The control of nonholonomic wheeled mobile robots (WMRs) has gained a lot of attention in the field of robotics over the past few decades as WMRs provide an increased range of motion resulting in a larger workspace. This research focuses on the application of Model Predictive Control (MPC) for real-time trajectory tracking of a nonholonomic WMR. MPC is a control strategy in which the control law is designed based on optimizing a cost function. The input and output constraints that may arise in practical situations can be directly incorporated into the control system using MPC. Computation time is the biggest hurdle in adapting MPC strategies for trajectory tracking. This research applies a non-feasible active set MPC algorithm developed in [1] which is faster than the traditional active set methods (ASMs). A discrete-time linear model of a general WMR is used for the simulation. MATLAB simulations are performed for tracking circular as well as square trajectories using the discretized WMR model and the non-feasible ASM (NF-ASM). The performance of NF-ASM is compared to two other well-known traditional algorithms, i.e. Fletcher’s ASM and MATLAB’s Quadratic Programming algorithm. It is shown that, although all these algorithms are capable of providing satisfactory trajectory tracking performance, NF-ASM is a better choice in terms of the simulation time and required number of iterations for realtime trajectory tracking of any type as long as the constraints on the inputs stay active for a long period during the simulation. / UOIT

In this paper we give a general framework for isotone optimization. First we discuss a generalized version of the pool-adjacent-violators algorithm (PAVA) to minimize a separable convex function with simple chain constraints. Besides of general convex functions we extend existing PAVA implementations in terms of observation weights, approaches for tie handling, and responses from repeated measurement designs. Since isotone optimization problems can be formulated as convex programming problems with linear constraints we then develop a primal active set method to solve such problem. This methodology is applied on specific loss functions relevant in statistics. Both approaches are implemented in the R package isotone. (authors' abstract)

Denna kandidatrapport studerar en projektuppgift som har utförts av en grupp studenter på Linköpings universitet. Uppgiften har givits av en industridoktorand på Saab och härstammar från reglering av styrsystem i stridsflygplan. Det som har tagits fram av kandidatgruppen är en kvadratisk optimeringslösare som även kan köras från MATLAB. Undersökningar har gjorts om det går att implementera optimeringsalgoritmen i programspråket C med projektets tidsbegränsning, om lösaren kan bli lika snabb som den kommersiella produkten Gurobi och om projektet går att utföra utan någon speciell utvecklingsmetodik. I resultat går det att se att det gick att implementera optimeringsalgoritmen, att lösaren inte kunde bli lika snabb som Gurobi och att kandidatgruppen inte använde någon speciell utvecklingsmetodik. Slutsatser kandidatgruppen har dragit ar att valet av optimeringsalgoritm inte var helt genomtänkt, att mer mer tid och resurser hade lösaren kanske kunnat blivit lika snabb som Gurobi och att arbetet fungerade tillfredsställande utan någon speciell utvecklingsmetod.

La présente étude concerne l’étude d’optimisation thermique d’une machine électrique. Un modèle nodal est utilisé pour la simulation du champ de température. Ce modèle résout l’équation de la chaleur en trois dimensions, en coordonnées cylindriques et en régime transitoire ou permanent. On prend en compte les deux mécanismes de transport les plus importants : La conduction et la convection. L’évaluation de ce modèle est effectuée par l’intermédiaire de 13 valeurs de débits de référence. C’est en faisant varier ces variables qu’on évalue la performance du refroidissement dans la machine. Avant de partir sur l’étude d’optimisation de cettegéométrie, on a lancé une étude d’optimisation d’un cas plus simple afin de mieux comprendre les différents outils d’optimisation disponibles. L’expérience acquise avec les cas simples est utilisée dans l’optimisation thermique de la machine. La machine est thermiquement évaluée sur la combinaison de deux critères : la température maximale et la température moyenne. Des contraintes ont été additionnées afin d’obtenir des résultats physiquement acceptables. Le problème est résolu à l’aide des méthodes de gradient (Active-set et Point-Intérieur) et des Algorithmes Génétiques. / This work relates the thermal optimization of an electrical machine. The lumped method is used to simulate the temperature field. This model solves the heat equation in three dimensions, in cylindrical coordinates and in transient or steady state. We consider two transport mechanisms: conduction and convection. The evaluation of this model is performed by means of 13 design variables that correspond to the main flow rates of the equipment. We analyse the machine cooling performance by varying these 13 flow rates. Before starting the study of such a complicated geometry, we picked a simpler case in order to better understand the variety of the available optimization tools. The experience obtained in the simpler case is applyed in the resolution of the thermal optimization problem of the electrical machine. This machine is evaluated from the thermal point of view by combining two criteria : the maximum and the mean temperature. Constraints are used to keep the problem consistent. We solved the problem using the gradient based methods (Active-set and Interior-Point) and the Genetic Algorithms.

This research studies how to efficiently predict optimal active constraints of an inequality constrained optimization problem, in the context of Interior Point Methods (IPMs). We propose a framework based on shifting/perturbing the inequality constraints of the problem. Despite being a class of powerful tools for solving Linear Programming (LP) problems, IPMs are well-known to encounter difficulties with active-set prediction due essentially to their construction. When applied to an inequality constrained optimization problem, IPMs generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination, as well as increasing ill-conditioning of the solution process. We show that, existing techniques for active-set prediction, however, suffer from difficulties in making an accurate prediction at the early stage of the iterative process of IPMs; when these techniques are ready to yield an accurate prediction towards the end of a run, as the iterates approach the solution set, the IPMs have to solve increasingly ill-conditioned and hence difficult, subproblems. To address this challenging question, we propose the use of controlled perturbations. Namely, in the context of LP problems, we consider perturbing the inequality constraints (by a small amount) so as to enlarge the feasible set. We show that if the perturbations are chosen judiciously, the solution of the original problem lies on or close to the central path of the perturbed problem. We solve the resulting perturbed problem(s) using a path-following IPM while predicting on the way the active set of the original LP problem; we find that our approach is able to accurately predict the optimal active set of the original problem before the duality gap for the perturbed problem gets too small. Furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the active set for the perturbed problem or for the original one if no perturbations are used. Proof-of-concept algorithms are presented and encouraging preliminary numerical experience is also reported when comparing activity prediction for the perturbed and unperturbed problem formulations. We also extend the idea of using controlled perturbations to enhance the capabilities of optimal active-set prediction for IPMs for convex Quadratic Programming (QP) problems. QP problems share many properties of LP, and based on these properties, some results require more care; furthermore, encouraging preliminary numerical experience is also presented for the QP case.

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points al,...,am C Rn . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30, 000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.

La présente étude concerne l'étude d'optimisation thermique d'une machine électrique. Un modèle nodal est utilisé pour la simulation du champ de température. Ce modèle résout l'équation de la chaleur en trois dimensions, en coordonnées cylindriques et en régime transitoire ou permanent. On prend en compte les deux mécanismes de transport les plus importants : La conduction et la convection. L'évaluation de ce modèle est effectuée par l'intermédiaire de 13 valeurs de débits de référence. C'est en faisant varier ces variables qu'on évalue la performance du refroidissement dans la machine. Avant de partir sur l'étude d'optimisation de cettegéométrie, on a lancé une étude d'optimisation d'un cas plus simple afin de mieux comprendre les différents outils d'optimisation disponibles. L'expérience acquise avec les cas simples est utilisée dans l'optimisation thermique de la machine. La machine est thermiquement évaluée sur la combinaison de deux critères : la température maximale et la température moyenne. Des contraintes ont été additionnées afin d'obtenir des résultats physiquement acceptables. Le problème est résolu à l'aide des méthodes de gradient (Active-set et Point-Intérieur) et des Algorithmes Génétiques.

This thesis develops efficient optimization methods for Model Predictive Control (MPC) to enable its application to constrained systems with fast and uncertain dynamics. The key contribution is an active set method which exploits the parametric nature of the sequential optimization problem and is obtained from a dynamic programming formulation of the MPC problem. This method is first applied to the nominal linear MPC problem and is successively extended to linear systems with additive uncertainty and input constraints or state/input constraints. The thesis discusses both offline (projection-based) and online (active set) methods for the solution of controllability problems for linear systems with additive uncertainty. The active set method uses first-order necessary conditions for optimality to construct parametric programming regions for a particular given active set locally along a line of search in the space of feasible initial conditions. Along this line of search the homotopy of optimal solutions is exploited: a known solution at some given plant state is continuously deformed into the solution at the actual measured current plant state by performing the required active set changes whenever a boundary of a parametric programming region is crossed during the line search operation. The sequence of solutions for the finite horizon optimal control problem is therefore obtained locally for the given plant state. This method overcomes the main limitation of parametric programming methods that have been applied in the MPC context which usually require the offline precomputation of all possible regions. In contrast to this the proposed approach is an online method with very low computational demands which efficiently exploits the parametric nature of the solution and returns exact local DP solutions. The final chapter of this thesis discusses an application of robust tube-based MPC to the nonlinear MPC problem based on successive linearization.

Problemas de otimização em caixas são de grande importância, não só por surgirem naturalmente na formulação de problemas da vida prática, mas também por aparecerem como subproblemas de métodos de penalização ou do tipo Lagrangiano Aumentado para resolução de problemas de programação não-linear. O objetivo do trabalho é estudar um algoritmo de restrições ativas para problemas de otimização em caixas recentemente apresentado chamado ASA e compará-lo à versão mais recente de GENCAN, que é também um método de restrições ativas. Para tanto, foi elaborada uma metodologia de testes robusta e minuciosa, que se propõe a remediar vários dos aspectos comumente criticados em trabalhos anteriores. Com isso, puderam ser extraídas conclusões que levaram à melhoria de GENCAN, conforme ficou posteriormente comprovado por meio da metodologia aqui introduzida. / Box-constrained optimization problems are of great importance not only for naturally arising in several real-life problems formulation, but also for their occurrence as sub-problems in both penalty and Augmented Lagrangian methods for solving nonlinear programming problems. This work aimed at studying a recently introduced active-set method for box-constrained optimization called ASA and comparing it to the latest version of GENCAN, which is also an active-set method. For that purpose, we designed a robust and thorough testing methodology intended to remedy many of the widely criticized aspects of prior works. Thereby, we could draw conclusions leading to GENCAN\'s further development, as it later became evident by means of the same methodology herein proposed.

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a₁,..., am ∈ Rn. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30,000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer. / Singapore-MIT Alliance (SMA)