Bounds for the remainders of uncertain matrix exponential and sampled-data control of polytopic linear systems

Abstract

This paper determines an explicit upper bound to the norm of any given degree of the Taylor’s expansion remainder for the matrix exponential function. It depends on the spectral norm and the corresponding measure of a square matrix. The generalization to cope with uncertain polytopic matrices follows from the definition of a norm and a measure for this mathematical entity and the determination of the corresponding upper bound for the expansion remainder. Naturally, the results are applied to robust stability analysis and state feedback control synthesis of sampled-data polytopic systems. It is shown that a sampled-data uncertain system obtained from a continuous-time polytopic one can be expressed (through a nonconservative sufficient condition) by a feedback interconnection of a discrete-time polytopic system and a norm bounded linear operator. Academical examples illustrate the theoretical results.