Stabilization of Linear Systems with Input Amplitude and Rate Constraints

For discrete-time linear systems with inputs amplitude and rate constraints, we show that there exists a finite horizon such that the infinite horizon constrained LQR problem can turn to a finite horizon constrained LQR problem and give an upper bound of this horizon for any given initial state. We also give an upper bound of this finite horizon for all initial states inside the stability region, finally, we give a mixed controller which stabilizes the system