CHATTERING !!!
R
and relative degree is equal to 1.
R
can the similar effect be obtained
with control as a continuous state
function?
if control is a non-Lipschitzian function
R
For the system with
and continuous control
s   s sign ( s)
It means that state trajectories belong to the surface s(x)=0 after a finite
time interval.
Sliding Mode
Second Order Sliding Mode
and Relative Degree
v
∫
u
Control is a continuous function as an output of integrator
with a discontinuous state function as an input
Then sliding mode can be enforced with v as a discontinuous function of and
For example if sliding mode exists on line
then s tends to zero asymptotically and sliding mode exists in the origin of two
dimensional subspace
It is hardly reasonable to call this conventional sliding mode as the second order sliding
mode. For slightly modified switching line
,, s>0 the state reaches the
origin after a finite time interval. The finiteness of reaching time served for several authors
as the argument to label this motion in the point
“second order sliding mode”.
x2
x1  x2
x 2  u
1
u   Msign( s), s  x2 
x1
3
2
s=0
x1 sign( x1 )
1-2 reaching phase
2-3 sliding mode of the 1st order
Point 3 sliding mode of the 2nd order
Finite times of 1-2 and 2-3
x1  x 2
x 2  x3
System of the 3rd order
1st phase - reaching surface S=0
2nd phase - reaching curve s=0 in S=0
sliding mode of the 1st order
x 3  u
u   Msign( S ),
S  s 
s  x2 
s sign( s),
x1 sign( x1 )
3rd phase – reaching the origin
sliding mode of the 2nd order
4th phase – sliding mode of the 3rd order
the origin
Finite times of the first 3 phases
in
TWISTING ALGORITHM
Again control is a continuous function
as an out put of integrator
Of course relative degree between discontinuous input v and output s is still equal to
and the conventional sliding mode can be enforced, since ds/dt is used.
1
Super TWISTING ALGORITHM
Control
u
is continuous, no
, relative degree of the open loop system
from v to s is equal to 2!
Finite time convergence
and
Bounded disturbance
can be rejected
However it works for the systems for special
continuous part with non-lipschizian function.
ASYMPTOTIC STABILITY
AND ZERO DISTURBANCES
FINITE TIME CONVERGENCE
Homogeneity property
FINITE TIME CONVERGENCE (cont.)
Convergence time:
Examples of systems with no disturbances
HOMOGENEITY PROPERTY
for the systems with zero disturbances and
constant Mi. Motion Equations:
A. Levant, A. Polyakov and A.Poznyak, Yu. Orlov twisting algorithms with time varying disturbances
In what follows
TWISTING ALGORITHM
)
Beyond domain D with
Lyapunov function decays at finite rate
Trajectories can penetrate into D through SI=0
and leave it through
SII =0
only
TWISTING ALGORITHM
finite time convergence
The average rate of decaying of
Lyapunov function is finite and
negative, which means
Finite Convergence Time.
Super-Twisting Algorithm
Upper estimate of the disturbance
F<M/2
DIFFERENTIATORS
The first-order
system
+
f(t)
z
x
u
-
Low pass
filter
The second-order
system
v+
u
-
x -
+
s
f(t)
Second-order sliding
mode
u is continuous, low-pass filter is not needed.
• Objective: Chattering
reduction
• Method: Reducing the
magnitude of the
discontinuous control to
THE minimal value preserving
sliding mode under
uncertainty conditions.
0  1    1
[sign( x)]eq    a/k
 (t )  0  k
First, it was shown that
1.  (t )  0 <k is necessary condition for convergence
2.
For any 0 <k
there exists 0  0
such that finite-time convergence takes place for
Then
Similarly
.
In sliding mode
  0
k  k (t ) sign( (t )),
 (t )  sign( x(t )) eq   , 0  1    1.
sign( x(t )) eq 
 (t )
, 
k
 (t ) is close to k (t ).
 (t )
k
,
Challenge:
to generalize twisting algorithm to
get the third order sliding mode adding two
integrators with input similar to that for the 2nd order:
Unfortunately the 3rd order sliding mode without sliding modes of lower order can not be
implemented, indeed time derivative of sign-varying Lyapunov function