Kleene algebra to compute with invariant sets of dynamical

Transcription

1 to compute with invariant sets of dynamical systems Lab-STICC, ENSTA-Bretagne MRIS, ENSTA-Paris, March 15, 2018

2 [5][4] Motivation

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5 Consider the system Denote by ϕ(t, x) the ow map. S : ẋ(t) = f(x(t))

6 Positive invariant set Motivation

7 A set A is positive invariant [1] if x A,t 0 = ϕ(t, x) A. Or equivalently ϕ([0, ],A) A. The set of all positive invariant sets is a complete lattice.

8 Motivation

9 Lattice Motivation

10 A lattice (L, ) is a partially ordered set, closed under least upper and greatest lower bounds [2]. A machine lattice (L M, ) of L is complete sublattice of (L, ) which is nite. Moreover both L and L M have the same top and bottom.

11 Machine lattice

12 Motivation We consider a set F of automorphism f :P (X) P (X) such that f (X) = X f (A B) = f (A) f (B) Note that f is inclusion monotonic.

13 Kleene(+,, ) Kleene(,, ) Addition a + b f g Product a b f g Associativity a + (b + c) = (a + b) + c f (g h) = (f g) h a(bc) = (ab)c f (g h) = (f g) h Commutativity a + b = b + a f g = g f Distributivity a(b + c) = (ab) + (ac) f (g h) = (f g) (f h) (b + c)a = (ba) + (ca) (g h) f = (g f ) (h f ) zero a + 0 = a f = f One a1 = 1a = a f Id = Id f = f Annihilation a0 = 0a = 0 f = Idempotence a + a = a f f = f Partial order a b a + b = b f g f g = g Kleene star a = 1 + a + aa + aaa +... f = Id f f 2 f 3...

14 Reducers Motivation To an automorphism f F, we can associate the reducer R = Id f.

15 We have A B R (A) R (B) R (A) A monotonicity degrowth

16 Theorem. We have (Id f ) = f Proof. Since f is such that f (A B)) = f (A) f (B), we have and (Id f ) 2 (A) = (Id f )(A f (A)) = = A f (A) f (A f (A)) = A f (A) f 2 (A) (Id f ) (A) = A f (A) f 2 (A) f 3 (A) = f (A).

17 We dene Fix(f ) = {A f (A) = A} = Fix(Id f ) From the KnasterTarski theorem, it is a complete sublattice of L.

18 (a) : Red nodes : A, (b):a f (A), (c):a f (A) f 2 (A), (d):f (A).

19 Goal. Compute with closure sets f i,i {1,2,...}, i.e., compute with expressions such as f (A) (g (A) h (A)) We want to factorize the xed point operators as much as possible.

20 Factorization properties [3] f f = f (f ) = f (f g ) = (f g) f (f g ) = (f g)

21 Dealing state equations Motivation

22 Dene f (A) = ϕ([ 1,0],A) f (A) = ϕ([0,1],a) We have f (R n ) = R n f (R n ) = R n f (A B) = f (A) f (B) f (A B) = f (A) f (B)

23 The sets f (A), f (A) correspond to the largest positive and negative invariant sets included in A.

24 The largest invariant set included in A is ( f f ) (A)

25 Illustration Motivation

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28 Kleene intervals Motivation

29 Given an automorphism f, we want to compute f (a) where a is in (L, ) (for instance (R n, )). Machine sublattice L M of L (maze for instance).

30 Interval automorphism Motivation

31 An interval automorphism [f,f + ] containing f is a pair of two machine automorphism f,f + with such that Lemma. We have a L M f (a) f (a) f + (a). Fix ( (f ) ) L M Fix(f ) Fix ( (f + ) )

32 Fixed points Fix ( (f ) ) in magenta, Fix ( (f + ) ) in blue

33 Theorem. If a [a,a + ], where a,a + both belong to L M, then (i) f (a) [ (f ) (a ),(f + ) (a + ) ] (ii) f (f ) (a ) = (f ) (a ) (iii) f (a) (Id f + ) i (a + ), i 0

34 Algorithm Motivation

35 Computation of f (a),a [a]

36 Motivation

37 A L is a complemented distributive lattice. Every element a has a unique complement a, satisfying a a = and a a =.

38 We have a b b a a b = b a a b = b a (De Morgan's laws)

39 Interval arithmetic Motivation

40 [a,a + ] = [a +,a ] f ([a,a + ]) = [f (a ),f (a + )] [a,a + ] [b,b + ] = [a b,a + b + ] [a,a + ] [b,b + ] = [a b,a + b + ]

41 Monotonic case Motivation Compute x = f 1 (a) (f 2 (b) f 3 (c)). We have x [ ( ) f 1 (a ) (f 2 (b ) f 3 (c )) Id f i 1 (a + ) (( ) Id f2 i (b + ) ( ) Id f3 i (c + ) ) ].

42 Non monotonic case Motivation

43 We want to compute x = f 1 (a) f 2 arithmetic rules, we get [ x f 1 ( ) ) a f 2 (b + ( ) b. Applying interval ),f 1 (a + ( ) ] f 2 b, i.e., we need to go up to the xed point for both bounds.

44 Forward reach set Motivation

45 Forward reach set of A dened by Forw(f,A) = {x t 0, x 0 A,ϕ(t, x 0 ) = x}. We get Forw(f,A) = f ( A ).

46 Monotonic path planning

47 The set of paths that start in the set A and reach B is given by Path(A,B) = Forw(A) Back(B) = f ( A ) f ( B ).

48 A to B problem Motivation

49 Consider two sets A,B such that B A. We want to compute the set X = Capt A B = {x t 0,ϕ(t, x) B and t 1 [0,t],ϕ(t 1, x) A}.

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51 Non monotonic path planning

52 Find the set X of all paths that start in A, avoid B and reach C.

53 Franco Blanchini and Stefano Miani. Set-Theoretic Methods in Control. Springer Science & Business Media, October B. A. Davey and H. A. Priestley. Introduction to Lattices and Order. Cambridge University Press, (ISBN ), Dexter Kozen. A completeness theorem for kleene algebras and the algebra of regular events. In Giles Kahn, editor, Proceedings of the Sixth Annual IEEE Symp. on Logic in Computer Science, LICS 1991, pages IEEE Computer Society Press, July T. Le Mézo, L. Jaulin, and B. Zerr. Inner approximation of a capture basin of a dynamical system.

54 In Abstracts of the 9th Summer Workshop on Interval Methods. Lyon, France, June 19-22, T. Le Mézo, L. Jaulin, and B. Zerr. An interval approach to compute invariant sets. IEEE Transaction on Automatic Control, 62: , 2017.