Time Sub-Optimal Control of Triple Integrator

The time sub-optimal control is studied in this paper. The nonlinear controller that respects input saturations is derived for the simple linear system represented by the triple integrator. In comparison with the pure time optimal controller the designed sub-optimal controller changes its limit values smoothly with exponential behaviour. Similarly to the time optimal control the design is based on switching surfaces but these are shifted and modified according to the original ones in the time optimal control. This can assure the decrease of high sensitivity of time optimal control. New parameters introduced during the design correspond in linear cases to the poles of the closed loop system. They enable to tune the control changes. The time sub-optimal control is compared with model predictive control. The resulting formulas for the control value are complicated but they have an explicit form so they can be evaluated fast enough to be used in real time systems.