Stability analysis of sampled-data systems via affine canonical Bessel-Legendre inequalities

Abstract

The looped-functional approach has been extensively used for stability analysis of sampled-data systems. For this approach, this paper introduces two new functionals to minimise the conservatism of stability conditions. The first one is called a non-negative integral functional (NIF) that is added to the derivative of a common Lyapunov functional. The second one is a general looped-functional (GLF), whose values at sampling instants are traditionally not well-defined, and are defined here in terms of their limits. Leveraging further affine canonical Bessel-Legendre inequalities expressed by simplified polynomials, a polynomial-based NIF, a polynomial-based GLF, and a polynomial-based zero equality are tailored for obtaining hierarchical types of stability conditions in the form of linear matrix inequalities. Finally, numerical examples show that: (i) the proposed functionals lead to less conservatism when compared with several state-of-the-art methods; and, (ii) the resulting stability conditions exhibit a hierarchical characteristic in the sense that the higher level of hierarchy, the less conservatism of the criteria.