Characterizing Sliding Surfaces of Cyber-Physical Systems
L. Jaulin
August 31, 2020, 11:30-12:00 SNR 2020, Vienna (virtually) https://youtu.be/ntiru1sxnxc
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
Easy-boat
https://youtu.be/bNqiwW4p6WE
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
d ˙ = sin u cos (ψ − u) + cos π 5 > 0
Controller in: (d, ψ, q) ; out:u if d 2 − 1 > 0 then q := sign (d)
if cos (ψ + atan (d)) + cos π 4 ≤ 0 ∨ d 2 ≤ 1∧ cos ψ + cos π 4 ≤ 0 then u := π + ψ − q π 4 .
else u := −atan (d) .
Simulation
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
Simulation in the (t, d)-space
Simulation in the ( R t
cos u, d)-space
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
Formalism
Given Q − , Q + disjoint and closed, two smooth functions f a , f b . We define [3]
S (A) :
˙
x = f (x, q) =
f a (x, q) if x ∈ A f b (x, q) if x ∈ B = A
q = −1 as soon as x ∈ Q −
= +1 as soon as x ∈ Q +
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
Sliding surface
The sliding surface S ( A ) for S ( A ) is the subset of ∂ A such that the state can slide inside for a non degenerated interval of time.
Since Q − , Q + are disjoint, we have
S ( A ) = S q=−1 ( A ) ∪ S q=+1 ( A )
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
We can thus assume that q is fixed and find the sliding surfaces for
S ( A ) : x ˙ =
f a (x) if x ∈ A
f b (x) if x ∈ B = A
The Lie derivative of c : R n → R with respect to f is
L c f (x) = dc
dx (x) · f (x) .
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
If A :c (x) ≤ 0, then
S ( A ) = ∂ A ∩ {x | L c a (x) ≥ 0 ∧ L c b (x) ≤ 0} .
Sliding set S ( A ) (red) for A = {x|c (x) ≤ 0}
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
Proposition. If we have two closed sets A 1 and A 2 . We have [3]
(i) S (A 1 ∩ A 2 ) = (S (A 1 ) ∩ A 2 ) ∪ (S (A 2 ) ∩ A 1 ) (ii) S ( A 1 ∪ A 2 ) = S ( A 1 ) ∩ A 2
∪ S ( A 2 ) ∩ A 1
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
S ( A 1 ∪ ( A 2 ∩ A 3 ))
With easy-boat
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
Three problems:
Safety : the sailboat never goes against the wind. [1]
Capture : The boat will be captured by its corridor [2]
Characterize of the sliding surface.
We take x = (d, ψ) ,
Function f (x, q)
if cos (x 2 + atanx 1 ) + cos π 4 ≤ 0 ∨ x 1 2 − 1 ≤ 0∧ cos x 2 + cos π 4 ≤ 0 then u := π + x 2 − q π 4
else u := −atanx 1 . Return (sin u, 0)
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
x = (d, ψ) f a (x, q) =
sin π + x 2 − q π 4 0
f b (x) =
sin (−atanx 1 ) 0
A 1 =
x | cos (x 2 + atanx 1 ) + cos π 4 ≤ 0 A 2 =
x | x 2 1 − 1 ≤ 0 A 3 =
x | cos x 2 + cos π 4 ≤ 0 A = A 1 ∪ ( A 2 ∩ A 3 )
Q − = {x | x 1 + 1 ≤ 0}
Q + = {x | 1 − x 1 ≤ 0}
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
L c a
1(x, q) = dc dx
1(x) · f a (x, q) = − sin(
qπ
4
−x
2)·sin(atan(x
1)+x
2) x
21+1
L c b
1(x) = dc dx
1(x) · f b (x, q) = sin(atanx √
1+x
2)·x
1x
21+1
3L c a
2(x, q) = dc dx
2(x) · f a (x) = 2 sin( qπ 4 − x 2 ) · x 1 L c b
2(x) = dc dx
2(x) · f b (x, q) = √ −2x
21x
21+1
L c a
3(x, q) = dc dx
3(x) · f a (x, q) = 0 L c b
3(x) = dc dx
3(x) · f b (x, q) = 0
S ( A 1 ) = ∂ A 1 ∩ L 1 b ∩
L 1 a (1) ∩ Q − ∪ L 1 a (−1) ∩ Q +
S ( A ) = ∂ A ∩ L 2 ∩
L 2 (1) ∩ Q − ∪ L 2 (−1) ∩ Q +
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
Easy-boat Formalism With easy-boat
L. Jaulin and F. Le Bars.
A simple controller for line following of sailboats.
In 5th International Robotic Sailing Conference, pages 107–119, Cardiff, Wales, England, 2012. Springer.
L. Jaulin and F. Le Bars.
An Interval Approach for Stability Analysis; Application to Sailboat Robotics.
IEEE Transaction on Robotics, 27(5), 2012.
L. Jaulin and F. Le Bars.
Characterizing sliding surfaces of cyber-physical systems.
Acta Cybernetica (submitted), 2019.
L. Jaulin Characterizing Sliding Surfaces of Cyber-Physical Systems
