# Basic Theory of Sliding Mode Control

Basic Theory of Sliding Mode Control By: Song Yoong Siang (PhD Student)

Sliding mode control is one of the available robust controller designs. The main advantages of sliding mode control are finite-time convergence and reduced order compensated dynamics. In this article, the basic theory of sliding mode control is discussed.

Strategy of sliding mode control is to drive the system state variables toward a specified surface called sliding surface and maintain the trajectory of the state variables on this surface. Once the state variables reach sliding surface, asymptotic convergence of the state variables to zero can be achieved.

There are two steps approach to design a conventional sliding mode controller. The first step is to define the sliding surface, s, where the system state error, x converge to zero at this surface. This surface should be invariant of the controlled dynamics. The common sliding surface used is shown in Equation 1, where λ is a positive constant and n is the system order. When the system state reaches sliding surface, s will equal to zero, then the system state error, x will converge to zero asymptotically. There are no effect of disturbance observed in Equation 1. (1)

The second step is to define the control law, u which can drive the system state to the sliding surface in finite time. Control law is designed to guarantee that the reachability condition described by Equation 2 is satisfied. There are two component inside the control law, u, which are corrective control, uc and equivalent control, ue . Corrective control is used to compensate the deviations from the sliding surface to reach the sliding surface whereas equivalent control is used to make the derivative of the sliding surface equal zero to stay on the sliding surface. The typical choice for control law is shown in Equation 3, where , , and k represent scalars yet to be designed. (2) (3)

In an ideal sliding motion, there are high frequency switching between two different control laws as the system state trajectory repeatedly cross the sliding surface. This infinite frequency switching will trap the system state trajectory on the sliding surface.

However, the control law that satisfices reachability condition are discontinuous across the sliding surface. The imperfection in the sign function implementation leads to finite amplitude and finite frequency “zigzag” motion in the sliding surface. This effect is called chattering. Chattering results in high wear of moving mechanical part and degrades performance of the system. Therefore, it is important to avoid chattering by providing continuous control signals but at the same time maintain the robustness of the control system.

Reference(s):

1. Y. Shtessel, C. Edwards, L. Fridman, A. Levent, “Sliding Mode Control and Observation”, Springer New York, 2013.

2. E. Gallestey, P. Al-Hokayem, “Nonlinear Systems and Control“, Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology, 2015

3. C. Edwards, S. K. Spurgeon, “Sliding Mode Control: Theory and Applications”, Taylor & Francis, 1998

4. M. S. Fadali, “Nonlinear Control”, Electrical & Biomedical Engineering Dept., University of Nevada, Reno, 2011