Error bounds of piecewise-linear approximation

A description is given of upper bounds on the error caused by piecewise-linear approximation of a nonlinear element T in a feedback loop, where T represents a nonlinear operator whose domain is the set of the z-transform of the sampled sequences. Let W´ (z) be the solution of W´ (z)=X (z)+H (z) T´ [W´ (z)] which approximates the exact input-output relation W (z)=X (z)+H (z) T [W (z)]. In order to discuss an upper bound on the norm of the approximation error || W´ (z)-W (z) ||, the generalized parametric norm of an operator is proposed. The obtained upper bound is sufficiently small compared with a bound by the conventional norm in many cases