JOHNSON AND MURPHEY: SECOND-ORDER SWITCHING ..., SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL.1 Second-Order Switching Time Optimization for Non-linear Time-varying Dynamic Systems Elliot R. Johnson, Student Member, IEEE, Todd D. Murphey, Member, IEEE, Abstract This paper gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second-differentiable dynamics. Differential equa- tions governing the linear and bilinear operators required for calculating the derivatives are presented. Example optimizations of linear and non-linear systems are presented as evidence for the value of second-order optimization methods. One example converges in 33 iterations using second-order methods whereas the first-order algorithm requires over 30,000 iterations. Index Terms optimization, switched systems I. I NTRODUCTION Switched dynamic systems discontinuously switch from one dynamic function to the next in a known sequence as certain switching times are reached. Each dynamic function (i.e. ˙ x = f (x,u,t)) itself is continuous and differentiable. Switching time optimization is the problem of determining a set of switching times that minimize a cost function. In this paper the switching times are parameterized as a half space of partially ordered times (i.e., the switching times between two pre-specified modes satisfy an order, so the set of parameterized switching times does not form a vector space). Optimizations are implemented elliot.r.johnson@u.northwestern.edu t-murphey@northwestern.edu Mechanical Engineering, McCormick School of Engineering at Northwestern University, Evanston, IL This work was supported by the National Science Foundation under grant CCF-0907869. March 11, 2011 DRAFT