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<div>Reinforcement Learning (RL) is a data-driven technique which is finding increasing application in the development of controllers for sequential decision making problems. Their wide adoption can be attributed to the fact that the development of these controllers is independent of the</div><div>knowledge of the system and thus can be used even when the environment dynamics are unknown. Model-Based RL controllers explicitly model the system dynamics from the observed (training) data using a function approximator, followed by using a path planning algorithm to obtain the optimal control sequence. While these controllers have been proven to be successful in simulations, lack of strong safety guarantees in the presence of noise makes them ill-posed for deployment on hardware, specially in safety critical systems. The proposed work aims at bridging this gap by providing a verification framework to evaluate the safety guarantees for a Model-Based RL controller. Our method builds upon reachability analysis to determine if there is any action which can drive the system into a constrained (unsafe) region. Consequently, our method can provide a binary yes or no answer to whether all the initial set of states are (un)safe to propagate trajectories from in the presence of some bounded noise.</div>

Safety verification of continuous dynamical systems require the computationof the reachable set. The reachable set comprises those states the systemcan reach at a specific point in time. The present work aims to compute thisreachable set for the marine vessel, in the presence of uncertainties in thedynamic modeling of the system and in the presence of external disturbancesin the form of wind, waves and currents. The reachable set can then be usedto check if the vessel collides with an obstacle. The dynamic model used isthat of a nonlinear maneuvering model for the marine vessel. The dynamicson the azipod actuators are also considered.Several methods are considered to solve the reachability problem for themarine vessel. The first method considered is that of the Hamilton JacobiReachability analysis, where a dynamic game between the control input andthe disturbance input is played. This results in a dynamic programmingproblem known as the Hamilton Jacobi Bellman Isaacs (HJBI) equation. Itis solved using the Level-Set method, but it suffers from the curse of dimensionality.The other method considered is the use of set-theoretic approach,where an over-approximation of the reachable set is computed, in the contextof safety verification. But on the downside, large sets of admissible controlyields highly over-approximated reachable sets, which cannot be usedIn order to overcome the disadvantages posed by the first two methods,emphasizing on the real-time computation, a third method is developed, wherea supervised classification algorithm is used to compute the reachable setboundary. The dataset required for the classification algorithm is computedby solving a 2 Point Boundary Value Optimal Control Problem for the marinevessel. The features for classification algorithm can be extended, so as toinclude the uncertainties and disturbances in the system. The computationtime is greatly reduced and the accuracy of the method is comparable to theexact reachable set computation. / Säkerhetsverifiering av kontinuerliga dynamiska system kräver beräkningav mängden av tillstånd som kan nås vid en specifik tidpunkt, givet dess initialtillstånd.Detta arbete fokuserar påatt bestämma denna mängd av nåbaratillstånd för ett marint fartyg under modellosäkerheter och externa störningari form av vind, vågor och strömmar. Den nåbara mängden av tillstånd användssedan för att kontrollera om fartyget riskerar att kollidera med hinder.Den dynamiska modell som används i våra studier är en icke-linjär modelldär även dynamiken hos azipod-ställdonen betraktas.Arbetet studerar flera metoder för att lösa problemet: en klassisk Hamilton-Jacobi nåbarhetsanalys, en mängd-teoretisk teknik, samt en ny metod baseradpåmaskininlärning. Numeriska simuleringsstudier bekräftar att den föreslagnamaskininlärningsmetoden är snabbare än de tvåalternativen.

Assessing existing bridges is an important task in the sustainable management ofinfrastructure. In practice, structural bridge assessments are usually conducted usingtraditional and standardised methods, despite knowledge that these methods oftenprovide conservative estimates. In addition, more advanced methods are available, suchas nonlinear finite element (FE) analysis, that are used for research purposes and cansimulate the structural behaviour of bridges more accurately. Therefore, it would beuseful to develop practical and reliable procedures for refined assessments using theseadvanced techniques.Focusing on the ultimate load-carrying capacity of existing concrete bridges, this thesispresents a procedure for structural assessments. The fundamental idea is to improve theassessment successively, as necessary to predict bridges’ structural behaviour adequately.The procedure involves a multi-level assessment strategy with four levels of structuralanalysis, and an integrated framework for safety verification. At the initial level (Level 1)of the multi-level strategy, traditional standardised methods are used, no failures arecovered implicitly in the structural analysis and action effects are verified using localresistances calculated using analytical models. In the subsequent enhanced levels (Levels2 – 4), nonlinear FE analysis is used for stepwise integration of the verification of flexural,shear-related and anchorage failures into the structural analysis. The framework for safetyverifications includes partial safety factor (PSF), global resistance safety factor (GRSF) andfull probabilistic methods. Within each of these groups, verifications of desired safetymargins can be conducted with varying degrees of complexity.To demonstrate and evaluate the proposed structural assessment procedure, comparativestudies have been carried out, based on full-scale tests of a prestressed concrete bridge.This was the Kiruna Bridge, located in the northernmost city in Sweden, which was duefor demolition as part of a city transformation project, necessitated by large grounddeformations caused by the large nearby mine. Thus, it was available for destructiveexperimental investigation within the doctoral project presented in this thesis. The bridgehad five continuous spans, was 121.5 m long and consisted of three parallel girders with a connecting slab at the top. Both the girders and slab were tested to failure to investigatetheir structural behaviour and load-carrying capacity. Non-destructive and destructivetests were also applied to determine the residual prestress forces in the bridge girders andinvestigate the in situ applicability of methods developed for this purpose. The so-calledsaw-cut method and decompression-load method were used after refinement to enabletheir application to structures of such complexity. The variation of the experimentallydetermined residual prestress forces was remarkably high, depending on the sectioninvestigated. There were also high degrees of uncertainty in estimated values, and thusare only regarded as indications of the residual prestress force.Level 1 analysis of the multi-level assessment strategy consistently underestimatedcapacity, relative to the test results, and did not provide accurate predictions of the shearrelatedfailure observed in the test. With linear FE analysis and local resistance modelsdefined by the European standard, Eurocode 2, the load-carrying capacity wasunderestimated by 32 % for the bridge girder and 55 % for the bridge deck slab. At theenhanced level of structural analysis (Level 3), nonlinear FE analyses predicted thecapacities with less than 2 % deviation from the test results and correctly predicted thefailure mode. However, for existing bridges there are many uncertainties, for instance,the FE simulations were sensitive to the level of residual prestressing, boundaryconditions and assumed material parameters. To accurately take these aspects intoaccount, bridge-specific information is crucial.The complete structural assessment procedure, combining the multi-level strategy andsafety verification framework, was evaluated in a case study. Experiences from theprevious comparative studies were used in an assessment of the Kiruna Bridge followingthe Swedish assessment code. The initial assessment at Level 1 of the multi-level strategyand safety verification, using the PSF method, indicated that the shear capacity of one ofthe girders was critical. The most adverse load case (a combination of permanent loads,prestressing and variable traffic loads) was further investigated through enhancedstructural analyses implicitly accounting for flexural and shear-related failures (Level 3).Nonlinear FE analysis and safety evaluation using the PSF method, several variants of theGRSF method and the full probabilistic analysis for resistance indicated that the permittedaxle load for the critical classification vehicle could be 5.6 – 6.5 times higher than thelimit obtained from the initial assessment at Level 1. However, the study also indicatedthat the model uncertainty was not fully considered in these values. The modeluncertainty was shown to have strong effects on the safety verification and (thus)permissible axle loads. The case study also highlighted the need for a strategy forsuccessively improving structural analysis to improve understanding of bridges’ structuralbehaviour. The refined analysis indicated a complex failure mode, with yielding of thestirrups in the bridge girders and transverse flexural reinforcement in the bridge deck slab,but with a final shear failure of the slab. It would be impossible to capture suchcomplexity in a traditional standardised assessment, which (as mentioned) indicated thatthe shear capacity of the girder limited permissible axle loads. However, nonlinear FEanalyses are computationally demanding, and numerous modelling choices are required.Besides a strategy for rationally improving the analysis and helping analysts to focus oncritical aspects, detailed guidelines for nonlinear FE analysis should be applied to reduce the analyst-dependent variability of results and (thus) the model uncertainty. Clearly, toensure the validity of bridge assessment methods under in situ conditions, theirevaluations should include in situ tests. This thesis presents outcomes of such tests, therebyhighlighting important aspects for future improvements in the assessment of existingbridges.

<p>Set-based methods have been leveraged in many engineering applications from robust control and global optimization, to probabilistic planning and estimation. While useful, these methods have most widely been applied to analysis over sets that are convex, due to their ease in both representation and calculation. The representation and analysis of nonconvex sets is inherently complex. When nonconvexity arises in design and control applications, the nonconvex set is often over-approximated by a convex set to provide conservative results. However, the level of conservatism may be large and difficult to quantify, often leading to trivial results and requiring repetitive analysis by the engineer. Nonconvexity is inherent and unavoidable in many applications, such as the analysis of hybrid systems and robust safety constraints. </p> <p>In this dissertation, I present a new nonconvex set representation named the hybrid zonotope. The hybrid zonotope builds upon a combination of recent advances in the compact representation of convex sets in the controls literature with methods leveraged in solving mixed-integer programming problems. It is shown that the hybrid zonotope is equivalent to the union of an exponential number of convex sets while using a linear number of continuous and binary variables in the set’s representation. I provide identities for, and derivations of, the set operations of hybrid zonotopes for linear mappings, Minkowski sums, generalized intersections, halfspace intersections, Cartesian products, unions, complements, point containment, set containment, support functions, and convex enclosures. I also provide methods for redundancy removal and order reduction to improve the compactness and computational efficiency of the represented sets. Therefore proving the hybrid zonotopes expressive power and applicability to many nonconvex set-theoretic methods. Beyond basic set operations, I specifically show how the exact forward and backward reachable sets of linear hybrid systems may be found using identities that are calculated algebraically and scale linearly. Numerical examples show the scalability of the proposed methods and how they may be used to verify the safety and performance of complex systems. These exact methods may also be used to evaluate the level of conservatism of the existing approximate methods provided in the literature.  </p>