A Nonconservative LMI Condition for Stability of Switched Systems With Guaranteed Dwell Time

Ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems. Several methods have been proposed in the literature to address this problem, but unfortunately they provide sufficient conditions only. This technical note proposes the use of homogeneous polynomial Lyapunov functions in the non-restrictive case where all the subsystems are Hurwitz, showing that a sufficient condition can be provided in terms of an LMI feasibility test by exploiting a key representation of polynomials. Several properties are proved for this condition, in particular that it is also necessary for a sufficiently large degree of these functions. As a result, the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well. Some examples illustrate the proposed approach.