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27 Feb 2016

Constrained Trajectory Generation and Fault Tolerant Control Based on Differential Flatness and B-splines

Fajar Suryawan, August 2010
The University of Newcastle

This thesis provides a unified treatment of the notions of differential flatness, for the characterisation of continuous-time linear systems, and B-splines, a mathematical concept commonly used in computer graphics. Differential flatness is a property of some controlled (linear or nonlinear) dynamical systems, often encountered in applications, which allows for a complete parameterisation of all system variables (inputs and states) in terms of a finite number of variables, called flat outputs, and a finite number of their time derivatives. The notion of differential flatness for a system is especially useful in situations when explicit trajectory generation is required. In fact, under the differential flatness formalism the motion planning problem, as far as the differential equation is concerned, is trivialised. However, a very important limitation, ubiquitous in all practical applications, is the presence of constraints. The problem of constrained trajectory generation is intimately related to that of optimal control, where one wants to achieve certain objectives with limited resources, and time-optimal control, in which one seeks to perform a task as fast as possible while, at the same time, satisfying all system constraints. In the literature, trajectory generation and [time-] optimal control often use some parameterisation to represent the system's signals. Polynomials and B-splines are a natural choice since they have several desirable properties. However, there has not been much work exploiting the combined properties of differential flatness for linear systems and B-splines. The first focus of this thesis is, hence, to investigate the use of B-splines for constrained trajectory generation of continuous-time linear flat systems in such a way that their respective properties are jointly exploited and complemented. This synthesis offers new methods and insights to the fields of constrained trajectory optimisation, optimal control, and minimum-time trajectory generation. The differential flatness parameterisation also offers analytical redundancy relations. That is, the value of some variables can be algebraically inferred from some other measured variables. This fact can be used to perform algebraic estimation and fault detection in linear and nonlinear systems. The second focus of this thesis is, thus, to develop a method to perform algebraic estimation and fault detection, based structurally on the differential flatness notion, for linear and nonlinear systems, and using a numerical method based on B-splines. The methodology to tackle the focal problems of constrained trajectory generation and fault tolerant control, based on differential flatness and B-splines, is primarily developed for linear systems. Then, experimental validations of the methods, using a laboratory-scale magnetic levitation system, are provided. Finally, some extensions of the ideas to nonlinear systems are discussed.

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