On distributed delay in linear control Laws-part I: discrete-delay implementations

This note proposes two approaches to approximate distributed delay in linear control laws and, furthermore, to implement it in the z-domain and in the s-domain. The H^∞-norm of the approximation error converges to 0 when the number N of approximation steps approaches +∞. Hence, the instability problem due to the approximation error, which has been widely studied in recent years, does not exist provided that N is large enough. Moreover, the static gain is guaranteed so that no extra efforts are needed to retain the steady-state performance. As by-products, two new formulas for the forward and backward rectangular rules are obtained. These formulas are more accurate than the conventional ones when the integrand has an exponential term.