Angular representations of nonlinear control systems
The aim of this thesis is to examine and extend the angular approach, which is a new approach for designing control and analysis of nonlinear systems. In this approach, the system is converted into two subsystems. One subsystem, the so-called spherical subsystem, is defined on a sphere, whilst the other subsystem, the radial subsystem, is one dimensional. A suitable control is designed using the radial subsystem. Various new methods are presented for the stabilisation and control design of a general class of nonlinear systems, based upon the angular approach. The advantage of this approach is that the stabilisation and control design problem of a nonlinear system is replaced by a one-dimensional control design and stabilisation problem. In addition, the control design using the angular approach is a straightforward, systematic method which is applicable to a wide class of nonlinear systems with and without uncertainty. Whenever the input map of the radial subsystem is zero, the radial control is not accessible and the control should be modified such that the defined control is accessible everywhere within the entire operating region. Several methods are considered for modifying the radial control including dynamical radial method. An adaptive angular method is also proposed to design an angular control for stabilisation of a nonlinear system with unknown parameters. The optimal control of nonlinear systems based upon the associated angular approach is also studied in this thesis. After decoupling the two associated (radial and spherical) subsystems and considering only the radial system, a finite-horizon radial optimal control is designed which minimises the appropriate radial cost function. Then the successive approximation technique is introduced in which the equations are replaced by a sequence of linear, time-varying approximations. The resulting optimal control is then applied to the original angular system. This control forces the original angular system to an equilibrium point. In addition, the control design and stabilisation problem for multi-input nonlinear systems using the angular approach is also studied.