Feedback design for discontinuous time-delay systems based on functional differential inclusions

This paper investigates a feedback stabilization problem for discontinuous nonlinear systems with time-delay. To deal with the time-delay systems with discontinuity, an extension of Filippov solution, which is originally proposed for differential equations with discontinuous right-hand side, is given to functional differential inclusion with discontinuous right-hand side. With the definition of solution, a sufficient condition for stability is presented which is nothing but Lyapunov-Krasovskii stability criterion for evaluating all Filippov solutions. Furthermore, a special case is discussed, where the Lyapunov-Krasovskii functional candidate is constructed with particular structure. Finally, with the proposed fundamental tools, a feedback design example is shown for the speed tracking control problem of internal combustion engines which addresses the intake-to-power delay and the disturbance torque with switching properties according to engine operation modes, and the control law is obtained by introducing a Lyapunov-Krasovskii functional with the structure in the special case.