in Sliding Mode Control Arie Levant School of Mathematical - PowerPoint PPT Presentation

IEEE CDC 2013 Excerpt from the presentation Fixed and Finite Time Stability in Sliding Mode Control Arie Levant School of Mathematical Sciences, Tel-Aviv University, Israel Homepage: http://www.tau.ac.il/~levant/ 1

Feasibility of fixed-time stable control-1 � ∈ F ( x ) is fixed-time or practically fixed-time stable. x ∀τ γ > ∀ > Proposition. , 0 R 0 ≥ ∈ there are x x , � , || x || R , x F x ( ) , such that � 0 0 0 0 0 τ ≥ γ || x || || x || � . 0 0 Take very small τ , and very large R and γ … 2

One Euler step can be incomparably larger than the current (arbitrarily large!) distance from the origin. Any small delay (sampling step) is also not allowed. 3

τ -solutions (Euler approximations) � ∈ F ( x ), x ∈ R n – Filippov conditions x = ξ ∈ x t ( ) F x t ( ( )) � (~ sampling) k k ∈ < − ≤ τ t [ , t t + ] , 0 t t + k k 1 k 1 k Over any time segment τ -solutions converge to Filippov solutions with τ → 0 . t → ∞ , τ -solutions of r -sliding Proposition. With any { } t , k k homogeneous controllers, linear asymptotically stable controllers, etc. converge into small vicinity of 0. ≥ ⇒ τ = τ || x || R || x || / || x || O ( ) � 4

Feasibility of fixed-time stable control-2 � ∈ F ( x ) is fixed-time or practically fixed-time stable. x T ( ) R = sup of the times needed for the solutions to enter Let enter ≤ ∃ ≥ || x || R . ⇒ lim T ( ) R 0 enter →∞ R = (Lyap. FxTS functions, uniform Usually lim T ( ) R 0 enter →∞ R differentiators, homogeneity at infinity, etc.) = . For any τ > 0 for any large Theorem. lim T ( ) R 0 enter →∞ R initial conditions there are τ -solutions which escape to infinity faster than any exponent. 5

Simulation example 6

Classical scalar fixed-time stable system = = − − 1/3 3 x u x x � 7

Fixed-time stabilization 10 − x = τ = 3 Initial values 20 , − ≤ ⋅ 5 Accuracy: | x | 1.12 10 , 8

10 − = τ = 3 Initial value (0) x 30 , − ≤ ⋅ 5 Accuracy: | x | 1.12 10 9

10 − = τ = 3 Initial value (0) x 40 , . Overshoot! − ≤ ⋅ 5 Accuracy: | x | 1.12 10 10

− − − − − = τ = 3 4 5 6 7 Explosion with (0) x 50 , 10 ,10 ,10 ,10 ,10 . 10 − τ = 3 Results with : 2 steps 11

10 − τ = 3 Results with : 3 steps 12

10 − τ = 3 Results with : 4 steps 13

10 − τ = 3 Results with : 5 steps 14

10 − τ = 3 Results with : 6 steps 15

10 − τ = 3 : OVERFLOW at t = 0.007 16

Conclusions • Computer realization of fixed-time stable systems over unbounded operational regions is not possible. • Such systems are realizable for any bounded region of initial conditions, provided the sampling/integration step is taken small enough. Sampling step should be very carefully chosen. 17