Strong stability and the Cayley transform

The general notion of 'strong' stability for internal autonomous system descriptions has been recently introduced for continuous and discrete-time systems. This is a stronger notion of stability compared with alternative definitions (asymptotic, Lyapunov), which prohibits systems described by natural coordinates to have overshooting responses, for arbitrary initial conditions in state space. The paper reviews three refined notions of strong stability, along with the necessary and sufficient conditions corresponding to each notion. Using the Cayley transformation, it is shown that the notions in the two domains are essentially equivalent and that the strong stability conditions can be transformed from one domain to the other in a straightforward way. © 2013 John Wiley & Sons, Ltd.