Experimental Benchmarking of Energy-saving Sub-Optimal Sliding Mode Control

TL;DR

The paper addresses energy-efficient robust control for second-order sliding mode systems subject to matched disturbances. It compares an energy-saving sub-optimal SM controller, which includes $u \in \{-U,0,U\}$ with deliberate $u=0$ phases, to a standard terminal second-order SM controller, both operating on the same quadratic sliding surface. The authors perform experimental benchmarking on a voice-coil actuator with sensor noise and a parasitic first-order actuator dynamic, evaluating convergence and energy use via $E = \int |u(t)| dt$. Results show that the energy-saving SM reduces overall energy consumption while preserving finite-time convergence, validating its practical applicability in hardware with nonideal sensing and actuator dynamics.

Abstract

The recently introduced energy-saving extension of the sub-optimal sliding mode control allows for control-off phases during the convergence to second-order equilibrium. This way, it enables for a lower energy consumption compared to the original sub-optimal sliding mode (SM) algorithm, both commutating a discontinuous control signal. In this paper, the energy-saving sub-optimal SM control is experimentally benchmarked against a standard second-order SM controller which also has a discontinuous control action. Here the so-called terminal second-order SM algorithm is used. The controlled plant is affected by the matched bounded disturbances which are unknown, and the output is additionally subject to the sensor noise. Moreover, a first-order actuator dynamics can lead to chattering, which is parasitic for SM applications. For a fair comparison, the same quadratic terminal surface is designed when benchmarking both SM controllers. Both experimentally compared SM algorithms have the same (bounded) control magnitude and states initial conditions.

Figures (6)

• Figure 1: Parametric constraints \ref{['eq:2:1:8']}--\ref{['eq:2:1:10']} of the energy-saving sub-optimal SM control ruderman2023energy drawn in the $(\beta_1,\beta_2)$ plane.
• Figure 2: Convergence time cost function $\hat{J}$ in dependency of $\beta_1$.
• Figure 3: Constrained objective function ($J$ -- $\hat{J}$) of the $\beta_1$, $\beta_2$ parameters, for an exemplary perturbation-to-control ratio $D/U=0.3$, ref. ruderman2023energy.
• Figure 4: Voice-coil based linear actuator with one DOF in $x$ coordinates.
• Figure 5: Noisy system signals: distribution of the measured output $x(t)$ at zero input $u(t) = 0$ in (a), fragment of the measured output $x(t)$ at applied constant input $u(t)=\mathrm{const}$ in (b), fragment of the low-pass filtered $dx/dt$ obtained by the discrete time differentiation of the measured $x(t)$ in (c).