Nonlinear and hierarchical hybrid control systems

In [12, 13] a theory of hierarchical hybrid control (HHC) was introduced via the formulation of a dynamical consistency (DC) relation between aggregated sets of system states. In the differential control systems case, the state space D of any given continuous base control system S is decomposed by a so-called finite analytic partition pi into a finite set of disjoint open connected blocks. For any such S and pi, a high level finite state control system Mp (called a partition machine of S ) is then obtained from the DC dynamics of the blocks of pi. It is shown that in the class of hybrid in-block controllable (HIBC) partitions the controllability of the base system S is equivalent to the controllability of the partition machine Mp .; Within this context, the converse question naturally arises as to whether any given finite state machine has a representation as the partition machine of some continuous control system S . In Chapter 2, we show that this is indeed the case and we present a constructive solution to this problem for all finite state machines.; The question of finding HIBC partitions leads us to the general study of controllability for nonlinear systems. In Chapter 3, a form of open local accessibility for nonlinear control systems is introduced called the continuous fountain condition. Subject to the condition that (i) the states of a system are continuous fountains, and (ii) one of various recurrence conditions holds; it is established that the system state space is (globally) controllable. It is shown that these controllability results imply the controllability of certain subsets of the state space of Hamiltonian control systems called energy slices. Several algebraic conditions for verification of the fountain property without analysing the actual geometry of accessible sets are established. Finally, these results are shown to have application to HHC theory in that they give conditions for a finite analytic partition to satisfy HIBC hypothesis,; In practice, an application of HHC theory would encounter problems in which the dynamics of the system, the nature of the controllable flows, and the positions of the boundaries of a partition themselves could only be approximately specified, Consequently, it is of interest to establish conditions under which the dynamics of Mp and its associated set of control laws are robust (that is to say insensitive) with respect to sufficiently small perturbations of the partitions defining Mp and small perturbations of the controllable flow. Chapter 4 addresses this issue. Two notions of the deformation of a partition pi are specified as, respectively, maps of the boundaries and the blocks of pi. The robustness properties of the partition machine Mp with respect to deformations of pi, and with respect to deformations of the controllable flow are investigated.; In Chapter 5, HHC theory is generalised to hybrid systems with disturbances. A disturbance rejection hierarchical hybrid control theory is introduced and then applied to a multi-tank example and to a highly simplified air traffic management example.