  We study the problem of jointly designing a sparse sensor and actuator
schedule for linear dynamical systems while guaranteeing a control/estimation
performance that approximates the fully sensed/actuated setting. We further
prove a separation principle, showing that the problem can be decomposed into
finding sensor and actuator schedules separately. However, it is shown that
this problem cannot be efficiently solved or approximated in polynomial, or
even quasi-polynomial time for time-invariant sensor/actuator schedules;
instead, we develop deterministic polynomial-time algorithms for a time-varying
sensor/actuator schedule with guaranteed approximation bounds. Our main result
is to provide a polynomial-time joint actuator and sensor schedule that on
average selects only a constant number of sensors and actuators at each time
step, irrespective of the dimension of the system. The key idea is to sparsify
the controllability and observability Gramians while providing approximation
guarantees for Hankel singular values. This idea is inspired by recent results
in theoretical computer science literature on sparsification.
