On state space representation for linear discrete-time systems in Hilbert spaces
Résumé
For a linear continuous-time control system in Hilbert space with state x(t) is associated a discrete-time system where the state variable is z k = (x((k + 1)h) + x(kh))/2, with small h. This allows to introduce a discrete derivative ∆z k = (x((k + 1)h) − x(kh))/h. The obtained discrete-time system has structural properties with a similar formulation as continuous sys-tem. Stability is equivalent to the fact that the spectrum of the state oper-ator of discrete-time system is in the left half plane, Lyapunov and Riccati equation are similar.
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