# Modelling and Analysis of Switched Physical Systems Using Mixed Boolean Singular Representation

Student thesis: Masters ThesisMPhil

### Abstract

Traditionally engineering systems have been analysed as linear systems. However, nowadays most engineering systems presents discrete behaviours (e.g. switches, clutches, contacts, to name some), necessitating a different approach for the analysis of such systems. For this reason, a hybrid systems approach is used. Hybrid systems contain continuous and discrete behaviours. These systems are also known as switched systems.

Hybrid systems behave in a continuous mode most of the time with discrete
behaviour only appearing when a switching element commutes. These behaviours can occur depending on the commutation conditions (either controlled commutations or arbitrary commutation).

Bond graph models are useful to represent multidomain engineering systems. This is due to the unified representation of the elements for the different engineering domains involved (electrical, mechanical, hydraulics, thermal), meaning that elements from different domains have equivalent representations in the model. Therefore, their interactions can be observed in the entire model instead of being analysed as independent subsystems, which later will be placed under specific constraints in order to comply with some physical laws that allows the analysis of their interaction as a whole system.

Hybrid bond graphs are introduced in order to represent continuous and discrete behaviours. This is usually made by having different models, some for the continuous cases, and different ones for the discontinuous cases. In order to differentiate hybrid systems from linear time invariant systems a representation for commuting elements is introduced.

There are several approaches to represent commuting elements, either for the purpose of structural analysis or efficient numerical simulation of the systems. In this case switched junctions are used to introduce a standard notation for bond graph hybrid systems. The use of switched junctions affects the behaviour of the causality in some elements. This is known as Dynamic causality.

Dynamic causality represents the changes of power transfer on the system when a commuting element changes its state (either from ON configuration to OFF configuration or the other way around). These changes are usually ignored in traditional approaches in order to simplify the analysis of the systems for simulation purposes but at the cost of loss of information from the system.

Another consequence of the use of switched junctions is the introduction of Boolean parameters on the model equations. Boolean parameters help towards the continuous analysis of the system, meaning that the analysis of the system is from a single general equation containing all the available configurations, rather than analysing all of the available configurations independently in order to obtain a particular implicit equation for each configuration.

The focus of this thesis is to propose a generalized notation for the hybrid bond graph systems, and a set of rules to obtain the hybrid bond graph model. This generalized notation is based on Boolean parameters in order to simplify the analysis and show all the available configurations in one general equation.

The main contribution of this thesis is the generalization of the notation of hybrid bond graph models, and the steps that needs to be followed in order to obtain the system's implicit equation. Also, the formulation of the general implicit equation, and the necessary conditions to identify the valid configurations are presented. The results are also used for the analysis of implicit equations that were obtained using any traditional mathematical approach.
Date of Award 29 May 2019 English University of Bath Roger Ngwompo (Supervisor) & Patrick Keogh (Supervisor)

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