Periodic behavior of locally stabilizing saturated linear controllers for the discrete-time double integrator

The main contribution of this paper is to point out the vast difference between a discrete-time double integrator subject to actuator saturation and a continuous-time double integrator subject to actuator saturation. In continuous-time, any linear state feedback control law which locally stabilizes the double integrator, also globally stabilizes the double integrator in the presence of actuator saturation. In discrete-time, the equivalent of the double integrator has intrinsically different behavior. Some linear feedbacks which locally stabilize the double integrator, also globally stabilize the double integrator in the presence of actuator saturation. However, other linear feedbacks which locally stabilize the double integrator, do not globally stabilize the double integrator in the presence of actuator saturation. We classify a class of linear feedback laws which achieve local stability but also yield nonzero periodic solutions in the presence of actuator saturation and therefore cannot achieve global asymptotic stability