Controller Design

Consider a nonlinear system presented by a n-th order scalar differential equation with uncertainty:

We define the trajectory tracking error as:

where our goal is to drive .

We "Replace" the n-th order problem with an equivalent 1st order problem. Create an intermediate variable s, such that:

In finite time, if s remains bounded close to zero, the trajectory error is bounded as follows:

The Sliding Condition is defined as:

and outside the bounds of s, it may be defined as:

We determine the control law as follows:

which suffers from control chattering due to the switching function. Example implementation shown in SlidingMode_sign.mdl.

The chattering may be eliminated by using a saturation function instead as follows:

Example implementation shown in SlidingMode_sat.mdl.

Note: For previous versions of MATLAB (tested MATLAB 2018 and older), please use the .mdl files with 'prev' extension.