Eulerian state estimation

(1)

Maze Eulerian filter

Eulerian state estimation

T. Le Mézo, L. Jaulin, B. Zerr

Swim-Smart 2017, Manchester

(2)

Eulerian filter

Eulerian state estimation

(3)

˙

x(t) = f(x(t))

Leaves: Lagrangian view of the wind; Flags: an Eulerian view [3]

(4)

Eulerian filter

Eulerian state estimation can be formalized as:

(i) x(t) = ˙ f(x(t)) (evolution)

(ii) x(t _i ) ∈ X i ⊂ R ⁿ (event)

(iii) ∀(i, j ) ∈ J, t i ≤ t j (precedence)

Bracket the set X ⊂ R ⁿ of all feasible x(t).

(5)

Invariant sets

(6)

Eulerian filter

Denote by ϕ the flow map of our system, i.e., with x ₀ = x(0), the

system reaches ϕ(t ,x ₀ ) at time t.

(7)

A set A is positive invariant if

x ∈ A,t ≥ 0 = ⇒ ϕ(t, x) ∈ A.

The set of all positive invariant sets is a lattice, i.e., the union and the intersection are closed.

Thus, the notion of largest positive invariant set contained in X can

be defined.

(8)

Eulerian filter

The largest positive invariant set included in X is:

Inv ⁺ (f, X) = {x ₀ | ∀t ≥ 0, ϕ(t, x ₀ ) ∈ X } .

(9)

Mazes allow us to compute an inner and an outer approximation for

Inv ⁺ (f, X ).

(10)

Eulerian filter

As an illustration, consider the Van der Pol system:

x ˙ ₁ = x ₂

˙

x 2 = 1 − x ₁ ²

· x 2 − x 1

(11)

6 4 2 0 2 4 6

6

4

2

0

2

4

6

(12)

Eulerian filter

X

x

1

x

₂

Largest positive invariant set Inv ⁺ (f,X)

(13)

Largest negative invariant set.

Inv ⁻ (f,X) = {x ₀ | ∀t ≤ 0,ϕ (t, x ₀ ) ∈ X } . We have

Inv ⁻ (f, X ) = Inv ⁺ (−f, X ).

(14)

Eulerian filter

X

x

1

x

₂

Largest negative invariant set Inv ⁻ (f,X)

(15)

Largest invariant set

Inv(f,X) = {x ₀ | ∀t ∈ R,ϕ (t,x ₀ ) ∈ X } .

We have

Inv(f, X ) = Inv ⁺ (−f, X ) ∩ Inv ⁺ (f, X ).

Thus Inv(f, X ) can be defined in terms of largest positive invariant

sets.

(16)

Eulerian filter

Forward reach set

Forw (f, X) = {x | ∃t ≥ 0, ∃x ₀ ∈ X, ϕ(t, x ₀ ) = x} . We have

Forw (f,X) = Inv ⁺ (−f,X) .

(17)

X

x

1

x

₂

Forw (f, X ) for X = [0.4, 1.0] × [1.4,1.8]

(18)

Eulerian filter

Backward reach set.

Back(f,X) = {x ₀ | ∃t ≥ 0,ϕ(t,x ₀ ) ∈ X } . Since

Back(f, X) = Inv ⁺ (f, X) .

(19)

X

x

1

x

₂

Back(f, X ) for X = [0.4, 1.0] × [1.4,1.8].

(20)

Eulerian filter

Maze

(21)

An interval is a domain which encloses a real number.

A polygon is a domain which encloses a vector of R ⁿ .

A maze is a domain which encloses a path [2][1].

(22)

Eulerian filter

A maze is a set of paths.

(23)

Mazes can be made more accurate:

(24)

Eulerian filter

Here, a maze L is composed of A paving P

A polygon for each box of P

Doors between adjacent boxes

(25)

The set of mazes forms a lattice with respect to ⊂.

L a ⊂ L b means :

the boxes of L a are subboxes of the boxes of L b . The polygons of L a are included in those of L b

The doors of L a are thinner than those of L b .

(26)

Eulerian filter

The left maze contains less paths than the right maze.

Note that yellow polygons are convex.

(27)

Inner approximation

(28)

Eulerian filter

Target contractor. If a box [x] of P is outside X (it is outside

Inv ⁺ ( X )) then remove [x] and close all doors entering in [x].

(29)

Flow contractor. For each box [x] of P , we contract the polygon

using the constraint x ˙ = f(x).

(30)

Eulerian filter

Propagation

(31)

(32)

Eulerian filter

[ a ] [ b ] [ c ]

[ d ]

[ e ]

[ f ]

Yellow area: X

(33)

The red parts have been deleted

(34)

Eulerian filter

The yellow area is contracted

(35)

At each step, the yellow area encloses Inv ⁺ ( X )

(36)

Eulerian filter

At each step, the red area is outside Inv ⁺ ( X )

(37)

The yellow area encloses Inv ⁺ ( X )

(38)

Eulerian filter

(39)

Inflation propagation

(40)

Eulerian filter [a] [b]

[c]

[d] [e]

[f]

An interpretation can be given only when the fixed point is reached.

The yellow area is an inner approximation of Inv ⁺ ( X )

(41)

Eulerian filter

(42)

Eulerian filter

Define ` sets X 0 , X 1 , . . . , X ` of the state space. Define Z ^forw _k the set of all state vectors x(t) inside X k that have visited

X 0 , X 1 , . . . ,X k−1 . We have

Z ^forw k+1 = Forw Z ^forw k

∩ X k+1

with Z ^forw 0 = X 0 .

(43)

The trajectories (b),(c) are consistent with the sets X k−1 , X k , X k+1

(44)

Eulerian filter

Set Z ^forw _k of all x(t) in X k that have already visited

X 0 , X 1 , . . . ,X k−1

(45)

Forw Z ^forw _k

corresponds to all states x(t) that have visited

X 0 , X 1 , . . . , X k

(46)

Eulerian filter

Set Z ^forw _k+1 of all states x(t) in X k +1 that have already visited

X 0 , X 1 , . . . , X k

(47)

Eulerian smoother

(48)

Eulerian filter

Define the set Z ^back _k of all states x(t) inside X k that have visited X 0 , X 1 , . . . , X k−1 in the past and will visit X k+1 , . . . , X ` in the future. We have

Z ^back _k = Back Z ^back k+1

∩ Z ^forw _k

with Z ^back _` = Z ^forw _` . The will be called the Eulerian smoother.

(49)

Set Z ^back _k+1 of all states x(t) inside Z ^forw _k+1 that will visit X k +2 , . . . , X `

(50)

Eulerian filter

Set Z ^back _k of all states x(t) inside Z ^forw _k that will visit X k +1 , . . . , X `

(51)

The set of trajectories that started inside X 0 and visited the sets X 1 , X 2 , . . . , X `−1 sequentially, and that ended in X ` can thus be enclosed by

Forw Z ^back 0

∩ Back

Z ^back `

.

(52)

Eulerian filter

Set Forw Z ^back ₀

∩ Back Z ^back _`

enclosing the trajectory consistent

with the past and future visits

(53)

Example. Take the Van der Pol system with

X 0 = [a] = [0, 0.6] × [0.8,1.8], X 1 = [b] = [0.7,1.5] × [−0.2,0.2] and

X 2 = [c] = [0.2, 0.6] × [−2.2,−1.5].

(54)

Eulerian filter

[a]

x

1

x

₂

[b]

[c]

Feasible states associated to the Eulerian state estimation problem

(55)

An application of Eulerian state estimation moving taking

advantage of ocean currents.

(56)

Eulerian filter

Visiting the three red boxes using a buoy that follows the currents

is an Eulerian state estimation problem

(57)

Eulerian state estimation

Eulerian state estimation

T. Le Mézo, L. Jaulin, B. Zerr

Swim-Smart 2017, Manchester

Eulerian state estimation

˙

x(t) = f(x(t))

Leaves: Lagrangian view of the wind; Flags: an Eulerian view [3]

Eulerian state estimation can be formalized as:

(i) x(t) = ˙ f(x(t)) (evolution)

(ii) x(t i ) ∈ X i ⊂ R n (event)

(iii) ∀(i, j ) ∈ J, t i ≤ t j (precedence)

Bracket the set X ⊂ R n of all feasible x(t).

Invariant sets

Denote by ϕ the flow map of our system, i.e., with x 0 = x(0), the

system reaches ϕ(t ,x 0 ) at time t.

A set A is positive invariant if

x ∈ A,t ≥ 0 = ⇒ ϕ(t, x) ∈ A.

The set of all positive invariant sets is a lattice, i.e., the union and the intersection are closed.

Thus, the notion of largest positive invariant set contained in X can

be defined.

The largest positive invariant set included in X is:

Inv + (f, X) = {x 0 | ∀t ≥ 0, ϕ(t, x 0 ) ∈ X } .

Mazes allow us to compute an inner and an outer approximation for

Inv + (f, X ).

As an illustration, consider the Van der Pol system:

x ˙ 1 = x 2

˙

x 2 = 1 − x 1 2

· x 2 − x 1

6 4 2 0 2 4 6

6

4

2

0

2

4

6

X

x

x

Largest positive invariant set Inv + (f,X)

Largest negative invariant set.

Inv − (f,X) = {x 0 | ∀t ≤ 0,ϕ (t, x 0 ) ∈ X } . We have

Inv − (f, X ) = Inv + (−f, X ).

X

x

x

Largest negative invariant set Inv − (f,X)

Largest invariant set

Inv(f,X) = {x 0 | ∀t ∈ R,ϕ (t,x 0 ) ∈ X } .

We have

Inv(f, X ) = Inv + (−f, X ) ∩ Inv + (f, X ).

Thus Inv(f, X ) can be defined in terms of largest positive invariant

sets.

Forward reach set

Forw (f, X) = {x | ∃t ≥ 0, ∃x 0 ∈ X, ϕ(t, x 0 ) = x} . We have

Forw (f,X) = Inv + (−f,X) .

X

x

x

Forw (f, X ) for X = [0.4, 1.0] × [1.4,1.8]

Backward reach set.

Back(f,X) = {x 0 | ∃t ≥ 0,ϕ(t,x 0 ) ∈ X } . Since

Back(f, X) = Inv + (f, X) .

X

x

x

Back(f, X ) for X = [0.4, 1.0] × [1.4,1.8].

Maze

An interval is a domain which encloses a real number.

A polygon is a domain which encloses a vector of R n .

A maze is a domain which encloses a path [2][1].

A maze is a set of paths.

Mazes can be made more accurate:

Here, a maze L is composed of A paving P

A polygon for each box of P

Doors between adjacent boxes

The set of mazes forms a lattice with respect to ⊂.

L a ⊂ L b means :

(ii) x(t _i ) ∈ X i ⊂ R ⁿ (event)

Bracket the set X ⊂ R ⁿ of all feasible x(t).

Denote by ϕ the flow map of our system, i.e., with x ₀ = x(0), the

system reaches ϕ(t ,x ₀ ) at time t.

Inv ⁺ (f, X) = {x ₀ | ∀t ≥ 0, ϕ(t, x ₀ ) ∈ X } .

Inv ⁺ (f, X ).

x ˙ ₁ = x ₂

x 2 = 1 − x ₁ ²

Largest positive invariant set Inv ⁺ (f,X)

Inv ⁻ (f,X) = {x ₀ | ∀t ≤ 0,ϕ (t, x ₀ ) ∈ X } . We have

Inv ⁻ (f, X ) = Inv ⁺ (−f, X ).

Largest negative invariant set Inv ⁻ (f,X)

Inv(f,X) = {x ₀ | ∀t ∈ R,ϕ (t,x ₀ ) ∈ X } .

Inv(f, X ) = Inv ⁺ (−f, X ) ∩ Inv ⁺ (f, X ).

Forw (f, X) = {x | ∃t ≥ 0, ∃x ₀ ∈ X, ϕ(t, x ₀ ) = x} . We have

Forw (f,X) = Inv ⁺ (−f,X) .

Back(f,X) = {x ₀ | ∃t ≥ 0,ϕ(t,x ₀ ) ∈ X } . Since

Back(f, X) = Inv ⁺ (f, X) .

A polygon is a domain which encloses a vector of R ⁿ .

Inv ⁺ ( X )) then remove [x] and close all doors entering in [x].

At each step, the yellow area encloses Inv ⁺ ( X )

At each step, the red area is outside Inv ⁺ ( X )

The yellow area encloses Inv ⁺ ( X )

The yellow area is an inner approximation of Inv ⁺ ( X )

Define ` sets X 0 , X 1 , . . . , X ` of the state space. Define Z ^forw _k the set of all state vectors x(t) inside X k that have visited

Z ^forw k+1 = Forw Z ^forw k

with Z ^forw 0 = X 0 .

Set Z ^forw _k of all x(t) in X k that have already visited

Forw Z ^forw _k

Set Z ^forw _k+1 of all states x(t) in X k +1 that have already visited

Define the set Z ^back _k of all states x(t) inside X k that have visited X 0 , X 1 , . . . , X k−1 in the past and will visit X k+1 , . . . , X ` in the future. We have

Z ^back _k = Back Z ^back k+1

∩ Z ^forw _k

with Z ^back _` = Z ^forw _` . The will be called the Eulerian smoother.

Set Z ^back _k+1 of all states x(t) inside Z ^forw _k+1 that will visit X k +2 , . . . , X `

Set Z ^back _k of all states x(t) inside Z ^forw _k that will visit X k +1 , . . . , X `

Forw Z ^back 0

Z ^back `

Set Forw Z ^back ₀

∩ Back Z ^back _`