Stability of Two-Dimensional Linear Systems With Singularities on the Stability Boundary Using LMIs

This paper gives results on stability and asymptotic stability of two-dimensional systems using linear matrix inequalities (LMIs). Despite a long history of research in this area, systems with singularities on the stability boundary (SSB) have received limited attention because they cannot produce a sign definite solution to the required LMI. However, 2D systems describing some classes of models of vehicle platoons generically involve an SSB. Therefore, commonly used definitions for (asymptotic) stability and strict LMI conditions are not suitable to discuss the stability of these systems. It is shown that the existence of a negative semidefinite solution together with simple additional conditions is sufficient to guarantee asymptotic stability. Thus, the stability conditions discussed here can be used to study a wider range of dynamical systems, including systems with singularities on the stability boundary (SSB), which cannot be exponentially stable. A unified framework is used to analyse continuous-continuous, continuous-discrete and discrete-discrete systems simultaneously.