Analytic Left Inversion of Multivariable LotkaVolterra Models W. - PowerPoint PPT Presentation

Analytic Left Inversion of Multivariable Lotka–Volterra Models α W. Steven Gray β Mathematical Perspectives in Biology Madrid, Spain February 5, 2016 β Joint work with Luis A. Duffaut Espinosa (GMU) and Kurusch Ebrahimi-Fard (ICMAT) α Research supported by the BBVA Foundation Grant to Researchers, Innovators and Cultural Creators (Spain). W. S. Gray February 5, 2016 – ICMAT

Problem Trajectory Generation Problem : explicitly compute the input to drive a nonlinear system to produce some desired output. • Fliess operators : F c : u �→ y are analytic multivariable input-output maps, which are described by coefficients ( c, η ) and corresponding iterated integrals (M. Fliess, 1983). • Left inversion problem : given a multivariable Fliess operator F c and a function y in its range, determine an input u such that y = F c [ u ] . • Hopf algebra antipode : group ( G, ◦ ) of unital Fliess operators and its corresponding Hopf algebra H of coordinate functions; G ∋ F ◦− 1 = F c ◦ S , S : H → H c S ⋆ id = id ⋆ S = ǫ z i = β i z i + � n • Lotka–Volterra Model : ˙ j =1 α ij z i z j , i = 1 , 2 , . . . , n Input-Output systems are obtained by introducing time dependence on the parameters β i ( t ) and α ij ( t ) (inputs u k ) , and assuming that y = h ( z ) (outputs y = F c [ u ] ) . W. S. Gray February 5, 2016 – ICMAT 1

Setting Fliess operator � y = F c [ u ]( t, t 0 ) = ( c, η ) E η [ u ]( t, t 0 ) η ∈ X ∗ X = { x 0 , x 1 , . . . , x m } alphabet: c := � η ∈ R ℓ �� X �� system: η ∈ X ∗ ( c, η ) � �� � ∈ R ℓ u : [ t 0 , t 1 ] → R m , controls: u 0 := 1 � t E xi ¯ η [ u ]( t, t 0 ) = u i ( s ) E ¯ η [ u ]( s, t 0 ) ds x i ← → u i t 0 E ∅ [ u ] := 1 W. S. Gray February 5, 2016 – ICMAT 2

System interconnections I F c u y × F d product connection: F c F d = F c ⊔ ⊔ d F c u y + F d parallel connection: F c + F d = F c + d W. S. Gray February 5, 2016 – ICMAT 3

System interconnections II Cascade connection v u F d F c y d := � ( d, η ) ∈ R m , d 0 := 1 η ∈ X ∗ ( d, η ) η, � � � ( F c ◦ F d )[ u ]( t, t 0 ) = ( c, η ) E η F d [ u ] ( t, t 0 ) η ∈ X ∗ � t E xi ¯ η [ F d [ u ]]( t, t 0 ) = F di [ u ]( s, t 0 ) E ¯ η [ F d [ u ]]( s, t 0 ) ds t 0 x i η ◦ ′ d := x 0 ⊔ ( η ◦ ′ d ) � � ( F c ◦ F d )[ u ] = F c ◦′ d [ u ] d i ⊔ W. S. Gray February 5, 2016 – ICMAT 4

System interconnections III Feedback loop v u y F c v = u + F d ◦′ c [ v ] F c • d [ u ] = F c ◦′ ( ǫ − d ◦′ c ) ◦− 1 [ u ] F d Involves an extension of Fliess operators: unital Fliess operators F cǫ [ u ] := u + F c [ u ] = ( I + F c )[ u ] c ǫ := ǫ + c F cǫ ◦ F dǫ [ u ] = F cǫ ◦ dǫ [ u ] This composition defines a group ( G, ◦ ) with unit ǫ on R �� X ǫ �� [G-DE]. W. S. Gray February 5, 2016 – ICMAT 5

Coordinate functions I Fa` a di Bruno type Hopf algebra a i η : G → R , a i η ( c ǫ ) := � c ǫ , a i η � = ( c ǫ , η ) i ∈ R Coordinate functions: � c ǫ ◦ d ǫ , a i � c ǫ ⊗ d ǫ , ∆( a i η � η ) � = � a i η ′ ⊗ a j � c ǫ ⊗ d ǫ , η ′′ � = ( η ) Theorem : Coordinate functions form a connected graded commutative non-cocommutative Hopf algebra ( H, ∆ , ǫ, S, m, ι ) . � c ◦− 1 , a i η � = � c ǫ , S ( a i S : H → H η ) � Antipode : ǫ � ′ � ′ S ( a i η ) = − a i S ( a i η ′ ) a j η ′′ = − a i a i η ′ S ( a j η − η − η ′′ ) ( η ) ( η ) W. S. Gray February 5, 2016 – ICMAT 6

Coordinate functions II Coproduct and antipode calculations ∆ : H → H ⊗ H S : H → H ∆( a i ∅ ) = a i ∅ ⊗ 1 + 1 ⊗ a i S ( a i ∅ ) = − a i ∅ ∅ ∆( a i xj ) = a i xj ⊗ 1 + 1 ⊗ a i S ( a i xj ) = − a i xj xj ∆( a i x 0 ) = a i x 0 ⊗ 1 + 1 ⊗ a i x 0 + a i xℓ ⊗ a ℓ S ( a i x 0 ) = − a i x 0 + a i xℓ a ℓ ∅ ∅ ∆( a i xjxk ) = a i xjxk ⊗ 1 + 1 ⊗ a i S ( a i xjxk ) = − a i xjxk xjxk � c ǫ ◦ d ǫ , a i � c ◦− 1 , a i xjxk � = ( c ◦− 1 xjxk � = ( c ǫ ◦ d ǫ , x j x k ) i , x j x k ) i ǫ ǫ = a i x 0 ( c ǫ ) + a i x 0 ( d ǫ ) + a i xℓ ( c ǫ ) a ℓ = − a i x 0 ( c ǫ ) + a i xℓ ( c ǫ ) a ℓ ∅ ( d ǫ ) ∅ ( c ǫ ) = ( c ǫ , x 0 ) i + ( d ǫ , x 0 ) i + ( c ǫ , x ℓ ) i ( d ǫ , ∅ ) ℓ = − ( c ǫ , x 0 ) i + ( c ǫ , x ℓ ) i ( c ǫ , ∅ ) ℓ W. S. Gray February 5, 2016 – ICMAT 7

Left Inversion of MIMO Fliess operators I Observe: c ∈ R �� X �� can be written as c = c N + c F , where c N := � k ≥ 0 ( c, x k 0 ) x k 0 and c F := c − c N . Definition : Given c ∈ R k �� X �� , let r i ≥ 1 be the largest integer such that supp( c F,i ) ⊆ ri − 1 ri − 1 X ∗ , i = 1 , 2 , . . . , m . Then c i has relative degree r i if x x j ∈ supp( c i ) , for x 0 0 j ∈ { 1 , . . . , m } , otherwise it is not well defined. In addition, c has vector relative degree r = [ r 1 r 2 · · · r m ] if each c i has relative degree r i and the m × m matrix   ( c 1 , x r 1 − 1 ( c 1 , x r 1 − 1 ( c 1 , x r 1 − 1 x 1 ) x 2 ) · · · x m ) 0 0 0 . . . . A = . . . .   . . . . ( c m , x rm − 1 ( c m , x rm − 1 ( c m , x rm − 1 x 1 ) x 2 ) · · · x m ) 0 0 0 has full rank. Otherwise, c does not have vector relative degree. This definition coincides with the usual definition of relative degree given in a state space setting. But this definition is independent of the state space setting. W. S. Gray February 5, 2016 – ICMAT 8

Left Inversion of MIMO Fliess operators II Lemma : The set of series R m × m �� X �� having invertible constant terms is a group under the shuffle product. In particular, the shuffle inverse of any such series C is ⊔ − 1 = (( C, ∅ )( I − C ′ )) ⊔ ⊔ − 1 = ( C, ∅ ) − 1 ( C ′ ) ⊔ C ⊔ ⊔ ∗ , where C ′ = I − ( C, ∅ ) − 1 C is proper, i.e., ( C ′ , ∅ ) = 0 , and ( C ′ ) ⊔ ⊔ ∗ := � k ≥ 0 ( C ′ ) ⊔ ⊔ k . For any C ∈ R m × m �� X �� with an invertible constant term, F C , which is Lemma : defined componentwise by [ F C ] i,j = F Ci,j , has a well defined multiplicative inverse given by ( F C ) − 1 = F C ⊔ ⊔ − 1 . Let R [[ X 0 ]] be all commutative series over X 0 := { x 0 } . When c ∈ R [[ X 0 ]] , Notation: F c [ u ]( t ) = � 0 [ u ]( t ) = � k ≥ 0 ( c, x k k ≥ 0 ( c, x k 0 ) t k /k ! . 0 ) E xk W. S. Gray February 5, 2016 – ICMAT 9

Left Inversion of MIMO Fliess operators III y ( r ) = F ( xr 0) − 1( c ) [ u ] + F C [ u ] u ∈ R m � u = − ( F C [ u ]) − 1 F ( xr ( c y , x k 0 ) t k /k ! 0) − 1( c − cy ) [ u ] , y ( t ) = F cy [ u ]( t ) = k ≥ 0 d = C ⊔ ⊔ − 1 ⊔ ⊔ ( x r 0 ) − 1 ( c − c y ) u = − F d [ u ] , ri − 1 ri ( x r 0 ) − 1 ( c − c y ) i = ( x 0 ) − 1 ( c i − c yi ) and C i,j = ( x x j ) − 1 ( c i ) 0 Theorem : Suppose c ∈ R m �� X �� has vector relative degree r . Let y be analytic at t = 0 ∗ LC �� X �� satisfying ( c y , x ( r ) = ( c, x ( r ) with generating series c y ∈ R m 0 ) 0 ) . Then the input ∞ 0 ) t k � c u = (( C ⊔ ⊔ − 1 ⊔ 0 ) − 1 ( c − c y )) ◦− 1 ) | N , ( c u , x k ⊔ ( x r u ( t ) = with k ! k =0 is the unique solution to F c [ u ] = y on [0 , T ] for some T > 0 . Note: the condition ∗ on c y ensures that y is in the range of F c . W. S. Gray February 5, 2016 – ICMAT 10

Multivariable I/O Lotka–Volterra Models I z i = β i z i + � n ˙ j =1 α ij z i z j , i = 1 , 2 , . . . , n     z 1 ˙ β 1 z 1 − α 12 z 1 z 2  = z 2 ˙ − β 2 z 2 + α 21 z 1 z 2 − α 23 z 2 z 3 2 Predators - 1 Prey    z 3 ˙ − β 3 z 3 + α 32 z 3 z 2 The systems within the first octant have: • periodic orbits around ( β 2 /α 21 , β 1 /α 12 , 0) if β 1 α 32 = β 3 α 12 • extinction of one population if β 1 α 32 < β 3 α 12 • unbounded growing if β 1 α 32 > β 3 α 12 . ANSATZ: Input-output models are obtained by introducing time dependence on the parameters β i ( t ) ’s or α ij ( t ) ’s (inputs), and assuming y = h ( z ) (outputs) . W. S. Gray February 5, 2016 – ICMAT 11

Multivariable I/O Lotka–Volterra Models II Vector relative degree r for three LV systems with y 1 = z 2 and y 2 = z 3 : I/O map r range restrictions � β 1 � y 1 � � F c : �→ not defined – β 2 y 2 � β 2 � y 1 � � ( c y 1 , ∅ ) = ( c 1 , ∅ ) �→ F c : [1 1] β 3 y 2 ( c y 2 , ∅ ) = ( c 2 , ∅ ) ( c y 1 , ∅ ) = ( c 1 , ∅ ) � β 1 � y 1 � � [2 1] F c : �→ ( c y 1 , x 0 ) = ( c 1 , x 0 ) β 3 y 2 (full) ( c y 2 , ∅ ) = ( c 2 , ∅ ) Consider case 2: r = [1 1] , u 1 := β 2 , u 2 := β 3         β 1 z 1 − α 12 z 1 z 2 z 1 ˙ 0 0 � y 1 � � z 2 �  =  −  u 1 −  u 2 , α 21 z 1 z 2 − α 23 z 2 z 3 = z 2 ˙ z 2 0     y 2 z 3 z 3 ˙ α 32 z 3 z 2 0 z 3 with z i (0) = z i, 0 > 0 , i = 1 , 2 , 3 . Normalizing all parameters to 1 . W. S. Gray February 5, 2016 – ICMAT 12