Divide and Conquer: Distributed Optimization and Robustness Analysis

Abstract

Linköping 2015Cover illustration: A design approach for distributed optimization alogirhtms, used in papers C and F. It states that given the sparstiy graph of a coupled optimization problem, defined in Section 5.1, we can group its variables and its constituent subprobelms so that the coupling structure can be represented using a tree. We can then solve the problem distributedly using message passing, which utilizes the tree as its computational graph. To my parents AbstractAs control of large-scale complex systems has become more and more prevalent within control, so has the need for analyzing such controlled systems. This is particularly due to the fact that many of the control design approaches tend to neglect intricacies in such systems, e.g., uncertainties, time delays, nonlinearities, so as to simplify the design procedure. Robustness analysis techniques allow us to assess the effect of such neglected intricacies on performance and stability. Performing robustness analysis commonly requires solving an optimization problem. However, the number of variables of this optimization problem, and hence the computational time, scales badly with the dimension of the system. This limits our ability to analyze large-scale complex systems in a centralized manner. In addition, certain structural constraints, such as privacy requirements or geographical separation, can prevent us from even forming the analysis problem in a centralized manner.In this thesis, we address these issues by exploiting structures that are common in large-scale systems and/or their corresponding analysis problems. This enables us to reduce the computational cost of solving these problems both in a centralized and distributed manner. In order to facilitate distributed solutions, we employ or design tailored distributed optimization techniques. Particularly, we propose three distributed optimization algorithms for solving the analysis problem, which provide superior convergence and/or computational properties over existing algorithms. Furthermore, these algorithms can also be used for solving general loosely coupled optimization problems that appear in a variety of fields ranging from control, estimation and communication systems to supply chain management and economics.v Populärvetenskaplig sammanfattningRegulatorer för styrning av system är ofta designade med hjälp av en beskrivning, typiskt en matematisk modell, av det man vill styra. Dessa modeller är oftast endast approximativa beskrivningar av det verkliga systemet och därför behäftade med osäkerheter. Detta medför att det beteende man förväntar sig av ett reglerat system baserat på den matematiska modellen inte alltid stämmer överens med det observerade beteendet. Att analysera hur stor denna avvikelse kan bli kallas robusthetsanalys. Detta är ett väl studerat område för mindre och medelstora system. Dock är kunskaperna begränsade för riktigt stora system, speciellt om man inte vill göra en alltför konservativ analys. Exempel på stora system är kraftnät och flygplan. Kraftnät är stora på gr...

On Motion Planning Using Numerical Optimal Control

“…However, due to the maneuvers only being loosely coupled through the added optimization variables x opt , the computational effort of solving the problem will still be tractable if solvers that utilize this sparsity is used, such as ipopt and worhp. Another possibility with the loosely coupled formulation is that it is likely that methods employing parallel computations can be utilized, such as the ones presented in [63].…”