This paper addresses input-to-state stability (ISS) and integral ISS (iISS) redesign to enhance robustness of feedback control systems in the face of actuator disturbances. The idea of the redesign is to modify control laws available for disturbance-free systems a priori in order to achieve robustness against the disturbance. In practice, control system design often encounters time-delays, discontinuities and actuator limitations at the same time. This paper proposes a unified treatment of these issues in the redesign for actuator disturbances. For the new class of systems, we pursue LgV-type formulas which are widely used for ordinary differential equations. In addition, we incorporate saturation into the LgV-type control for implementation in the presence of actuator limitations. To derive such a redesign formula, we develop a framework of ISS/iISS Lyapunov-Krasovskii functionals allowing for the Filippov solutions. This paper demonstrates that the notion of invariantly differentiable functionals is useful for deriving LgV-type control laws and dealing with discontinuities directly. The iISS formulation which includes the ISS as a special case allows us to address actuator limitations in the redesign problem in a flexible manner.
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