i i i i i i i i i i i i i i i i i i i i i i i i “73605_C000” — 2010/10/28 — 7:41 — page v — #5 i i i i MATLAB® and Simulink® are trademarks of The MathWorks, Inc and are used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number: 978-1-4200-7360-7 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have 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www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Control system applications / edited by William S Levine 2nd ed p cm (The electrical engineering handbook series) Includes bibliographical references and index ISBN 978-1-4200-7360-7 Automatic control Control theory I Levine, W S II Title TJ225.C66 2011 629.8 dc22 2010026364 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com i i i i i i Contents Preface to the Second Edition xi Acknowledgments xiii Editorial Board xv Editor xvii Contributors xix SECTION I Automotive Linear Parameter-Varying Control of Nonlinear Systems with Applications to Automotive and Aerospace Controls 1-1 Hans P Geering Powertrain Control 2-1 Davor Hrovat, Mrdjan Jankovic, Ilya Kolmanovsky, Stephen Magner, and Diana Yanakiev Vehicle Controls 3-1 Davor Hrovat, Hongtei E Tseng, Jianbo Lu, Josko Deur, Francis Assadian, Francesco Borrelli, and Paolo Falcone Model-Based Supervisory Control for Energy Optimization of Hybrid-Electric Vehicles 4-1 Lino Guzzella and Antonio Sciarretta Purge Scheduling for Dead-Ended Anode Operation of PEM Fuel Cells 5-1 Jason B Siegel, Anna G Stefanopoulou, Giulio Ripaccioli, and Stefano Di Cairano SECTION II Aerospace Aerospace Real-Time Control System and Software 6-1 Rongsheng (Ken) Li and Michael Santina Stochastic Decision Making and Aerial Surveillance Control Strategies for Teams of Unmanned Aerial Vehicles 7-1 Raymond W Holsapple, John J Baker, and Amir J Matlock Control Allocation 8-1 Michael W Oppenheimer, David B Doman, and Michael A Bolender Swarm Stability 9-1 Veysel Gazi and Kevin M Passino vii i i i i i i viii Contents SECTION III 10 Industrial Control of Machine Tools and Machining Processes 10-1 Jaspreet S Dhupia and A Galip Ulsoy 11 Process Control in Semiconductor Manufacturing 11-1 Thomas F Edgar 12 Control of Polymerization Processes 12-1 Babatunde Ogunnaike, Grộgory Franỗois, Masoud Soroush, and Dominique Bonvin 13 Multiscale Modeling and Control of Porous Thin Film Growth 13-1 Gangshi Hu, Xinyu Zhang, Gerassimos Orkoulas, and Panagiotis D Christofides 14 Control of Particulate Processes 14-1 Mingheng Li and Panagiotis D Christofides 15 Nonlinear Model Predictive Control for Batch Processes 15-1 Zoltan K Nagy and Richard D Braatz 16 The Use of Multivariate Statistics in Process Control 16-1 Michael J Piovoso and Karlene A Hoo 17 Plantwide Control 17-1 Karlene A Hoo 18 Automation and Control Solutions for Flat Strip Metal Processing 18-1 Francesco Alessandro Cuzzola and Thomas Parisini SECTION IV 19 Biological and Medical Model-Based Control of Biochemical Reactors 19-1 Michael A Henson 20 Robotic Surgery 20-1 Rajesh Kumar 21 Stochastic Gene Expression: Modeling, Analysis, and Identification 21-1 Mustafa Khammash and Brian Munsky 22 Modeling the Human Body as a Dynamical System: Applications to Drug Discovery and Development 22-1 M Vidyasagar SECTION V 23 Electronics Control of Brushless DC Motors 23-1 Farhad Aghili 24 Hybrid Model Predictive Control of the Boost Converter 24-1 Raymond A DeCarlo, Jason C Neely, and Steven D Pekarek i i i i i i ix Contents SECTION VI 25 26 Networks The SNR Approach to Networked Control 25-1 Eduardo I Silva, Juan C Agüero, Graham C Goodwin, Katrina Lau, and Meng Wang Optimization and Control of Communication Networks 26-1 Srinivas Shakkottai and Atilla Eryilmaz SECTION VII Special Applications 27 Advanced Motion Control Design 27-1 Maarten Steinbuch, Roel J E Merry, Matthijs L G Boerlage, Michael J C Ronde, and Marinus J G van de Molengraft 28 Color Controls: An Advanced Feedback System 28-1 Lalit K Mestha and Alvaro E Gil 29 The Construction of Portfolios of Financial Assets: An Application of Optimal Stochastic Control 29-1 Charles E Rohrs and Melanie B Rudoy 30 Earthquake Response Control for Civil Structures 30-1 Jeff T Scruggs and Henri P Gavin 31 32 Quantum Estimation and Control 31-1 Matthew R James and Robert L Kosut Motion Control of Marine Craft 32-1 Tristan Perez and Thor I Fossen 33 34 Control of Unstable Oscillations in Flows 33-1 Anuradha M Annaswamy and Seunghyuck Hong Modeling and Control of Air Conditioning and Refrigeration Systems 34-1 Andrew Alleyne, Vikas Chandan, Neera Jain, Bin Li, and Rich Otten Index Index-1 i i i i i i Preface to the Second Edition As you may know, the first edition of The Control Handbook was very well received Many copies were sold and a gratifying number of people took the time to tell me that they found it useful To the publisher, these are all reasons to a second edition To the editor of the first edition, these same facts are a modest disincentive The risk that a second edition will not be as good as the first one is real and worrisome I have tried very hard to insure that the second edition is at least as good as the first one was I hope you agree that I have succeeded I have made two major changes in the second edition The first is that all the Applications chapters are new It is simply a fact of life in engineering that once a problem is solved, people are no longer as interested in it as they were when it was unsolved I have tried to find especially inspiring and exciting applications for this second edition Secondly, it has become clear to me that organizing the Applications book by academic discipline is no longer sensible Most control applications are interdisciplinary For example, an automotive control system that involves sensors to convert mechanical signals into electrical ones, actuators that convert electrical signals into mechanical ones, several computers and a communication network to link sensors and actuators to the computers does not belong solely to any specific academic area You will notice that the applications are now organized broadly by application areas, such as automotive and aerospace One aspect of this new organization has created a minor and, I think, amusing problem Several wonderful applications did not fit into my new taxonomy I originally grouped them under the title Miscellaneous Several authors objected to the slightly pejorative nature of the term “miscellaneous.” I agreed with them and, after some thinking, consulting with literate friends and with some of the library resources, I have renamed that section “Special Applications.” Regardless of the name, they are all interesting and important and I hope you will read those articles as well as the ones that did fit my organizational scheme There has also been considerable progress in the areas covered in the Advanced Methods book This is reflected in the roughly two dozen articles in this second edition that are completely new Some of these are in two new sections, “Analysis and Design of Hybrid Systems” and “Networks and Networked Controls.” There have even been a few changes in the Fundamentals Primarily, there is greater emphasis on sampling and discretization This is because most control systems are now implemented digitally I have enjoyed editing this second edition and learned a great deal while I was doing it I hope that you will enjoy reading it and learn a great deal from doing so William S Levine xi i i i i i i 34-6 Control System Applications refrigerant loop Combinations of these can then be transformed to more industrially common output variables such as superheat: the temperature value of the refrigerant above saturated vapor at the exit of the evaporator Tsh = Tg − Tsat (Pe ) (34.2) Additionally, from Figure 34.4, cooling capacity can be a calculated as function of the difference between the refrigerant inlet (T4 ) and outlet (T1 ) temperatures for a given pressure (Pe ) in the evaporator Air-side measurements, such as the evaporator air inlet temperature, or heat exchanger wall temperatures can also be used in feedback These are often easier to obtain since they not involve sensors immersed within the refrigerant There are additional inputs and outputs that can be considered; for example, Jensen and Skogestad [8] utilized refrigerant charge as an input and condenser subcooling as an output for a particular optimal control approach Additionally, for chiller applications [9], the secondary loop temperature characteristics can be used as outputs with flow control valves as inputs A more extensive set of output descriptions can be found in [6,7] 34.3 Basic System Dynamics Understanding the complex dynamics of AC&R systems is vital to proper control In this section we give basic starting points for system modeling and separate the exposition by time scales The mass flow devices (compressor and valve) in Figure 34.4 are described algebraically The energy flow devices (evaporator and condenser heat exchangers) are described by dynamics of varying degrees of complexity depending on the approach taken 34.3.1 Mass Flow Devices Using the simplified system given in Figure 34.4, the basic behavior of the four components can be described Assuming a positive displacement compressor, two simple algebraic relationships are usually sufficient to create a model of its mass flow rate Mass flow rate is calculated in Equation 34.3, where ρk = ρ(Pk,in , hk,in ), and a volumetric efficiency, ηvol , is assumed Additionally, compression can be assumed to be an adiabatic process with an isentropic efficiency, and therefore the relationship between the entrance and exit enthalpies is given in Equation 34.4, where hout,isentropic = h(Pout , sk ) and sk = s(Pin , hin ) This can be rearranged to give Equation 34.5 Both the volumetric and isentropic efficiencies are assumed to change with operating condition and are given by semiempirical maps (Equations 34.6 and 34.7), where Pratio = Pout Pin mk = ωk Vk ρk ηvol , ˙ hout hout,isentropic − hin = ηk , hout − hin hout,isentropic + hin (ηk − 1) , = ηk (34.3) (34.4) (34.5) ηvol = f1 (Pratio , ωk ), (34.6) ηk = f2 (Pratio , ωk ) (34.7) An example of a volumetric efficiency mapping is shown in Figure 34.5 along with the data used to create it If necessary, the static nonlinear compressor model can be linearized around a particular system operating condition This results in a linear compressor model of the form mk = Gaink · ωk that can be ˙ used for control design and to study the parametric sensitivity of the system Similar to the compressor, two algebraic relationships can be used to model a fixed opening expansion device Mass flow rate is calculated assuming standard orifice flow (Equation 34.8) and using a i i i i i i 34-7 Modeling and Control of Air Conditioning and Refrigeration Systems ηvol 1.0 0.8 0.6 0.4 0.2 3000 2500 2000 1500 1000 RPM 500 0 Pressure ratio FIGURE 34.5 A two input performance map of volumetric efficiency with data superimposed semiempirical map for the discharge coefficient (Equation 34.9) The discharge coefficient is assumed to be a function of valve area opening input, uv , and pressure differential, ΔP = (Pin − Pout ) Additionally, expansion is assumed to be an isenthalpic process (Equation 34.10) mv = Cd ρ(Pin − Pout ), ˙ (34.8) Cd = f3 (uv , ΔP), (34.9) hv,in = hv,out (34.10) This basic approach can represent static orifice tubes and controlled area electronic expansion valves (EEVs) quite well The area opening model can be linearized around a particular system operating condition This results in a linear model of the form mv = Gainv · uv that can be used for control design ˙ Components such as thermostatic expansion valves (TXVs) have significant dynamics based on their construction Equations 34.8 through 34.10 could then be augmented with an appropriate lag filter [10] forced by an evaporator superheat temperature uv = KTXV Tsh τTXV s + (34.11) 34.3.2 Heat Exchanger Models In general, the three modeling paradigms that have been applied to modeling of heat exchangers for AC&R systems are lumped parameter, finite volume (or discretized), and moving boundary models The most important task in modeling AC&R systems is effectively capturing the behavior of the heat exchangers [9,12], since they dominate the dynamic behavior of the system Refs [9,11,12] provide literature reviews of the relevant vapor compression system modeling efforts The first paradigm is termed here as the lumped parameter model Lumped parameter heat exchanger models attempt to capture the behavior of a heat exchanger with a single lumped heat transfer parameter These models are commonly presented in textbooks, as illustrated in [13] Often, lumped parameter models are used to model vapor compression systems in conjunction with some other component (e.g., the cabin of a car or a room in a building) In this case, the focus of the modeling effort is not on the dynamics of the vapor compression system but on the cooling of the conditioned environment The simplicity of lumped parameter models tends to be insufficient to capture the dynamic response of some important system outputs (e.g., superheat), and therefore their use in control design is limited Finite volume and discretized approaches to the dynamic modeling of vapor compression systems decompose the heat exchanger geometry to a finite set of small regions, allowing spatial effects to be i i i i i i 34-8 Control System Applications mair_in Tair_in Qw,1 Tw, N Wall Refrigerant k hin N mout hout ΔL Ltotal FIGURE 34.6 Schematic representation of a discretized finite volume modeling paradigm captured by the model Figure 34.6 illustrates the concept of discretizing the heat exchanger into many (N) finite volume regions where the accuracy of the system model improves with an increased number of regions The governing partial differential conservation equations for mass, momentum, and energy are applied to each region, resulting in high-order dynamic models The complexity of the models is primarily used to capture the spatially varying fluid flow and heat transfer phenomena that occur in compact heat exchangers Commercial software packages are available that use a finite volume approach to modeling multiphase flow within heat exchangers (e.g., E-Thermal [14], Modelica [15], or SINDA/FLUENT [16]) The third modeling paradigm is termed the moving boundary approach Moving boundary models attempt to capture the dynamics of multiple phase flows within a heat exchanger by allowing the effective position(s) of phase change to vary as a function of time The parameters for each fluid phase region in the heat exchanger are lumped, resulting in a model of fairly low dynamic order This approach was first presented by Wedekind et al., who proposed using a mean void fraction to develop a transient model of evaporating and condensing flows [17] As detailed below, the dominant dynamics associated with the multiphase flow within the heat exchanger are captured by the varying interface between fluid phase regions represented in the moving boundary approach As a result, the moving boundary framework provides models that can accurately predict the behavior of important system outputs that must be controlled to obtain efficient system operation (i.e., superheat, heat exchanger pressure) The lumped parameter nature of each fluid phase region ensures that the overall dynamic model complexity remains low enough to permit the application of known control design techniques For controls related activity, the low-order dynamic model afforded by this approach usually makes it the model of choice [18] The moving boundary approach is based on the assumption of 1-dimensional fluid flow with effective diameters, flow lengths, and surface areas In essence, this treats any heat exchanger as a long thin tube The approach also assumes uniform pressure throughout the heat exchanger As shown schematically in Figure 34.7, the heat exchanger is divided into regions based on the fluid phase, and the effective parameters are lumped in each region The interface between fluid phase regions is allowed to be a dynamic variable and vary throughout the length of the heat exchanger The following discussion provides an overview of moving boundary models of an evaporator A comprehensive derivation of the models, as well as other heat exchanger configurations such as condensers and internal heat exchangers, is given in [19,20] The evaporator model assumes a two-phase flow condition at the heat exchanger inlet that transitions to a single-phase flow at a specific point within the heat exchanger The location of the interface between these two phase regions is allowed to be a dynamic variable The governing ordinary differential equations ODEs are obtained by integrating the governing partial differential equations PDEs along the length of the heat exchanger and assuming lumped parameters in each fluid region [18,19] Several assumptions are made regarding the lumped parameters of the evaporator model The air temperature used to determine the heat transfer between the walls of the heat exchanger and the air is assumed to be a weighted average of the inlet and outlet air temperatures across each lumped region, Ta = Ta,in (μ) + Ta,out (1 − μ), μ ∈ [0, 1] In the two-phase region, the fluid properties are determined by i i i i i i Modeling and Control of Air Conditioning and Refrigeration Systems 34-9 mair_in Tair_in Tw2(t) Tw1(t) mout Pe(t) hin z hout(t) Superheat Two-phase L1(t) Ltotal FIGURE 34.7 Diagram of the moving boundary evaporator with two fluid regions assuming a mean void fraction; for example, ρ1 = ρf (1 − γ) + ρg (¯ γ ∈ [0, 1] In the superheat region, ¯ γ), ¯ average properties between the inlet and outlet refrigerant state are used, that is, h2 = (hg + hout ) 2, Tr2 = T(Pe , h2 ), and ρ2 = ρ(Pe , h2 ) For the evaporator model, the time derivative of the mean void fraction is neglected This assumption is valid not only because the change in mean void fraction tends to be small during transients considered, but also because its time dependence is related to dynamic modes that are much faster than the dominant system dynamics Thus mean void fraction dynamics can usually be replaced with their instantaneous, algebraic equivalents As a consequence of assuming uniform pressure throughout the heat exchanger, the momentum of the fluid is assumed to be the same upon exit as it is upon entrance Therefore, full conservation of momentum equations are not applied; instead there is a simple pressure drop augmentation given at the exit of the heat exchanger The remaining governing partial differential equations for the conservation of refrigerant mass, refrigerant energy, and heat exchanger wall energy in a fluid region are given by Equations 34.12 through 34.14 ∂ ρAcs ∂t + ∂ (m) ˙ = 0, ∂z ∂ ρAcs h − Acs P ∂ (mh) ˙ + = pi αi (Tw − Tr ), ∂t ∂z ∂ (Tw ) (Cp ρA)w = pi αi (Tr − Tw ) + po αo (Ta − Tw ) ∂t (34.12) (34.13) (34.14) The integration of Equations 34.12 through 34.14 over the two-phase and superheat regions of the evaporator results in the relevant ODEs governing system behavior [18,19] The resulting ODEs for conservation of refrigerant mass, refrigerant energy, and wall energy for the ˙ ˙ ˙ ˙ ˙ two-phase and superheat regions contain only five explicit time derivatives: L1 , Pe , hout , Tw1 , and Tw2 The equations can be combined to result in the descriptor system as Z(x, u) · x = f (x, u) ˙ (34.15) The states are shown in Figure 34.7 as x = [L1 Pe hout Tw1 Tw2 ]T , and the elements of the Z(x, u) matrix and f (x, u) vector are given in [19] Here, the inputs to the evaporator model in T Equation 34.15 are u = mout hin mair_in Tair_in ˙ ˙ ˙ The nonlinear model presented in Equation 34.15 can be linearized around a particular system operating condition This results in a linear evaporator model that can be used for control design and to study the parametric sensitivity of the system The full derivation of the linear model is presented in [19] The procedure would then be repeated for the condenser heat exchanger with increased states due to the additional refrigerant zone present [19] A similar approach can be done for other heat exchangers; for example, a counterflow liquid–liquid heat exchanger The modeling approach can also be augmented in i i i i i i 34-10 Control System Applications various ways For example, a hybrid or switched model can be used when the heat exchanger models given above encounter conditions that would remove one of the zones in the model Ref [21] demonstrates a switched system model for a condenser heat exchanger based on the notion of bumpless transfer techniques that allows for large transients, such as compressor on–off cycling, to be accommodated in the models 34.3.3 Other Component Models The previous two subsections gave modeling approaches for the four basic components in Figure 34.4 To achieve an accurate representation of practical vapor compression cycle systems, several other components may be needed These include chiller loops, counterflow heat exchangers, cooling towers, oil separators, filter/driers, and expansion tanks Additionally, piping elements that model flow splits and flow convergence may also be needed, for example, if there are multiple evaporators for a single condensing unit Furthermore, it may be necessary to combine functionality of individual components as would be the case of treating heat transfer through pipes as a combination of both mass transport and the heat exchanger model Refs [22,23] detail the dynamic modeling of several other types of vapor compression cycle component models along with validation results Figure 34.8 summarizes the modeling approach for the moving boundary method The two mass flow devices can be treated as static nonlinear maps The two heat exchangers vary in their complexity depending on the amount of state information that is necessary for the task; the condenser has types of fluid regions necessitating additional states Between them, they contain the dominant dynamics of the systems The forced air flow across the heat exchangers, similar to the refrigerant mass flow devices, can be represented by static nonlinear maps based on fan charts [24] This approach has been utilized to develop MATLAB - and Simulink -based simulation tools for control-oriented modeling of AC&R systems [19,21–23] 34.3.4 Simplified System Models The more complex models given above are good for codesign of new controllers and plants in that they allow for parametric variations in system design parameters Additionally, they are good for embedded applications involving diagnostics and residual generation However, there is a challenge to direct parameterization of a system representation due to the descriptor form of Equation 34.15 Should the goal be to close a loop around an existing physical system, system identification (ID) techniques are often very suitable Via ID and model reduction [25] demonstrated quantitatively that the dominant system dynamics correspond to the thermal capacitances of the heat exchanger wall sections Therefore, the evaporator shown in Figure 34.7 could be represented by two states, the two wall temperatures, with the other characteristics represented by algebraic relations to these states This order reduction indicates that low-order models can be sufficient for control-based approaches and could be identified using available ID approaches For single-input single-output (SISO) system models, such as expansion device opening (dynamic) Condenser state variables Valve (static) (dynamic) Compressor (static) Evaporator state variables FIGURE 34.8 Summary of moving boundary dynamic modeling structure for AC&R system i i i i i i 34-11 Modeling and Control of Air Conditioning and Refrigeration Systems to superheat, a simple transfer function such as Equation 34.16 would be able to capture local system behavior Ke e−s·tdelay Tsh = G(s) = (34.16) uv (s + pe ) It should be noted that the system identification approach could be utilized in conjunction with the higher fidelity simulation models described above Detailed simulation models would be created and parameterized; then ID techniques would be used on those models to create appropriate, and possibly multivariable, input–output models These could then be used for control design and system evaluation without having to build physical hardware Additionally, individual components of a system, such as shown in Figure 34.7, could be provided by system identification while the rest can be constructed from first principles This “gray box” approach to system modeling can be successfully implemented for hardware-in-the-loop testing as well as system diagnostics 34.3.5 System Nonlinearity The system given in Equation 34.16 would be simple to design SISO controllers for if it were linear and time invariant (LTI) However, the parameters in Equation 34.16 change significantly as the system changes operating condition This is due to the significant system nonlinearity associated with the multiphase fluid flow and heat transfer conditions An example of this system behavior is given in Figure 34.9 that shows changes in superheat owing to 100 rpm step variations in compressor speed at different operating speeds The system response varies significantly depending on the operating speed (i.e., cooling capacity) of the compressor As evidenced, there is up to a factor of change in the steady-state response as the operating condition changes by a factor of 2.5 This level of nonlinearity makes it challenging to design robust control algorithms that meet performance requirements over the range of conditions these systems are likely to encounter Additionally, the complex descriptor-based nature of the dynamics described above and in [19,23] makes it difficult to use techniques such as linear parameter varying (LPV) control The ability to handle large dynamical plant variations in a robust fashion is one of the current and future challenges for AC&R control 34.4 Basic Control Approaches 34.4.1 Hysteretic On–Off Control By far, the most common approach to the control of individual AC&R systems is to use the compressor in a cyclic on–off fashion to modulate cooling capacity while engaging a mechanical expansion device Superheat (°C) 16 600 RPM 1000 RPM 1500 RPM 14 12 10 100 200 300 Time (s) 400 500 FIGURE 34.9 Step response data depicting nonlinearity in AC&R system dynamics i i i i i i 34-12 Control System Applications Condenser fan Condenser Evaporator Expansion valve Compressor Evaporator fan on-off Supply air Return air Conditioned environment temperature Conditioned environment FIGURE 34.10 Schematic representation of an AC&R system interacting with a conditioned environment (e.g., orifice, capillary tube, TXV) to control the amount of superheated vapor exiting the evaporator Some amount of evaporator superheat is necessary to protect the compressor but too much leads to inefficient evaporator operation since the heat transfer is much greater with two-phase fluid than with vapor The cooling capacity is modulated either by turning on or off a motor driving the compressor or by engaging/disengaging a clutch mechanism being driven by some prime mover as shown in Figure 34.2 For different systems, the fans may cycle with the compressor or may operate on a separate schedule A schematic representation of such a system is given in Figure 34.10 with a block diagram schematic given in Figure 34.11 The typical performance for such an approach is shown in Figure 34.12 whereby a hot truck environment, such as the one shown in Figure 34.2, is cooled to a prescribed setpoint As can be seen in Figure 34.12, the temperature oscillates about a given setpoint after convergence from its initial condition The size of the oscillation and the symmetry about the setpoint are functions of the hysteresis parameters The benefit of this approach to capacity control is its simplicity and low cost The on–off compressor function is much cheaper than the power electronics needed to drive a variable speed compressor or the mechanical system needed to drive a variable speed transmission off of a prime mover Additionally, the fan speeds can be cued to the compressor; for example, via a set of belt drives for compact systems Therefore, a simple temperature sensor and rule-based logic is sufficient to close the loop Moreover, since the thermal time constants are usually long, the cycling of the compressor is filtered, much like the current pulse width modulation of a DC motor driver The drawback to this approach is the inability to adapt to changing conditions, both for the enclosed space and for the ambient external conditions Should tighter temperature control or higher efficiency be desired, it may be necessary to consider other control approaches ON – Setpoint + OFF Relay Compressor Air conditioning & Refrigeration system Conditioned environment temperature FIGURE 34.11 On–off hysteretic control system for AC&R compressor capacity control i i i i i i Modeling and Control of Air Conditioning and Refrigeration Systems 34-13 34.4.2 Variable Input Control: PID The amount of energy consumed by an AC&R system varies considerably and depends on the desired temperature of the conditioned environment, the ambient conditions, and the level of internal heat generation within the conditioned environment Capacity control methods, such as the on–off approach, allow these systems to meet varying cooling loads A summary of various studies to determine the best capacity control method was conducted by [26] which found that variable speed compressor control provided the greatest flexibility to match heat loads, resulting in the best overall system efficiency The variable speed compressor control strategies resulted in 20–40% reductions in seasonal power consumption, albeit with a significant increase in cost In addition to variable speed compressors, it is possible to utilize variable orifice EEVs and even variable speed heat exchanger fans All these increase the cost and complexity of both the physical system and the control system but offer the potential for improved performance in terms of temperature regulation and energy consumption The simplest approach to control an AC&R system with variable inputs, and one that is often used for these system, is to utilize individual ProportionalIntegral-Derivative (PID) loops for particular input–output pairs Figure 34.13 illustrates valve control of superheat by estimating the refrigerant temperature and pressure and compressor control of capacity by using the evaporator air inlet temperature as a feedback variable There are several alternative feedback variables that can be used in addition to those shown here For example, Refs [9,27] utilize additional valve control to capture refrigerant in the receiver and isolate it from the rest of the loop Consequently, the results in [9,27] used refrigerant charge as an input variable to control subcooling in the condenser Other approaches include control of frost buildup on the evaporator coil It is possible to extend the operating range of SISO approaches by utilizing more advanced loop shaping or optimal H-infinity techniques However, the dynamics of the SISO control loop are usually of sufficiently low order that PID is sufficient The drawback to closing individual loops is that the coupling among the different input–output pairs can lead to controller fighting [28,29] For example, both the valve and the compressor are regulating mass flow throughout the refrigerant circuit and are therefore coupled Figure 34.14 illustrates the coupled nature for a particular automotive system model; tight regulation of the valve-controlled evaporator superheat loop acts as a major disturbance to the compressor controlled evaporator pressure (i.e., capacity) loop One alternative, to be described later, is to use multivariable control to compensate for the system coupling However, this would limit the accessibility of the control approach to the majority of AC&R design and calibration engineers The PID approach is one that is well understood and accepted by a broad range of users A more favorable approach would be to utilize decoupling techniques for the different feedback loops and then apply PID control to the decoupled system There are several different decoupling techniques ranging from static decoupling of a given input–output representation [30] to the Temperature Initial condition Setpoint Time FIGURE 34.12 On–off hysteretic control system performance i i i i i i 34-14 Control System Applications Condenser fan Condenser EEV Evaporator Compressor Evaporator fan Evaporator superheat PID1 Evaporator air inlet temperature PID2 FIGURE 34.13 Individual PID control loops for an AC&R system reconfiguration of the system by redefining control objectives so as to realize a more naturally decoupled system [31] suitable for simple PID control 34.4.3 Gain Scheduling Evaporator pressure (kPa) 350 Evaporator superheat (°C) As illustrated in Figure 34.9, AC&R systems vary significantly under different operating conditions Oftentimes, this plant variation is not explicitly compensated for and the controller is conservatively tuned to the best it can This is particularly true for the hysteretic on–off control approach given above One approach to compensate for the plant variation is to schedule the controller as a function of operating conditions [32] There are different methods of scheduling but, if one assumes the most common structure of a PID controller, a straightforward approach is to directly interpolate PID gains as a function of the scheduling variable Figure 34.15 shows a candidate interpolation strategy based on Takagi–Sugeno models [33] for three different nominal controllers where the scheduling variable is the evaporator inlet air temperature This variable would be a suitable indicator of the conditioned environment temperature for the system Each control gain (Kp , Ki , Kd ) is multiplied by the appropriate weighting or control 20 300 250 200 150 Output Reference 10 20 30 Time (s) 40 50 60 Output Reference 15 10 10 20 30 Time (s) 40 50 60 FIGURE 34.14 Dual SISO loop proportional-integral control of an automotive AC&R system [28] i i i i i i Modeling and Control of Air Conditioning and Refrigeration Systems Control percentage 34-15 Controller Controller Controller 0.5 12 16 20 Evaporator inlet temperature [°C] 24 FIGURE 34.15 Scheduling algorithm for multiple local controllers percentage Stability results for a class of this type of system used in AC&R applications can be found in [34] Figures 34.16 and 34.17 show evaporator superheat regulation for a system with an EEV and constant compressor/fan speeds The scheduling of PID gains with evaporator inlet air temperature provides a much more consistent system response to a change in the desired setpoint The superheat setpoint value chosen was 9◦ C for this particular experimental system Typical superheat setpoints for automotive AC&R systems are in the range of 3–5◦ C, while many home and commercial systems can be as high as 10◦ C A similar scheduling approach could be used for the hysteretic capacity control approach above whereby the relay parameters can be interpolated as a function of operating condition Comparing Figures 34.16 and 34.17, the scheduled PID controller provides a more uniform superheat control performance than a fixed PID controller over a range of operating conditions In closing discussions of basic control approaches, it is useful to mention that typical augmentations to standard SISO feedback control approaches include the use of feedforward controllers; for example, to minimize the effect of one control input on another For electronic systems, it may be possible to feed the compressor speed reference commands to either the fan or the valve controllers that may utilize them in an anticipatory sense 34.5 Advanced Control Design The next level of control sophistication is the utilization of multi-input, multi-output (MIMO) approaches aimed at coordinating multiple outputs and actuators This can be vital if there is very tightly coupled dynamic behavior between the different input–output variables Additionally, it becomes more important Superheat (°C) 9.5 9.0 8.5 Setpoint 12°C 35°C 8.0 7.5 50 100 Time (s) 150 200 FIGURE 34.16 Fixed PID superheat regulation with varying evaporator air inlet temp i i i i i i 34-16 Control System Applications Superheat (°C) 9.5 9.0 8.5 Setpoint 12°C 35°C 8.0 7.5 50 100 Time (s) 150 200 FIGURE 34.17 Gain-scheduled PID superheat regulation with varying evaporator air inlet temp for complicated systems where there may be multiple sets of heat exchangers, valves, and compressor racks as may be found in large distributed systems such as supermarket refrigeration display cases [35] There have been previous MIMO control efforts and, with the move toward electrification of AC&R systems, it is promising that MIMO control approaches will become prevalent Some of the earliest MIMO work was that of [36] which examined a linear quadratic regulator (LQR) and linear quadratic gaussian (LQG) approach to coordinating inputs affecting mass flow such as compressor speed and valve opening The control performance was significantly superior to SISO control system performance in terms of response speed and disturbance minimization Also, the utilization of model predictive control (MPC) has emerged [37,38] as a means to satisfy input and state constraints on the system while still maintaining good output tracking performance The use of linear optimal methods such as LQR and LQG with Loop Transfer Recovery are very suitable for AC&R systems, provided that the system maintains its operation within a prescribed range of operating conditions where the assumption of linearity is valid Excursions outside the range of validity would necessitate the type of scheduling arrangements mentioned in Section 34.4 For MPC schemes, it is possible to implicitly schedule by continuously adjusting the plant model as the control horizon recedes Within the context of AC&R research, several advances beyond PID control loops have focused on optimal setpoint generation for minimizing some cost function Figure 34.18 illustrates an outer loop optimization routine that would feed setpoints to an inner loop feedback controller The outer optimization loop would monitor the overall system, including the conditioned environment and the ambient environment, and then determine the appropriate cooling capacity needed This would then be fed to the inner closed-loop controller that would regulate to these setpoints In addition to capacity, some of the AC&R system User demand Setpoint optimization + – Inner loop controller Inner loop Outer loop FIGURE 34.18 Inner and outer loop approach to AC&R control design i i i i i i Modeling and Control of Air Conditioning and Refrigeration Systems 34-17 elements of a cost function include the power consumed by the components, particularly compressors and fans The cost functions can also contain temperature deviation errors and even direct estimates for the state of items being refrigerated [35] The optimization can be performed offline and stored in tabular form and then retrieved during operation Online optimization approaches, in a receding-horizon MPC sense, have shown some promise and are likely to gain acceptance with time 34.6 Concluding Remarks The field of AC&R is a fertile one for controls There is a clear impact that can be made on the operation and efficiency of these devices through the use of modern control design tools Throughout the 1990s and 2000s the industry evolved from a purely mechanical control approach using clutches and pressure-balanced TXV’s to a more electronic approach with sensors and embedded systems An analogy could be made between the internal combustion engine’s conversion from mechanical governance (e.g., carburetors) in the 1970s to networked integrated powertrain management systems at present If the analogy holds, we can expect future AC&R systems to provide much more accurate temperature control while significantly cutting energy use and also improving machine reliability The systems and descriptions given here should be considered a starting point for understanding the dynamics and control of AC&R systems There are significant complicating factors and configurations for different types of systems For example, the vapor compression cycle system can be run in reverse to act as a heat pump which can be desirable when trying to remove the frost that would build up on an evaporator coil Moreover, as shown in Figure 34.3, there can be additional subsystems that are present on a real system that would complicate the implementation of controllers designed for an idealized system However, the basic feedback concepts described in this chapter for idealized systems would hold for the more complex physical systems There remain some key barriers to achieving advanced system control A primary one is the system nonlinearity coupled with the absence of a well-parameterized nonlinear system model in a convenient form such as LPV Related to this is the need for further modeling work that is explicitly focused on modeling for control design Finally, the cost of the overall electrification of the system should be sufficiently reduced such that it becomes economically feasible, given the cost of available energy, to implement advanced system hardware Nomenclature Variable α Δ γ ¯ η ρ τ ω A Cd Cp G h K i Description heat transfer coefficient change mean void fraction efficiency density time constant rotational speed area flow coefficient specific heat transfer function specific enthalpy control gain i i i i i 34-18 L m ˙ N P p Q RPM s t T u V W x z Z Control System Applications length mass flow rate finite volume regions pressure perimeter, pole location heat revolutions per minute entropy, Laplace variable time temperature input volume work dynamic state spatial coordinate descriptor form matrix Subscript 1,2,3,4 1,2 a cs d e f g i in k o out p r sat sh TXV v vol w Description 1st, 2nd, 3rd, 4th transition point 1st, 2nd region air cross-sectional derivative evaporator liquid vapor inner, integral inlet compressor outer outlet proportional refrigerant saturation superheat thermostatic expansion valve valve volumetric wall References Energy Information Administration, Annual Energy Review 2006, Washington, DC, June 2007 Also, http://www.eia.doe.gov DOE Report No DOE/EIA-0573, 2007 Also, http://www.eia.doe.gov/oiaf/1605/ggrpt Koomey, J and Brown, R.E., The role of building technologies in reducing and controlling peak electricity demand, LBNL Technical Report 49947, September 2002 Buildings Energy Data Book 2008 Also, http://buildingsdatabook.eere.energy.gov Constable, G and Somerville, R., A Century of Innovation: Twenty Engineering Achievements that Transformed our Lives, National Academies Press, Washington, DC, 2003 i i i i i i Modeling and Control of Air Conditioning and Refrigeration Systems 34-19 Althouse, A.D., Turnquist, C.H., and Bracciano, A.F., Modern Refrigeration and Air Conditioning, Tinley Park, IL: The Goodheart-Willlcox Co., 1995 Stoeker, W.F., Industrial Refrigeration Handbook, New York: McGraw-Hill, 1998 Jensen, J and Skogestad, S., Optimal operation of simple refrigeration cycles: Part I: Degrees of freedom and optimality of sub-cooling, Computers & Chemical Engineering, 31(5–6), 712–721, 2007 Bendapudi, S., Braun, J.E., and Groll, E.A., A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers, International Journal of Refrigeration-Revue Internationale Du Froid, 31(8), 1437–1452, 2008 10 James, K.A and James, R.W., Transient analysis of thermostatic expansion valves for refrigeration system evaporators using mathematical models, Transactions of Institution of Measurement and Control, 9(4), 198–205, 1987 11 Lebrun, J and Bourdouxhe, J.P., Reference Guide for Dynamic Models of HVAC Equipment, ASHRAE Project 738-TRP, Atlanta, GA, 1998 12 Bendapudi, S and Braun, J.E., A review of literature on dynamic models of vapor compression equipment, ASHRAE Report #4036–5, May 2002 13 Incropera, F and deWitt, D.P., Introduction to Heat Transfer, New York: John Wiley & Sons, 2002 14 Anand, G., Mahajan, M., Jain, N., Maniam, B., and Tumas, T.M., e-thermal: Automobile air conditioning module, Society of Automotive Engineers 2004 World Congress, SAE Paper 2004-01-1509, Detroit, MI, 2004 15 Eborn, J., Tummescheit, H., and Prolss, K., Air conditioning—a Modelica library for dynamic simulation of AC systems, 4th International Modelica Conference, pp 185–192, Hamburg-Harburg, Germany, March 7–8, 2005 16 Cullimore, B.A and Hendricks, T.J., Design and transient simulation of vehicle air conditioning systems, Society of Automotive Engineers 5th Vehicle Thermal Management Systems Conference, Paper VTMS 2001-01-1692, 2001 17 Wedekind, G.L., Bhatt, B.L., and Beck, B.T., A system mean void fraction model for predicting various transient phenomena associated with two-phase evaporating and condensing flows, International Journal of Multiphase Flow, 4, 97–114, 1978 18 He, X.D., Liu, S., and Asada, H., Modeling of vapor compression cycles for multivariable feedback control of HVAC systems, ASME Journal of Dynamic Systems, Measurement and Control, 119(2), 183–191, 1997 19 Rasmussen, B.P Dynamic modeling and advanced control of air conditioning and refrigeration systems, PhD Thesis, Department of Mechanical and Industrial Engineering, University of Illinois, UrbanaChampaign, IL, 2005 20 Eldredge, B.D., Rasmussen, B.P., and Alleyne, A.G., Moving-boundary heat exchanger models with variable outlet phase, ASME Journal of Dynamic Systems, Measurement, and Control, 130(6), Article ID 061003, 2008 21 Li, B and Alleyne, A., A dynamic model of a vapor compression cycle with shut-down and start-up operations, International Journal of Refrigeration, 33(3), 538–552, May 2010 22 Alleyne, A.G., Rasmussen, B.P., Keir, M.C., and Eldredge, B.D., Advances in energy systems modeling and control, Proceedings of 2007 American Controls Conference, pp 4363–4373, New York, NY, July 2007 23 Li, B and Alleyne, A.G., A full dynamic model of a HVAC vapor compression cycle interacting with a dynamic environment, Proceedings of 2009 American Controls Conference, pp 3662–3668, St Louis, MO, June 2009 24 Chen, H., Thomas, L., and Besant, R.W., Fan supplied heat exchanger fin performance under frosting conditions, International Journal of Refrigeration, 26(1), 140–149, 2003 25 Rasmussen, B., Musser, A., and Alleyne, A Model-driven system identification of transcritical vapor compression systems, IEEE Transactions on Control Systems Technology, 13(3), 444–451, 2005 26 Qureshi, T.Q and Tassou, S.A., Variable-speed capacity control in refrigeration systems, Applied Thermal Engineering, 16(2), 103–113, 1996 27 Jensen, J and Skogestad, S., Optimal operation of simple refrigeration cycles: Part II: Degrees of freedom and optimality of sub-cooling, Computers & Chemical Engineering, 31(12), 1590–1601, 2007 28 Shah, R., Rasmussen, B.P., and Alleyne, A.G., Application of a multivariable adaptive control strategy to automotive air conditioning systems, International Journal of Adaptive Control and Signal Processing, 18(2), 199–221, 2004 29 Keir, M.C., Dynamic modeling, control, and fault detection in vapor compression systems, MS Thesis, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana-Champaign, IL, 2006 i i i i i i 34-20 Control System Applications 30 Astrom, K.J., Johansson, K.H., and Wang, Q., Design of decoupled PID controllers for MIMO systems, Proceedings of the 2001 American Control Conference, pp 2015–2020, Arlington, VA, June 2001 31 Jain N., Li, B., Keir, M., Hencey, B., and Alleyne A., Decentralized feedback structures of a vapor compression cycle system, IEEE Transactions on Control Systems Technology, 18(1), 185–193, January 2010 32 Shamma, J.S and Athans, M., Gain scheduling: Potential hazards and possible remedies, IEEE Control Systems Magazine, 12(3), 101–107, 1992 33 Murray-Smith, R and Johansen, T.A (Eds), Multiple Model Approaches to Modelling and Control, Bristol, PA, Taylor & Francis, 1997 34 Rasmussen, B.P and Alleyne, A.G., Gain scheduled control of an air conditioning systems using the Youla parameterization, Proceedings of the 2006 American Control Conference, pp 5336–5341, Minneapolis, MN, June 2006 35 Cai, J., Jensen, J.B., Skogestad, S., and Stoustrup, J., On the trade-off between energy consumption and food quality loss in supermarket refrigeration systems, Proceedings of the 2008 American Control Conference, pp 2880–2885, Seattle, WA, June 2008 36 He, X.D., Asada, H.H., Liu, S., and Itoh, H., Multivariable control of vapor compression systems, HVAC & R Research, 4(3), 205–230, 1998 37 Elliott, M.S and Rasmussen, B.P., Model-based predictive control of a multi-evaporator vapor compression cooling cycle, Proceedings of the 2008 American Control Conference, pp 1463–1468, Seattle, WA, 2008 38 Larsen, L.F., Model based control of refrigeration systems, PhD Thesis, TU Aalborg, Denmark, 2006 i i ... the publisher, these are all reasons to a second edition To the editor of the first edition, these same facts are a modest disincentive The risk that a second edition will not be as good as the. .. the model-based feedback control, and in Section 1.6.2, the model-based feedforward control As usual, in control, the requirements for the accuracy of the mathematical models upon which the control. .. insure that the second edition is at least as good as the first one was I hope you agree that I have succeeded I have made two major changes in the second edition The first is that all the Applications