Sliding modes in multidimensional systems with variable structure

This paper treats variable structure systems with vector control inputs. Each component of the control vector is discontinuous on a corresponding surface in the system coordinate space. In such systems, sliding modes may arise and then trajectories lie on the discontinuity surfaces or their intersections. A so-called method of equivalent control is suggested which allows formal development of equations of such motions. The validity of the sliding mode equations derived through the equivalent control approach is established by introduction into the system of small nonidealities (of hysteresis, time-lag, and a periodic type) which are then made to tend to zero. It is suggested that Liapunov stability concepts be applied to derive the conditions for existence of sliding modes. It follows from the equations describing system evolution in the sliding mode that trajectory characteristics are dependent on the discontinuity surfaces equations. Therefore, proper selection of the surfaces on which the components of the control vector are discontinuous enables one to introduce desired properties into the system response. A desired response can be obtained if the trajectory reaches the intersection of the discontinuity surface starting from any initial state, and that at each point of the intersection the existence conditions for the sliding mode hold. The paper considers different methods of variable structure systems synthesis on the basis of the sliding modes being deliberately introduced into the system. Algorithms are described for controlling plants with both constant and variable parameters, and also plants with external disturbances. A class of systems is indicated in which evolution in the sliding mode possesses invariance with respect to variable parameters and disturbances.