Nonlinear Dyn https://doi.org/10.1007/s11071-018-4521-6 ORIGINAL PAPER Continuous sliding mode controllers for multi-input multi-output systems Hancheol Cho · Thanapat Wanichanon · Firdaus E. Udwadia Received: 5 September 2016 / Accepted: 13 August 2018 © Springer Nature B.V. 2018 Abstract This paper develops new continuous sliding mode controllers for multi-input multi-output mechan- ical 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 fundamen- tal equation of constrained motion that causes the sys- tem 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 contin- H. Cho (B ) Water Power Technologies Program, Sandia National Laboratories, Albuquerque, NM 87123, USA e-mail: hancho@sandia.gov T. Wanichanon Department of Mechanical Engineering, Mahidol University, 25/25, Phutthamonthon, Nakorn Pathom 73170, Thailand e-mail: thanapat.wan@mahidol.edu F. E. Udwadia Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, University of Southern California, Los Angeles, CA 90089-1453, USA e-mail: feuusc@gmail.com uous 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. Keywords Sliding mode control · Chattering alleviation · Fundamental equation of constrained motion · Multi-input multi-output systems 1 Introduction The precision reference-tracking control problem in the presence of system uncertainties has historically attracted a number of researchers due to its great prac- tical importance in real life. Over the past few decades, there have been successful attempts to develop high- performance linear or nonlinear controllers that track given reference trajectories with high accuracy. Among them, nonlinear adaptive control [1], model predictive control [2], backstepping control [3], and sliding mode control [4, 5] are worthy of attention. Sliding mode con- trol (SMC) is especially widely used to cope with such uncertainties due to its simplicity and robustness. How- ever, the robustness and high accuracy of SMC are obtained at the expense of chattering which results in high-frequency oscillations. This chattering problem, which leads to high-speed switching about the so-called 123