2. CHATTERING !!!

3. R R and relative degree is equal to 1.

4. can the similar effect be obtained with control as a continuous state function? if control is a non-Lipschitzian function

5. 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

6. Second Order Sliding Mode and Relative Degree vu ∫ 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”.

7. 1 2 3 1-2 reaching phase 2-3 sliding mode of the 1 st order Point 3 sliding mode of the 2nd order s=0 System of the 3 rd order Finite times of 1-2 and 2-3 1 st phase - reaching surface S=0 2 nd phase - reaching curve s=0 in S=0 sliding mode of the 1st order 3 rd phase – reaching the origin sliding mode of the 2nd order 4 th phase – sliding mode of the 3 rd order in the origin Finite times of the first 3 phases

8. 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 1 and the conventional sliding mode can be enforced, since ds/dt is used.

9. 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.

10. ASYMPTOTIC STABILITY AND ZERO DISTURBANCES

11. FINITE TIME CONVERGENCE Homogeneity property

12. Convergence time: FINITE TIME CONVERGENCE (cont.)

13. Examples of systems with no disturbances

14. HOMOGENEITY PROPERTY for the systems with zero disturbances and constant M i. Motion Equations: In what follows A.Levant, A. Polyakov and A.Poznyak, Yu. Orlov - twisting algorithms with time varying disturbances

18. TWISTING ALGORITHM ) Beyond domain D with Lyapunov function decays at finite rate Trajectories can penetrate into D through S I =0 and leave it through S II = 0 only

19. TWISTING ALGORITHM finite time convergence The average rate of decaying of Lyapunov function is finite and negative, which means Finite Convergence Time.

20. Super-Twisting Algorithm Upper estimate of the disturbance F<M/2

21. DIFFERENTIATORS The first-order system + - f(t) x u z Low pass filter The second-order system + - - + f(t) s x v u Second-order sliding mode u is continuous, low-pass filter is not needed.

22. Objective: Chattering reduction Method: Reducing the magnitude of the discontinuous control to THE minimal value preserving sliding mode under uncertainty conditions.

24. a/k

25. Similarly In sliding mode 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

26. Challenge: to generalize twisting algorithm to get the third order sliding mode adding two integrators with input similar to that for the 2 nd order: Unfortunately the 3 rd order sliding mode without sliding modes of lower order can not be implemented, indeed time derivative of sign-varying Lyapunov function