Receptance-based stability criterion for second-order linear systems with time-varying delay

This paper presents a simple receptance-based criterion to analyze the robust stability of second-order systems with time-varying delay. The proposed approach is based on the closed-loop receptance which is directly related to the open-loop one by using the Sherman-Morrison equation. In this kind of problem, stability analysis cannot be performed from the closed-loop eigenvalues due to the time-variant nature of the system. In this context, a new robust stability condition is proposed by using the Small-Gain Theorem for second-order systems with either single or multiple inputs. The main contribution can be interpreted as a receptance-based generalization of the Single-Input Single-Output (SISO) first order Small-Gain Theorem condition. Moreover, the proposed stability criterion is combined with a detuning strategy to deal with the trade-off between performance and robustness with respect to delay variation. No limitation is imposed to the time-varying delay derivative which is a general result. Moreover, the proposed approach can also be used to analyze delay uncertainty due to the implementation simplicity since closed-loop poles are not computed in this criterion. Numerical examples are given to illustrate the effectiveness of the proposed approach.