Homogeneity of differential inclusions: Application to sliding modes

Homogeneity theory of differential equations is naturally extended to differential inclusions (DI). Stabilization accuracy of disturbed finite-time stable DIs is directly determined by the coordinate weights. The developed theory is applied to sliding-mode (SM) control. SMs are used to control uncertain dynamic systems by establishing and exactly keeping properly chosen connections between the system coordinates. The corresponding control and observation problems are usually reducible to finite-time stabilization of controlled DIs. Universal homogeneous SM controllers, asymptotically optimal robust exact differentiators and universal homogeneous output-feedback black-box controllers are constructed. The control can be made as smooth as needed, and dangerous vibrations (i.e. high energy 'chattering') can be effectively removed. Computer simulation demonstrates the applicability of the results.