  Lyapunov exponents are a widely used tool for studying dynamical systems.
When calculating Lyapunov exponents for piecewise-smooth systems with
time-delayed arguments one faces a lack of continuity in the variational
problem. This paper studies how to build a variational equation for the
efficient construction of Jacobians along trajectories of the delayed nonsmooth
system. Trajectories of the piecewise-smooth system may encounter a so-called
grazing event where the trajectory approaches a discontinuity surface in the
state space in a non-transversal manner. For this event we develop a grazing
point estimation algorithm to ensure the accuracy of trajectories for the
nonlinear and the variational equations. We show that the eigenvalues of the
Jacobian matrix computed by the algorithm converge with an order consistent
with the order of the numerical integration method, therefore guaranteeing the
reliability of our proposed numerical method. Finally, the method is
demonstrated on a periodically forced impacting oscillator under the
time-delayed feedback control.
