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High Order Sliding Mode, Relative Degree, Finite Time Convergence and Disturbance Rejection

High Order Sliding Mode, Relative Degree, Finite Time Convergence and Disturbance Rejection. CHATTERING !!!. Second Order Sliding Mode, Relative Degree, Finite Time Convergence and Disturbance Reject. R. R. and relative degree is equal to 1. The Problem:.

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High Order Sliding Mode, Relative Degree, Finite Time Convergence and Disturbance Rejection

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  1. High Order Sliding Mode, Relative Degree, Finite Time Convergence and Disturbance Rejection

  2. CHATTERING !!!

  3. Second Order Sliding Mode, Relative Degree, Finite Time Convergence and Disturbance Reject R R and relative degree is equal to 1.

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

  5. EXAMPLE R and continuous control For the system with It means that state trajectories belong to the surface s(x)=0 after a finite time interval. Sliding Mode

  6. 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 vas 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. Short Discussion 1 1-2 reaching phase 2-3 sliding mode of the 1st order Point 3 sliding mode of the 2nd order 3 2 s=0 Finite times of 1-2 and 2-3 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 3rd phase – reaching the originsliding mode of the 2nd order 4th phase – sliding mode of the 3rd 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 inputv and output sis still equal to1 and the conventional sliding mode can be enforced, since ds/dtis used.

  9. Super TWISTING ALGORITHM Control u is continuous, no , relative degree of the open loop system from v to sis 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. FINITE TIME CONVERGENCE (cont.) Convergence time:

  13. Examples of systems with no disturbances

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

  15. TWISTING ALGORITHM ) Beyond domain D with Lyapunov function decays at finite rate Trajectories can penetrate into D throughSI=0and leave it through SII =0 only

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

  17. Super-Twisting Algorithm may be positive (!!!) Again finite convergence time Upper estimate of the disturbance F<M/2

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

  19. Adaptive second order sliding mode • Objective: Chattering reduction • Method: Reducing the magnitude of the discontinuous control to THE minimal value preserving sliding mode under uncertainty conditions.

  20. a/k

  21. Adaptive super-twisting algorithm First, it was shown that 1. is necessary condition for convergence 2. For any there exists such thatfinite-time convergence takes place for Then Similarly . In sliding mode

  22. Remark Instead of Conclusion Challenge: to generalize twisting algorithm to get the third ordersliding 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

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