Strong Structural Controllability of Signed Networks

In this paper, we discuss the controllability of a family of linear time-invariant (LTI) networks defined on a signed graph. In this direction, we introduce the notion of positive and negative signed zero forcing sets for the controllability analysis of positive and negative eigenvalues of system matrices with the same sign pattern. A sufficient combinatorial condition that ensures the strong structural controllability of signed networks is then proposed. Moreover, an upper bound on the maximum multiplicity of positive and negative eigenvalues associated with a signed graph is provided

Modelling & analysis of hybrid dynamic systems using a bond graph approach

Hybrid models are those containing continuous and discontinuous behaviour. In constructing dynamic systems models, it is frequently desirable to abstract rapidly changing, highly nonlinear behaviour to a discontinuity. Bond graphs lend themselves to systems modelling by being multi-disciplinary and reflecting the physics of the system. One advantage is that they can produce a mathematical model in a form that simulates quickly and efficiently. Hybrid bond graphs are a logical development which could further improve speed and efficiency. A range of hybrid bond graph forms have been proposed which are suitable for either simulation or further analysis, but not both. None have reached common usage. A Hybrid bond graph method is proposed here which is suitable for simulation as well as providing engineering insight through analysis. This new method features a distinction between structural and parametric switching. The controlled junction is used for the former, and gives rise to dynamic causality. A controlled element is developed for the latter. Dynamic causality is unconstrained so as to aid insight, and a new notation is proposed. The junction structure matrix for the hybrid bond graph features Boolean terms to reflect the controlled junctions in the graph structure. This hybrid JSM is used to generate a mixed-Boolean state equation. When storage elements are in dynamic causality, the resulting system equation is implicit. The focus of this thesis is the exploitation of the model. The implicit form enables application of matrix-rank criteria from control theory, and control properties can be seen in the structure and causal assignment. An impulsive mode may occur when storage elements are in dynamic causality, but otherwise there are no energy losses associated with commutation because this method dictates the way discontinuities are abstracted. The main contribution is therefore a Hybrid Bond Graph which reflects the physics of commutating systems and offers engineering insight through the choice of controlled elements and dynamic causality. It generates a unique, implicit, mixed-Boolean system equation, describing all modes of operation. This form is suitable for both simulation and analysis

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

The Mechanics and Control of Undulatory Robotic Locomotion

In this dissertation, we examine a formulation of problems of undulatory robotic locomotion within the context of mechanical systems with nonholonomic constraints and symmetries. Using tools from geometric mechanics, we study the underlying structure found in general problems of locomotion. In doing so, we decompose locomotion into two basic components: internal shape changes and net changes in position and orientation. This decomposition has a natural mathematical interpretation in which the relationship between shape changes and locomotion can be described using a connection on a trivial principal fiber bundle. We begin by reviewing the processes of Lagrangian reduction and reconstruction for unconstrained mechanical systems with Lie group symmetries, and present new formulations of this process which are easily adapted to accommodate external constraints. Additionally, important physical quantities such as the mechanical connection and reduced mass-inertia matrix can be trivially determined using this formulation. The presence of symmetries then allows us to reduce the necessary calculations to simple matrix manipulations. The addition of constraints significantly complicates the reduction process; however, we show that for invariant constraints, a meaningful connection can be synthesized by defining a generalized momentum representing the momentum of the system in directions allowed by the constraints. We then prove that the generalized momentum and its governing equation possess certain invariances which allows for a reduction process similar to that found in the unconstrained case. The form of the reduced equations highlights the synthesized connection and the matrix quantities used to calculate these equations. The use of connections naturally leads to methods for testing controllability and aids in developing intuition regarding the generation of various locomotive gaits. We present accessibility and controllability tests based on taking derivatives of the connection, and relate these tests to taking Lie brackets of the input vector fields. The theory is illustrated using several examples, in particular the examples of the snakeboard and Hirose snake robot. We interpret each of these examples in light of the theory developed in this thesis, and examine the generation of locomotive gaits using sinusoidal inputs and their relationship to the controllability tests based on Lie brackets

Control in moving interfaces and deep learning

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 14-05-2021This thesis has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex

Linear systems and control structure selection

This thesis is concerned with the development of concepts and results to facilitate study in two areas of control methodology. The two notions investigated are measures of controllability and observability and eigenstructure assignment. The link between these two areas is exposed, and it is demonstrated how the eigenvectors of a system play an important role in determining the degree of controllability and observability. The main concerns are issues dealing with the complexity of the instrumentation, and in particular the development of techniques that may assist in the development of methodology for sensor and actuator placement. The research involves the development of notions that help to structure a system on which control design is based. There are two areas of investigation. The first is the development of concepts and tools that aid in the selection and placement of sensors and actuators based on properties related to degrees of controllability and observability. The second is the investigation of the eigenstructure of a system and its properties, which enable the development of design procedures based on eigenstructure properties. A study of existing measures of controllability and observability leads to new techniques which take into consideration the problem of coordinate transformations, which is often overlooked. It is shown that the degree of controllability is influenced by changes in the structure of the state feedback matrix, as well as how controllability properties can be determined from Pliicker matrices of transfer function matrices. It is also shown that the energy required to move a system from one state to another is linked to the singular values of the output controllability grammian. A review of the problem of eigenstructure assignment paves the way for the development of a new technique of assigning the closed loop eigenstructure. This is based on matrix fraction description algorithms, and stems from an algebraic description of the total system behaviour, leading to a systematic study of closed loop eigenvectors by using a parametric approach. A new algebraic characterisation of the family of closed loop eigenvectors and related input and output directions is shown. Closed loop system robustness to parameter variations is also considered, where it is shown that there is a link with the orthogonality of the matrix of eigenvectors. As a result, the notion of strong stability is introduced, where it is shown that the shape of the eigenframe plays a role in the system response by way of overshoots. The work develops concepts and results which are important steps in the development of an integrated methodology for input, output structure selection