Glycolysis is a feedback process through which living cells use glucose to generate adenosine triphosphate (ATP), necessary for cell operations. A linearized glycolysis model (Chandra, 2011) is given by

where δ is the perturbation (disturbance input) on ATP production, Δy is the change in ATP level (output). α > 0 is the cooperativity of ATP binding to PFK, g > 0 is the feedback strength of ATP on PK (PFK and PK are two different types of glycolytic enzymes), k > 0 is the intermediate reaction rate, q > 0 is the autocatalytic stoichiometry, and h > 0 is the feedback strength of ATP on the PFK enzyme.

a. Since in this system δ is a disturbance input, zero steady-state error is achieved when Δy/δ = 0. Show that in steady state (Δ̇ x = 0; Δ̇ y = 0), Δy/δ = 1/a - h.

b. Use the Routh-Hurwitz stability criterion to show that the system will be closed-loop stable as long as

c. Assuming that h is the only parameter of choice for steady-state error adjustments, show that zero steady-state error is not achievable.

a+ g+h (q+ 1)k -qa - g(q +1)+ qh][Ay. -k