Separator Algebra for State Estimation

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1 Separator Algebra for State Estimation Luc Jaulin To cite this version: Luc Jaulin. Separator Algebra for State Estimation. SMART 2015, Sep 2015, Manchester, United Kingdom. 2015,. HAL Id: hal Submitted on 1 Dec 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Separator Algebra for State Estimation Luc Jaulin, ENSTA-Bretagne, LabSTICC. 1 Introduction Consider the following state estimation problem[jau15] (i) ẋ(t)=f(x(t)),t R (ii) g(t k ) Y(k),k N (1) Ourobjective is to find an inner and an er approximation of the setx(t) of all state vectors thatareconsistentwith(1)attimet.ifwedefinebyflowmapϕ t1,t 2 asfollows: x(t1 )=x 1 andẋ(t)=f(x(t)) x 2 =ϕ t1,t 2 (x 1 ). (2) Thesetofall causal feasible states attimetisdefinedby X(t)= ϕ tk,t g 1 (Y(k)). (3) t k t In this paper, we show how it is possible to find both an inner and an er approximations for X(t). Some existing methods are able to find an er approximation[kjwm99][grma13], but, tomyknowledge,noneofthemisabletogetaninnerapproximation.themainideaistocopya classical contractor approach [CJ09] for state estimation, but to use separators [JD14] instead of contractors. 2 Separators Inthissection,wepresentseparatorsandshowhowtheycanbeusedbyapaverinordertobracket thesolutionsets.an interval ofrisaclosedconnectedsetofr.abox[x]ofr n isthecartesian productofnintervals.thesetofallboxesofr n isdenotedbyir n.acontractor Cisanoperator IR n IR n suchthatc([x]) [x]and[x] [y] C([x]) C([y]).AsetSis consistent withthe contractorc (wewillwrites C)ifforall[x],wehaveC([x]) S=[x] S.Aseparator S ispair ofcontractors S in,s suchthat,forall[x] IR n,wehaves in ([x]) S ([x])=[x].asetsis consistent withtheseparators (wewrites S),ifS S ands S in.wheres={x x/ S}. Usingaseparatorinsideapaver wecaneasilytoclassifypartofthesearchspacethatareinside orsideasolutionsetsassociatedwiths. Thealgebraforseparatorsisadirectextensionofcontractoralgebra[CJ09].IfS i = Si in,s i,i {1, 2} are separators, we define S 1 S 2 = S in 1 S in 2,S S 1 S 2 = S in 1 S in 2,S (intersection) (union) 1 ) (inverse) 1 S2 1 S2 f 1 (S 1 )= f 1 (S in 1 ),f 1 (S (4)

3 IfS i aresetsofr n,wehave[jsd14] (i) S 1 S 2 S 1 S 2 (ii) S 1 S 2 S 1 S 2 (iii) f 1 (S 1 ) f 1 (S 1 ). (5) Interval analysis[moo66][kk96] combined with contractors[cj09] has been shown to be able to give an er approximation of set. For the inner subpaving, the De Morgan rules make it possible toexpressthecomplementarysetxofx.then,basiccontractortechniquescanbeusedtogetan inner characterizationx. Now, the task is not so easy and the role of separators is to make it automatic. 3 Transformation of separators A transformation is an invertible function f such as an analytical expression if known for both f andf 1. Theset of transformationfromr n tor n isagroupwithrespect tothecomposition. Symmetries, translations, homotheties, rotations,... are linear transformations. Theorem.ConsiderasetXandatransformationf.Denoteby[f]and f 1 twoinclusionfunctions forf andf 1.IfS X isaseparatorforxthenaseparators Y fory=f(x)is [y] [f] S in X f 1 ([y]) [y], [f] S X f 1 ([y]) [y] (6) or equivalently f(x) [f] SX in f 1 Id, [f] SX f 1 Id (7) where Id is the identity contractor. Remark. The separator defined by (6) corresponds to what we call the transformation of a separatorbyf andwewrites Y =f(s X ).Asaconsequence,thankstothetheorem,wecanaddto (5) the property (iv) f(x) f(s X ). which will be used later for our state estimation problem. Proof.TheseparatorS Y isequivalenttoy=f(x)if S Y ([y]) Y=[y] Y SY in([y]) Y =[y] Y. (8) SinceSY ([y]) [y]ands Y ([y]) [y],itsufficestoprovethat (i) S Y ([y]) [y] Y (ii) SY in([y]) [y] Y. (9) Letusfirstprove(i).Wehave [y] Y=f f 1 ([y]) f 1 (Y) f isbijective =f f 1 ([y]) X X=f 1 (Y) f f 1 ([y]) X f 1 isaninclusionfunctionforf 1 f(s X f 1 ([y]) ) SX isacontractorforx [f] SX f 1 ([y]) [f] isaninclusionfunctionforf (10)

4 Thus[y] Y [f] S X f 1 ([y]) [y] =S Y ([y]).letusnowprove(ii).wehave [y] Y=f f 1 ([y]) f 1 Y f isbijective =f f 1 ([y]) X X=f 1 Y f f 1 ([y]) X f 1 isaninclusionfunctionforf 1 f(s X in f 1 ([y]) ) SX in isacontractorforx [f] SX in f 1 ([y]) [f] isaninclusionfunctionforf (11) Thus[y] Y [f] SX f 1 ([y]) [y] Y=SY in([y])whichterminatestheproof. Example. Consider the constraint 2 0 cosα sinα y1 1 [1,3]. (12) 0 1 sinα cosα y 2 2 If we apply an efficient forward-backward contractor in a paver, we get the contractions illustrated bythepavingoffigure1,left.now,ifwetake x1 2 0 cosα sinα y1 1 = =f x sinα cosα y (y) (13) orequivalently weget y1 = y 2 cosα sinα sinα cosα x1 1 + =f(x), (14) 0 1 x 2 2 y=f(x), and x [1,3]. (15) An optimal separators X can be built forxand the separatortransformprovidesusaseparator S Y for Y. As illustrated by Figure 1, right, the resulting separator S Y is more efficient than the classical one based on forward-backward contractors. Note that in case we are not able to have an innerapproximationforf 1,theproblemoffindinganinnerapproximationoftheimageofaset f(x) becomes much more difficult. See, e.g.,[vjvs05][gj10]. 4 State estimation IfS X(0) isaseparatorforx(0) and ifs Y(k) areseparators fory(k), thenaseparatorfor theset X(t)definedby(3)is S X(t) = ϕ g 1 tk,t S Y(k). (16) t k t Inthisformula,g 1 S Y(k) isaseparator.duetothefactthatϕtk,t isbijectiveandthatweare able to find an inclusion function for ϕ tk,t andϕ 1 t [RN11], the separator ϕ k,t t k,t g 1 S Y(k) is clearly defined using the separator transform. To illustrate the method, let us consider a robot described by v(t)cosθ(t) ẋ(t)= (evolution) v(t)sinθ(t) (17) x(t k ) y(t k )+[ 0.3,0.3],t k =0.1 k, k N (observation)

5 Fig. 1. Left. Contractions obtained using a classical forward-backward propagation; Right. Contractions obtained usingtheseparatortransform.theframecorrespondstothebox[ 6,6] 2. wherev(t)andθ(t)aremeasuredwithanaccuracyof±0.03.theobservationequationisdueto thefactthattherobotmeasuresevery0.1secitsdistancetotheoriginwithanaccuracyof±0.3. The actual(but unknown) trajectory for the robot is x(t)= 2+3cost 2sint. (18) Fort 0.2 k, k=0,...,7,thesetsx(t)obtainedbyourobserverarerepresentedonfigure2. BlackboxesareinsideX(t),greyboxesaresideandthewhiteboxescovertheboundary.For t=0,x(t)isaringwhichbecomesasmallsetfort=1.4oncetherobothasmovedsufficiently. Thefactthatthewhiteareacoveringtheboundarybecomesthickismainlyduetothestateerrors inside the evolution equation. References [CJ09] G. C L. J. Contractor Programming. Artificial Intelligence 173, (2009). [GJ10] A. G L. J. Inner Approximation of the Range of Vector-Valued Functions. Reliable Computing pages 1 23(2010). [GRMA13] A. G, B. R, L. M, F. A. An Introduction to Box Particle Filtering. IEEE Signal Processing Magazine 30(1), (2013). [Jau15] L. J. Automation for Robotics. ISTE editions(2015). [JD14] L. J B. D. Separators: a new interval tool to bracket solution sets; application to path planning. Engineering Applications of Artificial Intelligence 33, (2014). [JSD14] L. J, A. S, B. D. Inner and er approximations of probabilistic sets. In ICVRAM 2014 (2014). [KJWM99] M. K"", L. J, E. W, D. M. Guaranteed mobile robot tracking using interval analysis. In Proceedings of the MISC 99 Workshop on Applications of Interval Analysis to Systems and Control, pages , Girona, Spain(1999).

6 Fig.2. Inner and er approximations of the setof all feasible state vectorsx(t),fort 0,0.2,...,1.4. Theframeboxesare[ 6,6] 2.

7 [KK96] R. B. K" V. K, editors. Applications of Interval Computations. Kluwer, Dordrecht, the Netherlands(1996). [Moo66] R. E. M. Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ(1966). [RN11] N. R' N. N(. Computing Reachable Sets for Uncertain Nonlinear Hybrid Systems using Interval Constraint Propagation Techniques. Nonlinear Analysis: Hybrid Systems 5(2), (2011). [VJVS05] P. H V, L. J, J. V, M. A. S. Inneranderapproximationof the polar diagram of a sailboat. In workshop on Interval Analysis and Constraint Propagation for Applications(IntCP 05), Barcelona, Spain (2005).