Test Matrices for Componentwise Asymptotic Stability of Polytopic Systems

Abstract

The concept of “componentwise asymptotic stability” (abbreviated CWAS) was initially introduced as a special type of stability for single-model linear systems, with continuous- or discrete-time dynamics. Subsequently, the framework was enlarged to encompass linear systems with interval-type uncertainties. For both these classes of systems with continuous-time dynamics, previous works show that an essentially-nonnegative matrix, generically denoted U, can be built as a CWAS testing instrument, which exploits the equivalence between the statements “system is CWAS” and “Perron-Frobenius eigenvalue of U is negative”. The current article extends the aforementioned result to the general case of linear systems with arbitrary polytopic uncertainties. A family of essentially nonnegative matrices is constructed by considering the row-representative theory and the matrices associated with the polytope's vertices; the CWAS testing principle relies on the representative matrices (matrix) that possess (possesses) the maximum Peron-Frobenius eigenvalue (meaning the generalization of U's role). An example based on the operation of a mechanical system with polytopic uncertainties illustrates the construction of the test matrix / matrices and the applicability in CWAS analysis. The paper focuses on continuous-time dynamics as corresponding to the traditional mathematical scenario of differential equations defined on compact sets, but the new result can be easily adapted to discrete-time dynamics.