Bifurcation of Limit Cycles from a Fold-Fold Singularity in Planar Switched Systems

We use a bifurcation approach to investigate the dynamics of a planar switched system that alternates between two smooth systems of ODEs denoted as (L) and (R), respectively. For an x epsilon R that plays a role of bifurcation parameter, a switch to (R) occurs when the trajectory hits the switching line {($) over barx} x R and a switch to (L) occurs when the trajectory hits the switching line {-($) over barx} x R. This type of switching is known as relay, hysteresis, or hybrid switching in control. The main result of the paper gives sufficient conditions for bifurcation of an attracting or repelling limit cycle from a point O epsilon{O} x RR when x ($) over bar crosses O. The result is achieved by identifying a region where the dynamics of the system is described by the map P-x ($) over bar(y) = y +alpha y(2)+beta x ($) over bar /y + o(y(2)). The map P-x ($) over bar is obtained as a local return map induced by the switching line {($) over barx} x R. The fixed points of P-x ($) over bar correspond to small amplitude limit cycles surrounding O. Motivated by applications to antilock braking systems, we focus on a particular class of switched systems where, for x ($) over bar = 0, the point 0 is a so-called fold-fold singularity, i.e., the vector fields of both systems (L) and (R) are parallel to {0} x R at O. The result of the paper can be used for the design of switched control strategies that ensure limit cycling around a given point of the phase space. In particular, we illustrate the main theorem by establishing limit cycling behavior in relay affine systems (that model, e.g., power converters). Furthermore, we design a two-rule switched control to guarantee the existence of such an attracting limit cycle in antilock braking systems, whose magnitude can be made as small as necessary. This diminishes the need of additional switching rules used in other papers to not exceed the actuator technical capabilities.