Browsing by Subject "Piecewise linear systems"

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Open Access
Stability of planar piecewise linear systems :a geometric approach
(2015-09) Abdullahi, Adamu
This thesis focuses on the stability analysis of piecewise linear systems. Such systems consist of linear subsystems, each of which is active in a particular region of the state-space. Many practical and theoretical systems can be modelled as piecewise linear systems. Despite their simple structure, analysis of piecewise linear systems can be rather complex. For instance, most of the results for stability can be based on a Lyapunov approach. However, a major drawback of applying this method is that, it usually only provides su cient conditions for stability. A geometric approach will be used to derive new stability criteria for planar piecewise linear systems. Any planar piecewise linear (multi-modal) system is shown to be globally asymptotically stable just in case each linear mode satis es certain conditions that solely depend on how its eigenvectors stand relative to the cone on which it is de ned. The stability conditions are in terms of the eigenvalues, eigenvectors, and the cone. The improvements on the known stability conditions are the following: i) The condition is directly in terms of the \givens" of the problem. ii) Non-transitive modes are identi ed. iii) Initial states and their trajectories are classi ed (basins of attraction and repulsion are indicated). iv) The known condition for bimodal systems is obtained as an easy corollary of the main result. Additionally, using our result on stability, we design a hybrid controller for a class of second order LTI systems that do not admit a static output feedback controller. The e ectiveness of the proposed controller is illustrated on a magnetic levitation system.
Open Access
Well-posedness and stability of planar conewise linear systems
(2021-09) Namdar, Daniyal
Planar conewise linear systems constitute a subset of piecewise linear systems. The state space of a conewise linear system is a nite number of convex polyhedral cones lling up the space. Each cone is generated by a positive linear combination of a nite set of vectors, not all zero. In each cone the dynamics is that of a linear system and any pair of neighboring cones share the same dynamics at the common border, which is itself a cone of one lower dimension. Each cone with its linear dynamics is called a mode of the conewise system. This thesis focuses on the simplest case of planar systems that is composed of a nite number of cones of dimension two; with borders that are cones of dimension one, that is rays. Stability of such conewise linear systems is well understood and there are a number of necessary and su cient conditions. Somewhat surprisingly, their well-posedness is not so well understood or studied except for the special case where there are two modes only, i.e, the bimodal case. A graphical necessary and su cient condition is here derived for the wellposedness of a planar conewise linear system of arbitrary number of modes and the well-known condition for stability is re-stated on this same graph. This graphical result is expected to provide some guidance to well-posedness studies of conewise systems in a higher dimension.