Exact robust stability analysis of uncertain systems with a scalar parameter via LMIs

This paper provides an exact linear matrix inequality (LMI) condition for robust asymptotic stability of uncertain systems depending polynomially on a scalar parameter in both continuous-time and discrete-time cases. Specifically, this condition exploits sum of squares (SOS) techniques and is based on the construction of polynomials of known degree that detect the presence of eigenvalues on the boundary of the stability region. It is shown that this condition requires a much smaller computational burden than existing exact LMI conditions which might be prohibitive even for small scale systems.