We have implemented the sliding mode algorithm as part
of a hybrid system simulation engine using a variable step,
forward Euler numerical integration scheme. When a mode switch
between fields 
and 
is detected (see Fig. 19),
a binary search determines the point of switching to
within a pre-specified accuracy. After the mode switch is executed
and the state vector is transferred between modes to 
,
it is checked whether the new mode, , persists for at least
one maximum time step, 
. If not, sliding mode simulation
is activated.
The sliding mode simulation algorithm (Algorithm 1)
first computes
based on 
with as
initial point. Discontinuous changes may take place, and the resulting
is inferred.
Next, 
is computed to
infer 
and 
, which may differ from
when jumps in the state vector occur between
and . Next,
is computed based on and
, and jumps between and are taken into
account when 
is inferred.
Now,
is taken as the inital point of a vector
. A
binary search is performed along this vector
given a pre-specified number of steps
to determine 
on the sliding surface. After 
and its
corresponding field are determined, it is checked whether
chattering persists and the process continues.
Figure 19: The sliding mode numerical simulation algorithm.
