Bounds on limit cycles in two-dimensional bang-bang control systems with an almost time-optimal switching curve

In the mechanization of the switching curve for single-input, time-optimal control of a linear plant, unavoidable physical limitations lead to an actual switching curve which deviates significantly from the optimal. If the desired system operation is the reaching of the origin of the state plane, nonoptimal operation can result in either a constant steady state different from the origin, a chatter point, or a limit cycle. In this paper, only the limit cycle is assumed to constitute tolerable system operation. Hence, a class of nonoptimal switching curves is defined, any member of which is classified as a chatterfree switching curve. Both an upper and a lower bound are derived for the resultant limit cycle when a chatterfree switching curve is used for the bang-bang control of either a two-dimensional plant with distinct real poles or a two-dimensional plant with complex conjugate poles. Existence of a nontrivial lower bound demonstrates the existence of a limit cycle. The optimal switching curve with a switching time delay is cited as an example of a chatterfree switching curve. The upper and lower bounds may be evaluated numerically in practice by a simple graphical construction.