New Optimal Periodic Control Policy for the Optimal Periodic Performance of a Chemostat Using a Fourier-Gegenbauer-Based Predictor-Corrector Method

Abstract

In its simplest form, the chemostat consists of microorganisms or cells which grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions on the cells' growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an "over-yielding" state in which the performance becomes higher than that at the steady-state operation. In this paper we show that an optimal control policy for maximizing the chemostat performance can be accurately and efficiently derived numerically using a novel class of integral-pseudospectral methods and adaptive h-integral-pseudospectral methods composed through a predictor-corrector algorithm. Some new formulas for the construction of Fourier pseudospectral integration matrices and barycentric shifted Gegenbauer quadratures are derived. A rigorous study of the errors and convergence rates of shifted Gegenbauer quadratures as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T -periodic functions is presented. We introduce also a novel adaptive scheme for detecting jump discontinuities and reconstructing a discontinuous function from the pseudospectral data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.

Optimal Periodic Control of Unmanned Aerial Vehicles Based on Fourier Integral Pseudospectral and Edge-Detection Methods

“…The following notations are used throughout this paper to abridge and simplify the mathematical formulas. Many of these notations appeared earlier in [39,44]; however, for convenience and to keep the paper self-explanatory, we summarize them below together with the new notations. Logical Symbols.…”

“…We refer to Θ 2π by the "basic/principle" FIM, because of its simplest form, the "generating" FIM due to Formula (4), or the "natural" FIM, since its merits place it as the natural choice among all possible FIMs. For further information about FPS interpolation and quadratures, the reader may consult the recent works [39,42,44].…”

“…Later, optimal periodic control theory found its way in performance optimization of satellites, aircraft flights, ships and passenger cars during cruising, diesel engines, bioreactors, drug delivery, etc. [34][35][36][37][38][39].…”

“…Fourier collocation and quadrature induced by the accurate and efficient Fourier integration matrix (FIM) are then applied to discretize the problem successfully into a constrained nonlinear programming problem (NLP), which can be treated using standard optimization software. A novel edge-detection technique is also introduced to accurately locate the discontinuity points of the bang-bang thrust and effectively reconstruct it directly from the Fourier pseudospectral (FPS) data; the proposed technique, which we call the FPSED method, is a robust development of an earlier version which appeared recently in [39]. We show further that smoothing techniques combined with Fourierbased methods of low mode number or mesh densities do not generally simulate 2D-dimensional UAV flights accurately, because they can smear the discontinuities of bang-bang thrusts, and the errors in the designed thrust may cause deviations in the optimal state variables of the flight.…”

“…Moreover, the extreme thrust values are given, but the thrust optimal switching times are not known a priori. For a comprehensive survey on the excellent virtues of integral reformulations and FIMs used in our work, we refer to [39][40][41][42][43][44] and the references therein. Clear expositions of the IPS methods can also be found in [45][46][47][48][49][50] and the references therein.…”