Global stability of relay feedback systems

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit c ycles for these systems are well known, Global stability conditions, howeve r, are practically nonexistent. This paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with Linear time-invariant (LTI) stable systems. The analysis cons ists in finding quadratic surface Lyapunov functions for Poincare maps asso ciated with RFS, These results are based on the discovery that a typical Po incare map induced by an LTI flow between two hyperplanes can be represente d as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodolo gy yields only a sufficient criterion of stability, it has proved very succ essful in globally analyzing a large number of examples with a unique local ly stable symmetric unimodal limit cycle. In fact, it is still an open prob lem whether there exists an example with a globally stable symmetric unimod al limit cycle that could not be successfully analyzed with this new method ology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension, Such results lead us to bel ieve that globally stable limit cycles of RFS frequently have quadratic sur face Lyapunov functions.