Exactness Verification of Sum-of-Squares Approximations to Robust Semidefinite Programs with Functional Variables

Abstract-Robust semidefinite programs (robust SDPs in short) with functional variables are revisited in this paper. We firstly consider the approximate approach suggested by Jennawasin and Oishi (in Proceedings of the 17th IFAC World Congress, Seoul, Korea, July 2008), and then provide a numerically computable condition to verify when the optimal value of an approximate problem is actually equal to that of the original robust SDP. The idea is based on capturing some special structure of a dual feasible solution of the approximate problem

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

This paper investigates linear systems with polynomial dependence on time-invariant uncertainties constrained in the simplex via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). It is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem. Moreover, this condition is also necessary by using a sufficiently large degree of the HPD-QLF. © 2009 Elsevier Ltd. All rights reserved.postprin

Control of Constrained Dynamical Systems with Performance Guarantees: With Application to Vehicle motion Control

In control engineering, models of the system are commonly used for controller design. A standard control design problem consists of steering the given system output (or states) towards a predefined reference. Such a problem can be solved by employing feedback control strategies. By utilizing the knowledge of the model, these strategies compute the control inputs that shrink the error between the system outputs and their desired references over time. Usually, the control inputs must be computed such that the system output signals are kept in a desired region, possibly due to design or safety requirements. Also, the input signals should be within the physical limits of the actuators. Depending on the constraints, their violation might result in unacceptable system failures (e.g. deadly injury in the worst case). Thus, in safety-critical applications, a controller must be robust towards the modelling uncertainties and provide a priori guarantees for constraint satisfaction. A fundamental tool in constrained control application is the robust control invariant sets (RCI). For a controlled dynamical system, if initial states belong to RCI set, control inputs always exist that keep the future state trajectories restricted within the set. Hence, RCI sets can characterize a system that never violates constraints. These sets are the primary ingredient in the synthesis of the well-known constraint control strategies like model predictive control (MPC) and interpolation-based controller (IBC). Consequently, a large body of research has been devoted to the computation of these sets. In the thesis, we will focus on the computation of RCI sets and the method to generate control inputs that keep the system trajectories within RCI set. We specifically focus on the systems which have time-varying dynamics and polytopic constraints. Depending upon the nature of the time-varying element in the system description (i.e., if they are observable or not), we propose different sets of algorithms.The first group of algorithms apply to the system with time-varying, bounded uncertainties. To systematically handle the uncertainties and reduce conservatism, we exploit various tools from the robust control literature to derive novel conditions for invariance. The obtained conditions are then combined with a newly developed method for volume maximization and minimization in a convex optimization problem to compute desirably large and small RCI sets. In addition to ensuring invariance, it is also possible to guarantee desired closed-loop performance within the RCI set. Furthermore, developed algorithms can generate RCI sets with a predefined number of hyper-planes. This feature allows us to adjust the computational complexity of MPC and IBC controller when the sets are utilized in controller synthesis. Using numerical examples, we show that the proposed algorithms can outperform (volume-wise) many state-of-the-art methods when computing RCI sets.In the other case, we assume the time-varying parameters in system description to be observable. The developed algorithm has many similar characteristics as the earlier case, but now to utilize the parameter information, the control law and the RCI set are allowed to be parameter-dependent. We have numerically shown that the presented algorithm can generate invariant sets which are larger than the maximal RCI sets computed without exploiting parameter information.Lastly, we demonstrate how we can utilize some of these algorithms to construct a computationally efficient IBC controller for the vehicle motion control. The devised IBC controller guarantees to meet safety requirements mentioned in ISO 26262 and the ride comfort requirement by design

Two-Stage Stochastic Semidefinite Programming: Theory, Algorithms, and Application to AC Power Flow under Uncertainty

In real life decision problems, one almost always is confronted with uncertainty and risk. For practical optimization problems this is manifested by unknown parameters within the input data, or, an inexact knowledge about the system description itself. In case the uncertain problem data is governed by a known probability distribution, stochastic programming offers a variety of models hedging against uncertainty and risk. Most widely employed are two-stage models, who admit a recourse structure: The first-stage decisions are taken before the random event occurs. After its outcome, a recourse (second-stage) action is made, often but not always understood as some "compensation''. In the present thesis, the optimization problems that involve parameters which are not known with certainty are semidefinite programming problems. The constraint sets of these optimization problems are given by intersections of the cone of symmetric, positive semidefinite matrices with either affine or more general equations. Objective functions, formally, may be fairly general, although they often are linear as in the present thesis. We consider risk neutral and risk averse two-stage stochastic semidefinite programs with continuous and mixed-integer recourse, respectively. For these stochastic optimization problems we analyze their structure, derive solution methods relying on decomposition, and finally apply our results to unit commitment in alternating current (AC) power systems. Furthermore, deterministic unit commitment in AC power transmission systems is addressed. Beside traditional unit commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming (SDP) approach is used which leads to large-scale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an all-at-once manner by general SDP solvers, it requires an efficient and reliable method to tackle them. To this end, an algorithm based on Benders decomposition is proposed. With power demand (load) and in-feed from renewables serving as sources of uncertainty, two-stage stochastic programs are set up heading for unit commitment schedules which are both cost-effective and robust with respect to data perturbations. The impact of different, risk neutral and risk averse, stochastic criteria on the shapes of the optimal stochastic solutions will be examined. To tackle the resulting two-stage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to two-stage stochastic mixed-integer semidefinite programs having a special structure. To solve the latter, the L-shaped method and dual decomposition have been applied and compared.Betrachtet man reale Entscheidungsprobleme, die also der Wirklichkeit entstammen, so ist man fast immer mit Unsicherheiten und Risiken konfrontiert. Für konkrete Optimierungsprobleme äußert sich dies sowohl in Form von ungewissen Parametern in den Eingangsdaten, als auch durch eine unzureichende Kenntnis über die Systembeschreibung selbst. Handelt es sich um zufallsbehaftete Eingangsdaten, dessen Verteilung bekannt ist, so stellt die Stochastische Optimierung eine Vielzahl von Modellen bereit - allesamt mit dem Ziel sich gegen Unsicherheiten und Risiken abzusichern. Die am Häufigsten verwendeten stochastischen Modelle sind zweistufige Modelle. Diese gestatten folgende Kompensationsstrategie: Eine Erststufenentscheidung wird getroffen bevor das Zufallsereignis eintritt. Nach Realisierung des Zufalls können Korrekturmaßnahmen (zweite Stufe) ergriffen werden, welche häufig, aber nicht immer, als "Kompensation" verstanden werden. Die vorliegende Arbeit behandelt Semidefinite Programme, dessen Parameter nicht mit Sicherheit bekannt sind. Der Zulässigkeitsbereich dieser Optimierungsprobleme entsteht aus dem Durchschnitt affiner oder auch allgemeinerer Gleichungen mit dem Kegel der symmetrisch und positiv semidefiniten Matrizen. Die Zielfunktion kann relativ allgemein sein, wird aber häufig, wie es auch in dieser Arbeit der Fall ist, als linear angenommen. Es werden risikoneutrale und risikoaverse zweistufige stochastische semidefinite Optimierungsprobleme mit jeweils stetiger und gemischt-ganzzahliger Kompensation betrachtet. Wir analysieren die Struktur dieser stochastischen Optimierungsprobleme, leiten dekompositionsbasierte Lösungsverfahren her und wenden unsere Resultate auf das Problem der optimalen Kraftwerkseinsatzplanung in Wechselstromnetzen an. Ferner beschäftigt sich diese Arbeit mit der deterministischen Kraftwerkseinsatzplanung in Wechselstromnetzen. Neben den traditionellen technischen Bedingungen an die einzelnen Kraftwerke wird auch die Physik des Wechselstroms berücksichtigt. Um global optimale Lösungen zu erhalten wird eine auf Semidefinite Programmierung (SDP) basierende Lösungsstrategie benutzt. Dieser Ansatz resultiert in einem umfangreichen semidefiniten Programm, welches zusätzlich diskrete Entscheidungsvariablen enthält. Da selbst die SDP Relaxierung dieses Optimierungsproblems zu groß ist um es mittels gängiger SDP Löser auf einmal zu lösen, wird eine effiziente und zuverlässige Methode benötigt. Es wird ein Algorithmus basierend auf dem Dekompositionsprinzip von Benders vorgeschlagen. Ausgehend vom Energiebedarf (Last) und der Einspeisung der erneuerbaren Energien als Unsicherheitsquelle, wird ein zweistufiges stochastisches Optimierungsproblem formuliert. Das Ziel ist es, einen Kraftwerkseinsatzplan zu finden, der wirtschaftlich effektiv und robust gegenüber Veränderungen in den Daten ist. Es werden die Auswirkungen des risikoneutralen und risikoaversen Ansatzes auf die stochastische Lösung untersucht und miteinander verglichen. Um die resultierenden zweistufigen Programme zu lösen wird das Wechselstromnetz mit Hilfe des SDP Ansatzes approximiert. Dies führt zu zweistufigen stochastischen gemischt-ganzzahligen semidefiniten Programmen mit spezieller Struktur. Als Lösungsmethoden wurden die L-shaped Methode und die duale Dekomposition verwendet

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios