Comparative Performance of Decoupled Input–Output Linearizing Controller and Linear Interpolation PID Controller: Enhancing Biomass and Ethanol Production in Saccharomyces cerevisiae

Abstract

A decoupled input–output linearizing controller (DIOLC) was designed as an alternative advanced control strategy for controlling bioprocesses. Simulation studies of its implementation were carried out to control ethanol and biomass production in Saccharomyces cerevisiae and its performance was compared to that of a proportional–integral–derivative (PID) controller with parameters tuned according to a linear schedule. The overall performance of the DIOLC was better in the test experiments requiring the controllers to respond accurately to simultaneous changes in the trajectories of the substrate and dissolved oxygen concentration. It also exhibited better performance in perturbation experiments of the most significant parameters q _S,max, q _O2,max, and k _s , determined through a statistical design of experiments involving 730 simulations. DIOLC exhibited a superior ability of constraining the process when implemented in extreme metabolic regimes of high oxygen demand for maximizing biomass concentration and low oxygen demand for maximizing ethanol concentration.

Appendix

The equations for the Sonnleitner and Kappeli model are given below (Eqs. A1–A4)

$$ \frac{{dx}}{{dt}} = Y_{{{\rm{bio}}/{\rm{glu}}}}^{{{\rm{oxid}}}}\frac{{{{q}_{{{{O}_{2}},glu\max }}}}}{a}\left( {\frac{{{{c}_{L}}}}{{{{c}_{L}} + {{k}_{o}}}}} \right)x + Y_{{{\rm{bio}}/{\rm{glu}}}}^{{{\rm{red}}}}\left( {{{q}_{S}} - q_{{{{O}_{2}}}}^{{{\rm{oxid}}}}} \right)x + {{Y}_{{{\rm{bio}}/{\rm{ethanol}}}}}{{q}_{{{\rm{ethanol}}}}}x - Dx $$

(A1)

$$ \frac{ds }{dt }=-{q_{{S,\max }}}\frac{s}{{s+{k_s}}}x+D\left( {{s_F}-s} \right) $$

(A2)

$$ \frac{de }{dt }=Y_{\mathrm{ethanol}/\mathrm{glu}}^{\mathrm{red}}\left( {{q_S}-q_{{{O_2}}}^{\mathrm{oxid}}} \right)x-\frac{{{q_{{{O_2},\mathrm{ethanol},\max }}}}}{k}\frac{{{c_L}}}{{{c_L}+{k_o}}}x-De $$

(A3)

$$ \frac{{d{c_L}}}{dt }={k_L}a\left( {c_L^{*}-{c_L}} \right)-{q_{{{{\mathrm{O}}_2}\text{,}\mathrm{glu}\text{,}\max }}}\frac{{{c_L}}}{{{c_L}+{k_o}}}x-{q_{{{{\mathrm{O}}_2}\text{,}\mathrm{ethanol}\text{,}\max }}}\frac{{{c_L}}}{{{c_L}+{k_o}}}x-D{c_L} $$

(A4)

Equation 1 was modified to take into account the minimum of the oxidative flux—flux used by glucose and the ethanol as follows:

$\frac{dx }{dt }=Y_{\mathrm{bio}/\mathrm{glu}}^{\mathrm{oxid}}\frac{{{q_{{{O_2},glu\max }}}}}{a}(\frac{{{c_L}}}{{{c_L}+{k_o}}})x+Y_{\mathrm{bio}/\mathrm{glu}}^{\mathrm{red}}\left( {{q_S}-q_{{{O_2}}}^{\mathrm{oxid}}} \right)x+{Y_{\mathrm{bio}/\mathrm{eth}}}\frac{{{q_{{{O_2}\text{,}\mathrm{eth},\max }}}}}{k}\left( {\frac{{{c_L}}}{{{c_L}+{k_o}}}} \right)x-Dx$ These equations are then rewritten in the following structural form:

$$ \left[ \begin{array}{*{20}c} {\dot{x}} \hfill \\ {\dot{s}} \hfill \\ {\dot{e}} \hfill \\ {{\dot{c}}_L} \hfill \\\end{array} \right]=\left[ \begin{array}{*{20}c} ({\alpha_1}-{\alpha_3}){\mu_1}(s,{c_L})+{\alpha_2}{\mu_2}(s)+{\alpha_4}{\mu_3}(s,e,{c_L}) \hfill \\ -{\alpha_4}{\mu_2}(s) \hfill \\ -{\alpha_6}{\mu_1}(s,{c_L})+{\alpha_5}{\mu_2}(s)-{\alpha_7}{\mu_3}(s,e,{c_L}) \hfill \\ -{\mu_1}(s,{c_L})-{\mu_3}(s,e,{c_L}) \hfill \\\end{array} \right]x+\left[ {\begin{array}{*{20}c} {-x} \hfill & 0 \hfill \\ {{s_F}-s} \hfill & 0 \hfill \\ {-e} \hfill & 0 \hfill \\ {-{c_L}} \hfill & {c_L^{*}-{c_L}} \hfill \\ \end{array}} \right]\left[ {\begin{array}{*{20}c} D \\ {{k_L}a} \\ \end{array}} \right] $$

(A5)

Where, the kinetic structures are given by ${\mu_1}\left( {s,{c_L}} \right)={q_{{{O_2},glu,\max }}}\frac{{{c_L}}}{{{c_L}+{k_o}}}$, ${\mu_2}(s)=\frac{s}{{s+{k_s}}}$ and ${\mu_3}(s,e,{c_L})={q_{{{O_2},\mathrm{ethanol},\max }}}\frac{{{c_L}}}{{{c_L}+{k_o}}}$. Similarly, the parameters were defined as given below:

$$ {\alpha_1}=\frac{{Y_{\mathrm{bio}/\mathrm{glu}}^{\mathrm{oxid}}}}{a},{\alpha_2}=Y_{\mathrm{bio}/\mathrm{glu}}^{\mathrm{red}}q_{\mathrm{sub}}^{\max },{\alpha_3}=\frac{{Y_{\mathrm{bio}/\mathrm{glu}}^{\mathrm{red}}}}{a},{\alpha_4}=\frac{{{Y_{\mathrm{bio}/\mathrm{eth}}}}}{k},{\alpha_5}=Y_{\mathrm{eth}/\mathrm{glu}}^{\mathrm{red}}q_{\mathrm{sub}}^{\max },{\alpha_6}=\frac{{Y_{\mathrm{eth}/\mathrm{glu}}^{\mathrm{red}}}}{a} $$

And ${\alpha_7}=\frac{1}{k}$