Sliding Mode Control Theory

Variable structure control with sliding modes is illustrated in the general framework of variable structure systems. Some of the characteristics of variable structure systems are first presented to highlight their potential use in control design. In particular, the Zeno phenomenon, which appears when a variable structure system is constrained onto a switching surface by the local vector fields, is reinterpreted as a goal for a control feedback system where it is desired to steer and maintain a proper error variable at zero. Then the system dynamics of the variable structure system is interpreted as a differential inclusion whose solution on the switching surface is understood in the Filippov’s sense. This suggests the design of a variable structure control that forces a properly designed Zeno behavior in spite of uncertainty, exploiting the attractive features of differential inclusions. Various approaches to the design of variable structure control systems with sliding modes are presented starting from the more classical ones, where the sliding mode dynamics is linear, to the more recent and challenging approaches based on nonlinear sliding surfaces and higher-order sliding modes. The treatment is carried out using several examples and a few simple dynamical systems to limit as much as possible the use of advanced mathematical treatments. The aim is to explore the main ideas and the pros and cons of sliding mode control approach rather than to discuss in detail the underlying technicalities, which can be found in the given references.