Manipulating exponential products Instead of working with complicated concatenations of flows

Transcription

1 Manipulating exponential products Instead of working with complicated concatenations of flows z(t) = e t9 t2 t8 (f +f 1 +f 2 ) dt... e t1 (f +f 1 f 2 ) dt t1 e (f +f 1 +f 2 ) dt (p) it is desirable to rewrite the solution curve using a minimal number of vector fields f πk that span the tangent space (typically using iterated Lie brackets of the system fields f, f 1,... f m ) Coordinates of the first kind z(t) = e b 1(t,u)f π1 +b 2 (t,u)f π1 +b 3 (t,u)f π b n(t,u)f π n (p) Coordinates of the second kind z(t) = e c 1(t,u)f π1 e c 2(t,u)f π1 e c 3(t,u)f π3... e cn(t,u)fπ n (p) Using the Campbell-Baker-Hausdorff formula, this is possible, but a book-keeping nightmare. Moreover, the CBH formula does not use a basis, but uses

2 Series expansions: Lift to universal, free system Starting with an affine, real analytic system on R n ẋ = m u i (t)f i (x(t)) i=1 or (chronological calculus), work with induced system on the algebra C (M) of smooth functions Basically, from ẋ = f u (x) to f u : Φ (f u Φ) = Φ, f u Formally, on free associative algebra Â(Z ) over a set Z = {X 1,... X m } of m indeterminates consider system Ṡ = S(t) m u i (t)x i i=1

3 Formal solution Chen-Fliess series Ṡ = S(t) m u i (t)x i, i=1 S() = I on algebra Â(Z ) of formal power series in the noncommuting indeterminates (letters) X 1,... X m has the unique solution CF(T, u) = I T t1 tp 1 u ip (t p )... u i1 (t 1 ) dt 1... dt p } {{ } Υ I (T,u) Use as asymptotic expansion for evolution of output y = ϕ(x) along solution of ẋ = u 1 f 1 (x) +... u m f m (x). ϕ(x(t, u)) I T t1 tp 1 X i1... X ip }{{} X I u ip (t p )...u i1 (t 1 ) dt 1... dt p (f i1... f ip ϕ)(x )

4 Series expansions, intro Splitting into geometric state-dependent and time-varying parts ẋ = u 1 (t)f 1 (x) u m (t)f m (x) = F(t, x) y = ϕ(x) f i : M TM smooth vector fields on manifold M n, u : [, T ] U R m measurable controls/perturbations, and φ C ω (M) measured output.

5 Series solution by iteration φ(x(t, u)) = 1 φ(x ) + t u a(s)ds (f a φ)(x ) + t u b(s)ds (f b φ)(x ) t s1 s1 t s1 t s1 t s1 t s1 u a(s 1 )u a (s 2 )ds 2 ds 1 (f a f a φ)(x ) u a(s 1 )u b (s 2 )ds 2 ds 1 (f a f b φ)(x ) u b(s 1 )u a (s 2 )ds 2 ds 1 (f b f a φ)(x ) u b(s 1 )u b (s 2 )ds 2 ds 1 (f b f b φ)(x ) s2 u a(s 1 )u a (s 2 )u a (s 3 )ds 3 ds 2 ds 1 (f a f a f a φ)(x ) +... Objective: Collect first order differential operators, and minimize number of higher order differential operators involved

6 Integrate by parts: The wrong way to do it t s1 u a(s 1 )u b (s 2 )ds 2 ds 1 f a f b + t s1 u b(s 1 )u a (s 2 )ds 2 ds 1 f b f a = s1 = t u a(s 1 )u b (s 2 )ds 2 ds 1 f a f b t u a(s 1 )u b (s 2 )ds 2 ds 1 f b f a + t s1 u a(s 1 )u b (s 2 )ds 2 ds 1 f b f a + t s1 u b(s 1 )u a (s 2 )ds 2 ds 1 f b f a = t s1 + t s1 u a(s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( u a (s 1 ) s 1 u b(s 2 ) + u b (s 1 ) ) s 1 u a(s 2 )ds 2 ds 1 f b f a = t s1 u a(s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( ) ( ) t + u t a(s) ds u b(s) ds f b f a Lie brackets together w/ iterated integrals in right order higher order deriv s (wrong order) w/ pointwise prod s of int s

7 Integrate by parts, smart way Do not manipulate iterated integrals and iterated Lie brackets of vector fields by hand work on level of words (their indices) I {a,b} I I = a a + b b aa aa ab ab ba ba bb bb aaa aaa +... = a a + b b aa aa ab (ab ba) (ab + ba) ba bb bb aaa aaa +...

8 Drop everything except the indices - maps The iterated integral Υ i1 i 2...i n = t t1 tn 1 u i1 (t 1 )u i2 (t 2 ) u in (t n ) dt n dt n 1 dt 1 is uniquely identified by the multi-index ( word ) i 1 i 2... i n The n-th order partial differential operator f in f in 1... f i1 is uniquely identified by the multi-index ( word ) i 1 i 2... i n The Chen series is identified with the identity map on free associative algebra A(Z ) over set of indices Z = {1,... m} CF Id A(Z ) = w w Â(Z ) A(Z ) n w Z n with shuffle product on left and concatenation on right

9 Recall: definition of the shuffle Combinatorially: for words w, z Z and letters a, b Z ( w a ) X ( z b ) = (( w a ) X z ) b + ( w X ( z b )) a Example: (ab) X (cd) = a b c d + a c b d + c a b d + a c d b + c a d b + c d a b Algebraically: transpose of the coproduct where < v X w, z > = < v w, (z) > : A(Z ) A(Z ) A(Z ) by (a) = 1 a+a 1 for a Z

10 Shuffles and simplices On permutations algebras Duchamp and Agrachev consider partially commutative and noncommutative shuffles. Illustration: σ 12 σ 2 σ 1 x 2 = σ 21 σ 1 In the case of three letters {1, 2, 3} = σ (12) x 3 = σ 312 σ 132 E.g. σ (12) x 3 = {t : t 1 t 2 1, t 3 1} For multiplicative integrands f (x, y, z) = f 1 (x) f 2 (y) f 3 (z)

11 Homomorphisms I For fixed smooth vector fields f i F : A(Z ) partial diff operators on C (M) F : (i 1 i 2... i n ) f i1 f i2... f in associative algebras: concatenation composition For fixed control u U Z Υ(u): A(Z ) AC([, T ], R) Υ(u): (i 1 i 2... i n ) T t1 tp 1 u ip (t p )... u i1 (t 1 ) dt 1... dt p associative algebras (Ree s theorem): shuffle of words pointwise multiplication of functions

12 CF-coefficients satisfy shuffle-relations Sketch of proof (by induction on the combined lengths of the coefficients) Υ 1 (t, u) 1 Υ a x 1 (t, u) = Υ a (t, u) = Υ a (t, u) 1 = Υ a (t, u) Υ 1 (t, u) for any letter a X

13 Induction step Υ (wa) x (zb) (T, u) = = Υ ((wa) x z)b+(w x (zb))a (T, u) = Υ ((wa) x z)b (T, u) + Υ (w x (zb))a (T, u) = T Υ (wa) x z(t, u) u b (t)dt + T Υ w x (zb)(t, u) u a (t)dt = T (Υ wa(t, u) Υ z (t, u) u b (t) + Υ w (t, u) Υ zb (t, u) u a (t)) = T ( Υwa (t, u) d dt Υ zb(t, u) + d dt (Υ wa(t, u)) Υ zb (t, u) ) dt = Υ wa (T, u) Υ zb (T, u)

14 Homomorphisms II Restriction is Lie algebra homomorphism F : L(Z ) A(Z ) Γ (M) (vector fields) Do not fix controls: iterated integral functionals Υ: A(Z ) IIF(U Z ) ( Υ: (i 1 i 2... i n ) u T t1 tp 1 associative algebras: shuffle of words pointwise multiplication of iterated integral functionals u ip (t p )... u i1 (t 1 ) dt 1... dt p ) Much better: Theorem: If U = L 1 ([, T ], [ 1, 1]) then Υ: (A(Z ), ) IIF(U Z ) is a Zinbiel algebra isomorphism.

15 Zinbiel product and algebra Abstractly, right Zinbiel identity U (V W ) = (U V ) W + (V U) W Concrete examples in control: polynomials X n X m = m n+m X n+m and X n X m = 1 n X n+m AC([, )): and iterated integrals functionals (U V )(t) = t U(s) V (s) ds (U V )(t) = t U(s)ds V (t) subsets, e.g. exponentials e imt e int = m n+m ei(m+n)t and e imt e int = 1 m ei(m+n)t

16 Solving DEs by iteration and Zinbiel products The integrated form of the universal control system Ṡ = S Φ, S() = 1 with Φ = m u i X i i=1 is compactly rewritten using chronological products S = 1 + S Φ Iteration yields the explicit series expansion S = 1 + (1 + S Φ) Φ = 1 + Φ + ((1 + S Φ) Φ) Φ = 1 + Φ + (Φ Φ) + (((1 + S Φ) Φ) Φ) Φ = 1 + Φ + (Φ Φ) + ((Φ Φ) Φ) + ((((1 + S Φ) Φ) Φ) Φ) Φ. = 1 + Φ + (Φ Φ) + ((Φ Φ) Φ) + (((Φ Φ) Φ) Φ)...

17 Rewriting the Chen series Recall: Chen series is an exponential Lie series ( ) w w = exp ζ b b = exp(ξ b b) w Z b B b B where B is ordered basis of free Lie algebra L(Z ) A(Z ) Coord s of 2 nd kind (classic) Coord s of the 1 st kind (new) ξ HK = ξ H ξ K ζ = π 1 ξ Use in control / geometric integration: explicit formula for iterated integral functionals Υ ζb and Υ ξb.

18 Intro Controllability Series expansions Coordinates of the second kind For a Hall set H over Z and letters a Z and H, K, HK H ξa = a, ξhk = µhk ξh ξk (multifactorial µhk ) Schützenberger (1958) Agrachev/Gamkrelidze (?) (1979) chronological prod Sussmann (1985) exp prod Reutenauer/Melançon (1989) Grayson/Grossmann (199) Exp-prod

19 Intro Controllability Series expansions Coordinates ξh of 2nd kind, historically Exp-prod

20 Intro Controllability Series expansions Coordinates ξh of 2nd kind, historically Exp-prod

21 Zinbiel product in 1958

22 Illustration: Normal form for free nilpotent system Normal from for a free nilpotent system (of rank r = 5) using a typical Hall set on the alphabet Z = {a, b} ξ a = u a ξ b = u b ξ ab = ξ a ξ b = ξ a u b ξ aab = ξ a ξ ab = ξa 2 u b using ψ b (aab) = (a(ab)) ξ bab = ξ b ξ ab = ξ b ξ a u b using ψ b (bab) = (b(ab)) ξ aaab = ξ a ξ aab = ξa 3 u b using ψ b (aaab) = (a(a(ab))) ξ baab = ξ b ξ aab = ξ b ξa 2 u b using ψ b (baab) = (b(a(ab))) ξ bbab = ξ b ξ bab = ξb 2ξ a u b using ψ b (bbab) = (b(b(ab))) ξ aaaab = ξ a ξ aaab = ξa 4 u b using ψ b (aaaab) = (a(a(a(ab)))) ξ baaab = ξ b ξ aaab = ξ b ξa 3 u b using ψ b (baaab) = (b(a(a(ab)))) ξ bbaab = ξ b ξ baab = ξb 2ξ2 a u b using ψ b (bbaab) = (b(b(a(ab)))) ξ abaab = ξ ab ξ aab = ξ ab ξa 3 u b using ψ b (abaab) = ((ab)(a(ab))) ξ abbab = ξ ab ξ bab = ξ ab ξb 2ξ a u b using ψ b (abbab) = ((ab)(b(ab))) Cost: accept label coord s not by integers, but by Hall words Benefit: One-line formula for all iterated integrals! ξ HK = ξ H ξ K

23 Dendriform algebras Introduced by Loday[21], described in control by Rocha [24], exploited by Ebrahimi-Fard and Manchon [26,7] Axioms: A dendriform (di)algebra is a k-vector space with product x and two bilinear operations, satisfying where u (v w) = (u x v) w (u v) w = u (v x w) u (v w) = (u v) w u x v = (u v) + (u v) we disregarded, did not use, the 3rd axiom! Dendriform structure shows relation between Magnus and Fer expansions [Ebrahimi-Fard and Manchon, 28]

24 Selected references L. Loday Dialgebras, Lect. Notes Math. 1763, Springer, Berlin pp (21). M. Schützenberger Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un probléme de mathématiques appliquées, Séminaire P. Dubreil, Algèbres et Théorie des Nombres, Faculté des Sciences de Paris (1958/59). K. Ebrahimi-Fard, D. Manchon, and F. Patras New identities in dendriform algebras [arxiv: v2] K. Ebrahimi-Fard, and D. Manchon A Magnus- and Fer-type formula in dendriform algebras [arxiv:77.67v2] M. Kawski and H. J. Sussmann. Control Theory from the Geometric Viewpoint [Springer, 24] Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in: Operators, Systems, and Linear Algebra, U. Helmke, D. Prätzel-Wolters and E. Zerz, eds., Teubner (1997), pp M. Kawski, Chronological algebras in nonlinear control, Itogi Nauki i Techniki, vol.68 (2) pp English translation in J. Math.Sciences, vol. 13 (21) pp M. Hazewinkel, Endomorphisms of Hopf algebras and a little bit of control LNCIS 321 (25) pp M. Hazewinkel, The Leibniz-Hopf Algebra and Lyndon Words Centrum voor Wiskunde en Informatica Report AM-R9612, (1996). A.Agrachëv and R. Gamkrelidze, The shuffle product and symmetric groups, Lect. Notes Pure Appl. Math., 152, (1994) pp