Robust control framework in the weighted Sobolev space

Abstract

This Ph.D. thesis proposes new formulations of the H2 and H∞ controllers in the weighted Sobolev spaces. The novel approaches, here called W2 and W∞, are developed taking into account the weighted Sobolev norm of the cost variable, aiming to achieve an improved transient performance with a faster disturbance attenuation. Initially, the nonlinear W2 and W∞ control problems are formulated for closed-loop autonomous nonlinear systems, and the control problems are developed via dynamicprogramming, resulting in complex Hamilton-Jacobi (HJ) equations to be solved. Due to the difficulty of solving analytically these HJ equations, the Successive Galerkin Approximation Algorithm is extended to the W2 and W∞ controllers, and used to approximate solutions. Numerical experiments are performed with a Two-wheeled Self-balanced vehicle, and a comparative analysis with the classic H∞ controller is presented. The results demonstrate that the controllers resulting from the weighted Sobolev approach achieve a better transient performance with a faster disturbance attenuation. This doctoral thesis also proposes new formulations of the nonlinear H2 and H∞ optimal control approaches in the weighted Sobolev spaces, in order to handle two classes of underactuated mechanical systems: the class of reduced underactuated mechanical systems, with the objective of achieving trajectory tracking for a reduced number of Degrees Of Freedom (DOF), called controlled DOF; and the class of underactuated mechanical systems with input coupling, with the objective of driving the controlled DOF along a desired trajectory while stabilizing the remaining ones. For these systems, the optimal nonlinear W2 and W∞ control problems are formulated via dynamic-programming, and particular solutions are presented for the resulting HJ equations with the corresponding stability analysis. In addition, the concepts of Wm,p,σ-gain and Wm,p,σ-stability are established and applied to the particular case studies. It is shown that, for the particular class of underactuated mechanical systems, whose control inputs and disturbances vector span the same space in the vector space of generalized forces, the W2 and W∞ controllers become equivalent. The efficacy of the proposed W2 and W∞ control strategies for mechanical systems are demonstrated via numerical experiments conducted with a fully actuated manipulator, a Two-wheeled Self-balanced vehicle, and a Quadrotor unmanned aerial vehicle (UAV). It is verified that these controllers provide a better transient perfor- mance with a faster response against external disturbances, in comparison with a more traditional nonlinear H∞ controller, in addition to be of simpler design. This thesis also formulates the linear W∞ controller in the weighted Sobolev space for linear time-invariant systems. The design of state and output feedback controllers is addressed, and a new approach in which the dynamic behavior of the disturbances is taken into consideration in the control design stage by means of a disturbance model is introduced. Pole placement constraints are also developed, allowing the synthesis of linear W∞ controllers with the closed-loop poles allocated in a predefined region of the complex plane. Numerical experiments are performed with a simple linear system, a Twowheeled Self-balanced vehicle, and a Quadrotor UAV, which demonstrate that the state and output feedback linear W∞ controllers achieve a better transient performance with a faster disturbance attenuation in comparison with a linear H∞ controller. Lastly, the W∞ controller is employed in the case study of convertible Tilt-rotor UAVs. A detailed modeling of the nonlinear multi-body dynamics of the Tilt-rotor UAV is conducted using the Euler-Lagrange formalism, and the nonconservative forces and torques generated by the propellers, servomotors, fuselage, wings, tail surfaces, and aerodynamic interference are computed and mapped to the vector of generalized forces. State and output feedback linear W∞ controllers are synthesized to solve the trajectory tracking problem of the convertible Tilt-rotor UAV in the helicopter flight mode, being the Von Kármán wind turbulence model used to emulate the environment wind and considered as disturbance model in the linear control design. Besides, a nonlinear W∞ controller is designed to solve the full flight envelope trajectory tracking problem of the convertible Tilt-rotor UAV. To design the nonlinear controller, the approach proposed for underactuated mechanical systems with input coupling, previously mentioned, is extended, the mechanical system is partitioned with respect to stabilized, regulated, and controlled DOF, aiming to achieve trajectory tracking of the controller DOF, and set references to the regulated DOF, while stabilizing the remaining DOF. The nonlinear controller is implemented taking into account a control allocation scheme, which is proposed to handle the time-varying rank of the convertible Tilt-rotor UAV input coupling matrix, taking into account the magnitude and orientation of the relative wind-speed to properly map the generalized inputs to the control signals. Numerical experiments are conducted in a high fidelity simulator, they demonstrate that the proposed W∞ controllers achieve trajectory tracking in the helicopter flight mode, regarding the linear W∞ controllers, and trajectory tracking throughout the full flight envelope, regarding the nonlinear W∞ controller, while attenuating effects of external disturbances and wind gusts.