Stability criteria and bounds for limit cycles of relay feedback systems

Local stability of limit cycles for time-delay relay-feedback systems

“…For = 0, if n = n = 0, then Theorem 3.1 still works. Indeed, n = n = 0 implies that the limit cycle is nontangent with the switching planes at traversing instant, like the case considered in [1], [8], [10]. However, the methods used in [1], [8], and [10] are not applicable to deal with the local stability of limit cycles in Form 1.…”

“…Let N = f0;1; ...;n 0 1g: (10) The first lemma specifies two integers n ; n 2 N, which will be used in the development.…”

“…This relies on the continuity at the intersecting points, and intuitively it is the "overshoot" effect. For delay-free systems, the nontangent condition at traversing points has to be assumed, like the case considered in [1], [8], and [10]. Such a condition makes the local stability analysis simpler.…”

“…In [5], the method of linear matrix inequalities is used to compute a local stability bound. Another discussion for the local stability is given in [10]. For the global stability of limit cycles, a recent paper [6] obtains sufficient conditions in terms of a set of linear matrix inequalities by finding the so-called surface Lyapunov function of Poincare maps.…”

“…Exact methods have also been reported. See [1], [6], [8]- [10] and the references therein. Astrom [1] gives elegant criteria for the local stability of limit cycles by considering the linear approximation of the Poincare map.…”