Higher-Order Corrections to the Pi Criterion Using Center Manifold Theory

The frequency-dependent pi criterion of Bittanti et al. [7] is a very important tool that has been used extensively in applications, to predict potential performance improvement under periodic forcing in a nonlinear system. The pi criterion is local in nature and provides an approximate formula for the performance index under small-amplitude periodic forcing. Motivated by basic results from Center Manifold theory, the present work develops higher-order approximations, suitable for periodic forcing functions of larger amplitude. The proposed method is based on solving the Center Manifold partial differential equation via power series. The end result of the proposed approach is the approximate calculation of the performance index in the form of a truncated series expansion, which provides accurate results under larger amplitudes. The proposed method is applied to a continuous stirred tank reactor, where the yield of the desired product must be maximized.