Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach
A recent work by Mazenc and Malisoff provides a trajectory-based approach for proving stability of time-varying systems with time-varying delays. Here, we provide several significant applications of their approach. In two results, we use a Lyapunov function for a corresponding undelayed system to provide a new method for proving stability of linear continuous-time time-varying systems with bounded time-varying delays. We allow uncertainties in the coefficient matrices of the systems. Our main results use upper bounds on an integral average involving the delay. The results establish input-to-state stability with respect to disturbances. We also provide a novel reduction model approach that ensures global exponential stabilization of linear systems with a time-varying pointwise delay in the input, which allows the delay to be discontinuous and uncertain. Finally, we provide an alternative to the reduction model method, based on a different dynamic extension. Our examples demonstrate the usefulness of our findings in several settings.
Publication Source (Journal or Book title)
SIAM Journal on Control and Optimization
Mazenc, F., Malisoff, M., & Niculescu, S. (2017). Stability and control design for time-varying systems with time-varying delays using a trajectory-based approach. SIAM Journal on Control and Optimization, 55 (1), 533-556. https://doi.org/10.1137/15M1027838