Design and Application of Energy-saving Sub-Optimal Sliding Mode Control

TL;DR

The chapter addresses energy efficiency in robust second-order sliding mode control by introducing ES-SOSMC, which injects a controlled off-phase to reduce fuel consumption without sacrificing finite-time convergence. Building on the SOSMC framework for relative-degree-two plants with bounded disturbances, it defines an energy cost function and optimizes two switching thresholds $\beta_1$ and $\beta_2$ to achieve energy savings while preserving performance. The approach is complemented by a chattering analysis using describing functions and applied to a stiff position-control scenario involving a moving rough surface, such as scanning or machining tasks, where ES-SOSMC achieves comparable tracking with reduced energy use for $\Delta/U$ up to about $0.35$. The results suggest substantial practical benefits for energy-conscious robotic and mechatronic systems operating under uncertain disturbances and actuator dynamics, with clear guidelines for parameterization and stability.

Abstract

The recently introduced energy-saving extension of the sub-optimal sliding mode control (SOSMC), which is known in the literature for the last two and half decades, incorporates a control-off mode that allows for saving energy during the finite-time convergence process. This novel energy-saving algorithm (denoted by ES-SOSMC) assumes the systems with relative degree two between the sliding variable and the switching control with a bounded magnitude, while the matched upper-bounded perturbations are not necessarily continuous. The design and practical application of the ES-SOSMC are the subject of this chapter. A method for parameterizing the ES-SOSMC through a constrained minimization of the energy cost function is recalled which guarantees the total energy consumption is lower than that of the conventional SOSMC. Also the residual steady-state oscillations (chattering), occurring when additional (actuator) dynamics are taken into account, are addressed. An application example for scanning and machining a rough surface, both of which require a stiff position control in contact with a moving surface, demonstrates practical suitability of the control. Here, ES-SOSMC is compared with SOSMC by showing an equivalent tracking and stabilization performance and evaluating the energy-saving operation with respect to a fuel consumption norm.

Figures (11)

• Figure 1: Fuel consumption metric (i.e. energy costs) of a continuously switching SMC.
• Figure 2: Phase-portrait of unperturbed second-order system with time-optimal and fuel-optimal control.
• Figure 3: Parametric constraints \ref{['eq:18']}, \ref{['eq:19']}, \ref{['eq:20']} of the global finite-time ES-SOSMC convergence in $(\beta_1,\beta_2)$ plane.
• Figure 4: Exemplary convergence cost function $\hat{J}$ of SOSMC in dependency of $\beta_1$.
• Figure 5: Constrained objective function $(J-\hat{J})$ of the $\beta_1$, $\beta_2$ parameters of ES-SOSMC for the disturbance-to-control ratios: $\Delta/U=0.2$ in (a), $\Delta/U=0.3$ in (b), $\Delta/U=0.4$ in (c).