Describing Function-Based Analysis and Design of Approximated Sliding-Mode Controllers with Reduced Chattering

Sliding-mode control (SMC) is a powerful robust control design technique that, when appropriately implemented, ensures insensitivity to the so-called matched bounded disturbances and finite-time convergence. However, the insensitivity requires an ideal implementation of discontinuous signals, often based on the sign function that, in practice, results in the presence of parasitic oscillations called chattering. Chattering is unavoidable in systems with SMC, including continuous and higher-order SMC (HOSM) approaches. One of the simplest and commonly used solutions to alleviate chattering is the approximation of the SMC by substituting the sign function with its approximation by a sigmoid function or a saturation function, although this obviously transforms the insensitivity property into a reduction of the influence of the disturbances. In fact, the accuracy of approximating discontinuity creates a trade-off between the reduction of the influence of the disturbances and the amount of chattering. Hence, the following question appears: Is it possible to systematically design a SMC-approximation avoiding a blind search requiring a huge number of numerical and/or hardware experiments? This chapter presents a method to design the boundary-layer parameter of the saturation function. The design is based on the describing function (DF) and harmonic balance (HB) techniques for estimating the parameters of chattering, i.e., frequency and amplitude of the parasitic oscillations.