Fall 2004 Problem Set #2
The drug Hydrochlorothiazide ("Diazide") is sometimes used to regulate blood pressure. However, it is also a diuretic and can cause a substantial increase in urine production. To counter this side effect, the antispasmodic drug Oxybutynin ("Ditropan") is often prescribed as well. However, Oxybutynin is known to increase ocular pressure and can be quite dangerous for patients with glaucoma.
An alternative experimental drug, "OGZ", has been developed and is purported to have much milder side effects when used in conjunction with Hydrochlorothiazide. Trials indicate that although OGZ can counteract the increased production of urine, it also causes a slight increase in blood pressure.
A clinical testing procedure has produced the following transfer function model describing the effects of regular doses of Hydrochlorothiazide and OGZ:
- rate of Hydrochlorothiazide ingestion (mg/day)
(Problem originally from Ogunnaike and Ray, 1994).
Can both blood pressure and urine production be controlled using these two drugs?
Using the Relative Gain Array, discuss the seriousness of the OGZ interactions with the Hydrochlorothiazide.
A multi-loop control system is to be configured for a stirred tank chemical reactor making a vinyl
acetate (VA)-methyl-methacrylate (MMA) copolymer. The following transfer function model as been developed from a mechanistic model of the plant, and is to be used to determine a preliminary control configuration. The variables are all expressed as deviations from their respective steady state values.
The outputs are: polymer production rate (kg/hr), mole fraction of MMA in the copolymer, and weight average molecular weight respectively. The inputs are: flowrate of VA monomer (kg/hr), flowrate of initiator (kg/hr), and flowrate of chain transfer agent (kg/hr).
The reactor is also subject to fluctuations in the flowrate of MMA monomer to the reactor; the corresponding disturbance vector is also indicated below.
- flowrate of vinyl acetate monomer (kg/hr)
- flowrate of chain transfer agent (kg/hr)
This model was proposed in the literature by a well-known polymer research group at the University of Wisconsin. A polymer modeling software product from this group is now the basis of a polymer simulation function in the HYSIS® commercial simulation software package. (a)
Propose a control configuration for this process, and justify your choice using the various tools discussed in class. In particular, you should address interaction effects, controllability, and the loop dynamics in your analysis.
Note that the process has a special structure, and one of the loop pairing choices is in fact fixed. Why? You should be able to reduce the pairing problem to a 2 x 2 problem.
Note - if you get stuck at this point and can't see the 2 x 2 problem, please come and see me before continuing.
Identify the transmission interaction in this process.
For the remainder of this question, consider ONLY the 2 x 2 sub-problem identified in part (a). (c)
As mentioned above, the process is subject to a disturbance consisting of fluctuations in the flowrate of MMA monomer. The disturbance transfer function vector is summarized above. Using the Relative Disturbance Gain, determine whether the interaction associated with your proposed loop configuration in (a) will help or hinder the rejection of the disturbance. Explain your result.
Determine the characteristic equation for your closed-loop 2 x 2 process using Maple. In order to do this, it will be necessary to have tuning constants. Use PI controllers for each loop. A rough choice of proportional gain can be obtained as 1/Kp, where Kp is the process gain. Choose a nominal value for the integral time. Assess the stability of the closed-loop process under these preliminary tuning constants.
Once you have assessed the stability using the characteristic equation, determine the closed-loop servo transfer function matrix using Maple. The necessary commands are contained in the Maple worksheet available from the Maple help page on the course web.
Determine a simplified decoupler for your control configuration (the term “simplified” is per Marlin’s use; Seborg et al. use the term “ideal”). Are the components of the decoupler physically realizable? Will a decoupler help the disturbance rejection performance of your control system? Explain your answer, and identify additional concerns (if any).
In preparation for controller design, empirical transfer functions have been identified for the input and output variables in a heavy oil fractionation column. The next step is to identify the most feasible control configuration. The input variables are the top draw rate and the intermediate reflux duty, while the output variables are the top end point (a measure of the heaviest components in the overhead product), and the intermediate reflux temperature (an indication of the composition in the column). The fractionator is subject to disturbances in an intermediate reflux stream duty located lower in the column (the atmospheric gas oil “pumparound”). The duty disturbance introduces fluctuations up the column, and the disturbance transfer function shown below describes the effect of these fluctuations.
The gains are dimensionless, and the units of time are in minutes. This model was proposed by Shell Development Co. (based in Houston) as a test problem for control analysis.
Propose a control configuration for this process, and justify your choice using the various tools discussed in class. In particular, you should address interaction effects, possible instability, controllability, and the loop dynamics in your analysis.
As mentioned above, the process is subject to a disturbance consisting of fluctuations in the heating medium flowrate. The disturbance transfer function vector is summarized above. Using the Relative Disturbance Gain, determine whether the interaction associated with your proposed loop configuration in (a) will help or hinder the rejection of the disturbance. Explain your result.
Using the tuning method of your choice (please identify which method you are using!), determine single-loop controller tunings for your proposed configuration. The primary objective of these controllers is to reject the disturbance, so bear this in mind when tuning your controllers. Simulate the multivariable closed-loop performance of your control configuration in response to the disturbance described in (b), and include disturbance responses in your assignment. You should use the 2x2 Simulink worksheet that we worked with in the tutorials. Comment on the closed-loop performance for 2x2 problem using these single-loop tuning values. One rule of thumb for adjusting PID tunings in the presence of interaction has been proposed by McAvoy, and involves reducing the controller proportional gains depending on the relative gain. Specifically,
⎧⎪(λ − λ 2− λ)K* , λ > 1
ci is the proportional gain obtained from the single-loop tuning rules. Use this rule to adjust
your single-loop tunings, and discuss whether this approach improves or detracts from the performance of the 2x2 system with single-loop tuning.
Assume that the time delays are negligible, and determine the characteristic equation for your closed-loop process. Assess the stability of the closed-loop process with your tuning constants from (c). (Note - I suggest using Maple to do this task. You might find the “fsolve( , complex)” command useful – a sample worksheet will be available from the course web – Maple help page).
Determine a simplified decoupler for your control configuration (the term “simplified” is per Marlin’s use; Seborg et al. use the term “ideal”). Are the components of the decoupler physically realizable? Will a decoupler help the disturbance rejection performance of your control system? Explain your answer, and identify additional concerns (if any).

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