Fixed-Time Inflection Point Seeking

We extend the algorithms for fixed-time seeking of extrema to seeking inflection points of static maps. We present two algorithms—one for gradient-based fixed-time inflection point seeking (FxTGIPS) and the other for Newton-based fixed-time inflection point seeking (FxTNIPS). To clarify terminology, while in standard extremum seeking the gradient algorithm estimates the gradient and drives it to zero, in an inflection point seeking algorithm the gradient algorithm estimates the gradient of the gradient, i.e., the Hessian, and drives it to zero. In inflection point seeking we consider maps that are somewhat more general (as specified in the paper) than plainly saying that they have a third derivative, i.e., the ‘jerk,’ different from zero, at the point where their Hessian is zero. With the FxTGIPS algorithm we achieve a settling time that is independent (locally) of the initial distance of the estimate from the inflection point but this settling time depends on the unknown jerk coefficient at the inflection point, i.e., the settling time is not user-assignable. With the FxTNIPS algorithm, which estimates the jerk at the inflection point and the reciprocal of the jerk, we achieve a settling time that is independent of the unknown jerk and is user-assignable. Both FxT stability results are semiglobal and practica and both are established via averaging theory and singular perturbation theory for dynamical systems that are not necessarily Lipschitz continuous.