Model-Driven System Identiﬁcation of Transcritical Vapor Compression Systems

444 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005

Model-Driven System Identification of Transcritical Vapor Compression SystemsBryan P. Rasmussen, Andrew G. Alleyne, and Andrew B. Musser

Abstract—This brief uses an air conditioning system to illustratethe benefits of iteratively combining first principles and systemidentification techniques to develop control-oriented models ofcomplex systems. A transcritical vapor compression system isinitially modeled with first principles and then verified with exper-imental data. Both single-input–single-output (SISO) and multi-input–multi-output (MIMO) system identification techniquesare then used to construct locally linear models. Motivated bythe ability to capture the salient dynamic characteristics withlow-order identified models, the physical model is evaluated foressentially nonminimal dynamics. A singular perturbation modelreduction approach is then applied to obtain a minimal representa-tion of the dynamics more suitable for control design, and yieldinginsight to the underlying system dynamics previously unavailablein the literature. The results demonstrate that iteratively modelinga complex system with first principles and system identificationtechniques gives greater confidence in the first principles model,and better understanding of the underlying physical dynamics.Although this iterative process requires more time and effort,significant insight and model improvements can be realized.

Index Terms—Air conditioning, control engineering, identifica-tion, modeling, reduced order systems.

NOMENCLATURE

Variables

State–space matrices.Length.Pressure.Temperature.Matrix.Continuous function.Specific enthalpy.Mass flow rate.Time.Controllable inputs.Dynamic states.Outputs.Compressor speed.

Subscripts

First, second region.Air.Average.Cold; gas cooler.Evaporator.

Manuscript received October 9, 2003; revised June 21, 2004. Manuscript re-ceived in final form July 23, 2004. Recommended by Editor-in-Chief F. J. Doyle.This work was supported by the Air Conditioning and Refrigeration Center, Uni-versity of Illinois at Urbana-Champaign, under Project 123.

The authors are with the Department of Mechanical and Industrial Engi-neering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA(e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TCST.2004.839572

Hot.Heat exchanger.Identified.

in,out In, out.Compressor, valve.Reduced order.Superheat.Wall.

I. INTRODUCTION

CONTROLS engineers are often required to develop dy-namic models for a class of complex physical systems

with little prior knowledge of the system dynamics. Adding tothe challenge of this task is the desire to gain physical insightinto the underlying dynamics. This task can sometimes spanyears or decades before the dynamics of the given class of sys-tems are well understood and characterized with simple con-trol-oriented models.

The foremost difficulty in this process is balancing accuracywith simplicity. Although many processes can be accuratelymodeled with high-order ordinary or partial differential equa-tions (PDEs), the practicing control engineer often needs prac-tical low-order controllers [1]. This means that significant timemust be spent either: 1) developing simple, accurate, control-oriented models prior to the control design phase or 2) reducinghigh-order controllers after the design phase, but prior to imple-mentation. The distinct advantage of developing simple modelsis the physical insight gained. This insight is invaluable as a toolin developing general control strategies (i.e., selecting sensors,actuators) and in system design (e.g., sizing components).

Because of these advantages, engineers expend considerableeffort to develop low-order physical models. For example, ef-forts to model the two-phase fluid dynamics associated withdrum boilers spanned several decades before finding an appro-priate balance of simplicity and accuracy [2]. However, once anaccurate low-order model has been identified, the developmentof control strategies follows quickly. As an example, this wasevident with the publications that followed the Moore–Greitzercompressor model [3], [4].

The objective of obtaining a low-order model containingphysical insight seemingly would preclude the use of systemidentification techniques, and restrict the engineer to firstprinciples based modeling. However, this work uses an air con-ditioning system as an example to demonstrate that iterativelyusing both first principles modeling and system identificationtechniques can be mutually beneficial. Previously, subcritical airconditioning systems have been modeled using a lumped-pa-rameter, moving-boundary formulation [5]–[8]. When thismethodology is applied to a transcritical cycle, the resultingmodel is of comparable order to the subcritical cycle models.

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 3, MAY 2005 445

Fig. 1. Diagram of modeling process illustrating iterations between firstprinciples modeling and identification.

However, as will be shown, the resulting model is essentially anonminimal representation of the system dynamics.

Although the specific system discussed in this brief is a vaporcompression cycle, many of the benefits will be similar for othercomplex physical systems. Specifically, the identified modelscan guide the engineer in selecting the appropriate balance be-tween model fidelity and simplicity. Likewise, insight from thephysical model can guide the choice of appropriate inputs, out-puts, model order, and structure for system identification.

This iterative procedure is best explained by the diagramshown in Fig. 1. First principles are used to derive the governingdynamic equations for a transcritical air conditioning system,resulting in an eleventh-order model [9]. This model is validatedusing data obtained from an experimental system. The data fromthis system are also used to construct single-input–single-output(SISO) and multi-input–multi-output (MIMO) empiricalmodels using system identification techniques. While theSISO identification results in third-order input–output models,the MIMO identification necessitates a fifth-order model foradequate prediction, demonstrating the need for MIMO iden-tification of such systems. Moreover, the simple identifiedbehavior suggests the presence of low-order dominant dy-namics, and prompts investigation into possibilities for modelreduction of the first principles model. Subsequent analysisreveals that the eleventh-order model is singularly perturbed[10], which leads to physically-based model reduction. Anal-ysis also reveals that the system is highly coupled, furtherdemonstrating the inadequacy of SISO models in describingthe system dynamics. The final reduced order model is ofsimilar order to the MIMO identified model, giving greaterconfidence in the physical model and, more importantly, alow-order physically-based control-oriented model that waspreviously unavailable in the literature.

The remainder of this brief is organized as follows. Neces-sary background information is presented in Section II, anddetails regarding the experimental setup are given in Section III.Section IV presents the first principles model for a transcriticalvapor compression system, and Section V presents the systemidentification results. Section VI discusses the physically-basedmodel reduction using singular perturbation approximation.Concluding remarks are given in Section VII.

Fig. 2. Diagram of transcritical vapor compression cycle.

II. BACKGROUND

A majority of air conditioning and refrigeration systems op-erate using a vapor compression cycle. Modeling the complexthermofluid dynamics of these systems with distributed param-eters can result in high-order models that are of limited usefor control design [11]–[14]. However, the dominant dynamicsof these systems are observed to be low order, nonlinear, andnonminimum phase. Vapor compression systems typically havebeen controlled using simple SISO mechanical controllers. Theextensively coupled dynamics cause these control strategies toperform poorly, with occasional instability [5], [15], which sug-gests that the use of multivariable control strategies could bebeneficial [7].

This brief considers a transcritical vapor compression cycle aspictured in Fig. 2. This is an atypical vapor compression cyclethat uses carbon dioxide ( ) as the working fluid. This type ofsystem has attracted attention as a possible replacement of tra-ditional subcritical refrigerants because is a natural fluid,and does not have the negative environmental impacts of tradi-tional refrigerants. From a controls perspective, this system hasthe distinct advantage of having an extra degree of freedom totrade efficiency for capacity [16].

Beginning at the top of the diagram (Fig. 2), the high-pressure,supercriticalfluidflowsthroughamicrochannelgascooler,whereheat is rejected. From the gas cooler, the refrigerant flows to the“hot”sideofamicrochannelcounterflowheatexchanger.This in-ternal heat exchanger uses the cold fluid leaving the evaporatorto cool the hot refrigerant from the gas cooler. This increases thecapacity of the air conditioning system, as well as ensuring thatonly refrigerant vapor enters the compressor. After the internalheat exchanger the refrigerant flows through an expansion valve.Through this valve the fluid expands and transitions from a super-critical fluid to a two-phase mixture. The refrigerant then entersthe microchannel evaporator where heat is absorbed as the fluidevaporates.Fromtheevaporator, the refrigerantflows through the“cold” side of the internal heat exchanger, and then to the com-pressor where the fluid is compressed to a higher pressure.

III. EXPERIMENTAL SYSTEM

The test facility was located on the campus of the Universityof Illinois at Urbana-Champaign. The principal facility was asecond-generation prototype automotive air conditioningsystem (MAC2R744). This system was used previously to study


Fig. 3. Experimental testing facility (from [17]).

steady-state efficiency in [17], where a detailed description ofthe system components is available.

A schematic of the system is given in Fig. 3. The experi-mental system has two insulated environmental chambers thatare controlled to simulate desired indoor and outdoor condi-tions. An electronic heater and glycol chiller are used to heatand cool the air flow after leaving the evaporator and gas cooler,respectively, while a motor and clutch system is used to drivethe compressor. Extensive steady-state calibration of the sensorshas been conducted to ensure the validity of all measurements[17]. For transient tests, the sampling frequency was approxi-mately 1 Hz. This frequency maximized the sensor dwell time,thus, increasing the signal to noise ratio for the desired numberof sensors to be monitored.

The standard system actuators included an expansion valve,compressor, and air fans. For the transient tests an electronic ex-pansion valve and variable displacement compressor were used.The compressor was fixed at full displacement and mass flowrate was varied by altering the rotational speed. Air flow rateswere varied by changing the fan speeds, while inlet air temper-ature was maintained at a desired value.

Fig. 4. Evaporator with two-phase flow at entrance and superheated vapor atexit.

IV. FIRST PRINCIPLES MODEL

A control-oriented model for a transcritical air conditioningsystem is derived in detail in a related paper [9], and only the es-sential results are presented here. Each component is modeledseparately, and the system model is then created by appropri-ately relating the inputs and outputs of each of the componentmodels. The four controllable inputs to the system are assumedto be compressor speed, expansion valve opening, and mass flowrates of air across the evaporator and gas cooler. The outputs ofinterest are superheat temperature (measure of efficiency), evap-orator outlet air temperature (measure of comfort), and the op-erating pressures in the evaporator and gas cooler.

The actuating components such as the compressor and ex-pansion valve have dynamics that are considerably faster thanthe dominant dynamics of the system. These components aremodeled with nonlinear algebraic equations that calculate massflow as a function of the operating pressures, and outlet enthalpyas a function of the inlet enthalpy. The gas cooler, evaporator,and internal heat exchanger are modeled with nonlinear ordi-nary differential equations, resulting in third-order, fifth-order,and third-order models, respectively. Thus, the overall system isan eleventh-order model.

All fluid in the gas cooler is assumed to be in a supercriticalstate. Therefore, a lumped parameter model of the gas cooler as-sumes one single-phase region. The fluid in the evaporator is as-sumed to enter the evaporator as a two-phase fluid, and exit as asuperheated vapor, as depicted in Fig. 4. Therefore, the lumpedparameter model assumes two separate fluid regions. The in-ternal heat exchanger is modeled with the three ordinary differ-ential equations based on lumped capacitance assumptions anda counterflow configuration is assumed.

Each of the component models is given as nonlinearstate–space models, , where the elementsof and are nontrivial and presented in detail in[9]. The state and input vectors for each of the three componentmodels are defined in (1)–(6). The state variables are definedin terms of pressures, enthalpies, etc. and are the result of thederivation procedure. However, they are not the only possiblechoice of physical states as discussed in Section VI. For ex-ample, the states of the evaporator model ( ) are length oftwo-phase flow , evaporation pressure , outlet enthalpy

, and the two lumped wall temperatures , and .The inputs to each of the component models are generallyoutputs of other component models. For example, the inputs


Fig. 5. Evaporator pressure for step changes in compressor speed (data, nonlinear, and linear models).

to the evaporator model are the inlet and outlet refrigerantmass flow rates and (outputs of the valve andcompressor models), the inlet enthalpy (output of thevalve model), and the temperature and mass flow rate of air

and (inputs to the overall system).

(1)

(2)

(3)

(4)

(5)

(6)

The nonlinear model is linearized about a selected operatingcondition, and both the nonlinear and linear models are com-pared to data obtained using the test facility discussed in Sec-tion III. Although a more complete validation is given in [9], anexample comparing evaporator pressure is shown in Fig. 5 forthe highway driving condition. The model shows good agree-ment with data, and captures the steady-state response, and risetime of the system. Admittedly there is some mismatch, but toimprove the accuracy of the model would likely require a sig-nificant increase in model complexity, such as using finite dif-ference equations or PDEs . Given these complex alternatives,the ability of the model to capture a majority of the dynamic be-havior is viewed as a success.

This first principles model is the result of using an establishedmodeling approach [5]. Although the resulting model is suffi-ciently accurate, the approach results in several modes that arenonessential for dynamic prediction. This will be seen in thesystem identification results in the following sections.

V. SYSTEM IDENTIFICATION

A common approach for modeling air conditioning systems isusing system identification techniques [18], [19]. System iden-tification can yield very accurate models, but has the distinctdisadvantages of being dependent on the system considered andonly valid around the operating point considered. However, theresulting models can be used to verify the level of complexityof the first principles model.

For identification purposes, it was necessary to excite thesystem by varying each of the inputs. Because SISO modelidentification is concerned only with individual input-output be-havior, the data for this identification approach was generatedby varying each of the system inputs separately using a pseudo-random binary sequence (PRBS). Although not ideal for non-linear identification, this signal is persistently exciting of ade-quate order, and is sufficient for identifying approximate linearmodels. The choice of input amplitudes was approximately 10%of the actuator range, generally resulting in a 5%–10% changein the outputs, which was sufficient to clearly discern the dy-namic behavior of the system. For identifying MIMO models,all of the system inputs were varied simultaneously.

The individual input-output models were constructed of-fline assuming an ARMAX model structure,

. The coefficients of the , , andpolynomials are found using a standard nonlinear least

squares iterative search method (Gauss–Newton) [20], thatminimizes a quadratic prediction error criterion, asdefined in (7), while imposing constraints to ensure that onlymodels with stable predictors are used [21].

(7)

The minimum model order necessary to adequately model thedynamics, while ensuring whiteness and independence of themodel residuals, was third-order or lower. The data sets weredivided into estimation and validation sets and the models werecross-validated to ensure that the models were not over-fitted toa specific data set. This process was repeated for three commonoperating conditions (idle, city, and highway driving). For afew of the input–output combinations the resulting models weresecond order, while others were observed to be unaffected bychanges in the inputs. For example gas cooler exit air tempera-ture, , remained virtually unchanged for changes in expan-sion valve (Fig. 6) and evaporator air flow rate. The individualtransfer functions for each of the input–output pairs for all threeconditions can be found in [22]. Comparison between one ofthe SISO models and data are shown in Fig. 6. As discussed in[21], the initial input–output values are assumed to be close tothe equilibrium values and are unnecessary for identification ofa linear model. Therefore, the sample means are removed prior


Fig. 6. SISO system ID results: step changes in valve opening at highwaycondition.

to the offline identification and the results are plotted with a zeromean value.

Given the results of the SISO identification, it would betempting for someone unfamiliar with system identificationto assume that only a third-order physical model is needed.However, the states of the various SISO models need not bethe same, and, therefore, a MIMO model would be a moreappropriate description of the system. Although the individualSISO models could be combined, scaled, and then reducedusing a balanced realization/truncation approach, a more directapproach for constructing a MIMO model (subspace algo-rithms) is preferred.

A state–space model structure is selected of the form of (8),where is the sampling time, are constantmatrices, and , , , and are the time sequencesof states, inputs, outputs, and model residuals, respectively.This model structure differs notably from those in previousstudies that identified air conditioning systems using multivari-able ARX models [19]. The particular advantage of the modelstructure in (8) is its close relationship to physically-based con-tinuous time state–space models. Other multivariable methods(i.e., multivariable ARX, vector difference equation, etc.) areless easily related to their first principles counterparts [21].

(8)

The algorithm used to identify the matriceswas a combined prediction error method and subspace algo-rithm. The initial guess values of these matrices are determinedby an N4SID algorithm [23]. Contrary to the classical identifi-cation methods which determined the system matrices first andthen the system states, the N4SID subspace method identifiesthe state vector first, and then determines the system matricesusing a linear least squares approach. As discussed in [21], [24]the general steps of subspace algorithms are: 1) construct the ex-tended observability matrix from input-output data; 2) select ap-propriate weighting matrices (possibilities are listed in [24]) andperform a singular value decomposition (SVD); 3) determinethe state vector from the SVD; 4) determine the

Fig. 7. MIMO system ID results for random step changes in all inputs at idlecondition.

matrices using a linear least squares approach; and 5) determinethe matrix from and the covariance of theresiduals. For this application, an initial guess for the systemmatrices is obtained using an N4SID algorithm, and the modelis then adjusted by improving the prediction error fit using an ap-proach similar to the SISO algorithm. Models of order 1 through10 were estimated and cross-validated. The fifth-order modeladequately predicted all outputs and had the lowest correlationerrors of any of the models generated (Fig. 7). Numerical repre-sentations of the identified state–space matrices for one possibleoperating condition (highway driving) are given in (9)–(12).

As a demonstration of the model fit for both approaches, threebasic statistics about the model residuals are shown in Table I.The model residuals, , are defined as withas the predicted output and as the measured output. The max-imum residual and the average residual are calculated as givenin (13) and (14) with being the number of measurements.Additionally, a relative error measure is calculated as shown in(15), where is the mean value of . The percentage of modelfit can then be calculated as: 100 , shown in(9)–(15) at the bottom of the next page.

VI. REDUCED ORDER FIRST PRINCIPLES MODEL

There is a clear discrepancy in model complexity betweenthe eleventh-order first principles model and the lower orderidentified models. The large difference in model order suggeststhat the first principles model is more complex than is neces-sary to capture the low-order input–output behavior. Althoughnumerical model reduction techniques could be applied to thelinearized physical model to eliminate nonessential modes, thestates of the resulting model would lose physical significance,and little insight to the dynamics would be gained. This promptsa close examination of the physical model and the derivationprocedure.

Examining the eigenvalues of the linearized componentmodels, they are found to differ by several orders of magnitude,indicating that the various dynamic modes evolve on extremelydifferent time scales. However, the results of Section V do not


TABLE IMODEL FIT MEASURES FOR IDENTIFICATION

indicate which dynamic modes dominate the system response,and what are the physical significance of these modes.

A unique aspect of thermofluid systems is the redundancy ofinformation of the thermodynamic state variables. For example,given pressure and temperature all the remaining fluid proper-ties can be found (i.e., density, enthalpy, etc.). This results infreedom in selecting variables as dynamic states. In this case,we wish to select the combination of states that most directlydecouples the system dynamics into fast and slow modes. In [9],

an alternative choice of states is found that effectively separatesthe fast and slow dynamic modes. This choice of states has intu-itive appeal as well. The slow states are the total wall energy ofeach heat exchanger region, and total refrigerant mass in eachheat exchanger. The fast states are the total refrigerant energy ineach heat exchanger region. Additionally, [9] reveals a dynamicmode that is completely redundant. Because the total amount ofrefrigerant is fixed, there is no need to have separate states forthe refrigerant mass in each of the heat exchangers.

(9)

(10)

(11)

(12)

(13)

(14)

(15)


TABLE IISYSTEM EIGENVALUE APPROXIMATION

Fig. 8. Evaporator pressure for step changes in compressor speed (full order and reduced order models).

Since the eigenvalues of the different modes differ by morethan an order of magnitude, the component models are said to besingularly perturbed [10]. A reduced order linearized model canbe found by “residualizing” the fast dynamic modes [25]. Thisprocess of model reduction replaces the fast dynamic modeswith their algebraic equivalents, thus, reducing the model orderwhile preserving the physical nature of the state variables. [26]shows that for singularly perturbed systems, there is a connec-tion between the fast dynamic modes and the so-called weaklycontrollable/observable modes. Thus, by eliminating the redun-dant dynamic mode and the fast dynamic modes, we retain thedominant dynamic behavior of the system and achieve a min-imal representation of the system dynamics.

The final form of this reduced order model can be repre-sented in the standard state–space form, shown in (16), wherethe matrices are found in [22]. Thesystem inputs/disturbances are given in (17) (valve opening,compressor speed, inlet evaporator air temperature, evaporatorair mass flow rate, inlet gas cooler air temperature, and gascooler air flow rate). The system outputs are given in (18)(evaporator superheat, evaporator pressure, gas cooler pressure,evaporator exit air temperature, gas cooler exit air temperature).The reduced order system model closely approximates theeigenvalues of the full-order model, as seen in Table II.

A comparison of the full-order and reduced order linearmodels is shown in Fig. 8, where the full-order linear and thereduced order linear models are indistinguishable. Thus, themodel reduction prompted by the system identification effortsresulted in a model that was of lower order, but with negligibleloss in model accuracy.

(16)

(17)

(18)

VII. CONCLUSION

The principal contribution of this brief is a demonstration ofhow control-oriented modeling of complex systems can benefitfrom iteratively using first principles and system identification.A transcritical air conditioning system is modeled according tothe process shown in Fig. 1. The model is initially developedwith first principles using an approach established in the thermalsystems field. When compared to the results of system identifi-cation, the discrepancy in model complexity prompts a detailedevaluation of the physical modeling process resulting in insightsinto the system dynamics previously unavailable in the litera-ture. Furthermore, modes that are nonessential are identified andphysically-based model reduction techniques are subsequentlyemployed to achieve a minimal representation of the dynamicsmore suitable for control design. The final reduced order phys-ical model sufficiently captures the salient dynamic behavior ofthe system and results in negligible loss of accuracy comparedto the higher order model previously derived.

The development of minimal control-oriented models canbenefit from iteration between both first principals modelingand system identification. Although in hindsight the choiceof model order is clear, engineers are often required to modelcomplex systems with little a priori knowledge. For manycomplex nonlinear systems formed by the interconnection ofmany subsystems, the search for nonessential physical dynamicmodes could benefit from the identification results to providean estimate of the appropriate order. The iterative processdescribed in this brief requires more time and effort, but resultsin models with the appropriate balance between fidelity andsimplicity, and greater confidence in the final model.

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Model-Driven System Identiﬁcation of Transcritical Vapor Compression Systems