Abstract-This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional is proposed using augmented states which enriches and encompasses the classical functionals of the literature. It results in tractable stability conditions expressed in terms of linear matrix inequalities. This methodology follows from the application of the Bessel inequality to the projections over the Legendre polynomials. Numerical examples illustrate the potential of our approach through three scenari: a stable ODE perturbed by the PDE, an unstable open-loop ODE and an unstable closed-loop ODE stabilized by the PDE.
Summary
The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time‐delay systems, an exact stability result is firstly derived using pole locations. Then, based on the small‐gain theorem and on the quadratic separation framework, some robust stability criteria are provided. The latter follows from a projection of the infinite dimensional system states onto an orthogonal basis of Legendre polynomials. Numerical examples comparing these results with the ones in the literature are presented along with demonstrations of the effectiveness of the developed robust stability criteria.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.