Computation of multiple Lie derivatives by algorithmic differentiation

Abstract

Lie derivatives are often used in nonlinear control and system theory. In general, these Lie derivatives are computed symbolically using computer algebra software. Although this approach is well-suited for small and medium-size problems, it is difficult to apply this technique to very complicated systems. We suggest an alternative method to compute the values of iterated and mixed Lie derivatives by algorithmic differentiation.

An Algebraic Approach to Identifiability

“…see [36]. As indistinguishable initial conditions yield the same output trajectory for any input, all coefficients in this Lie series, i.e., all mixed Lie derivatives must be equal.…”

Computation of higher order Lie derivatives on the Infinity Computer

“…The techniques for computing the derivatives based on symbolic or automatic differentiation can be used to obtain the desired result [9]. Recall that (see [46] ) working with a system of differential equations, that is with m > 1 the analytical computation of the j-th derivative y (j) involves a tensor of order j and the computation becames, in general, more involved.…”

“…We stress that many of the existing computational platforms are not equipped with functions for computing Lie derivatives. One toolbox that allows the computation of Lie derivatives is ADOL-C, a c++ package for automatic differentiation of algorithms written in C/C++ [8,9,10].…”

Computation of mixed Lie derivatives in nonlinear control

“…Eq. (10) illustrates that the order of univariate derivatives is increasing significantly with the number of vector fields [9]. Using the algorithmic differentiation tool ADOL-C [1], univariate Taylor series can very efficiently be calculated with the routine forward.…”