Analysis of Closed-Loop Performance and Frequency-Domain Design of Compensating Filters for Sliding Mode Control Systems

Abstract

It is known that the presence of parasitic dynamics in a sliding mode (SM) system causes high-frequency vibrations (oscillations) or chattering [1]-[3]. There are a number of papers devoted to chattering analysis and reduction [4]-[10]. Chattering has been viewed as the only manifestation of the parasitic dynamics presence in a SM system. The averaged motions in the SM system have always been considered the same as the motions in the so-called reduced-order model [11]. The reduced-order model is obtained from the original equations of the system under the assumption of the ideal SM in the system. Under this assumption, the averaged control in the reduced-order model becomes the equivalent control [11]. This approach is well known and a few techniques are developed in details. However, the practice of the SM control systems design shows that the real SM system cannot ensure ideal disturbance rejection. Therefore, if the difference between the real SM and the ideal SM is attributed to the presence of parasitic dynamics in the former then the parasitic dynamics must affect the averaged motions. Yet, the effect of the parasitic dynamics on the closed-loop performance can be discovered only if a non-reduced-order model of averaged motions is used. Besides improving the accuracy, the non-reduced-order model would provide the capability of accounting for the effects of non-ideal disturbance rejection and non-ideal input tracking. The development of the non-reduced order model becomes feasible owing to the locus of a perturbed relay system (LPRS) method [12] that involves the concept of the so-called equivalent gain of the relay, which describes the propagation of the averaged motions through the system with self-excited oscillations. Further, the problems of designing a predetermined frequency of chattering and of the closed-loop performance enhancement may be posed. The solution of those two problems may have a significant practical impact, as it is chattering that prevents the SM principle from a wider practical use, and it is performance deterioration not accounted for during the design that creates the situation of “higher expectations” from the SM principle.