Separator Algebra for State Estimation

Separator Algebra for State Estimation

SMART 2015

Manchester, September 16-17, 2015 Luc Jaulin

ENSTA Bretagne, LabSTICC.

http://www.ensta-bretagne.fr/jaulin/

1 Context

2 Problem

(i) x˙ (t) = f (x (t)), t ∈ R (ii) g (x (t_k)) ∈ Y _{(k) , k ∈} N

Find an inner-outer approximation of the set X_{(t) of fea-} sible states

We define by flow map ϕ_t1,t2:

x(t₁) = x₁ and x˙ (t) = f (x(t)) ⇒ x₂ = ϕ_t

1,t₂ (x₁) .

The set of all causal feasible states at time t is X_{(t) =}

t_k≤t

ϕ_t

k,t ◦ g⁻¹ (Y_(k)).

3 Contractors

C([x_{]) ⊂ [}x_] (contractance) [x_{] ⊂ [}y_{] ⇒ C([}x_{]) ⊂ C([}y_]) (monotonicity)

Inclusion

C₁ ⊂ C₂ ⇔ ∀[x] ∈ IRⁿ_{, C1([}x]) ⊂ C₂([x]).

A set S _{is consistent with C (we write} S _{∼ C) if} C([x]) ∩ S _{= [}x] ∩ S_.

C is minimal if

S _{∼ C}

S _{∼ C1} ^{⇒ C ⊂ C1.}

The negation ¬C of C is defined by

¬C ([x]) = {x _∈ [x] | x ∈ C/ ([x_])}. It is not a box in general.

4 Separators

A separator S is pair of contractors Sⁱⁿ,S^out such that Sⁱⁿ([x]) ∪ S^out([x]) = [x] (complementarity).

A set S _{is consistent with S (we write} S _{∼ S), if} S _{∼ S}^out _and S _{∼ S}ⁱⁿ_.

The remainder of S is

∂S([x]) = Sⁱⁿ([x]) ∩ S^out([x]).

∂S is a contractor, not a separator.

We have

¬Sⁱⁿ([x]) ∪ ¬S^out([x]) ∪ ∂S([x]) = [x] . Moreover, they do not overlap.

¬Sⁱⁿ([x]), ¬S^out([x]) and ∂S([x])

Inclusion

S₁ ⊂ S₂ ⇔ S₁ⁱⁿ ⊂ S₂ⁱⁿ and S₁^out ⊂ S₂^out. Here ⊂ means more accurate.

S is minimal if

S₁ ⊂ S ⇒ S₁ = S. i.e., if Sⁱⁿ and S^out are both minimal.

5 Paver

We want to compute X⁻_,X⁺ _{such that} X⁻ _⊂ X _⊂ X⁺_.

Algorithm Paver(in: [x_{],S; out:} X⁻_, X⁺₎ 1 L := {[x_]};

2 Pull [x] from L;

3 [xⁱⁿ],[x^out] = S([x]);

4 Store [x_]\[xⁱⁿ_{] into} X⁻ and also into X⁺_; 5 [x] = [xⁱⁿ] ∩ [x^out];

6 If w([x]) < ε, then store [x] in X⁺_,

7 Else bisect [x] and push into L the two childs 8 If L = ∅, go to 2.

6 Algebra

Contractor algebra does not allow decreasing operations, i.e.,

∀i, C_i ⊂ C_i^′ ⇒ E (C₁,C₂, . . .) ⊂ E C₁^′ ,C₂^′ , . . . .

The complementary C of a contractor C, the restriction C₁\C₂, etc. cannot be defined.

Separators extend the operations allowed for contractors to non monotonic expressions.

The complement of S = Sⁱⁿ,S^out is S = S^out,Sⁱⁿ .

If S_i = S_iⁱⁿ,S_i^out , i ≥ 1, are separators, we define

S₁ ∩ S₂ = S₁ⁱⁿ ∪ S₂ⁱⁿ,S₁^out ∩ S₂^out (intersection) S₁ ∪ S₂ = S₁ⁱⁿ ∩ S₂ⁱⁿ,S₁^out ∪ S₂^out (union)

{q}

S_i =







{m−q−1}

S_iⁱⁿ,

{q}

S_i^out





 (relaxed intersection)

S₁\S₂ = S₁ ∩ S₂. (difference)

Theorem. If S_i are subsets of Rⁿ_{, we have} (i) S_{1 ∩} S_{2 ∼ S1 ∩ S2} (ii) S_{1 ∪} S_{2 ∼ S1 ∪ S2}

(iii) S_{i ∼ Si}

(iv) S_{i ∼ S}^k

i , k ≥ 0 (v)

{q}

S_{i ∼}

{q}

S_i (vi) S_1\S_{2 ∼ S1\S2.}

7 Transformation of separators

A transformation is an invertible function f such as an in- clusion function is known for both f _and f⁻¹_.

The set of transformation from Rⁿ _to Rⁿ is a group with respect to the composition ◦. Symmetries, translations, homotheties, rotations, . . . are linear transformations.

The flow ϕ_t1,t2 is a transformation. Indeed:

ϕ⁻¹_t

2,t₁ = ϕ_t

1,t₂.

Theorem. Consider a set X and a transformation f. If SX

is a separator for X then a separator SY for Y ₌ f ₍X_{) is} [f] ◦ SX ◦ f⁻¹ ∩ Id.

Example. Consider the constraint

2 0

0 1

cosα sinα

−sinα cosα

y₁ − 1

y₂ − 2 ∈ [1,3]. Define

f ₍x_{) =} ^{cosα −sinα} sinα cosα

12 0 0 1

x₁

x₂ + 1

2 . The constraint can be rewritten as

y = f (x) , and x ∈ [1,3].

8 State estimation

If SY_(k) are separators for Y(k), then a separator for the set X _{(t) is}

S_X(t) =

t_k≤t

ϕ_t

k,t ◦ g⁻¹ S_Y(k) .

Example







˙

x (t) = v (t) cosθ(t) v (t) sinθ (t)

x (t_k) ∈ y (t_k) + [−0.3,0.3], t_k = 0.1 · k, k ∈ N

t = 0

t = 0.1

t = 0.2

t = 0.3

t = 0.4

t = 0.5

t = 0.6

t = 0.7

t = 0.8

t = 0.9

t = 1.0

t = 1.1

t = 1.2

t = 1.3

t = 1.4