Characterization of Minimum Time-Fuel Optimal Control for LTI Systems

A problem of computing time-fuel optimal control for state transfer of a single input linear time invariant (LTI) system to the origin is considered. The input is assumed to be bounded. Since, the optimal control is bang-off-bang in nature, it is characterized by sequences of +1 , 0 and -1 and the corresponding switching time instants. All (candidate) sequences satisfying the Pontryagin's maximum principle (PMP) necessary conditions are characterized. The number of candidate sequences is obtained as a function of the order of system and a method to list all candidate sequences is derived. Corresponding to each candidate sequence, switching time instants are computed by solving a static optimization problem. Since the candidate control input is a piece-wise constant function, the time-fuel cost functional is converted to a linear function in switching time instants. By using a simple substitution of variables, reachability constraints are converted to polynomial equations and inequalities. Such a static optimization problem can be solved separately for each candidate sequence. Finally, the optimal control input is obtained from candidate sequences which gives the least cost. For each sequence, optimization problem can be solved by converting it to a generalized moment problem (GMP) and then solving a hierachical sequence of semidefinite relaxations to approximate the minima and minimizer [1]. Lastly, a numerical example is presented for demonstration of method.