Measuring the instability in continuous-time linear systems with polytopic uncertainty

Measuring the instability of dynamical systems is an important problem for control synthesis. This paper considers continuous-time linear systems affected by structured uncertainty, and addresses the computation of the robust instability measure defined as the largest sum of the real parts of the unstable eigenvalues over all admissible uncertainties. In particular, it is supposed that the coefficients of the system are affine linear functions of an uncertain vector constrained into a polytope. First, an equivalent reformulation of this robust instability measure into a suitable robust stability margin of a finite family of systems is proposed. Second, a linear matrix inequality (LMI) condition is provided to establish upper bounds of the robust instability measure through the use of homogeneous Lyapunov functions. Third, a sufficient and necessary condition for establishing the optimality of a computed upper bound is proposed. Some numerical examples illustrate the proposed results.