Matrix Measures in the Qualitative Analysis of Parametric Uncertain Systems
Octavian C. Pastravanu, Mihaela-Hanako Matcovschi · Mathematical Problems in Engineering · 2009
The paper considersparametric uncertain systemsof the form , whereℳis either aconvex hull, or apositive coneof matrices, generated by the set of vertices𝒱= {M1, M2, …,MK} ⊂ℝn×n. Denote byμ∥ ∥the matrix measure corresponding to a vector norm ∥ ∥. Whenℳis a convex hull, the conditionμ∥ ∥(Mk) ≤r< 0, 1 ≤k≤K, is necessary and sufficient for the existence ofcommon strong Lyapunov functionsandexponentially contractive invariant setswith respect to the trajectories of the uncertain system. Whenℳis a positive cone, the conditionμ∥ ∥(Mk) ≤ 0, 1 ≤k≤K, is necessary and sufficient for the existence ofcommon weak Lyapunov functionsandconstant invariant setswith respect to the trajectories of the uncertain system. Both Lyapunov functions and invariant sets are described in terms of the vector norm ∥ ∥ used for defining the matrix measureμ∥ ∥. Numerical examples illustrate the applicability of our results.