Nonconservative LMI approach to robust stability for systems with uncertain scalar parameters

We study the robust stability of linear systems with several uncertain (complex, real) scalar parameters. Using quadratic Lyapunov functions depending polynomially upon the parameters, one is able to exhibit a countable family of LMIs such that: 1) solvability of one of them is sufficient for robust stability; and 2) reciprocally, robust stability implies their solvability, except possibly for a finite number of them. This extends the characterization by the solvability of Lyapunov inequality of the asymptotic stability of usual systems. The related issue of delay-independent stability for linear systems with multiple delays is also treated, and it is shown how the approach used is linked to the search for quadratic Lyapunov-Krasovskii functional of a special type.