Attractive Ellipsoids in Sliding Mode Control

  • Alexander Poznyak
  • Andrey Polyakov
  • Vadim Azhmyakov
Part of the Systems & Control: Foundations & Applications book series (SCFA)


In this chapter, a new sliding mode control design algorithm for a linear and a class of nonlinear quasi-Lipschitz disturbed systems is presented. It is based on the appropriate selection of a sliding surface via the invariant ellipsoid method. The designed control guarantees minimization of unmatched disturbance effects to system motions in a sliding mode. The theoretical results are verified by numerical simulations. Additionally, a methodology for the design of sliding mode controllers for linear systems subjected to matched and unmatched perturbations is proposed. It is considered that the control signal is applied through a first-order low-pass filter. The technique is based on the existence of an attracting (invariant) ellipsoid such that the convergence to a quasiminimal region of the origin using the suboptimal control signal is guaranteed. The design procedure is given in terms of the solution of a set of matrix inequalities. Benchmark examples illustrating the design are given.


Sliding mode control Unmatched perturbations Disturbance reduction 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alexander Poznyak
    • 1
  • Andrey Polyakov
    • 2
  • Vadim Azhmyakov
    • 3
  1. 1.Automatic Control DepartmentCentro de Investigacion y Estudios AvanzadosMéxicoMexico
  2. 2.Non-AINRIA-LNEVilleneuve d’AscqFrance
  3. 3.Faculty of Electronic and Biomedical EngineeringUniversity of Antonio NariñoNeivaColombia

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