Nonlinear Dynamics

, Volume 94, Issue 4, pp 2727–2747 | Cite as

Continuous sliding mode controllers for multi-input multi-output systems

  • Hancheol ChoEmail author
  • Thanapat Wanichanon
  • Firdaus E. Udwadia
Original Paper


This paper develops new continuous sliding mode controllers for multi-input multi-output mechanical systems in the presence of unknown, but bounded uncertainties in the given forces and in the masses. Assuming the absence of the uncertainties, a reference control input is first calculated using the fundamental equation of constrained motion that causes the system trajectories to exactly track the reference while minimizing a weighted \(L^{2}\) norm of the control effort. Next, in the presence of realistic uncertainties in the given forces and in the masses, two continuous sliding mode controllers are derived according to whether the mass matrix is diagonal or not. In the diagonal case, each element of the control vector is independently designed, while in the nondiagonal case the control vector is handled as a whole because its elements are coupled to one another. The two controllers are continuous because no signum functions are used. It is also shown that various forms of control input are possible depending on the control requirements among which a simple proportional-integral-derivative-type controller is exemplified in this paper. Two numerical examples serve to demonstrate the accuracy and robustness of the control methodology suggested herein.


Sliding mode control Chattering alleviation Fundamental equation of constrained motion Multi-input multi-output systems 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Human and animal rights

The research carried out in this article did not involve any human participants or animals.

Informed consent

Consent to submit has been received explicitly from all co-authors. The research did not involve any human participants.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Water Power Technologies ProgramSandia National LaboratoriesAlbuquerqueUSA
  2. 2.Department of Mechanical EngineeringMahidol UniversityPhutthamonthonThailand
  3. 3.Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations ManagementUniversity of Southern CaliforniaLos AngelesUSA

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