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CHATTERING !!!
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RR
and relative degree is equal to 1.
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can the similar effect be obtained with control as a continuous state function?
if control is a non-Lipschitzian function
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For the system with and continuous control
It means that state trajectories belong to the surface s(x)=0 after a finite time interval.
Sliding Mode
R
)( ssignss
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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”.
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1x
2x1
2
3
)( ),( 1122
21
xsignxxssMsignuux
xx
1-2 reaching phase2-3 sliding mode of the 1st orderPoint 3 sliding mode of the 2nd orders=0
System of the 3rd order
)(
),(
),(
112
3
32
21
xsignxxs
ssignssS
SMsignu
ux
xx
xx
Finite times of 1-2 and 2-3
1st phase - reaching surface S=0
2nd phase - reaching curve s=0 in S=0 sliding mode of the 1st order
3rd phase – reaching the origin sliding mode of the 2nd order
4th phase – sliding mode of the 3rd order in the origin
Finite times of the first 3 phases
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TWISTING ALGORITHMAgain 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 1and the conventional sliding mode can be enforced, since ds/dt is used.
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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 rejectedHowever it works for the systems for special continuous part with non-lipschizian function.
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ASYMPTOTIC STABILITY
AND ZERO DISTURBANCES
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FINITE TIME CONVERGENCEHomogeneity property
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Convergence time:
FINITE TIME CONVERGENCE (cont.)
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Examples of systems with no disturbances
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HOMOGENEITY PROPERTY
for the systems with zero disturbances and constant Mi. Motion Equations:
In what follows
A. Levant, A. Polyakov and A.Poznyak, Yu. Orlov - twisting algorithms with time varying disturbances
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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
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TWISTING ALGORITHMfinite time convergence
The average rate of decaying of
Lyapunov function is finite and negative, which means
Finite Convergence Time.
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Super-Twisting Algorithm
Upper estimate of the disturbance
F<M/2
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DIFFERENTIATORSThe first-order system
+
-f(t)x
u
z
Low pass filter
The second-order system
+-
- + f(t)s
xv u
Second-order sliding mode u is continuous, low-pass filter is not needed.
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• Objective: Chattering reduction• Method: Reducing the magnitude
of the discontinuous control to THE minimal value preserving sliding mode under uncertainty conditions.
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0 1 1 [ ( )]eqsign x a/k
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0( )t k
( ) ( ( )),
( ) ( ( )) , 0 1 1.eq
k k t sign t
t sign x t
( ) ( )( ( )) , ,
( ) is close to ( ).
eq
t tsign x t
k kt k t
Similarly
In sliding mode
0 ( ) < t k
0 < k 0 0
0
First, it was shown that 1. is necessary condition for convergence
there exists
such that finite-time convergence takes place for
.
2. For any
Then
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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