Control design of hybrid systems via dehybridization

Hybrid dynamical systems are those with interaction between continuous and discrete dynamics. For the analysis and control of such systems concepts and theories from either the continuous or the discrete domain are typically readapted. In this thesis the ideas from perturbation theory are readapted for approximating a hybrid system using a continuous one. To this purpose, hybrid systems that possess a two-time scale property, i.e. discrete states evolving in a fast time-scale and continuous states in a slow time-scale, are considered. Then, as in singular perturbation or averaging methods, the system is approximated by a slow continuous time system. Since the hybrid nature of the process is removed by averaging, such a procedure is referred to as dehybridization in this thesis. It is seen that fast transitions required for dehybridization correspond to fast switching in all but one of the discrete states (modes). Here, the notion of dominant mode is defined and the maximum time interval spent in the non-dominant modes is considered as the 'small' parameter which determines the quality of approximation. It is shown that in a finite time interval, the solutions of the hybrid model and the continuous averaged one stay 'close' such that the error between them goes to zero as the 'small' parameter goes to zero. To utilize the ideas of dehybridization for control purposes, a cascade control design scheme is proposed, where the inner-loop artificially creates the two-time scale behavior, while the outer-loop exponentially stabilizes the approximate continuous system. It is shown that if the origin is a common equilibrium point for all modes, then for sufficiently small values of the 'small' parameter, exponential stability of the hybrid model can be guaranteed. However, it is shown that if the origin is not an equilibrium point for some modes, then the trajectories of the hybrid model are ultimately bounded, the bound being a function of the 'small' parameter. The analysis approach used here defines the hybrid system as a perturbation of the averaged one and works along the lines of robust stability. The key technical diffierence is that though the norm of the perturbation is not small, the norm of its time integral is small. This thesis was motivated by the stick-slip drive, a friction-based micropositioning setup, which operates in two distinct modes 'stick' and 'slip'. It consists of two masses which stick together when the interfacial force is less than the Coulomb frictional force, and slips otherwise. The proposed methodology is illustrated through simulation and experimental results on the stick-slip drive.