Combinatorial characterization of the Assur graphs from engineering

AbstractWe introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. This paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting the recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from ‘minimal’ linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work

Locality and Complexity in Path Search

The path search problem considers a simple model of communication networks as channel graphs: directed acyclic graphs with a single source and sink. We consider each vertex to represent a switching point, and each edge a single communication line. Under a probabilistic model where each edge may independently be free (available for use) or blocked (already in use) with some constant probability, we seek to efficiently search the graph: examine (on average) as few edges as possible before determining if a path of free edges exists from source to sink. We consider the difficulty of searching various graphs under different search models, and examine the computational complexity of calculating the search cost of arbitrary graphs

The design of decentralised controllers for large scale systems

Bibliography:leaves 203-205.Decentralised control schemes are becoming more common in industry as the advantages of decentralised control become more apparent. These advantages include fewer tuning parameters than centralised controllers, the simplification and cost reduction of hardware requirements and greater reliability. In addition the application of decentralised controller design to large scale systems allows established CAD methods to be implemented easily and efficiently. When the control engineer designs a distributed controller the system is divided up into a number of subsystems and a controller designed for each subsystem. The controllers are designed independently for each subsystem ignoring any interaction that may occur between the different subsystems. In terms of the input-output representation of the system this means that the matrix representing the controller will be in a block diagonal form. In general the interactions between the different subsystems will not be negligible. In some cases the interactions will be such that stabilising the individual subsystems will not be sufficient to stabilise the system as a whole. Stability theorems are required to enable the designer to check if the decentralised controller that he has designed will in fact stabilise the system as a whole. Such stability theorems have been devised although at present they are too conservative. However even with such theorems available the designer must still select the subsystems to be controlled in such a way as to satisfy the conditions laid down for stability. The stability theories usually are based on a particular matrix structure. If the matrix representing the system possesses a structure detailed by the stability theorem in question then, subject to various conditions, the system as a whole will be stable under decentralised control. In this thesis a number of different matrix structures are considered that give information as to the stability of the closed loop system. Methods are developed that allow the designer to rearrange the matrix in such a way as to obtain a particular structure, if this is possible