Affine characterizations of minimum and mode-dependent dwell-times for uncertain linear switched systems

An alternative approach for minimum and mode-dependent dwell-time characterization for switched systems is derived. The proposed technique is related to Lyapunov looped-functionals, a new type of functionals leading to stability conditions affine in the system matrices, unlike standard results for minimum dwell-time. These conditions are expressed as infinite-dimensional LMIs which can be solved using recent polynomial optimization techniques such as sum-of-squares. The specific structure of the conditions is finally utilized in order to derive dwell-time stability results for uncertain switched systems. Several examples illustrate the efficiency of the approach.

💡 Research Summary

The paper introduces a novel framework for characterizing both minimum dwell‑time and mode‑dependent dwell‑time in linear switched systems, especially when parametric uncertainties are present. Traditional minimum dwell‑time results rely on Lyapunov‑based inequalities that are nonlinear in the system matrices, which makes the analysis and synthesis cumbersome for uncertain or high‑order systems. To overcome this limitation, the authors propose the use of Lyapunov looped‑functionals, a new class of functionals that enforce a monotonic decrease of a Lyapunov‑like quantity over each dwell interval while explicitly accounting for the boundary conditions at switching instants.

By constructing the looped‑functional as a combination of a quadratic Lyapunov term (V_i(x)=x^{\top}P_i x) and a time‑varying polynomial weight (W_i(\tau)), the resulting stability conditions become affine (linear) in the system matrices (A_i). The conditions are expressed as infinite‑dimensional linear matrix inequalities (LMIs) that must hold for all (\tau) in the interval (