Energy-saving sub-optimal sliding mode control with bounded actuation

The second-order sub-optimal sliding mode control (SMC), known in the literature for the last two decades, is extended by a control-off mode which allows for saving energy during the finite time convergence. The systems with relative degree two between the sliding variable and switching control with bounded actuation are considered, while the matched upper-bounded perturbations are not necessarily continuous. Detailed analysis of the proposed energy-saving sub optimal SMC is performed with regard to the parametric conditions, reaching and convergence time, and residual steady oscillations if the parasitic actuator dynamics is added. Constraints for both switching threshold parameters are formulated with respect to the control authority and perturbations upper bound. Based on the estimated finite convergence time, the parameterization of the switching thresholds is solved as constrained minimization of the derived energy cost function. The total energy consuming control-on time is guaranteed to be lower than the upper-bounded convergence time of the conventional sub-optimal SMC. Numerical evaluations expose the properties of the proposed energy-saving sub-optimal SMC and compare it with conventional sub-optimal SMC in terms of the fuel consumption during the convergence.