and show their application on linear-quadratic control 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

Dynamic origin-destination demand estimation using automatic vehicle identification data

Journal ArticleAbstract-This paper proposes a dynamic origin-destination (OD) estimation method to extract valuable point-to-point splitfraction information from automatic vehicle identification (AVI) counts without estimating market-penetration rates and identification rates of AVI tags. A nonlinear ordinary least-squares estimation model is presented to combine AVI counts, link counts, and historical demand information into a multiobjective optimization framework. A joint estimation formulation and a one-sided linear-penalty formulation are further developed to take into account possible identification and representativeness errors, and the resulting optimization problems are solved by using an iterative bilevel estimation procedure. Based on a synthetic data set, this study shows the effectiveness of the proposed estimation models under different market-penetration rates and identification rates

Optimal parameter selection for nonlinear multistage systems with time-delays

In this paper, we consider a novel dynamic optimization problem for nonlinear multistage systems with time-delays. Such systems evolve over multiple stages, with the dynamics in each stage depending on both the current state of the system and the state at delayed times. The optimization problem involves choosing the values of the time-delays, as well as the values of additional parameters that influence the system dynamics, to minimize a given cost functional. We first show that the partial derivatives of the system state with respect to the time-delays and system parameters can be computed by solving a set of auxiliary dynamic systems in conjunction with the governing multistage system. On this basis, a gradient-based optimization algorithm is proposed to determine the optimal values of the delays and system parameters. Finally, two example problems, one of which involves parameter identification for a realistic fed-batch fermentation process, are solved to demonstrate the algorithm’s effectiveness