M7.3 Stable Steady-State Operating Point

Design an IMC-based PID controller to control the bioreactor at equilibrium point 3—the stable nontrivial point. The steady state (also use this as the initial condition for your simulations) is

At this operating point, the state space model is

graphics/m07equ07.gif

Use MATLAB to find that the eigenvalues are –0.3 and –2.2640 hr^-1, so the system is stable and the IMC-based PID method for stable systems can be used (see Table 9-1). Find the transfer function relating the dilution rate to the biomass concentration and use this for controller design. You may wish to use the MATLAB function ss2tf to find the process transfer function

After placing the process model in gain and time constant form and recognizing pole-zero cancellation, you should find

Notice that the time constant of 0.4417 hr is significantly shorter than the residence time (3.33 hr). Also notice from Table 9-1, that the IMC-based controller is a PI controller.

Show how the response to a small setpoint change varies with l. Suggested setpoint changes are from 1.53016 to 1.52 and from 1.53016 to 1.54 g/liter.
For a particular value of l, show how the magnitude of the setpoint change affects your response. This is where the nonlinearity comes into play. Try changing from the steady-state value of 1.53016 to 1.0, 1.4, 1.6, and 2.5 g/liter. You will find problems with control saturation and for larger setpoint changes.
Often measurements cannot be made instantaneously, and there will be a transport delay associated with the measurement. Use the transport delay function in SIMULINK. Use q = 0.25 hour and discuss how the time-delay affects your choice of l.
Consider now disturbance rejection. Do not make a setpoint change but do make a step change in substrate feed concentration (x₂_f) from 4 to 3.5 at t = 1 hour. Compare disturbance rejection results with the PI controller developed in row 2 of Table 9-1.

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