Fuel/time optimal control of the benchmark problem

Design of fuel/time optimal control of the benchmark two-mass/spring system is addressed in the frequency domain. The optimal control profile is represented as the output of a time-delay filter, where the amplitude of the time-delayed signals are constrainted to satisfy the control bounds. The time delays of the filter are determined by solving a parameter optimization problem that minimizes a weighted fuel/time cost function subject to the constraint that the tune-delay filter cancel all the poles of the system and the control profile satisfies the rigid-body boundary conditions. It is shown that three control structures exist: a three-switch profile corresponding to the time optimal control problem that changes to a six-switch profile corresponding to a cost function that includes a small weight on the fuel. As the weight on the fuel increases beyond a critical value, the control profile changes to a two-switch profile. The value of the critical weight that represents the transition of the control profile from a six-switch to a two-switch control profile is determined.