Regions of stability for limit cycles of piecewise linear systems (original) (raw)
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In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections and 2k + 1 singular points can have at most k + 2 limit cycles, k + 1 of which surround the foci one by one and the last, k + 2, limit cycle surrounds all of the singular points of this system.
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In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections and 2k + 1 singular points can have at most k + 2 limit cycles, k+1 of which surround the foci one by one and the last, k+2, limit cycle surrounds all of the singular points of this system.
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Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions
IEEE Transactions on Automatic Control, 2003
In this paper we develop an entirely new constructive global analysis methodology for a class of hybrid systems known as Piecewise Linear Systems (PLS). This methodology consists in inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface, by constructing quadratic Lyapunov functions on switching surfaces. We found that an impact map induced by an LTI flow between two switching surfaces can be represented as a linear transformation analytically parameterized by a scalar function of the state. This representation of impact maps allows the search for quadratic surface Lyapunov functions to be done by simply solving a set of LMIs. Global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS can this way be efficiently checked. These new results were successfully applied to certain classes of PLS. Although this analysis methodology yields only sufficient criteria of stability, it has shown to be very successful in globally analyzing a large number of examples with a locally stable limit cycle or equilibrium point. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied.
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2008
In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number (but finite) of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections and 2k + 1 singular points can have at most k + 2 limit cycles, k + 1 of which surround the foci one by one and the last, (k + 2)-th, limit cycle surrounds all of the singular points of this system.
2003
This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS.
About Limit Cycles in Continuous Piecewise Linear Differential Systems
2017
In 2012, Lima and Llibre in [3] have studied a class of planar continuous piecewise linear vector fields with three zones. This class can be separated in four other classes and they proved, using the Poincaré map, that this particular class admits always a unique hyperbolic limit cycle. Here, we extended this study for other classes. We proved that some of them also admit always a unique hyperbolic limit cycle, moreover, we find a class that does not have limit cycles and prove the appearance of two limit cycles with one of these cycles appear by perturbations of a period annulus.
On the limit cycles of a class of piecewise linear differential systems in with two zones
Mathematics and Computers in Simulation, 2011
We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems with two zones. Our main result shows that three is an upper bound for the number of limit cycles that bifurcate from a center, up to first order expansion of the displacement function. Moreover, this upper bound is reached. The main technique used is the averaging method.