# Analysis of sliding modes in the frequency domain

The sliding mode (SM) control principle is a tool to design robust controllers for nonlinear dynamic plants operating under uncertain conditions. It has become one of the most popular research areas in automatic control and has a number of industrial applications. The main advantage of the sliding mode principle is low sensitivity of the sliding mode control system to plant parameter variations and disturbances. The sliding mode principle involves a discontinuous control, which can easily be implemented as a conventional “on-off” control. This makes the controller a relatively simple device.

Usually, SM control is considered a high-speed switching feedback control because of the variable structure control type. The purpose of the switching control law is to drive the plant state trajectory onto a predetermined surface in the state space and then to maintain the state trajectory on that surface. This surface is called the switching surface or the sliding surface. The control function is designed in such a way that when the plant state trajectory is “above” the surface, the control drives the plant state “down” to the surface, and when the trajectory is “below” the surface, the control drives the plant state “up” to the surface. To realize this principle, the control is designed as a certain discontinuous function of the plant state. As a result, the plant trajectory is always driven toward the sliding surface by the control, so that, once intercepted, the switching control maintains the plant state trajectory on the surface. Ideally speaking, the trajectory cannot deviate from the switching surface and “slides” along it. As a result, the closed-loop system dynamics are determined by the plant state trajectory restricted to the sliding surface. The whole design of a SM system breaks down into two steps. The first step is to properly choose or design a sliding surface so that the closed-loop system has desired dynamics. The second step is to design a switching control that can drive the plant state trajectory onto the sliding surface and maintain it on that surface.

## Keywords

Slide Mode Control Fractal Dynamic Slide Mode Disturbance Attenuation Switching Surface
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