CHATTERING !!!

RR

and relative degree is equal to 1.

can the similar effect be obtained with control as a continuous state function?

if control is a non-Lipschitzian function

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

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”.

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

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.

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.

ASYMPTOTIC STABILITY

AND ZERO DISTURBANCES

FINITE TIME CONVERGENCEHomogeneity property

Convergence time:

FINITE TIME CONVERGENCE (cont.)

Examples of systems with no disturbances

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

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 ALGORITHMfinite 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

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.

• Objective: Chattering reduction• Method: Reducing the magnitude

of the discontinuous control to THE minimal value preserving sliding mode under uncertainty conditions.

0 1 1 [ ( )]eqsign x a/k

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

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