Decoupling in Systems with Discontinuous Controls

Abstract

Since we are going to “unify” the design principles by introducing sliding modes, let us confine ourselves with only those discontinuous systems whose sliding equations may be written quite unambiguously. From the entire variety of nonlinear systems, this limitation isolates a subclass which, generally speaking, may be presented by equations linear in control

$$x = f(xj) + B(x,t)u,$$

((6.1))

where x∈ℝ _n and u∈ℝm. Formally, a standard statement of the control theory problems is in choosing such a control u, functionally or operationally dependent upon the system state, time and disturbances, that brings an appropriate transformation of the solution to the initial system (6.1). The idea of the “appropriate” transformation is treated in such a broad sense by the present-day control theory authors that it will be useful to give the most important problems statements in this book.