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Student #_________________

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UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

Final Exam

April 26, 2018

C11E3225 - Process Dynamics and Control

Instructor - W.R. Cluett

This is a closed book exam.

Put all work to be marked on the front of each page. Use back of page for rough work only.

Review the GIVEN INFORMATION on the next page before starting the exam.

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GIVEN INFORMATION

Linearization based on Taylor Series Expansion:

dx =f(x,u)

dt

Of Li f(x, u) f(x5 , u) + — I (x - x 5) + (u - u5 ) T XSS,USS au Xss,Uss

Laplace Transforms:

f(t) - F(s)

S(t) (unit step)

Final Value Theorem:

'f—)oo f(t) = 1im...+0[sF(s)]

Relative Gain: 2 I = 1

1— K12K21

'11'22

Unit Step Response of—j: K(1 - e)

Inverse of a 2x2 Matrix: [a bj = 1 [d Ic d ad—bc—c a'

Roots of ax + bx + c = 0: x = —b±/b2-4ac

2a

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1. (25 marks) Consider the continuous stirred-tank reactor (CSTR) shown in the figure below:

F

CAO

The reactor is operating with a constant volume and at a constant temperature. The flowrate is also constant. The reaction taking place in the reactor A - B is first order with rate expression rA = —kcA. The knowledge-driven model for the material balance on component A is given by Equation 1.

Equation 1:

d CA = F(CAO - CA) - VkCA

dt

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If there was no reaction taking place in the reactor, the model for the material balance on component A would instead be given by Equation 2:

Equation 2: dcA

V=F(CAo CA) dt

The units associated with the concentrations are mol/m3 , flowrate has units of m3/min, volume has units of m3, and the units of the rate constant k are min-'.

If we treat Equation 2 (no reaction) as the original system behavior, we can see that the presence of the reaction in Equation 1 adds a secondary effect that modifies both the steady-state and the dynamic behavior of the original system. In lecture this effect was referred to as an internal, feedback-type of interconnection.

i) Based on Equation 1, derive the transfer function relating CAO' (S) to CA(S),

express this transfer function in its standard form, and include units for the parameters.

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Based on Equation 2, derive the transfer function relating CAO' (s) to CA' (s), express this transfer function in its standard form, and include units for the parameters.

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By comparing the two transfer functions derived in parts (i) and (ii), what is the effect of the reaction on the dynamic behavior of the system? Use sketches of the unit step responses to support your written answer.

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iv) How are your results in part (iii) consistent with the effect of the reaction being a feedback-type of interconnection?

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v) Why is the effect of the reaction considered to be an internal interconnection?

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2. (15 marks) Now let's assume that the reaction taking place in the CSTR in Question 1 is second order with a rate expression of the form TA = —kcA2 with the units of the rate constant k

being [() min] 1. In this case, the knowledgeMO1

-driven model for the material balance

on component A is given by Equation 3.

Equation 3:

V- dt =F(cAo—cA)—Vkc

i) Based on Equation 3, derive the transfer function relating CAO' (s) to CA' (s), express this transfer function in its standard form, and include units for the parameters.

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Blank page for 2 i) if needed.

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By comparing the two transfer functions derived from Equations 1 and 3, what is the effect of the reaction being second order as compared to being first order on the dynamic behavior of the system?

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3. (35 marks) Now consider the system of four interconnected CSTRs shown in the figure below:

CAO

FI F

CAt

A2

V cA3

Let's assume that the reaction taking place in each reactor A -+ B is first order. Therefore, the knowledge-driven model for the material balance on component A for each reactor is given by Equation 1 with CAO and CA replaced by the appropriate inlet and outlet concentration terms associated with each reactor. For this system of four interconnected CSTRs, provide the following information with explanation:

i) The type of interconnection.

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ii) The transfer function relating CAO' (s) to CA4' (s).

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iii) The order of this system.

iv) The stability of this system.

v) The damping of this system.

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vi) The gain of this system.

vii) The approximate shape of the response in CA4 to a step change in CAO.

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4. (20 marks) A combination of two drugs, hydrochiorothiazide and oxybutynin, is commonly used to control both blood pressure and urine production rate in elderly patients. The goal is to control both variables using a multi-loop control approach. A model of this system has been developed and is given by Equation 4.

Equation 4:

—0.04e°

[Y11 - 0.11s+1 Y2J - 0.22

0.12s + 1

0.0005e_OSs

021s +1 [Ui

—0.02 lu2

0.21s + 1

Where yi and Y2 denote the blood pressure and urine production rate, respectively, and U1 and u2 denote the rate of hydrochiorotliiazide ingestion and rate of oxybutynin ingestion, respectively. All four variables shown in Equation 4 represent the Laplace transform of the corresponding deviation variables.

A definition for controllability of this system in the absence of any disturbances is as follows: a linear, dynamic system is controllable if the manipulated variables can be adjusted in such a way as to move the controlled variables from their initial setpoint values to an arbitrary set of final setpoint values with zero offset. Mathematically speaking, the system described by Equation 4 is controllable if there is a unique solution to Equation 5.

Equation 5:

[Yi1 - [Ku K121 [Iu1 [Y2J IK21 K22 ] [Lu2

where Lx denotes the final value of variable x minus the initial value of variable x.

See next page for part i) of this question.

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i) Is this system controllable? Explain your answer.

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What changes in the two drug ingestion rates would be required to change the

blood pressure by +0.1 and the urine production rate by -0.1, i.e. Ay,

=

[+0.1]9 I--0.1J.

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Determine the relative gain A11 and use its value to recommend the best pairing decision for the multi-loop control approach. Explain your answer.

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5. (30 marks) i) Draw the standard block diagram for a dynamic system consisting of one

manipulated input (u), one disturbance input (d), and one controlled output (y) that is under feedforward control. Use G, Gd, and GFF in your block diagram.

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Derive the overall transfer function for this feedforward-controlled system that relates the disturbance input (d) to the controlled output (y). Use G, Gd, and GFF in your derivation.

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If G and Gd are known to be stable transfer functions, what are the requirements on GFF for the overall system to be stable? Explain your answer.

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iv) Draw the standard block diagram for a dynamic system consisting of one manipulated input (u), one disturbance input (d), and one controlled output (y) that is under feedback control. Use G, Gd, and GFB in your block diagram.

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v) Derive the overall transfer function for this feedback-controlled system that relates the disturbance input (d) to the controlled output (y). Use G, Gd, and GFB in your derivation.

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t\ vi) Consider the case where G = K - and GFB = K

( 1 + -) = K(—). In this

case both G and GFB are unstable transfer functions. What are the requirements on GFB, in particular on the controller parameters K and tV,, for the overall system to be stable, assuming K> 0? Explain your answer.

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6. (30 marks) As part of the design project in this course, students had the opportunity to design feedback controllers for three different types of process dynamics, namely first order (FC), integrating (LC12 and LC1 1) and processes with delay (DCC and BCC). Design of feedback controllers for first order and integrating processes was relatively 'easy" in the sense that these controlled systems could be designed to be arbitrarily tight. However, this was not the case with DCC and BCC because of the presence of time delays, larger time constants, and loop interaction.

An example of a type of process dynamic that we studied in the first part of the course but did not have the opportunity to study in the second part of the course is given by the transfer function found in Equation 6. Feedback control 91 systems like the one found in Equation 6 are limited in terms of achievable performance because of the presence of a right-half plane zero.

Equation 6:

5(—s+1) G(s)

(2-)(3s+ 1)

i) What are the poles and zeros of this process?

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If a unit step change was made to the input to this process, what would be the final value of the process output?

iii) Sketch the unit step response of this system.

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iv) Assume that we put this process under feedback control using a proportional- only controller (GFB = ICE). Draw the root locus diagram associated with this feedback-controlled system assuming we use a positive value for K. To help you develop this diagram, determine the two Kc values at which the closed-loop poles are real and equal (b2 - 4ac = 0) and determine the Kc value at which the closed-loop poles are on the imaginary axis (—b = 0), where a, b and c denote the coefficients of the equation ax2 + bx + c = 0.

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Blank page for 6 iv) if needed.

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v) Assume that a value of Kc = 0.25 is selected. Derive the closed loop transfer function for the servo control problem. Sketch the step response of this closed loop system to a unit step change in the setpoint showing as much detail as possible.

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