Sliding mode control with a lemniscate-based sliding surface

In this paper, a novel sliding mode control method with a lemniscate-based sliding surface is introduced. For an underdamped second-order system that contains uncertainty, sliding mode control with the ellipsoidal sliding surface previously proposed by the authors has advantages over conventional sliding mode control with a linear sliding surface, in that it is smoother, has faster error convergence, and has lower energy consumption. However, in the case of adopting the ellipsoidal surface for an overdamped second-order system, the energy efficiency worsens. However, the lemniscate resembles the overdamped second-order response. Since the proposed lemniscate-based sliding surface is a smooth function without a cusp, chattering is reduced, and it is more robust than the conventional linear sliding surface because it has no reaching phase. It converges to the equilibrium point faster than the conventional linear sliding surface. To verify the advantages of the proposed method, we compared the proposed sliding surface with the conventional ellipsoidal and linear sliding surfaces through simulations of a simple second-order uncertain dynamic system.