LMIs in Control/Stability Analysis/Strong Stabilizability

Continuous-Time Strong Stabilizability Definition Consider a continuous-time LTI system, , with state-space realization (, , , ), where , , and , and it is assumed that (, ) is stabilizable, (, ) is detectable, and the transfer matrix has no poles on the imaginary axis. The system is strongly stabilizable if there exist , , and , where , such that LMIs where and , is the solution to the Lyapunov equation given by Moreover, a controller that strongly stabilizes is given by the state-space realization Discrete-Time Strong Stabilizability Definition Consider a discrete-time LTI system, , with state-space realization (, , , ), where , , and , and it is assumed that (, ) is stabilizable, (, ) is detectable, and the transfer matrix has no poles on the unit circle. The system is strongly stabilizable is there exist , , and , where , such that LMIs , , where , , , , is the solution to the discrete-time Lyapunov equation given by . Moreover, a discrete-time controller that strongly stabilizes is given by the state-space realization , . Proof this ensures that the feedback controller defined by and , renders the closed-loop system asymptotically stable and ensures that the feedback controller defined by and has a finite norm, and thus is asymptotically stable.