Higher-Order Sliding Mode Controllers and Differentiators

  • Yuri Shtessel
  • Christopher Edwards
  • Leonid Fridman
  • Arie Levant
Part of the Control Engineering book series (CONTRENGIN)


In this chapter we further develop the ideas of Chap. 4 where second-order sliding modes were formulated. As we have seen, second-order sliding modes make the sliding variables vanish in finite time, when the relative degree of the variable equals two, and are able to solve the same problem by means of continuous control, if the relative degree is one. This helps to remove dangerous high-energy vibrations (the dangerous types of chattering). So-called higher-order sliding modes (HOSMs) solve these problems for arbitrary relative degrees. The realization of the scheme requires more information: usually one needs to calculate or measure a number of successive time derivatives of the sliding variables. However that problem is also solved within a similar framework. As a result, arbitrary-order exact robust differentiators are developed, having their own significance in terms of general observation theory. In particular, tracking problems are solved in finite time and with ideal accuracy, by means of smooth control, if the output relative degree is known. The accuracy remains high in the presence of small noises, switching inaccuracies and delays, etc.


Relative Degree Blood Glucose Concentration Differential Inclusion Mode Controller Insulin Pump 
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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yuri Shtessel
    • 1
  • Christopher Edwards
    • 2
  • Leonid Fridman
    • 3
  • Arie Levant
    • 4
  1. 1.Department of Electrical and Computer EngineeringUniversity of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.College of Engineering, Mathematics and Physical ScienceUniversity of ExeterExeterUK
  3. 3.Department of Control Division of Electrical EngineeringFaculty of Engineering National Autonomous University of MexicoFederal DistrictMexico
  4. 4.Department of Applied Mathematics School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael

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