Using bases to simplify asymptotic expansions

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2 Using bases to simplify asymptotic expansions Matthias Kawski 1 Amit Sanyal 2 1 Department of Mathematics and Statistics Arizona State University 2 Department of Mechanical and Aerospace Engineering Arizona State University LHMNLC 6 : 3 rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control Nagoya 26 1 This work was partially supported by the National Science Foundation through the grant DMS This presentation was prepared with LaTeX using the beamer package.

3 Overview: Preliminary / progress report General problem Noncommuting flows / vector fields Asymptotic series expansions Linear dependencies / lack of well-definedness Specialization to simple mechanical systems Series expansions using the symmetric product (( Identities and simplifications )) Lie algebras that are never free (( Specialization to specific mechanical systems simulation, comparison of computational accuracy and efficiency )) Successful and ongoing simplification efforts Exponential product expansion of Chen-series: Zinbiel algebra and Hall bases Continuous CBH formula / logarithm of Chen-series Outlook and future plans

4 Why asymptotic expansions? Rarely have closed-form solution formulas for time-varying / nonlinear systems of DEs

5 Why asymptotic expansions? Rarely have closed-form solution formulas for time-varying / nonlinear systems of DEs Commonly: Right hand side of DE ẋ = F + G naturally splits into summands ẋ = F and ẋ = G that BOTH can integrated easily/exactly but need to account for infinity of interaction terms

6 Why asymptotic expansions? Rarely have closed-form solution formulas for time-varying / nonlinear systems of DEs Commonly: Right hand side of DE ẋ = F + G naturally splits into summands ẋ = F and ẋ = G that BOTH can integrated easily/exactly but need to account for infinity of interaction terms Commonly: Right hand side of DE ẋ = u i (t)f i (x) naturally splits into time and state dependent factors naturally want to separate off-line/exact integration of fixed geometric features, and just-in-time integration of controls/disturbances u i (t)

7 Why asymptotic expansions? Rarely have closed-form solution formulas for time-varying / nonlinear systems of DEs Commonly: Right hand side of DE ẋ = F + G naturally splits into summands ẋ = F and ẋ = G that BOTH can integrated easily/exactly but need to account for infinity of interaction terms Commonly: Right hand side of DE ẋ = u i (t)f i (x) naturally splits into time and state dependent factors naturally want to separate off-line/exact integration of fixed geometric features, and just-in-time integration of controls/disturbances u i (t) Perturbation analysis, averaging, vibrational control

8 Magnus (1954) expansion and trees Given Y = A(t)Y seek Ω(t) such that Y (t) = e Ω(t) Y () e.g. Iserles & Nørsett (1999), Butcher trees

9 Magnus (1954) expansion and trees Given Y = A(t)Y seek Ω(t) such that Y (t) = e Ω(t) Y () Ω(t) = t A(τ) dτ 1 2 t τ1 t τ1 τ t τ1 τ [A(τ 2 ), A(τ 1 )] dτ 2 dτ 1 [A(τ 3 ), [A(τ 2 ), A(τ 1 )]] dτ 3 dτ 2 dτ 1 [[A(τ 3 ), A(τ 2 )], A(τ 1 )] dτ 3 dτ 2 dτ 1 e.g. Iserles & Nørsett (1999), Butcher trees

10 Magnus (1954) expansion and trees Given Y = A(t)Y seek Ω(t) such that Y (t) = e Ω(t) Y () Ω(t) = t A(τ) dτ 1 2 t τ1 t τ1 τ t τ1 τ [A(τ 2 ), A(τ 1 )] dτ 2 dτ 1 [A(τ 3 ), [A(τ 2 ), A(τ 1 )]] dτ 3 dτ 2 dτ 1 [[A(τ 3 ), A(τ 2 )], A(τ 1 )] dτ 3 dτ 2 dτ 1 e.g. Iserles & Nørsett (1999), Butcher trees

11 Magnus (1954) expansion and trees Given Y = A(t)Y seek Ω(t) such that Y (t) = e Ω(t) Y () Ω(t) = t A(τ) dτ 1 2 t τ1 t τ1 τ t τ1 τ [A(τ 2 ), A(τ 1 )] dτ 2 dτ 1 [A(τ 3 ), [A(τ 2 ), A(τ 1 )]] dτ 3 dτ 2 dτ 1 [[A(τ 3 ), A(τ 2 )], A(τ 1 )] dτ 3 dτ 2 dτ 1 Effective encoding using rooted trees e.g. Iserles & Nørsett (1999), Butcher trees

12 Volterra series and chronological calculus Agrachev and Gamkrelidze (1979), Sarychev... Functional analysis: operators on C (M), i.e. f (p) p(f ) consistent and effective notation allows for exponential representation of flows of time-varying vector fields basic product of time varying vector fields X τ, Y τ (X Y ) τ = τ [X t, d dt Y t] dt Different algebra: left chronological identity ((X Y ) Z ) ((Y X) Z ) = (X (Y Z )) (Y (X Z )) linear dependent terms in Volterra-like series expansion using basis for free chronological algebra to collect terms?

13 Series expansions for mechanical systems Lewis and Murray (1995), Bullo (1999), Burdick and Vela,... Controlled system defined by affine connection, in coordinates q i + Γ i jk (q) q j q k = X i (t, q)= u l (t)y i l (q)

14 Series expansions for mechanical systems Lewis and Murray (1995), Bullo (1999), Burdick and Vela,... Controlled system defined by affine connection, in coordinates q i + Γ i jk (q) q j q k = X i (t, q)= u l (t)y i l (q) affine connection usually from Riemannian metric (energy) ( ) Γ i jk = 1 Mlj 2 Mlk + M q i li M ij q j q l

15 Series expansions for mechanical systems Lewis and Murray (1995), Bullo (1999), Burdick and Vela,... Controlled system defined by affine connection, in coordinates q i + Γ i jk (q) q j q k = X i (t, q)= u l (t)y i l (q) affine connection usually from Riemannian metric (energy) ( ) Γ i jk = 1 Mlj 2 Mlk + M q i li M ij q j q l to rewrite as first order system on TQ, vertically lift vector fields X on a manifold Q to vector fields X lift on TQ. Geometrically: X lift (v q ) = d dt (υ q + tx(q)) t= In coordinates, if X(q) = X i (q) q i then X lift (υ q ) = X i (q) υ i

16 Symmetric products and series expansions Lewis and Murray (1995), Bullo (1999), Burdick and Vela,... use Lie brackets of lifted vector fields together with geodesic spray Z = υ i Γ i q i jk υj υ k υ i to define symmetric bracket of vector fields X and Y on Q X : Y lift = [X lift, [Z, Y lift ]]

17 Symmetric products and series expansions Lewis and Murray (1995), Bullo (1999), Burdick and Vela,... use Lie brackets of lifted vector fields together with geodesic spray Z = υ i Γ i q i jk υj υ k υ i to define symmetric bracket of vector fields X and Y on Q notation Y (q, T ) = T X : Y lift = [X lift, [Z, Y lift ]] Y (q, t)dt series expansion via iterated integrals and symm. products typical terms Y : Y : Y : Y and Y : Y : Y : Y

18 Symmetric products and series expansions Lewis and Murray (1995), Bullo (1999), Burdick and Vela,... use Lie brackets of lifted vector fields together with geodesic spray Z = υ i Γ i q i jk υj υ k υ i to define symmetric bracket of vector fields X and Y on Q notation Y (q, T ) = T X : Y lift = [X lift, [Z, Y lift ]] Y (q, t)dt series expansion via iterated integrals and symm. products typical terms Y : Y : Y : Y and Y : Y : Y : Y Question: linear independence of terms in series?

19 Identities in the symmetric algebra? If L(X 1,... X m, Z ) is free, is Sym Z (X 1,... X m ) ever free?

20 Identities in the symmetric algebra? If L(X 1,... X m, Z ) is free, is Sym Z (X 1,... X m ) ever free? For vector fields f i on M n can Sym Z (f 1,... f m ) ever be free? If not, what identities are always satisfied? Bases? Use these to further collapse the series.

21 Identities in the symmetric algebra? If L(X 1,... X m, Z ) is free, is Sym Z (X 1,... X m ) ever free? For vector fields f i on M n can Sym Z (f 1,... f m ) ever be free? If not, what identities are always satisfied? Bases? Use these to further collapse the series. When is it a (well-studied) Jordan algebra? i.e. commutative w/ Jordan identity ((xx)xy)) = (x((xx)y))

22 Identities in the symmetric algebra? If L(X 1,... X m, Z ) is free, is Sym Z (X 1,... X m ) ever free? For vector fields f i on M n can Sym Z (f 1,... f m ) ever be free? If not, what identities are always satisfied? Bases? Use these to further collapse the series. When is it a (well-studied) Jordan algebra? i.e. commutative w/ Jordan identity ((xx)xy)) = (x((xx)y)) Specialize to Z quadratic in velocities... (c.f. homogeneity, Murray, Lewis, Bullo, Vela, Burdick,... )

23 Identities in the symmetric algebra? If L(X 1,... X m, Z ) is free, is Sym Z (X 1,... X m ) ever free? For vector fields f i on M n can Sym Z (f 1,... f m ) ever be free? If not, what identities are always satisfied? Bases? Use these to further collapse the series. When is it a (well-studied) Jordan algebra? i.e. commutative w/ Jordan identity ((xx)xy)) = (x((xx)y)) Specialize to Z quadratic in velocities... (c.f. homogeneity, Murray, Lewis, Bullo, Vela, Burdick,... ) Specialize to distinguished curvatures i.e. assuming additional identities on Γ i jk and q l Γ i jk

24 Search for cubic identities in the symmetric algebra An early false lead a cubic operad? V : X : Y : W + Y : W : V : X + X : W : V : Y = = V : X : Y : W + V : Y : X : W + X : Y : V : W

25 Search for cubic identities in the symmetric algebra An early false lead a cubic operad? V : X : Y : W + Y : W : V : X + X : W : V : Y = = V : X : Y : W + V : Y : X : W + X : Y : V : W Another sample, triple brackets of coordinates vector fields < dx α, x i : x j : x k : + (Γ α is + Γα si )( Γs kl x j + (Γ m kl + Γm lk ) ( Γ α mj x i x l >= (Γ α is +Γα si )(Γm kl +Γm lk )(Γs mj +Γs jm ) + Γs lk x j ) + ) + Γα jm x i ( Γ m kl x i + ( 2 Γ α kl x i x j ) + Γm lk (Γ α x i mj + Γα jm ) + 2 Γ α lk x i x j )

26 Search for cubic identities in the symmetric algebra An early false lead a cubic operad? V : X : Y : W + Y : W : V : X + X : W : V : Y = = V : X : Y : W + V : Y : X : W + X : Y : V : W Another sample, triple brackets of coordinates vector fields < dx α, x i : x j : x k : + (Γ α is + Γα si )( Γs kl x j + (Γ m kl + Γm lk ) ( Γ α mj x i x l >= (Γ α is +Γα si )(Γm kl +Γm lk )(Γs mj +Γs jm ) + Γs lk x j ) + ) + Γα jm x i ( Γ m kl x i + ( 2 Γ α kl x i x j ) + Γm lk (Γ α x i mj + Γα jm ) + 2 Γ α lk x i x j ) Take differences, kill second derivatives. Use curvature...

27 Compare: Lie algebra of two fields on R Molev (1983, Russian): Lie algebra L(X, Y ) generated by two vector fields X and Y on R is never free Kirillov, Kontsevich, and Molev (199): For Lie algebra L generated by two vector fields on R in general position calculated dim s of hom. subspaces and entropy, and established identity X σ(1), X σ(2), X σ(3), X σ(4), Z L σ S 4 sgn(σ) [X σ(1), [X σ(2), [X σ(3), [X σ(4), Z ]]]] = McLachlan and Ryland (23): starting at order 13, the Lie algebras of classical mechanics in general position on Q = R n w/ Euclidean metric M = I and on general Q differ!

28 Compare: Lie algebra of classical mechanics the Lie algebra of simple mechanical systems is never free: with canonical Poisson bracket {A, B} = A B A B q i p i p i q i i kinetic energy T = 1 2 pt M(q)p and potential energy V (q) always satisfy identities such as {V, {V, {V, H}}} (since e.g. the modified potential {V, {V, H}} = V V q i M ij q j is independent of p) i,j in The algebraic entropy of classical mechanics (23) McLachlan and Ryland calculate the entropy lim (dim n Ln (T, V )) 1/n = Witt s formula (free Lie alg) dim L n (X, Y ) = 1 n µ(d) 2 n d bases ( by hand, for small dim ) by Murua (1999) also Blanes, Casas, and Ros (2, 21) d n 2n n

29 Compare: factoring the Chen Fliess series Affine analytic control system on M n, initialized at x() = p ẋ = m u j (t)f j (x), with output ϕ: M n R j=1 asymptotic series x(t ) = T t1 I=(i 1...i s) ts u i1 (t 1 )... u is (t s ) dt s... dt 1 (f i1... f is ϕ )(p)

30 Compare: factoring the Chen Fliess series Affine analytic control system on M n, initialized at x() = p ẋ = m u j (t)f j (x), with output ϕ: M n R j=1 asymptotic series x(t ) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 (f i1... f is ϕ )(p) formal series in free associative algebra A = A(X 1,... X m ) over set of indeterminates X 1,... X m S(T, u) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 X i1... X is

31 Compare: factoring the Chen Fliess series Affine analytic control system on M n, initialized at x() = p ẋ = m u j (t)f j (x), with output ϕ: M n R j=1 asymptotic series x(t ) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 (f i1... f is ϕ )(p) formal series in free associative algebra A = A(X 1,... X m ) over set of indeterminates X 1,... X m S(T, u) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 X i1... X is redundant, linearly dependent, truncations not systems

32 Coordinates of the first / second kind The series actually takes values in the free Lie algebra L = L(X 1,... X m ) A S(T, u) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 X i1... X is

33 Coordinates of the first / second kind The series actually takes values in the free Lie algebra L = L(X 1,... X m ) A S(T, u) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 X i1... X is For any ordered basis B L, there exist uniquely determined iterated integral functionals assuming that the space of coefficients U is sufficiently rich, e.g. U = L loc ([, T ], R m ) ξ H, ζ H : U C[, T ] such that S(T, u) = exp( ζ H (T, u)h) = H B H B exp(ξ H (T, u)h)

34 Coordinates of the first / second kind The series actually takes values in the free Lie algebra L = L(X 1,... X m ) A S(T, u) = I=(i 1...i s) T t1 ts u i1 (t 1 )... u is (t s ) dt s... dt 1 X i1... X is For any ordered basis B L, there exist uniquely determined iterated integral functionals assuming that the space of coefficients U is sufficiently rich, e.g. U = L loc ([, T ], R m ) ξ H, ζ H : U C[, T ] such that S(T, u) = exp( ζ H (T, u)h) = H B H B Wanted: suitable bases B for L that yield simple formulas for ξ H and ζ H exp(ξ H (T, u)h)

35 Hall sets A Hall set over a set X is any strictly ordered subset H of parenthesixed words satisfying (i) X H (ii) Suppose a X. Then (t, a) H iff t H, t < a and a < (t, a). (iii) Suppose u, v, w, (u, v) H. Then (t, (t, t )) H iff t t (t, t ) and t < (t, (t, t )). Example: X = {a, b} ((a(ab))(b(ab))) ((ab)(b(b(ab)))) ((ab)(b(a(ab)))) ((ab)(a(a(ab)))) (b(b(b(b(ab))))) (b(b(b(a(ab))))) (b(b(a(a(ab))))) (b(a(a(a(ab))))) ((ab)(b(ab))) ((ab)(a(ab))) (b(b(b(ab)))) (b(b(a(ab)))) (b(a(a(ab)))) (a(a(a(ab)))) (b(b(ab))) (b(a(ab))) (a(a(ab))) (b(ab)) (a(ab)) (ab) b a

36 Elegant formula for coordinates ξ H of 2 nd kind Use Hall bases for free Lie algebra

37 Elegant formula for coordinates ξ H of 2 nd kind Use Hall bases for free Lie algebra Use Zinbiel product on iterated integrals (U V )(t) = t U(s) V (s) ds

38 Elegant formula for coordinates ξ H of 2 nd kind Use Hall bases for free Lie algebra Use Zinbiel product on iterated integrals (U V )(t) = t U(s) V (s) ds which makes AC([, T ]) into a left Zinbiel algebra, satisfying the left Zinbiel identity (U (V W )) = ((U V ) W ) + ((V U) W )

39 Elegant formula for coordinates ξ H of 2 nd kind Use Hall bases for free Lie algebra Use Zinbiel product on iterated integrals (U V )(t) = t U(s) V (s) ds which makes AC([, T ]) into a left Zinbiel algebra, satisfying the left Zinbiel identity (U (V W )) = ((U V ) W ) + ((V U) W ) (Schützenberger 1958, Sussmann 1985, Grossmann 1988, Melançon and Reutenauer 199, MK) ξ HK = ξ H ξ K for H, K, HK H

40 In progress: The logarithm of the Chen series, or: A continuous Campbell-Baker-Hausdorff formula Wanted: similar simple formula for coord s ζ B of 1 st kind Advantages, especially in computational numerics: One exponential only! which bases? what is the product structure for the ζs, if there is any? proposed (partial) solutions: Rocha, Murua, MK

41 The first iterated integrals Notational convenience: Zinbiel algebra isomomorphism Υ from free Zinbiel algebra to Zinbiel algebra of iterated integral functionals Υ(wa): (u, t) t Υ(w)(u, s) u a(s) ds ζ a = a Υ(a)(u, t) = t u a(s)ds ζ b = b Υ(ab)(u, t) = t s u a(σ) dσ u b (s)ds ζ ab = 1 2 (ab ba) ζ aab = 1 6 ζ bab = 1 6 ( aab 2aba + baa) ( abb + 2bab bba) ζ aaab = 1 6 (abaa aaba) ζ baab = 1 6 (abab aabb + bbaa baba) ζ bbab = 1 6 (bbab babb)

42 The next few iterated integrals ζ aaaab = 1 3 (aaaab + aaaba 4aabaa + abaaa + baaaa) ζ baaab = 1 3 ( 2aaabb + 3aabab + 3aabba 2abaab 2ababa + 3abbaa 2baaab 2baaba + 3babaa 2bbaaa) ζ abaab = 1 3 ( 3aaabb + 2aabab + 2aabba + 2abaab 3ababa + 2abbaa 3baaab + 2baaba + 2babaa 3bbaaa) ζ bbaab = 1 3 (2aabbb 3ababb + 2abbab + 2abbba 3baabb + 2babab + 2babba 3bbaba 3bbaab + 2bbbaa) ζ abbab = 1 3 ( aabbb ababb + 4abbab abbba baabb babab + 4babba bbaba bbaab bbbaa) ζ bbbab = 1 3 (abbbb + babbb 4bbabb + bbbab + bbbba)

43 The next few iterated integrals ζ aaaab = 1 3 (aaaab + aaaba 4aabaa + abaaa + baaaa) ζ baaab = 1 3 ( 2aaabb + 3aabab + 3aabba 2abaab 2ababa + 3abbaa 2baaab 2baaba + 3babaa 2bbaaa) ζ abaab = 1 3 ( 3aaabb + 2aabab + 2aabba + 2abaab 3ababa + 2abbaa 3baaab + 2baaba + 2babaa 3bbaaa) ζ bbaab = 1 3 (2aabbb 3ababb + 2abbab + 2abbba 3baabb + 2babab + 2babba 3bbaba 3bbaab + 2bbbaa) ζ abbab = 1 3 ( aabbb ababb + 4abbab abbba baabb babab + 4babba bbaba bbaab bbbaa) ζ bbbab = 1 3 (abbbb + babbb 4bbabb + bbbab + bbbba) Multiplicative structures?

44 The next few iterated integrals ζ aaaab = 1 3 (aaaab + aaaba 4aabaa + abaaa + baaaa) ζ baaab = 1 3 ( 2aaabb + 3aabab + 3aabba 2abaab 2ababa + 3abbaa 2baaab 2baaba + 3babaa 2bbaaa) ζ abaab = 1 3 ( 3aaabb + 2aabab + 2aabba + 2abaab 3ababa + 2abbaa 3baaab + 2baaba + 2babaa 3bbaaa) ζ bbaab = 1 3 (2aabbb 3ababb + 2abbab + 2abbba 3baabb + 2babab + 2babba 3bbaba 3bbaab + 2bbbaa) ζ abbab = 1 3 ( aabbb ababb + 4abbab abbba baabb babab + 4babba bbaba bbaab bbbaa) ζ bbbab = 1 3 (abbbb + babbb 4bbabb + bbbab + bbbba) Multiplicative structures? Maybe only apparent when using other bases but which?

45 The next few iterated integrals ζ aaaab = 1 3 (aaaab + aaaba 4aabaa + abaaa + baaaa) ζ baaab = 1 3 ( 2aaabb + 3aabab + 3aabba 2abaab 2ababa + 3abbaa 2baaab 2baaba + 3babaa 2bbaaa) ζ abaab = 1 3 ( 3aaabb + 2aabab + 2aabba + 2abaab 3ababa + 2abbaa 3baaab + 2baaba + 2babaa 3bbaaa) ζ bbaab = 1 3 (2aabbb 3ababb + 2abbab + 2abbba 3baabb + 2babab + 2babba 3bbaba 3bbaab + 2bbbaa) ζ abbab = 1 3 ( aabbb ababb + 4abbab abbba baabb babab + 4babba bbaba bbaab bbbaa) ζ bbbab = 1 3 (abbbb + babbb 4bbabb + bbbab + bbbba) Multiplicative structures? Maybe only apparent when using other bases but which? Special cases? Mechanical? extreme case SO(3) recover/compare trad formulas

46 Conclusion and continuing work Continuing efforts, esp.for computational purposes, to use geometric structures to simplify asymptotic expansions This presentation was prepared with LaTeX using the beamer package.

47 Conclusion and continuing work Continuing efforts, esp.for computational purposes, to use geometric structures to simplify asymptotic expansions Identify multiplicative structure of coordinates ζ of 1 st kind, and find bases that yield formulas as simple as ξ HK = ξ ξ K This presentation was prepared with LaTeX using the beamer package.

48 Conclusion and continuing work Continuing efforts, esp.for computational purposes, to use geometric structures to simplify asymptotic expansions Identify multiplicative structure of coordinates ζ of 1 st kind, and find bases that yield formulas as simple as ξ HK = ξ ξ K Specialize to general mechanical systems, analyze the geometry, find all structural identities, and use these to simplify solution formulas This presentation was prepared with LaTeX using the beamer package.

49 Conclusion and continuing work Continuing efforts, esp.for computational purposes, to use geometric structures to simplify asymptotic expansions Identify multiplicative structure of coordinates ζ of 1 st kind, and find bases that yield formulas as simple as ξ HK = ξ ξ K Specialize to general mechanical systems, analyze the geometry, find all structural identities, and use these to simplify solution formulas Specialize to specific mechanical systems, further collapse the formulas. This presentation was prepared with LaTeX using the beamer package.

50 Conclusion and continuing work Continuing efforts, esp.for computational purposes, to use geometric structures to simplify asymptotic expansions Identify multiplicative structure of coordinates ζ of 1 st kind, and find bases that yield formulas as simple as ξ HK = ξ ξ K Specialize to general mechanical systems, analyze the geometry, find all structural identities, and use these to simplify solution formulas Specialize to specific mechanical systems, further collapse the formulas. Demonstrate in simulations the accuracy and computational efficiency of simplified solution formulas This presentation was prepared with LaTeX using the beamer package.

Formal solution Chen-Fliess series

Formal solution Chen-Fliess series Ṡ = S(t) u a (t) a, S() = 1 Z = m a Z on algebra Â(Z ) of formal power series in aset Z of noncommuting indeterminates (letters) has the unique solution CF(T, u) = w