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Section: Software

Keywords : observability/identifiability (identifiability, observability), system of parametric ordinary differential/difference equations, simplification.

Expanded Lie Point Symmetry

Participants : Alexandre Sedoglavic [ correspondant ] , François Ollivier.

ELPS is a pilot implementation (coded as a maple package) that allows to reduce the number of parameter of parametric (ordinary) differential/difference/algebraic systems when the considered system have affine expanded Lie point symmetries (see http://www.lifl.fr/~sedoglav/Software and  [96] ). Given a model, ELPS allows to test its identifiability/observability and to reformulate the model if necessary.

Before analysing a parametric model described by a differential/difference system, it is useful to reduce the number of relevant parameters that determine the dynamics. Usually, presentation of this kind of simplification relies on rules of thumbs (for example, the knowledge of units in which is expressed the problem when dimensional analysis is used) and thus, can not be implemented easily. However, these reductions are generally based on the existence of Lie point symmetries of the considered problem. The package ELPS uses this strategy in order to reformulate the considered model if it is not observable/identifiable and thus simplify further computations. Example: let us consider the classical Verhulst's model:

dx/dt = (a- bx )x- cx , da/dt = db/dt = dc/dt = 0, dt/dt = 1.(23)

with output  y= btx . The package ELPS determines that there is a 4-dimensional Lie group of transformations that act on this model but leave its solutions set and its output invariant. Using these informations and assuming that  a$ \ne$c and  b$ \ne$0 , the code gives automatically a representation of the flow  ( t, x) of ( 23 ) using parametrization:  Im47 ${t=\#119853 /(a-c),~x=(a-c)\#119857 /b}$ , where  Im48 ${(\#119853 ,\#119857 )}$ is the flow of the following simpler differential equation  Im49 ${d\#119857 /d\#119853 =(1-\#119857 )\#119857 ,~y=\#119853 \#119857 }$ . In this formulation of ( 23 ), parameters  a and  c were lumped together into  a- c and its state variables  x and  t were nondimensionalise . The complexity of the whole process is polynomial time with respect to input's size and is based on the result  [63] .


observability

— In control theory, observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs.

identifiability

— When a process is described a by differential equations, the validation of a model implies to be able to compute a set of parameters allowing to product a theoretical behavior corresponding to experimental data. Before any identification of the parameters, a preliminary issue is to study identifiability which means that there is a unique set of parameters corresponding to a given behavior of the system.



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