LPRS Analysis of Singularly Perturbed Sliding Mode Control Systems

In this paper, we examine singularly perturbed sliding mode (SPSM) control systems with linear plants using the Locus of a Perturbed Relay System (LPRS) method. We derive necessary conditions for the existence of chattering, which depend on both the singular perturbation parameter and the plant dynamics. Moreover, concerning the singular perturbation parameter and plant dynamics, we demonstrate that the model of the averaged or slow motions in an SPSM system is of reduced order if an ideal sliding mode (SM) takes place and non-reduced if chattering occurs. Finally, starting from the definition of the LPRS, we derive first-order approximations of the chattering frequency and period, as well as the equivalent gain of the relay. The results are illustrated and applied on <inline-formula><tex-math notation="LaTeX">$1^{st}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$2^{nd}$</tex-math></inline-formula> order systems, as well as an example of a DC motor connected to a dynamic sensor.