Computation of Time-fuel Optimal Control for a Class of LTI System

A problem of computing minimum weighted time-fuel control for a single input second order linear time invariant system with bounded inputs (|u(t)| ≤ 1) is considered. Pontryagins maximum principle dictates that the optimal time-fuel control is necessarily bang-off-bang in nature i.e., it switches between +1, 0, -1 with atmost three switchings. As a consequence, the time-fuel cost functional reduces to a linear function of switching instants. Moreover, the minimum time fuel control is obtained by solving multiple nonconvex static optimization problems. The ensuing optimization problems that are required to be solved have a linear function in terms of switching instants as a cost function. This linear cost function is minimized subject to equality constraints which contain exponential terms in switching instants obtained by solving differential equation of LTI system for a known initial state. Additionally, switching instants must be in temporal order leading to several linear inequality constraints. The exponential terms are converted to polynomial terms by a simple substitution of variables. As a result, the linear cost function gets converted to a rational function in the substituted variables and linear inequality constraints get equivalently modified without loosing linearity. This optimization problem is solved by constructing a hierarchy of semidefinite relaxations to approximate the optimal value closely. Finally, we demonstrate the proposed method with the help of an illustrative example.