M7.2 Steady-State and Dynamic Behavior

We use the following parameters for a substrate inhibition model for this control study:

The steady-state dilution rate is D_s = 0.3 hr^-1 (and the residence time is 3.33 hours) and the feed substrate concentration is x_2fs = 4.0 g/liter.

Steady-State Conditions

The nonlinear process has the following three steady-state solutions (operating points) to Equation M7.1 for a dilution rate of 0.3 hr^-1:

Steady state	Biomass concentration	Substrate concentration stability
Equilibrium 1—wash out	x1s = 0	x2s = 4.0	stable
Equilibrium 2—nontrivial	x1s = 0.995103	x2s = 1.512243	unstable
Equilibrium 3—nontrivial	x1s = 1.530163	x2s = 0.174593	stable

Notice that we definitely do not want to operate a equilibrium point 1. Here there is no reaction occurring because the cells have been washed out of the reactor. The outlet substrate concentration is the same as the inlet substrate concentration under these conditions.

Linear Model

The state-space model matrices are

where the partial derivative of the specific growth rate with respect to the substrate concentration is

and where dilution rate is the manipulated input. Different control strategies have been used to control continuous biochemical reactors. One is based on measuring the biomass concentration and manipulating the dilution rate. Another is based on measuring the substrate concentration and manipulating the dilution rate.

In the following two problems, the biomass concentration is the measured output and the dilution rate is the manipulated input. You will be performing your control simulations using the nonlinear process, so the inputs and outputs from the process are in physical variable form, while the linear controller design is based on the use of deviation variables. Assume that the dilution rate is physically constrained between 0 and 0.6 hr^-1.