Reachability of nonlinear fed‐batch fermentation processes

Abstract

SUMMARYThe reachability of a simple nonlinear fed-batch fermentation process model is investigated in this paper. It is shown that the known difficulties of controlling such processes are primarily caused by the fact that the rank of the reachability distribution is always less than the number of state variables.Furthermore, a co-ordinates transformation is calculated analytically that shows the nonlinear combination of the state variables which is independent of the input. The results of the reachability analysis and that of the co-ordinates transformation are independent of the source function in the system model.The results are extended to the four state variable non-isotherm case, and to nonlinear fed-batch chemical reactors with general reaction kinetics.

An algorithm for determining a class of invariants in quasi-polynomial systems

“…Note that another algorithm based on the construction of Liealgebras has been applied to a similar fermenter model with the same structure but slightly different reaction kinetics in [22], giving the same final result. As a comparison, this new method provides a computationally more advantageous way of retrieval mainly because it does not require the analytic solution of partial differential equations which was the case in [22]. Now take a look at the four monomials U 1 , .…”

Controllability of cross-flow heat exchangers

“…The control problem is non-linear since the manipulated variable _ m m w appears as a product with the unknown temperature T w ðx; tÞ in Eq. (14). The previously used linear controllability ideas cannot be directly applied in this situation.…”

“…Similarly at the other extreme, for large flow rates Eqs. (15) and (14) give T out w ¼ T in w . Thus, in general, in the steady state T out w is between the two temperatures, T in a and T in w .…”

“…This is done by approximating first-and second-order derivatives by upwind and central differences, respectively, for Eqs. (15) and (14), and eliminating the algebraic Eq. (12) to give…”

“…The previously used linear controllability ideas cannot be directly applied in this situation. Furthermore, controllability of a linearized approximation does not imply controllability over the entire state space [13,14]. Finding the range of T out w by solving Eqs.…”