How Fast is Fast Enough?
The effect of analysis sample time comes down to the question of how much dead time can a sample add before feedback control noticeably deteriorates and causes instability. Intuitively you can understand that as the controller tuning slows down, the measurement speed required also slows down. Equation 1 documented at www.modelingandcontrol.com was developed to estimate when the sample dead time (ϴs) added to the original dead time (ϴo) makes the total loop dead time exceed the implied dead time from the controller tuning and causes the onset of degradation. The equation also estimates when the sample time gets so big it causes instability. The original equation was generalized to handle the pure integrating or “near-integrating” type of response of batch concentrations. Equation 1 shows that as you increase lambda (λ = closed loop time constant for self-regulating processes and the arrest time for integrating processes), the permissible amount of dead time (ϴs) from a sample increases for the standard PID algorithm.
ϴs ≤ Kx * (λ - ϴo ) (1)
ϴo = original dead time (sec)
ϴs = sample time (sec)
For self-regulating processes:
Kx = 0.5 for the start of degradation 2.0 for onset of instability
λ = closed loop time constant (sec)
For integrating processes:
Kx = 0.25 for the start of degradation 1.0 for onset of instability
λ = closed loop arrest time (sec)
To estimate the maximum allowable sample time that avoids the start of degradation and onset of instability, Kx is set equal to 0.5 and 2.0 for self-regulating and 0.25 and 1.0 for integrating processes, respectively in Equation 1. Lambda must be for the fastest response, which often depends upon its direction. For example, the glucose concentration response in a bioreactor is much faster for an increase in setpoint. The speed for a decrease in concentration depends upon the consumption rate. The attached plot from a virtual plant running 1000x real time shows that for glucose concentration control in a bioreactor, a sample time of about 11 hours causes oscillations with increasing amplitude (onset of instability) for a PID and an integrating response (Kx = 1.0) with a minimum closed-loop arrest time of 11 hours and a negligible original dead time. The figure shows that a PID algorithm enhancement for wireless control stabilizes the loop. The figure also illustrates how a feed-forward added to the PID output to pre-position the glucose flow for inoculation and increase the glucose flow with cell growth rate can considerably improve the glucose concentration control for both the standard and wireless PID, but feed-forward control could only delay, but not prevent, the eventual onset of instability for the standard PID.
The Max Slope is the Hope
Reactant and glucose concentration control also facilitates a more proficient design of experiments to identify kinetic parameters and an optimization of batch profiles by means of model-predictive control (MPC) since reaction, growth and formation rates depend upon these concentrations. For the identification of kinetic parameters, it is essential to use the slope instead of the actual product or cell concentration.
Figure 2. Reduction in batch cycle time by MPC production rate control
Table 1 lists candidates for controlled variables and manipulated variables for a bioreactor.
For many of the manipulated variables, the process action does not change between direct and reverse in the operating region. However, the pH and temperature optimum (peak in the plot of growth or formation rate versus pH or temperature) is narrow, so care must be taken to stay on one side of the peak to avoid a reversal in process action.
As the technology and availability of analyzers advances, you would think that exposing and controlling the secret profiles of batch reactors would be a top priority, since these reactors set the concentrations for downstream processes. In most cases, a slightly less-than-optimum profile would pay for a probe-type analyzer, and just one bad batch would pay for an at-line analyzer and sample system. So why maintain the status quo?
Top Ten Reasons to Keep Batch Reactor Concentration Profiles Secret
(10) Temperature and pressure is something you can feel
(9) Smart analyzers are too smart for their own good
(8) Everyone likes a good mystery
(7) Myths and war stories are interesting
(6) More meetings with catered lunches
(5) More creativity in performance reviews
(4) College texts on reaction kinetics are boring
(3) You can keep on trucking with PID control
(2) You can place bets on actual truck concentrations
(1) Purchasing is free to go with the lowest-cost supplier.
Greg McMillan is the coauthor of ControlTalk.