Contributions to quantitative feedback theory design and preliminary application to a variable-pitch quadcopter - Pretorius, Arnold (Department of Electrical Engineering, 2026)
Abstract: Contributions to quantitative feedback theory design and preliminary application to a variable-pitch quadcopter Arnold Pretorius (26/02/2023) This thesis details the mathematical and mechanical modelling and design, state estimation, and preliminary control of a novel variable-pitch quadcopter. The experimental framework is first developed, which includes the physical quadcopter platform, as well as a vision-based motion capture system. Modelling and system identification methods are applied to the quadcopter low-level subsystems, in order to understand and verify the fundamental dynamics of the system. Following this, a state estimation algorithm, based on the extended Kalman filter, is developed, which fuses camera information from the motion capture system to provide pose estimation of the quadcopter platform. A novel rotor thrust observer scheme is also presented, which enables on-board estimation of the quadcopter rotor thrusts during operation. Preliminary low-level control of the rotor speed and thrust is demonstrated, which is facilitated by the on-board rotor speed and thrust estimates. Finally, a simulated demonstration of the position control scheme is provided, which makes use of the previously modelled subsystems and designed control schemes. The simulation environment includes nonlinearities and noise effects that emulate that of the real-world experimentation, and the variable-pitch quadcopter is shown to perform as expected. This thesis investigates the state of the art of quantitative feedback theory, with a particular focus on reducing feedback controller design conservatism. Starting with the generalised single-input-single-output problem formulation, we introduce a novel means of synthesizing a per-plant closed-loop model specification that caters to the signal and phase limitations of every plant instance in the plant set. This information is then incorporated into a univariate constraint set on the feedback controller element, which is predicated on the existence of a valid, non-empty pre-filter solution space in the arithmethic-complex plane. Using a simple example case, we are able to show that the method is effective in balancing the tracking performance across the entire plant set, subject to the inherent signal limits. In a subsequent contribution, we introduce a new approach to the 2x2 model-error tracking problem that combines a plant-inverting design routine with a novel non plant-inverting method, with the aim of reducing the controller design conservatism. We show that geometric-based existence conditions can be exploited to arrive at a univariate design constraint set on the particular feedback controller element of interest, whilst reducing design conservatism at all pertinent frequencies of interest. This method is shown to substantially outperform traditional plant-inverting 2x2 methods, especially at the gain-phase crossover range. Serving as the main QFT contribution of this thesis, we develop a generalised multi-variable refinement approach to the tracking error problem that is intended to ease the feedback control design at all frequencies. Assuming a valid a priori feedback design exists, a feedforward filter is synthesized using optimisation, with the intention of loosening the strictures on a subsequent differential feedback design. The resulting prototype control solution is then used to provide additional gain and phase information that aids in reducing the design conservatism when applying the triangle inequality. This process can be applied iteratively in order to refine the loop transfer behaviour and reduce the feedback controller gain. The method is shown to surpass current multivariable QFT design routines in specific benchmarking examples in terms of expanding the admissible feedback controller per-frequency solution space, especially in the gain-phase crossover region.