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GUARANTED SET COMPUTATION Ellipsoidal approach

GUARANTED SET COMPUTATION Ellipsoidal approach. S. Lesecq GIPSA-Lab, Département Automatique INPG-UJF-CNRS Suzanne.lesecq@gipsa-lab.inpg.fr. Motivation Notations and definitions Outer bounding ellipsoid Algorithm(s) Parameterised family Simple example Factorisation.

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GUARANTED SET COMPUTATION Ellipsoidal approach

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  1. GUARANTED SET COMPUTATIONEllipsoidal approach S. Lesecq GIPSA-Lab, Département Automatique INPG-UJF-CNRS Suzanne.lesecq@gipsa-lab.inpg.fr

  2. Motivation Notations and definitions Outer bounding ellipsoid Algorithm(s) Parameterised family Simple example Factorisation Convergence conditions Common difficulties Choice of the “bound” Industrial context: input? Extra Inner bounding Uncertainty in the model References Outline

  3. Motivation - 1 • Discrete-time dynamic system • Observation ykmeasured output vector uk known input vector unknown: state, process perturbation, measurement noise • Identification (output linear in the parameters)

  4. Define two parallel hyperplanes S3 S1 2 2 Feasible set Compute a simpler set enclosing S S2 S dk 1 1 Motivation - 2 • Consider a SISO model dk : regressor,  : parameter vector Error bound ek

  5. 2 S S 1 Motivation - 3 • Possible simpler sets • Polytopes • Parallelotopes • Orthotopes • Ellipsoids… • Outer bounding ellipsoid Mininum size • Inner bounding ellipsoid Maximum size 2  Criterion to be optimised 1

  6. det trace det trace Motivation - 4 • Usual criteria • “Determinant criterion” (actually log(det)…) • “Trace criterion”

  7. Notations & definitions - 1 • Norm : Euclidian • Unit ball centred on the origin: • Bounded ellipsoid E(c,P) with non empty interior c: centre P = PT > 0 : shape and orientation

  8. 2 1 Notations & definitions - 2 • Particular cases • Strip = unbounded ellipsoid • Empty interior ellipsoid Centre c not unique, just satisfy: y = Cc 2 1

  9. Notations & definitions - 3 • Sum of K ellipsoids (prediction) • Intersection of K ellipsoids (correction)

  10. Motivation Notations and definitions Outer bounding ellipsoid Algorithm(s) Parameterised family Simple example Factorisation Examples : Identification Convergence conditions Common difficulties Choice of the “bound” Industrial context: input? Extra Inner bounding Uncertainty in the model References Outline

  11. 2 Sk Ek-1 1 Ek Weighted sum of Ek-1 and Sk Outer bounding ellipsoid - 1 • Recursive algorithm (Fogel and Huang, 1982) not normalised form (intersection): P = PT > 0

  12. Outer bounding ellipsoid - 2 • OBE algorithm Choice ofk, k ?

  13.  No update Outer bounding ellipsoid - 3 • “Basic OBE algorithm not optimal” (Belforte and Bona, 1985 and 1990) • Void intersection  must be detected  add intersection test (e.g. Pronzato and Walter, 1994) • Other classical situation

  14. l2 l1 Outer bounding ellipsoid - 4 • Usual criteria • In the litterature, two sets of programs (Favier and Arruda, 1996; Tran Dinh, 2005) • Set 1: minimize the geometrical size of Ek • Set 2 : insure the convergence of  2

  15. Fogel and Huang, 1982, (FH) Parametrised family (Durieu, et al., 1996) 2 families  Set Membership Set Approximation SMSA (Nayeri, et al., 1994) Same criterion (tr or det)  same ellipsoids Outer bounding ellipsoid - 5 • Set 1 (size)

  16. Dasgupta and Huang, 1987 (DH) If then Criterion: If then Lozano-Leal and Ortega, 1987 (LO) Criterion: If then Tan, et al., 1997 (TAN) Criterion : Outer bounding ellipsoid - 6 • Set 2

  17. Outer bounding – 7 (parameterised family) • Normalised problem (Durieu, et al., 2001) • MIMO models • State estimation (identification)  Sum, Intersection • Analytical results K ellipsoids • Optimisation problem addressed in details Empty interior Unbounded (strip)

  18. Outer bounding – 8 (parameterised family) • Sum of K ellipsoids (Durieu, et al., 2001) Problem: find • Theorem 4.1 The centre of the optimal ellipsoid E* for both problems is given by: Empty interior possible

  19. Outer bounding – 9 (parameterised family) Parameterized family  optimisation can be done • Theorem 4.2 • c*: independent of  • Solve for  the problem:  generally, suboptimal solution of:

  20. Outer bounding – 10 (parameterised family) • Necessary condition (Lemma 4.1) for E* to be optimal solution of • Theorem 4.4: Trace criterion  explicit solution • Theorem 4.5: recursive  nonrecursive approximating ellipsoid • Determinant criterion  no explicit solution

  21. Unbounded ellipsoid possible Outer bounding – 11 (parameterised family) • Intersection of K ellipsoids (Durieu, et al., 2001) Problem: find

  22. Outer bounding – 12 (parameterised family) • Theorem 5.1 • Proposition 5.1

  23. x2 H1 H2 S H3 x1 Outer bounding ellipsoid - 13 • Ellipsoid with parallel cuts algorithm (Goldfarb and Todd, 1982) • Sequential algorithm • Equivalent to (modified) OBE (Pronzato and Walter, 1994) • “Recursively optimal”  i.e. minimal volume ellipsoid containing E(ck-1,Pk-1)  Bk x2 Hk Ek Ek-1 x1

  24. uk yk Outer bounding ellipsoid - (example) • Simulated data ykmeasured output vector unknown: measurement noise (uniform distribution) SNR  50 dB • Identification

  25. 2 c 1  = * = 0.003 Outer bounding ellipsoid - (example) • Results: xtheoretic = [0.95, 0.05] c0 = 0, 02P = 106I Set 1 determinant Set 2

  26. Evolution of Det (k2P) LO = Lozano-Leal and Ortega, 1987 TAN = Tan, et al., 1997 DH = Dasgupta and Huang, 1987 FH = Fogel and Huang, 1982 Evolution of k2 1 4 2 3 3 2 4 k 1 k Outer bounding ellipsoid - (example)

  27. Outer bounding ellipsoid - (example) • Parameter a evolution Figure 2.4 : Evolution des paramètres estimés.

  28. Outer bounding ellipsoid - (example) • Parameter b evolution

  29. Outer bounding ellipsoid : summary • Two sets of algorithms (Favier and Arruda, 1996; Tran Dinh, 2005): • Minimise the geometrical size of E • Minimise 2 • Different formulations of the algorithm • Equivalence provable for some of them (OBE-EPC…) (Pronzato and Walter, 1994, Tran Dinh, 2005) • (Durieu, et al., 1996 and 2001): • Parameterised family • K ellipsoids • Convexity of criteria

  30. Motivation Notations and definitions Outer bounding ellipsoid Algorithm(s) Parameterised family Simple example Factorisation Examples : Identification Convergence conditions Common difficulties Choice of the “bound” Industrial context: input? Extra Inner bounding Uncertainty in the model References Outline

  31. M L LT = U 0 A H = Factorisation - 1 • Prerequisite • Orthogonal matrix: HT = H-1 • Numerically highly suitable • Orthogonal factorisation • Factorisation of a product of matrices

  32. Factorisation - 2 • Factorisation of a sum of matrices • Least Square problem solution • Recursive Least Square problem: also factorised…

  33. Factorisation - 3 • Parameterised family (Durieu, et al., 1996) hypothesis (not restrictive): y R Sum of 2 symmetric matrices  factorise! Intersection • Goldfarb and Todd, 1982 • LDL factorisation • Cholesky suggested

  34. LS: = XT M X Factorisation - 4 • Reformulation  optimisation problem (Lesecq and Barraud, 2002) Let then

  35.  = s2 > 0 ck, Mk: independent Factorisation - 5 • Factorised algorithm (Lesecq and Barraud, 2002) • Theoretical property   [0, 1[ : simpler demonstration

  36. = P Factorisation - 6 • Directly with P (general formulation) (Tran Dinh, 2005) ck, Mk: dependent

  37. Factorisation - summary • Absolutely necessary to ensure numerical stability • Academic example (Lesecq and Barraud, 2002) n = 8, M = hilb(8)=[hij = 1/(i+j-1)], c = ones (8,1), y = 1 temp = invhilb(9), d = temp(1:8,9)  = 0.001  not factorised = - 47.6 and factirised = 1.7 10-2 • Practical problem (identification) • Theoretical properties: easier demonstration • Parameterised family (Durieu, et al., 1996)  P and M algorithms (Lesecq and Barraud, 2002) • General formulation  P and M algorithms (Tran Dinh, 2005)

  38. Motivation Notations and definitions Outer bounding ellipsoid Algorithm(s) Parameterised family Simple example Factorisation Examples:Identification Convergence conditions Common difficulties Choice of the “bound” Industrial context: input? Extra Inner bounding Uncertainty in the model References Outline

  39. Examples - 1 • Industrial Data: • 1st example: industrial Looks like 1st order, 2 parameters aim: model identification  diagnosis • 2nd example: LIRMM robot 14 parameters 14 000 regressors! aim: model identification, large problem

  40. Examples - 2 • 1st example: recorded on a process (valve) output input

  41. No ellipsoid update Examples - 3 • 1st example • Data re-used several times • Determinant criterion •  = 0.002 • Measurement and regressor known Determinant criterion 1st circulation of data

  42. 2 c 1 Examples - 4 Ellipsoid updating • 1st example: parameters

  43. Examples - 5 • 1st example det(10) det(1) Gain properly identified Trace(10) Trace(1)

  44. Examples - 7 • 2nd example: LIRMM parallel robot N = 3500

  45. Examples – 8 • 2nd example: Recorded data (for instance) Sequential algorithm

  46. criterion Examples - 9 No empty intersection  = 6 Nm • 2nd example: parameters (60 circulations)

  47. Parallel robot, det, family of ellipsoids Factorisation interest No factorisation Thanks to N. Ramdani Work done by N. Ramdani P. Poignet LIRMM Factorisation

  48. Examples - 10 • 2nd example: Model reconsidered  split in several models  = 6 Nm 1st model

  49. Examples - 11 Center, obtained with  = 6 Nm and 60 circulations • 2nd example: Adaptation of the bound Far from hypothesis! • Analysis of data • Reject “outliers” • “heavy tail”

  50. Examples - 12 60 circulations • 2nd example: Adaptation of the bound:  “heavy tail”

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