Derivatives of the Matrix Exponential and Their Computation

AbstractMatrix exponentials and their derivatives play an important role in the perturbation analysis, control, and parameter estimation of linear dynamical systems. The well-known integral representation of the derivative of the matrix exponential exp(tA) in the direction V, namely ∫t0exp((t − τ)A)V exp(τA) dτ, enables us to derive a number of new properties for it, along with spectral, series, and exact representations. Many of these results extend to arbitrary analytic functions of a matrix argument, for which we have also derived a simple relation between the gradients of their entries and the directional derivatives in the elementary directions. Based on these results, we construct and optimize two new algorithms for computing the directional derivative. We have also developed a new algorithm for computing the matrix exponential that is more efficient than direct Padé approximation, which is based on a rational representation of the exponential in terms of the hyperbolic function A coth(A). Finally, these results are illustrated by an application to a biologically important parameter estimation problem which arises in nuclear magnetic resonance spectroscopy