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1
2
Feasibility of fixed-time stable control-1
x ∈ F(x) is fixed-time or practically fixed-time stable. Proposition. , ∀τ γ > R ∀ > there are , x x , || || x R ≥ , ( ) x F x ∈
- , such that
|| || || || x x τ ≥ γ
- .
in Sliding Mode Control Arie Levant School of Mathematical - - PowerPoint PPT Presentation
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
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3
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.
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4
τ-solutions (Euler approximations)
x ∈ F(x), x ∈ Rn – Filippov conditions
( ) ( ( ))
k k
x t F x t = ξ ∈
- (~ sampling)
1
[ , ]
k k
t t t + ∈ ,
1 k k
t t
+
< − ≤ τ
Over any time segment τ-solutions converge to Filippov solutions with
τ →
.
- Proposition. With any { }
k
t
,
k
t → ∞, τ-solutions of r-sliding
homogeneous controllers, linear asymptotically stable controllers, etc. converge into small vicinity of 0.
|| || || || / || || ( ) x R x x O ≥ ⇒ τ = τ
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5
Feasibility of fixed-time stable control-2
x ∈ F(x) is fixed-time or practically fixed-time stable.
Let
( )
enter
T R = sup of the times needed for the solutions to enter || || x R ≤
. ⇒ lim
( )
enter R
T R
→∞
∃ ≥ Usually lim ( )
enter R
T R
→∞
= (Lyap. FxTS functions, uniform
differentiators, homogeneity at infinity, etc.)
- Theorem. lim
( )
enter R
T R
→∞
= . For any τ > 0 for any large initial conditions there are τ-solutions which escape to infinity faster than any exponent.
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Simulation example
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Classical scalar fixed-time stable system
1/3 3
x u x x = = − −
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Fixed-time stabilization
Initial values 20 x = ,
3
10− τ = Accuracy:
5
| | 1.12 10 x
−
≤ ⋅ ,
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9
Initial value (0) 30 x = ,
3
10− τ = Accuracy:
5
| | 1.12 10 x
−
≤ ⋅
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10
Initial value (0) 40 x = ,
3
10− τ = . Overshoot! Accuracy:
5
| | 1.12 10 x
−
≤ ⋅
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Explosion with (0) 50 x = ,
3 4 5 6 7
10 ,10 ,10 ,10 ,10 .
− − − − −
τ = Results with
3
10− τ = : 2 steps
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Results with
3
10− τ = : 3 steps
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Results with
3
10− τ = : 4 steps
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14
Results with
3
10− τ = : 5 steps
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15
Results with
3
10− τ = : 6 steps
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16
3
10− τ = : OVERFLOW at t = 0.007
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Conclusions
- Computer realization of fixed-time stable systems over unbounded
- perational 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.