  It is due to the modularity of the analysis that results for cascaded systems
have proved their utility in numerous control applications as well as in the
development of general control techniques based on ``adding integrators''.
Nevertheless, the standing assumptions in most of the present literature on
cascaded systems is that, when decoupled, the subsystems constituting the
cascade are uniformly globally asymptotically stable (UGAS). Hence existing
results fail in the more general case when the subsystems are uniformly
semiglobally practically asymptotically stable (USPAS). This situation is often
encountered in control practice, e.g., in control of physical systems with
external perturbations, measurement noise, unmodelled dynamics, etc. This paper
generalizes previous results for cascades by establishing that, under a uniform
boundedness condition, the cascade of two USPAS systems remains USPAS. An
analogous result can be derived for USAS systems in cascade. Furthermore, we
show the utility of our results in the PID control of mechanical systems
considering the dynamics of the DC motors.
