Control Design of Hybrid Systems via Dehybridization

Abstract Hybrid dynamical systems are those with interaction b etween 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 p erturbation theory are readapted for approximating a hybrid system using a continuous one. To this purp ose, hybrid systems that p ossess a two-time scale prop erty, i.e. discrete states evolving in a fast time-scale and continuous states in a slow time-scale, are considered. Then, as in singular p erturbation 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 corresp ond 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 sp ent 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 b etween them goes to zero as the ‘small’ parameter goes to zero. To utilize the ideas of dehybridization for control purp oses, a cascade control design scheme is prop osed, where the inner-loop artificially creates the two-time scale b ehavior, while the outer-loop exp onentially stabilizes the approximate continuous system. It is shown that if the origin is a common equilibrium p oint for all modes, then for sufficiently small values of the ‘small’ parameter, exp onential stability of the hybrid model can b e guaranteed. However, it is shown that if the origin is not an equilibrium p oint for some modes, then the tra jectories of the hybrid model are ultimately b ounded, the b ound b eing a function of the ‘small’ parameter. The analysis approach used here defines the hybrid system as a p erturbation of the averaged one and works along the lines of robust stability. The key technical difference is that though the norm of the p erturbation is not small, the norm of its time integral is small. This thesis was motivated by the stick-slip drive, a friction-based micro-p ositioning setup, which op erates 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 prop osed methodology is illustrated through simulation and exp erimental results on the stick-slip drive. 1

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