Foundations of a Bicoprime Factorisation Theory: A Robust Control Perspective

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Abstract

This thesis investigates Bicoprime Factorisations (BCFs) and their possible uses in robust control theory. BCFs are a generalisation of coprime factorisations, which have been well known and widely used by the control community over the last few decades. Though they were introduced at roughly the same time as coprime factorisations, they have been largely ignored, with only a very small number of results derived in the literature.BCFs are first introduced and the fundamental theory behind them is developed. This includes results such as internal stability in terms of BCFs, parametrisation of the BCFs of a plant and state space constructions of BCFs. Subsequently, a BCF uncertainty structure is proposed, that encompasses both left and right coprime factor uncertainty. A robust control synthesis procedure is then developed with respect to this BCF uncertainty structure. The proposed synthesis method is shown to be advantageous in the following two aspects: (1) the standard assumptions associated with H-infinity control synthesis are directly fulfilled without the need of loop shifting or normalisation of the generalised plant and (2) any or all of the plant's unstable dynamics can be ignored, thus leading to a reduction in the dimensions of the Algebraic Riccati Equations (AREs) that need to be solved to achieve robust stabilisation.Normalised BCFs are then defined, which are shown to provide many advantages, especially in the context of robust control synthesis. When using a normalised BCF of the plant, lower bounds on the achievable BCF robust stability margin can be easily and directly computed a priori, as is the case for normalised coprime factors. Although the need for an iterative procedure is not completely avoided when designing an optimal controller, it is greatly simplified with the iteration variable being scalar. Unlike coprime factorisations where a single ARE needs to be solved to achieve normalisation, two coupled AREs must be satisfied for a BCF to be normalised. Two recursive methods are proposed to solve this problem.Lastly, an example is presented where the theory developed is used in a practical scenario. A quadrotor Unmanned Aerial Vehicle (UAV) is considered and a normalised BCF controller is designed which in combination with feedback linearisation is used to control both the attitude and position of the vehicle.

Keyword(s)

Bicoprime factorisations; H-infinity control; Robust control synthesis

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