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`Exhibit 2007
`Exhibit 2007
`
`

`

`
`
`System Dynamics
`and Control
`
`Eronini Umez-Eronini
`Morgan State University
`
`
`
`PWS Pubiishing
`
`An imprint of Brooks/Cole Publishing Company
`ICE-i?) An International Thomson Publishing Company
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`

`

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`
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`
`COPYRIGHT © 1999 by BrooksiCole Publishing Company
`A Division of International Thomson Publishing Inc.
`I®P The lTP logo is a registered trademark used herein under license.
`For more information. contact PWS Publishing at Brooks/Cole Publishing Company:
`
`BROOKSiCOLE PUBLISHING COMPANY International Thomson Editors
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`
`All rights reserved. No part of this work may be reproduced. stored in a retrieval
`system, or transcribed. in any form or by any means-—electronic. mechanical,
`photocopying, recording. or otherwise—-without the prior written permission of the
`Pacific Grove. California 93950.
`publisher. BrooksiCole Publishing Company.
`
`Printed in the United States of America.
`
`10987654321
`
`Library of Congress Cataloging-in-Publication Data
`
`Umez-Eronini. Eronini,
`System dynamics and control ;‘ Eronini Umez—Eronini.
`p.
`cm.
`Includes index.
`ISBN 0—534-94451—5
`2. Dynamics—Mathematical models.
`] . Automatic control.
`3. System analysis.
`1. Title.
`T1213.U453
`I998
`629.8—dc21
`
`98-18838
`CIP
`
`
`
`
`
`

`

`10
`
`Chapter 1
`
`Introduction
`
`time the aggregate volume of water used at the various sites is fairly uniform and pre-
`dictable, and that some variation in pressure and therefore performance can be tolerated
`in the applications is to operate the pump and fill the tank From the well on some sort
`of time schedule.
`
`Figure 1.4a shows the way the actual tank water level depends on the scheduled
`pump operation or the desired tank level. Although the pump operating schedule could
`have been deduced from, say, statistical analysis of past data on water use on the farm,
`there is nothing in the control scheme of Fig. 1.4a that prescribes any relationship be—
`tween one variable of the tank and the other (the variables in this case being Q", and
`it. The term Q"LII is related to it through the valve opening parameter C, which is taken
`here as a perturbation of the system or a disturbance input). Indeed if at any time the
`pattern of water use (equivalent valve opening) should deviate significantly from that
`on which the pump schedule is based, the control system could break down or perform
`very poorly. This type of control scheme is referred to as open-{00p amoral.
`The structure of an open—loop control system is shown in Fig. 1.4b. The specific
`characteristic or variable of the system. the output. that we wish to control is referred
`to as the mnrmiied trariaiiie. whereas the characteristic or variable that is determined
`
`by the control action is called the contmi input. The control elements can collectively
`be called the controller; although, as we shall see later, the control elements can be con-
`tained in other subgroups. The reference input generally implies the desired value of the
`controlled variable.
`In Fig.
`l.4b the loop is open in the sense that there is no path
`through which the control input can be determined in terms of the controlled variable,
`that is, the controller does not prescribe a relationship between the control input and the
`controlled variable.
`
`Figure 1.4
`Open-loop Control of Water
`Supply System
`
`Desired
`Tank Level
`
`Valve Opening
`
`Actual
`Tank Level
`
`9— Tank
`inflow
`
`
`
`—‘*—*
`
`Ptllnpi'Weil
`lSwilL‘li]
`
`—'—"'—"'
`
`Tank
`
`‘—* h
`
`(Pump Schedule}
`
`
`
`{a} Simple Control Scheme tor the Water Tank Level
`
`Disturbance Input
`
`Reference
`Input
`—»
`
`(.‘onlrol Elemenls
`ICnnlrolicri
`
`
`
`
`Control
`input
`"L—y
`
`Controlled
`Variable
`
`System
`
`lConIml Object]
`
`(1)} Open-Loop Control Structure
`
`
`
`

`

`
`
`Feedback Control
`Structure
`
`1.2
`
`Feedback Control Concept
`
`11
`
`Another approach to the control of the water tank level is to have the pump operator
`turn the pump on whenever the tank water level drops below the reference value and
`to turn it off again when the tank fills beyond the reference level. To facilitate the work
`of the operator, a level measuringftransmitting and indicating system (LT and LI} can
`be connected to the tank and used to display the tank level at a location where it can
`conveniently be observed (see Fig.
`l.5a). The result is a closed—loop control system,
`since there is a prescribed (albeit switching) relationship between the tank level and inw
`put flow. The operator observing the tank level provides the path that closes the loop.
`However, a control system with a human operator as an element of the controller is
`manual, that is, not automatic. On a busy farm or where it is not feasible to dedicate
`an employee to operation of the pump only. such manual control would clearly be in-
`
`/
`
`LSVCI
`Transmitter
`
`
`
`
`IIIIIII
`
`I
`II
`I
`
`
`Figure 1.5
`Manual and Automatic
`Control of Water Tank Level
`
`Level
`Indicator
`
`
`
`
`Operator
`
`
`
`_"' Quill
`
`{a} Manual Closed—Loop Control of Tank Level
`
`Level
`Transmitter
`
`
`+--"‘1
`
` Comparator
`
`
`
`
`and Power
`Setpoint
`(Desired Level} —r Switch
`{Threshold
`Logic)
`
`
`
`(b) Automatic C loscd—Loop Control of Tank Level
`
`Qout
`
`

`

`12
`
`Chapter 1
`
`Introduction
`
`adequate or unsatisfactory. An example of an automatic {closed-loop) control system
`for the tank level is shown in Fig.
`l.5b. Here the comparison of the tank level signal
`with the desired value of the tank level (entered into the system as a setpoint setting)
`and the turning of the pump on or off are all performed by appropriate hardware in the
`controller. However, note that both the manual and automatic level control systems. just
`described, involve
`
`I.
`
`2.
`
`the maintenance (in spite of external disturbances} of some prescribed relationship
`of one system variable to another using the result ofa comparison of these variables;
`the transmission of some signal or information from a later (output) stage to an ear-
`lier (input) stage of the system, that is, loop closure through feedback.
`
`The concept involved here is that of feedback control.
`In feedback control, a measurement of the output of the system is used to
`modify its input in such a way that the output stays near the desired value (see [2]).
`There are very many systems both naturally occurring and man-made that rely on
`feedback control for proper operation. In all cases there is usually a sensory instrument
`(sensor) orfeedback element that measures the output or system variable of interest and
`relays this measurement to a controller. The controller compares the signal with the de-
`sired value or sermon and sends appropriate instructions to the actuator or “effector"
`mechanism (orfinal control element), which then acts on the system or control object
`(or plant) to bring subsequent outputs into agreement {prescribed relationship) with the
`setpoint. What we have described is the typical feedback (or closed-loop) control struc-
`ture (see Fig. 1.6). In comparing the feedback control structure shown in Fig. 1.6 with
`the open-loop control structure shown in Fig.
`l.4b, note that the actuator could, de-
`pending on the circumstances, be considered part of the controller or part of the plant.
`The input to the actuator from the controller is called the iiiaaipulared variable.
`With the feedback control structure as a guide. we can analyze systems suspected
`of operating as feedback control processes. The idea is to identify components or
`groups of components of the system that function as the various elements in the feed-
`back control structure—controller. actuator. control object, and sensor—as well as to
`
`Conlro]
`Manipulated Variable
`
`Disturance
`
`Input
`
`
`Actuator
`
`
`System or
`
`
`
`or lE-fiec‘tor
`Comm]
`
`Ob'
`tor
`Mechanism
`
`
`
`Plait?
`or Final Comm]
`
`
`Element
`
`
`Output
`(Controlled
`.
`'.
`Variable)
`
`(Reference lnpul)
`+
`.
`.
`or (Setpolnt) —l—>
`)r (Desired Value)
`I
`l
`‘.'
`'
`I
`1
`
`Feedback Element
`
`
`
`or Sensor
`
`
`Measurement
`or Feedback Signal
`
`Figure 1.6
`
`Typical Feedback
`Control Structure
`
`
`
`

`

`
`
`1.2 Feedback Control Concept
`
`13
`
`establish the controlled variable and the setpoint. If all this cart be done, then the sys—
`tem under study is a feedback control system. Consider the case of the so-called human
`thermostat. This remarkable system maintains the body temperature at around 37°C
`(98.6”F} give or take about one degree. whether one is in Alaska in the middle of win—
`ter or in Florida in the middle of summer. In the latter case, that is, prevention ofover-
`heating, the actuation (control input) is the dilation of the blood vessels in the skin.
`which increases heat transport from the interior of the body to the surface; and sweat-
`ing, which increases the rate at which this heat is lost from the surface to the environ~
`menl, through evaporation {see [3]). The controller is located in the hypothalamus, an
`area at the base of the brain stem above the crossing of the optic nerves. But where are
`the sensors'? For a while it was thought that the skin was the predominant measurement
`device for overheat protection. An experiment was performed in which (I) the skin
`temperature. (2) the internal temperature {as measured by a thermocouple held against
`the ear drum, which is near the hypothalamus), and (3) the rate of sweating were all re-
`corded for a subject white at rest and during exercise. The results showed that the main
`sensors for overheat protection are located in the controller (in the hypothalamus) it—
`self! In Fig. [.73 the skin temperature (as a candidate feedback signal} shows no con—
`sistent relationship with the rate of sweating (a control input). Figure 1.7b. on the other
`I'
`
`80
`
`
`
`8U
`
`
`
`TU
`
`6t}
`
`5i}
`
`
`
`
`
`
`
`3|
`
`32
`
`33
`
`34
`
`3?
`
`
`.
`.
`15
`”56
`
`38
`
`
`
`
`
`
`
` 40
`
`
`
`
`
`
`
` Rateol‘Sweating(Calls)
` RateofSweating
`
`[Calm
`
`
`
`
`36.2
`
`36.4
`
`36.6
`
`36.8
`
`3?.0
`
`37.2
`
`3?.4
`
`37.6
`
`{a} SkinTcmperaturet"C)
`
`(h)
`
`InternalTemperature(”C)
`
`0 Rest
`0 Work to Calm
`it Work ( I2 Calls}
`
`Figure 1.1
`
`Heat Dissipation as a Function
`of Skin and Internal
`Temperatures. From
`“The Human Thermostat,” by
`T. H. Benzinger. Copyright ©
`1960 by Scientific American,
`Inc. All rights reserved.
`
`