A quadrotor is a type of small unmanned aerial vehicle (UAV) with four rotors. Various control techniques have been successfully applied to the quadrotor. In this thesis, two control methods, including linear quadratic regulator (LQR) and H2-optimal control, are applied to the autonomous navigation and control of a quadorotor named QBall-X4 that is developed by Quanser. The continuous-time dynamic model is established using the Euler-Lagrange approach. Due to the nonlinearities in the quadrotor dynamics, we propose a simplified linear model, which is further used for the controller design in this thesis. According to the simplified quadrotor dynamics, we design an LQR controller to regulate the quadrotor system from its initial position to the desired position. The effectiveness of the controller is verified by simulation studies. However, the LQR control system is operated in the nominal model, and it can not present guaranteed performance when system uncertainties exist. The main emphasis is placed on designing an H2-optimal controller that minimizes the H2-norm of the transfer function. The solution is obtained by using the state-space approach and linear matrix inequality (LMI) method, respectively. In contrast to LQR control method, which is normally applied to a system with no disturbance, the H2-optimal controller takes the form of an observer together with a state feedback control gain to deal with the system uncertainties and disturbances. The simulation results and experimental study verify that the proposed H2-optimal controller is an effective option for the quadrotor with the attendance of uncertainties and disturbances. / Graduate

With the fast development of communication systems, computers and consumer electronics, the power supplies for telecoms, servers, desktops, laptops, flat-panel TVs, LED lighting, etc. are required for more power delivery with smaller spaces. The LLC resonant converter has been widely adopted for these applications due to the advantages in high efficiency, high power density and holdup time operation capability. However, unlike PWM converters, the control of the LLC resonant converter is much more difficult because of the fast dynamic characteristic of the resonant tank. In some highly dynamic processes like the load transient, start-up, over-load protection and burst operation, it is hard to control the current and voltage stresses and oscillations in the resonant tank. Moreover, to meet the high power density requirement, the LLC is required to operate at a high switching frequency. Thus the driving of the synchronous rectifier (SR) poses a design challenge as well. To analyze the fast dynamic characteristic, a graphic state-plane technique has been adopted for a class of resonant converters. In this work, it has been extended to the LLC resonant converter. First of all, the LLC steady state and dynamic behaviors are analyzed in the state plane. After that, a simplified implementation of the optimal trajectory control is proposed to significantly improve the load transient response: the new steady state can be tracked in the minimal period of time. With the advantages of the state-trajectory analysis and digital control, the LLC soft start-up is optimized as well. The current and voltage stress is limited in the resonant tank during the start-up process. The output voltage is built up quickly and smoothly. Furthermore, the LLC burst mode is investigated and optimized in the state plane. Several optimal switching patterns are proposed to improve the light load efficiency and minimize the dynamic oscillations. During the burst on-time, the LLC can be controlled to track the steady state of the best efficiency load condition in one-pulse time. Thus, high light-load efficiency is accomplished. Finally, an intelligent SR driving scheme is proposed and its simple digital implementation is introduced. By sensing the SR drain to source voltage and detecting the paralleled body diode conduction, the SR gate driving signal can be tuned within all operating frequency regions. In conclusion, this work not only solves some major academic problems about analysis and control of the LLC resonant converter based on the graphic state plane, but also makes significant contributions to the industry by improving the LLC transient responses and overall efficiency. / Ph. D.

Optimal control design and implementation for nonlinear systems is a topic of much interest. However, unlike for linear systems, for nonlinear systems explicit analytical solution for optimal feedback control is not available. Numerical techniques, on the other hand, can be used to approximate the solution of the HJB equation to find the optimal control. In this research, a computational approach is developed for finding the optimal control for nonlinear systems with polynomial vector fields based on sum of squares technique. In this research, a numerical technique is developed for optimal control of polynomial nonlinear systems. The approach follows a four-step procedure to obtain both local and approximate global optimality. In the first step, local optimal control is found by using the linearization method and solving the Algebraic Riccati equation with respect to the quadratic part of a given performance index. Next, we utilize the density function method to find a globally stabilizing polynomial nonlinear control for the nonlinear system. In the third step, we find a corresponding Lyapunov function for the designed control in the previous steps based on the Hamilton Jacobi inequality by using semidefinite programming. Finally, to achieve global optimality, we iteratively update the pair of nonlinear control and Lyapunov function based on a state-dependent polynomial matrix inequality. Numerical examples illustrate the effectiveness of the design approach.

The duty cycle control (DCC) modulation scheme for the three-phase dual-active-bridge (3p-DAB) DC-DC converter is a promising three degree-of-freedom modulation scheme which can extend the converter’s soft-switching range and reduce conduction losses under partial loading and wide voltage variations. However, the prior suggested methods to implement DCC in 3p-DABs have drawbacks such as requiring a multi-frequency approximation and offline optimization process or achieving less than optimal efficiency. To overcome these challenges, this research first proposes an optimal DCC modulation strategy (OMS) for the 3p-DAB based on a novel piece-wise time-domain analysis (TDA) and optimization process that obtains the optimal control parameters for minimum RMS phase current. Secondly, this research proposes a novel closed-form minimum current stress optimization (MCSO) DCC scheme based on the theoretical findings of the TDA optimization. The MCSO reduces the transformer phase currents and extends soft-switching operation under partial loading and wide voltage variations. Experimental results via open-loop testing show that the proposed closed-form MCSO DCC scheme has virtually identical efficiency as the OMS, making this the first research to provide a closed-form DCC modulation scheme for a 3p-DAB that achieves efficiency results equivalent to a fully-optimized offline scheme, but without the drawbacks of the offline optimization process. / Thesis / Master of Applied Science (MASc)

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Sensitivity analysis is one of the most prominent gradient based optimization techniques for mechanical systems. Model sensitivities are the derivatives of the generalized coordinates defining the motion of the system in time with respect to the system design parameters. These sensitivities can be calculated using finite differences, but the accuracy and computational inefficiency of this method limits its use. Hence, the methodologies of direct and adjoint sensitivity analysis have gained prominence. Recent research has presented computationally efficient methodologies for both direct and adjoint sensitivity analysis of complex multibody dynamic systems. Multibody formulations with joint friction were developed in the recent years and these systems have to be modeled by highly non-linear differential algebraic equations (DAEs) that are difficult to solve using numerical methods. The sensitivity analysis of such systems and the subsequent design optimization is a novel area of research that has been explored in this work. The contribution of this work is in the development of the analytical methods for computation of sensitivities for the most commonly used multibody formulations incorporated with joint friction. Two different friction models have been studied, capable of emulating behaviors of stiction (static friction), sliding friction and viscous drag. A case study has been conducted on a spatial slider-crank mechanism to illustrate the application of this methodology to real-world systems. The Brown and McPhee friction model has been implemented using an index-1 formulation for computation of the dynamics and sensitivities in this case study. The effect of friction on the dynamics and model sensitivities has been analyzed by comparing the sensitivities of slider velocity with respect to the design parameters of crank length, rod length, and the parameters defining the friction model. Due to the highly non-linear nature of friction, it can be concluded that the model dynamics are more sensitive during the transition phases, where the friction coefficient changes from static to dynamic and vice versa. / Master of Science / Mechanisms have been in existence since the earliest days of technology and are more relevant than ever in this age of robotics, artificial intelligence and space exploration. Innovations like myoelectric and neural prosthetics, legged robotics, robotic surgeries, advanced manufacturing, extra-terrestrial vehicles and so on are the modern day manifestations of the traditional mechanisms that formed the backbone of the industrial revolution. All of these innovations implement precision controlled multibody dynamic systems as part of their function. This thesis explores the modelling of such dynamic systems using different mathematical formulations. The contribution of this work is the incorporation of friction in the formulation of such systems. The performance of any dynamical system depends on certain parameters, which can be optimized to meet a certain objective criteria. This is achieved by performing a sensitivity analysis with respect to those parameters on the mathematical formulation of the mechanism. The derivation of this approach has been explored in this thesis. For the benefit of the reader, the application of this method has been discussed using a case study of a simple 3-dimensional slider crank mechanism.

The exact controllability problems for systems modeled by linear parabolic differential equations and the Burger's equations are considered. A condition on the exact controllability of linear parabolic equations is obtained using the optimal control approach. We also prove that the exact control is the limit of appropriate optimal controls. A numerical scheme of computing exact controls for linear parabolic equations is constructed based on this result. To obtain numerical approximation of the exact control for the Burger's equation, we first construct another numerical scheme of computing exact controls for linear parabolic equations by reducing the problem to a hypoelliptic equation problem. A numerical scheme for the exact zero control of the Burger's equation is then constructed, based on the simple iteration of the corresponding linearized problem. The efficiency of the computational methods are illustrated by a variety of numerical experiments. / Ph. D.