Which Lp-norm for evaluating fractional system performances?*

Lp-norms, 1 ≤ p ≤ ∞, of stable fractional systems, which include or not a pure fractional integrator, are considered. It is proven, in this paper, that the Lp-norm is finite if and only if (i) transfer function relative degree is greater than (1 – 1/p) and (ii) the integrator order (if any) is less than (1 – 1/p). Consequently, when the stable fractional system has no pure integrator, its L1-norm is always finite independently of its relative degree. The established finiteness conditions may help choosing a performance index for evaluating different output feedback control laws.