Manipulating exponential products Instead of working with - PowerPoint PPT Presentation

Intro Controllability Series expansions Exp-prod Manipulating exponential products Instead of working with complicated concatenations of flows � t 9 � t 2 � t 1 t 8 ( f 0 + f 1 + f 2 ) dt ◦ . . . e t 1 ( f 0 + f 1 − f 2 ) dt ◦ e 0 ( f 0 + f 1 + f 2 ) dt ( p ) z ( t ) = e it is desirable to rewrite the solution curve using a minimal number of vector fields f π k that span the tangent space (typically using iterated Lie brackets of the system fields f 0 , f 1 , . . . f m ) Coordinates of the first kind z ( t ) = e b 1 ( t , u ) f π 1 + b 2 ( t , u ) f π 1 + b 3 ( t , u ) f π 3 + ... + b n ( t , u ) f π n ( p ) Coordinates of the second kind z ( t ) = e c 1 ( t , u ) f π 1 ◦ e c 2 ( t , u ) f π 1 ◦ e c 3 ( t , u ) f π 3 ◦ . . . ◦ e c n ( t , u ) f π 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 linear combinations of all possible iterated Lie brackets. Yet, by

Intro Controllability Series expansions Exp-prod Series expansions: Lift to universal, free system • Starting with an affine, real analytic system on R n m � ˙ x = 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 ˙ x = f u ( x ) to f u : Φ �→ ( f u Φ) = �∇ Φ , f u � • Formally, on free associative algebra ˆ A ( Z ) over a set Z = { X 1 , . . . X m } of m indeterminates consider system m � ˙ S = S ( t ) · u i ( t ) X i i = 1

Intro Controllability Series expansions Exp-prod Formal solution – Chen-Fliess series m � ˙ S = S ( t ) · u i ( t ) X i , S ( 0 ) = I i = 1 on algebra ˆ A ( Z ) of formal power series in the noncommuting indeterminates (letters) X 1 , . . . X m has the unique solution � T � t 1 � t p − 1 � CF ( T , u ) = · · · u i p ( t p ) . . . u i 1 ( t 1 ) dt 1 . . . dt p X i 1 . . . X i p 0 0 0 � �� � I � �� � X I Υ I ( T , u ) Use as asymptotic expansion for evolution of output y = ϕ ( x ) along solution of ˙ x = u 1 f 1 ( x ) + . . . u m f m ( x ) . � T � t 1 � t p − 1 � ϕ ( x ( T , u )) ∼ · · · u i p ( t p ) . . . u i 1 ( t 1 ) dt 1 . . . dt p ( f i 1 . . . f i p ϕ )( x 0 ) 0 0 0 I

Intro Controllability Series expansions Exp-prod Series expansions, intro Splitting into geometric state-dependent and time-varying parts ˙ x = u 1 ( t ) f 1 ( x ) + . . . + u m ( t ) f m ( x ) = F ( t , x ) = ϕ ( x ) y f i : M �→ TM smooth vector fields on manifold M n , u : [ 0 , T ] �→ U ⊂⊂ R m measurable controls/perturbations, and φ ∈ C ω ( M ) measured output.

Intro Controllability Series expansions Exp-prod Series solution by iteration φ ( x ( t , u )) = 1 · φ ( x 0 ) � t + 0 u a ( s ) ds ( f a φ )( x 0 ) � t + 0 u b ( s ) ds ( f b φ )( x 0 ) � t � s 1 � s 1 + 1 0 u a ( s 1 ) u a ( s 2 ) ds 2 ds 1 ( f a f a φ )( x 0 ) 2 0 0 � t � s 1 + 1 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 ( f a f b φ )( x 0 ) 2 0 � t � s 1 + 1 0 u b ( s 1 ) u a ( s 2 ) ds 2 ds 1 ( f b f a φ )( x 0 ) 2 0 � t � s 1 + 1 0 u b ( s 1 ) u b ( s 2 ) ds 2 ds 1 ( f b f b φ )( x 0 ) 2 0 � t � s 1 � s 2 + 1 0 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 0 ) 6 0 0 + . . . Objective: Collect first order differential operators, and minimize number of higher order differential operators involved

Intro Controllability Series expansions Exp-prod Integrate by parts: The wrong way to do it � t � s 1 � t � s 1 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 f a f b + 0 u b ( s 1 ) u a ( s 2 ) ds 2 ds 1 f b f a = 0 0 � t � s 1 � t � s 1 = 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 f a f b − 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 f b f a 0 0 � t � s 1 � t � s 1 + 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 f b f a + 0 u b ( s 1 ) u a ( s 2 ) ds 2 ds 1 f b f a 0 0 � t � s 1 = 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 ( f a f b − f b f a ) 0 � � � t � s 1 � s 1 + u a ( s 1 ) 0 u b ( s 2 ) + u b ( s 1 ) 0 u a ( s 2 ) ds 2 ds 1 f b f a 0 � t � s 1 = 0 u a ( s 1 ) u b ( s 2 ) ds 2 ds 1 ( f a f b − f b f a ) 0 �� t � �� t � + 0 u a ( s ) ds · 0 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

Intro Controllability Series expansions Exp-prod 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 = 1 ⊗ 1 = 1 ⊗ 1 + a ⊗ a + a ⊗ a + b ⊗ b + b ⊗ b + 1 + 1 aa ⊗ aa aa ⊗ aa 2 2 + 1 + 1 ab ⊗ ab ab ⊗ ( ab − ba ) 2 2 + 1 + 1 ba ⊗ ba ( ab + ba ) ⊗ ba 2 2 + 1 + 1 bb ⊗ bb bb ⊗ bb 2 2 + 1 + 1 aaa ⊗ aaa aaa ⊗ aaa 6 6 + + . . . . . .

Intro Controllability Series expansions Exp-prod Drop everything except the indices - maps • The iterated integral � t � t 1 � t n − 1 Υ i 1 i 2 ... i n = · · · u i 1 ( t 1 ) u i 2 ( t 2 ) · · · u i n ( t n ) dt n dt n − 1 · · · dt 1 0 0 0 is uniquely identified by the multi-index (“word”) i 1 i 2 . . . i n • The n -th order partial differential operator f i n ◦ f i n − 1 ◦ . . . f i 1 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 } � � ∈ ˆ CF ∼ Id A ( Z ) = w ⊗ w A ( Z ) ⊗ A ( Z ) w ∈ Z n n ≥ 0 with shuffle product on left and concatenation on right

Intro Controllability Series expansions Exp-prod 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 ( ab ) X ( cd ) = a b c d + a c b d + c a b d + Example: a c d b + c a d b + c d a b Algebraically: transpose of the coproduct ∆ < v X w , z > = < v ⊗ w , ∆( z ) > where ∆: A ( Z ) �→ A ( Z ) ⊗ A ( Z ) by ∆( a ) = 1 ⊗ a + a ⊗ 1 for a ∈ Z

Intro Controllability Series expansions Exp-prod 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 : 0 ≤ t 1 ≤ t 2 ≤ 1 , 0 ≤ t 3 ≤ 1 } For multiplicative integrands f ( x , y , z ) = f 1 ( x ) · f 2 ( y ) · f 3 ( z ) (using x y z , instead of t t t for better readability):

Intro Controllability Series expansions Exp-prod 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 i 1 ◦ f i 2 ◦ . . . f i n associative algebras: concatenation �→ composition • For fixed control u ∈ U Z Υ( u ): A ( Z ) �→ AC ([ 0 , T ] , R ) � T � t 1 � t p − 1 Υ( u ): ( i 1 i 2 . . . i n ) �→ · · · u i p ( t p ) . . . u i 1 ( t 1 ) dt 1 . . . dt p 0 0 0 associative algebras (Ree’s theorem): shuffle of words �→ pointwise multiplication of functions

Intro Controllability Series expansions Exp-prod 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

Intro Controllability Series expansions Exp-prod 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 � T = 0 Υ ( wa ) x z ( t , u ) · u b ( t ) dt + 0 Υ w x ( zb ) ( t , u ) · u a ( t ) dt � T = 0 (Υ wa ( t , u ) · Υ z ( t , u ) · u b ( t ) + Υ w ( t , u ) · Υ zb ( t , u ) · u a ( t )) dt � T � � Υ wa ( t , u ) · d dt Υ zb ( t , u ) + d = dt (Υ wa ( t , u )) · Υ zb ( t , u ) dt 0 = Υ wa ( T , u ) · Υ zb ( T , u )

Intro Controllability Series expansions Exp-prod 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 ) � � � T � t 1 � t p − 1 Υ: ( i 1 i 2 . . . i n ) �→ u �→ · · · u i p ( t p ) . . . u i 1 ( t 1 ) dt 1 . . . dt p 0 0 0 associative algebras: shuffle of words �→ pointwise multiplication of iterated integral functionals • Much better: Theorem: If U = L 1 ([ 0 , T ] , [ − 1 , 1 ]) then Υ: ( A ( Z ) , ∗ ) �→ IIF ( U Z ) is a Zinbiel algebra isomorphism.

Intro Controllability Series expansions Exp-prod Zinbiel product and algebra Abstractly, right Zinbiel identity U ∗ ( V ∗ W ) = ( U ∗ V ) ∗ W + ( V ∗ U ) ∗ W Concrete examples in control: X n ∗ X m = n + m X n + m m • polynomials X n ⋆ X m = 1 n X n + m and � t 0 U ( s ) V ′ ( s ) ds • AC ([ 0 , ∞ )) : ( U ∗ V )( t ) = � t and ( U ⋆ V )( t ) = 0 U ( s ) ds V ( t ) • iterated integrals functionals • subsets, e.g. exponentials e imt ∗ e int = e imt ⋆ e int = 1 n + m e i ( m + n ) t m m e i ( m + n ) t and