Analytic Left Inversion of SISO Lotka-Volterra Models

Transcription

1 Analytic Left Inversion of SISO Lotka-Volterra Models Luis A. Duffaut Espinosa George Mason University Joint work with W. Steven Gray (ODU) and K. Ebrahimi-Fard (ICMAT) This research was supported by the BBVA Foundation Grant to Researchers, Innovators and Cultural Creators (Spain).

2 Overview 1. Introduction. Preliminaries on Fliess Operators and Their Inverses 3. Left System Inversion of Lotka-Volterra Input-Output Systems 4. Numerical Simulations 5. Conclusions

3 1. Introduction Given an operator F, describing the dynamics of a system, and a function y in its range, the left inversion problem (LIP) is to determine a unique input u such that y = F[u]. A sufficient condition (but not necessary) for solving this problem in a state space setting is to have well defined relative degree (Isidori, 1995). The solution of LIP does not require a state space realization. Fliess operators provide an explicit analytical solution. Introduced by M. Fliess in 1983, Fliess operators are analytic input-output systems described by coefficients and iterated integrals of the inputs. Fliess operators can be viewed as a functional generalization of a Taylor series. For example, any Volterra operator with analytic kernels has a Fliess operator representation. 3

4 . Preliminaries Population models: Here we apply the method to the population dynamical system: n ż i = β i z i + α ij z i z j, i = 1,,...,n, (Lotka-Volterra model) j=1 where z i to the i-th population, β i is the growth rate for the i-th population, and α ij weights the effect of the j-th species on the i-th species. Input-output models are obtained by introducing time dependence on the β i (t) s or α ij (t) s (inputs), and assuming y = h(z) (outputs). For n =, ż1 ż = β 1z 1 α 1 z 1 z β z +α 1 z 1 z (Predator-Prey model) The vector fields are complete within the first quadrant giving concentric periodic trajectories about z e = (β /α 1,β 1 /α 1 ). 4

5 Fliess Operators: Let X = {x,x 1,...,x m } be an alphabet and X the set of all words over X (including the empty word ). A formal power series is any mapping c : X R l. Typically, c is written as a formal sum c = η X (c,η)η, and the set of all such series is R l X. For a measurable function u : [a,b] R m with finite L 1 -norm, define E η : L m 1 [t,t +T] C[t,t +T] by E [u] = 1, and E xi η [u](t,t ) = t where x i X, η X and u 1. t u i (τ)e η [u](τ,t )dτ, Note that to each letter x i has been assigned a function u i. For each c R l X F c [u](t) = η X (c,η)e η [u](t,t ), which is called a Fliess operator (Fliess, 1983). 5

6 Fliess Operator Inverses: u F c y F c F d = F c d, where denotes the F d shuffle product. Fig..1: Product connection. u F d Fig..: Cascade connection. v F c y F c F d = F c d, where c d denotes the composition product of c R l X and d R m X (Gray et al., 14) u v F c y Given c,d R m X, y satisfies y = F c [v] = F c [u+f d [y]]. F d If there exists e so that y = F e [u], then Fig..3: Feedback connection. F e [u] = F c [u+f d e [u]]. (contraction!) 6

7 On the other hand, v = u+f d c [v] (I +F d c )[v] = u. Apply the compositional inverse to both sides of this equation: v = (I +F d c ) 1 [u] := ( I +F ( d c) 1) [u]. In which case, F c@d [u] = F c [v] = F c (δ d c) 1[u], (explicit formula!) or equivalently, c@d = c (δ d c) 1, where F δ := I. The set of operators F δ = {I +F c : c R X }, forms a group under composition, in particular, a Faà di Bruno Hopf algebra with antipode, α, satisfying (δ +c) 1 := δ +c 1 = δ + η X (αa η )(c)η, where c 1 denotes the composition inverse of c and a η : R X R : c (c,η). Remark: The antipode has an explicit series representation (Gray & Duffaut Espinosa, 11, 14). 7

8 Now observe that any c R X can be written as c = c N +c F, where c N := k (c,xk )x k and c F := c c N. Definition.1: Given c R X, let r 1 be the largest integer such that supp(c F ) x r 1 X. Then c has relative degree r if the linear word x r 1 x 1 supp(c), otherwise it is not well defined. Remark: 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. Definition.: Given ξ X, the corresponding left-shift operator is { ξ 1 : X η R X : η : η = ξη : otherwise. Remark: The operation F c /F d = F c/d is given by c/d := c d 1, where c 1 := (c, ) 1 (c ) k, and c = 1 c/(c, ) is proper. k= 8

9 y = F c [u] y (1) = F x 1 (c)[u]. y (r 1) = F (x r 1 [u] ) 1 (c) y (r) = F (x r ) 1 (c)[u] +u F (x r 1 x 1 ) 1 (c) [u]. u = v F (x r (c)[u] ) 1 F (x r 1 [u] (y (r) = v) x 1 ) 1 (c) = F (x r ) 1 (c c y )[u] F (x r 1 x 1 ) 1 (c) [u] = F d [u], d = (xr ) 1 (c c y ) (x r 1 x 1 ) 1 (c). Theorem.3: Suppose c R X has relative degree r. Let y be analytic at t = with generating series c y R LC [[X ]] satisfying (c y,x k ) = (c,x k ), k =,...,r 1. (Here X := {x }.) Then the input u(t) = (c u,x k ) tk k!, k= where c u = ((x r ) 1 (c c y )/(x r 1 x 1 ) 1 (c)) 1, is the unique solution to F c [u] = y on [,T] for some T >. Remark: The condition on c y ensures that y is in the range of F c. 9

10 3. Left System Inversion of LV Input-Output Systems Four SISO predator-prey systems with output y = z 1 (prey): I/ map state space realization rel. degree [ ] α 1 z 1 z F c : β 1 y g (z) =, g 1 (z) = β z + α 1 z 1 z [ z1 ] 1 F c : α 1 y g (z) = [ β 1 z 1 β z + α 1 z 1 z ], g 1 (z) = [ z1 z ] 1 F c : β y g (z) = [ β1 z 1 α 1 z 1 z α 1 z 1 z ], g 1 (z) = [ z ] F c : α y g (z) = [ β1 z 1 α 1 z 1 z β z ], g 1 (z) = [ z 1 z ] 1

11 The population system under study: α ż1 = 1 z 1 z ż β z +α 1 z 1 z + z 1 u, y = z 1 with z 1 () = z 1, and z () = z,. Make α 1 = α 1 = β = Initial orbit 4 Final orbit 4 Vector field 3.5 Vector field Predator population 3.5 Predator population Equilibrium (1, 1.5) Equilibrium (1, 1) Prey population Prey population Fig. 3.1: u = 1. Fig. 3.: u = 1.5. c =z 1, α 1 z, z 1, x +z 1, x 1 + ( α 1 z, z 1, α 1 α 1 z, z 1, +α 1 β z, z 1, )x α 1 z, z 1, x x 1 α 1 z, z 1, x 1 x +z 1, x 1 +. Relative degree r = 1. 11

12 One must select an output function y(t) = (c y,x k ) tk k!, k= where c y is the generating series of y. It is sufficient to consider a polynomial of degree 3, so let (c y, ) = v, (c y,x i ) = v i for i = 1,,3. Thus, d := (xr ) 1 (c c y ) (x r 1 x 1 ) 1 (c) = α 1z, v 1 + z 1, α 1 α 1 z 1, z, v ) 1α 1 z, + ( v 1α 1 z, z 1, x + v 1 z 1, x 1 z 1, + v 1α 1 β z, z 1, v 1 α 1 α 1 z, ( α 1 β z, v z 1, v α 1 z, +α 1 α 1 z 1, z, +α 1 β α 1 z 1, z, z 1, v ) 3 α 1 β z, α 1 α 1 z 1, z, x + z 1, 1

13 In which case, c u = ( d 1) N = v 1 z 1, +α 1 z, + Design example: Given v 1 ( v 1 z 1, α 1 z, ) + v 1α 1 z, z 1, z 1, + v ) α 1 β z, +α 1 α 1 z 1, z, x + z 1, [t 1,t ] = [1.5,1.7] ( t =.), u(t 1 ) = 1, u(t ) = 1.5, initial orbit exit point [z 1 (1.5),z (1.5)] T = [3.8,.5] T, y(t ) =, find a smooth u(t) for t (t 1,t ) so that all constraints are satisfied. 13

14 Solution: Select the output as y(t) = (c y,x k ) tk k! = v 1t+v 3 t 3 /3!. k= From Theorem.3, c u = ( d 1) N is computed in terms of v 1 and v 3. Solving c y (v 1,v 3 ) c u (v 1,v 3 ) =. x k (.) k /k!,k> = 1.5 x k (.) k /k!,k> system of nonlinear algebraic equations gives the transfer input (up to order 6): u(t) = (t t 1 ) (t t 1 ) (t t 1 ) (t t 1 ) (t t 1 ) (t t 1 ) 6 14

15 4. Numerical Simulation Note that u(t ) = 1.5 and y(t ) =., and the error is y(t) ŷ(t) = 153.4(t t 1 ) (t t 1 ) (t t 1 ) 8 +. Time series for prey population Prey Initial prey cycle Transfer trajectory Final prey cycle t Time series for predator population Predator Initial predator cycle Transfer trajectory Final predator cycle t Fig. 4.1: Prey (top) and predator (bottom) populations. 15

16 Predator population Initial cycle Initial cycle directions Transition computed by proposers Final cycle Final cycle directions Prey population Fig. 4.: Orbit transfer. 16

17 5. Conclusions and Future Work The LIP for SISO Lotka-Volterra systems was solved using Fliess operators having well defined relative degree. The method provides an exact, explicit and analytic formula for the LIP. The MIMO version of the Lotka-Volterra trajectory design problem is under review for the CDC 15. Efficiency of the current software is currently being improved. 17