Instability Analysis of Uncertain Systems via Determinants and LMIs

This technical note investigates two instability measures in continuous-time (CT) and discrete-time (DT) uncertain systems, the first given by the spectral abscissa (CT case) or radius (DT case), and the second given by the sum (CT case) or product (DT case) of the unstable eigenvalues. It is supposed that the system depends polynomially on an uncertain vector constrained into a semi-algebraic set. The problem is to determine the largest instability measures over the admissible uncertainties. It is shown that a sufficient condition for establishing an upper bound of the sought measures can be obtained in terms of linear matrix inequality (LMI) feasibility tests by exploiting the determinants of some specific matrices, and that this condition is also necessary under some mild conditions on the semi-algebraic set by using multipliers with degree sufficiently large. Moreover, a condition is provided for establishing the tightness of the found best upper bounds. Lastly, it is shown that in the special case where the semi-algebraic set is an interval, the degree of the multipliers is known a priori. Some numerical examples illustrate the proposed results.