Input-to-State Stabilization in the Lp Space of Stabilizable Systems Described by Coupled Delay Differential and Difference Equations

This paper deals with the input-to-state stabilization, with respect to a disturbance acting on the control input, of stabilizable systems described by nonlinear coupled delay differential and difference equations. These equations describe, for instance, lossless propagation phenomena in electrical and hydraulic engineering, and include, as special cases, neutral functional differential equations in Hale's form and retarded functional differential equations. A recent Lyapunov-Krasovskii characterization of the global asymptotic stability, in the Lp norm, of these systems is exploited. Such a characterization is obtained by means of one only functional for the overall system, though both differential and difference equations are involved in the system model. In the spirit of Sontag's feedback control redesign method, it is shown that the disturbance can be attenuated, in the sense of input-to-state stability in the Lp norm, by adding to the control law a term obtained by the Lyapunov-Krasovskii functional for the global asymptotic stability, in the Lp norm, of the disturbance-free closed-loop system. An example is studied in order to show the effectiveness of the proposed methodology.