Control With Communications Constraints: Measuring the Instability in Parametric Linear Systems

This paper investigates the instability measure of linear systems defined as the sum of the unstable eigenvalues in the continuous-time case and the product of the unstable eigenvalues in the discrete-time case. The problem consists of determining the largest instability measure in systems depending polynomially on parameters constrained in a semialgebraic set. It is shown that upper bounds of the sought measure can be established via linear matrix inequality feasibility tests. Moreover, a priori and a posteriori conditions for establishing nonconservatism are proposed. Finally, two special cases of the proposed methodology are investigated—the first one concerns systems with a single parameter, and the second one concerns the determination of the largest spectral abscissa and radius. Three applications in control with communications constraints are discussed.