Equilibrium states of C∗-algebras associated to graphs of groups

A graph of groups consists of an undirected graph with each edge and vertex assigned a group. There is a one-to-one correspondence between graphs of groups and group actions on trees. In this way, the study of graphs of groups is equivalent to the study of group actions on trees. In 2017, Brownlowe, Mundey, Pask, Spielberg and Thomas developed a generators and relations construction of a C∗-algebra associated to a graph of groups, which they linked to a specific crossed-product C∗-algebra to show that the correspondence between graphs of groups and group actions on trees extends to their respective C∗- algebras. A Cuntz-Pimsner model for this graph of groups C∗-algebra was given by Mundey and Rennie in 2021. We compare a generators and relations approach to Toeplitz algebras to the Toeplitz-Pimsner construction. KMSβ states on a C∗-dynamical system with a real action are invariant states that characterise equilibria. In this thesis we provide a characterisation of KMSβ states on the Toeplitz and Cuntz-Pimsner algebras associated to a graph of groups. While a complete classification of KMSβ states on these C∗-algebras is an extremely large and difficult problem, we are able to classify a certain class of KMSβ states, namely those which arise from the Haar state on group C∗-algebras

Expropriating the Subject Reading Boltanski with Agamben

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Model Based and Robust Control Techniques for Internal Combustion Engine Throttle Valves

The performances of position controllers for a throttle valve used with internal combustion engines of heavy goods vehicles is investigated using different control techniques. The throttle valve is modelled including the hard stops and static friction (stick-slip friction), which are nonlinear components. This includes a new simple approach to the modelling of static friction. This nonlinear model was validated in the time domain using experimental results, parameterised by experimental data using a Matlab based parameter estimation tool. The resulting state space model was linearised for the purpose of designing various linear model based controllers. This linearised model was validated using experimental data in the frequency domain. The correct design of each model based controller is first confirmed by simulation using the linear throttle valve model, the specified step response being expected. Then the robustness is assessed in the frequency domain using the Matlab® Control System Design Toolbox and in the time domain by simulation using Monte Carlo based plant parameter mismatching between the simulated real plant and its model used for the control system design. Once satisfactory performance of a specific controller is predicted by simulation using the linear model, this is replaced by the nonlinear model to ascertain any deterioration in performance. Controllers exhibiting satisfactory performance in simulation with the nonlinear plant model are then investigated experimentally. The set of controllers investigated in this work includes types that are not currently employed commercially, as well as traditional ones, consisting of the IPD, PID, DPI controllers and the linear state feedback controller with and without an integrated observer. The other controllers are the sliding mode iii controller, observer based robust controller (OBRC) and the polynomial controller. The traditional controllers are designed using partial pole placement with the derived linear plant model. The other controllers have structures permitting full pole placement, of which robust pole placement is an important option. In the pole placement design, the locations of the closed loop poles are determined using the settling time formula. Despite the use of robust pole placement, the static friction caused a limit cycle, which led to the use of an anti-friction measure known as dither. The 14 different controllers were investigated for their ability to control the throttle valve position with nonlinear friction, parameter variations and external disturbances. This information was gathered, together with qualitative information regarding ease of design and practicability to form a performance comparison table. The original contributions emanating from the research programme are as follows: The successful application of new control techniques for throttle valves subject to significant static friction The first time investigation of partial and robust pole placement for throttle valve servo systems. A simplified static friction model which can be used for other applications