Written midterm 2 - 2014

Midterm date:	13 March 2014
(PDF)	Midterm questions

Date: Thursday, 13 March
Starting time: 18:30
Duration: 2 hours
Locations:
- T29, room 101: last names A to L
- T29, room 105: last names M to Z

What will be covered in the midterm?

The midterm will cover all material in the course, up to, and including 10 March 2014. This corresponds to material in Chapters 1 to 9 in Marlin's book. The midterm will focus mainly on the material since the previous midterm, however, all material will be required.

Highlights from the newer material include:

second order systems
controller objectives
representing real systems in deviation form on a block diagram with transfer functions
closed loop response to the CV from either the SP or from disturbance \(d\)
the PID controller: understanding what each mode in the controller does
using Simulink
how to tune the controller using the Ciancone rules, and what the tuning values do to the closed loop response
system identification using the Process Reaction Curve method

Answering questions in the midterm

You may bring in any printed materials to the midterm; any textbooks, any papers, etc.
You may use any calculator during the midterm.
You may answer the questions in any order in the answer booklet.
Time saving tip: never repeat the question back in your answer.
If anything seems unclear, or information appears to be incomplete, please make a reasonable assumption and continue with the question.

How to prepare for the midterm

Understand the concepts being learned. My courses are never about finding the right equation, plugging values in and solving. You will obviously need to use equations, but the end goal is demonstrate your understanding of the material, and that you can apply it to new situations.
Read the questions carefully: they are usually worded precisely. The biggest point where students lose marks is to answer only one part of the question.
Work through many of the practice questions on the website. They have full solutions.
Work through all the tutorial problems again, making sure you understand the core concepts. The tutorials are posted on the course home page.

The midterm

<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/> .. Note::

- You may bring in any printed materials to the midterm; any textbooks, any papers, *etc*. - You may use any calculator during the midterm. - **To help us with grading, please start each question on a new page, but use both sides of each page in your booklet**. - You may answer the questions in any order on all pages of the answer booklet. - This exam requires that you apply the material you have learned here in 3P to new, unfamiliar situations, which is the level of thinking we start to require from students that will be graduating or working in co-ops in a year from now. - Any ambiguity or lack of clarity in a question may be resolved by making a suitable and justifiable assumption, and continuing to answer the question with that assumption(s). - **Total marks**: 54 marks, 12.5% of course grade. - Total time: 2 hours. There are 4 pages on the exam, please ensure your copy is complete.

.. question:: :grading: 6 x 2 = 12

The questions below are related to a PID controller, with control law given in the time-domain. This is the form recommended for implementation.

.. math::

\text{MV}(t) = K_c \left(E(t) + \dfrac{1}{T_I}\int_0^t{E(t^\ast)\,dt^\ast}-T_D \frac{d\,\text{CV}}{dt} \right)

#. Clearly describe in one sentence what the purpose of the integral mode is, i.e. the :math:`T_I` term.

#. We learnt in class that the last term is derived in textbooks as

.. math::

+ T_D \dfrac{d\,\text{E}}{dt}

Why is the recommended form, shown at the start of this question, preferred over this form?

#. Under which condition is the derivative mode often set to zero?

#. Should the integral mode ever be turned off? Explain your answer in a single sentence.

#. We learnt about the Ciancone tuning rules (*see last page of this test*) to find initial values of :math:`K_c`, :math:`T_I` and :math:`T_D`. What main criterion is these Ciancone rules optimizing to provide tuning values for a first-order plus time delay (FOPTD) process?

#. If you were setting the tuning for a FOPTD process and did not have a copy of the Ciancone rules with you, what single value of :math:`\dfrac{T_I}{\theta+\tau}` would you remember for the rest of your career that would be a good enough guess for :math:`T_I`? Justify why.

.. answer::

#. The integral mode provides persistent manipulated variable input to the process, until the error is reduced to zero, ensuring offset-free behaviour.

#. The recommended form is derived by eliminating the derivative of the set point signal. That derivative is unbounded for step changes down or up in set point, and would otherwise lead to disruptive and sudden changes in the manipulated variable.

#. It is set to zero for noisy signals (noisy CV measurements), or in situations when it is not necessary, such as when dead time, :math:`\theta \approx 0`, as can be seen in the Ciancone correlation charts.

#. It is not usual to turn of the integral mode, as offset free behaviour is very desirable for almost all feedback control loops.

#. As described in the course textbook, the Ciancone turning rules are derived by optimizing several competing criteria. The main criterion is turning for minimum IAE (integral absolute error).

#. Using a value of :math:`\dfrac{T_I}{\tau + \theta} \approx` 0.6 to 0.7 would be suitable. Most loop tuning, at least for an initial starting guess, would work adequately at these values, for all values of :math:`\tau` and :math:`\theta`. It is not technically correct to say that :math:`T_I = 0.6`, since that is not the value on the :math:`y`-axis of the tuning plots.

.. question:: :grading: 18 = 5 + 4 + 5 + 4

#. Describe what the time-domain response to a step input for the following heat exchanger system would be, given by the transfer function below. Show all calculations. **[5]**

.. math::

\dfrac{T_\text{hot,out}(s)}{F_\text{steam,in}(s)} = \dfrac{2}{15s^2 + 8s + 1}

.. image:: ../figures/process-control/midterm-exams/2014-midterm-2-heat-exchanger.png :scale: 30 :width: 750px :align: center

#. For the above heat-exchanger system, the objective is to control the outlet temperature :math:`T_\text{hot,out}` by manipulating the :math:`F_\text{steam,in}`. Give two complete and specific examples of disturbances that will upset the process if there is no feedback control. **[4]**

#. You are responsible for operating the primary reactor in your plant. It is a CSTR with volume :math:`5.6\,\text{m}^3`. The current flow rate into the tank is :math:`0.4\,\text{m}^3\text{.min}^{-1}`.

You increase the feed concentration by 20% to alter the reaction kinetics and product purity. Approximately how long before the tank stabilizes again? **[5]**

#. The following output is from a process that is controlled by a PI controller.

.. image:: ../figures/process-control/Marlin-slides/Marlin-Fig-9-13.png :width: 750px :scale: 80

#. What two things can you change to reduce the IAE? **[2]** #. What will happen to the manipulated variable when you make those changes? **[2]**

.. answer::

#. The question clearly asks for a description only, so actual inversion to the time-domain, while not incorrect, is unnecessary.

The system will be an overdamped second-order response. We observe that by inspecting the roots of the denominator :math:`\left(-\dfrac{1}{5}\,\text{and}\,-\dfrac{1}{3} \right)`. Since both are real, and negative, we expect a second-order overdamped response.

You may chose to evaluate the final value, which will simply be double the magnitude of the input supplied to the system.

#. Two disturbances expected might be:

* the temperature of the heating steam will fluctuate (e.g. drop), which will change the amount of heat transferred (reduced), and cause the outlet temperature to change (decrease). * the flow of the cold stream entering will vary (e.g. increase), which will change the residence time in the heat exchanger (lower), and cause the outlet temperature change (drop).

#. You might have answered this question purely from an intuitive point of view, having seen the :math:`\tau = \dfrac{V}{F} = \dfrac{5.6\,\text{m}^3}{0.4\,\text{m}^3\text{.min}^{-1}} = 14\,\text{min}` term so many times in the prior course examples.

However, the main assumption you must clearly state, is that you are assuming a first order process, and that it is the time-constant, :math:`\tau`, for that first-order process.

If you were lost, you could do a quick symbolic calculation for tank system, such as

.. math::

V \dfrac{dC_A}{dt} &= F C_{A0} - FC_A - V (r_A) \\ \dfrac{V}{F} s C_A'(s) &= ....

to prove to yourself the :math:`\frac{V}{F}` terms appears in front of the :math:`s`, which is the time-constant :math:`\tau`.

Then to answer the question of "how long", we can state that it typically takes 5 time constants for the system to reach steady state and stabilize again, so :math:`5 \times 14 = 70` minutes would be a good approximation.

#. #. The IAE may be reduced by a *slight* increase in :math:`K_c` and a slight decrease in :math:`T_I`. We cannot make large changes, because at some point the loop will over-compensate, become oscillatory, and increase the IAE again. We showed the quadratic nature of these functions in class.

#. As the loop tuning is tightened, and made more aggressive, the manipulated variable will show greater variability in its trajectory. In other words, it will rise faster and more steeply, perhaps even overshooting its required value, and have to come down again.

</rst>

Written midterm 2 - 2014

The midterm

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